On the Categorical Approach to the Frobenius Trace Citation Hemo, Tamir (2023) On the Categorical Approach to the Frobenius Trace. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8d25-ky12. https://resolver.caltech.edu/CaltechTHESIS:05052023-230556627 Abstract Motivated by the study of the local and global Langlands correspondence from a geometric prespective, we establish two results of a general nature regarding categories of sheaves in algebraic geometry. The first result, motivated by the work of Drinfeld and Lafforgue on the Langlsnds correspondence over function fields, establishes a categorical enhancement of the Künneth formula for categories of Weil sheaves, generalizing a famous result of Drinfeld. In the second part, motivated by the geometric approach to the study of representations of reductive groups over local fields, we develop a method to calculate the categorical trace of monoidal categories arising from convolution pattern in algebraic geometry. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Mathematics, Representation Theory, Category Theory, Algebraic Geometry Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Apostol Award for Excellence in Teaching in Mathematics, 2021. Scott Russell Johnson Prize for Excellence in Graduate Studies, 2020. Thesis Availability: Public (worldwide access) Research Advisor(s): Zhu, Xinwen Thesis Committee: Flach, Matthias (chair) Mantovan, Elena Szumowicz, Anna Zhu, Xinwen Defense Date: 19 May 2023 Record Number: CaltechTHESIS:05052023-230556627 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05052023-230556627 DOI: 10.7907/8d25-ky12 ORCID: Author ORCID Hemo, Tamir 0000-0002-0239-6139 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 15154 Collection: CaltechTHESIS Deposited By: Tamir Hemo Deposited On: 05 Jun 2023 18:02 Last Modified: 16 Jun 2023 18:10 Thesis Files PDF - Final Version See Usage Policy. 714kB Repository Staff Only: item control page

On the Categorical Approach to the Frobenius Trace Thesis by Tamir Hemo In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2023 Defended May 19, 2023 ii © 2023 Tamir Hemo ORCID: 0000-0002-0239-6139 All rights reserved except where otherwise noted iii ACKNOWLEDGEMENTS I feel very fortunate and grateful to be in this position. I owe my PhD work, time, and process to many people. First, I would like to thank my advisor, Xinwen Zhu. For both his guidance and collaboration. He has taught me how to identify important content from noise, how to tackle unknown problems, and how to think about mathematics. The seminars and special topics classes organized by Prof. Zhu helped me find inspiration for research questions and help start our joint project on the unipotent categorical Langlands correspondence. I want to thank my collaborators Timo Richarz and Jakob Scholbach. For their time, intellectual generosity, inspiration, and companionship. I would also like to thank all my colleagues in the math department and the PMA division who made these years so much more enjoyable. Finally, I want to thank my family and friends who made it all possible. To my parents, whose love and support I can’t thank enough, and to my partner, Naomi, who was a guiding light during difficult times and whose presence makes every good moment infinitely better. iv ABSTRACT Motivated by the study of the local and global Langlands correspondence from a geometric prespective, we establish two results of a general nature regarding categories of sheaves in algebraic geometry. The first result, motivated by the work of Drinfeld and Lafforgue on the Langlsnds correspondence over function fields, establishes a categorical enhancement of the Künneth formula for categories of Weil sheaves, generalizing a famous result of Drinfeld. In the second part, motivated by the geometric approach to the study of representations of reductive groups over local fields, we develop a method to calculate the categorical trace of monoidal categories arising from convolution pattern in algebraic geometry. v TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iv v I Introduction 1 Chapter I: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Chapter II: Drinfeld’s lemma and the Global Function Field Lanlgands Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 The Langlands Correspondence over function fields . . . . . . . . . 5 2.2 Drinfeld’s lemma and the Künneth formula for Weil sheaves . . . . . 9 2.3 Application to the moduli of bundles and shtukas . . . . . . . . . . . 10 Chapter III: The Categorical Trace in the Categorical Local Lanlgands Program 12 3.1 Loop groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Convolution Patterns and the Iwahori-Hecke category . . . . . . . . 13 3.3 The Categorical trace construction . . . . . . . . . . . . . . . . . . 15 II A Categorical Künneth Formula For Weil Sheaves 17 Chapter IV: Recollections on infinity categories . . . . . . . . . . . . . . . . 4.1 Monoidal aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fixed points of infinity categories . . . . . . . . . . . . . . . . . . . Chapter V: Lisse and constructible sheaves . . . . . . . . . . . . . . . . . . . Chapter VI: Constructible Weil Sheaves . . . . . . . . . . . . . . . . . . . . 6.1 The Weil-proétale site . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Weil sheaves on products . . . . . . . . . . . . . . . . . . . . . . . 6.3 Partial-Frobenius stability . . . . . . . . . . . . . . . . . . . . . . . 6.4 Lisse and constructible Weil sheaves . . . . . . . . . . . . . . . . . 6.5 Relation with the Weil groupoid . . . . . . . . . . . . . . . . . . . . 6.6 Weil-étale versus étale sheaves . . . . . . . . . . . . . . . . . . . . . Chapter VII: The categorical Künneth formula . . . . . . . . . . . . . . . . . 7.1 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 A formulation in terms of prestacks . . . . . . . . . . . . . . . . . . 7.3 Full faithfulness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Drinfeld’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Factorizing representations . . . . . . . . . . . . . . . . . . . . . . 7.6 Essential surjectivity . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter VIII: Ind-constructible Weil sheaves . . . . . . . . . . . . . . . . . . 8.1 Ind-constructible Weil sheaves . . . . . . . . . . . . . . . . . . . . . 18 18 19 23 26 26 29 29 33 35 38 40 40 42 44 47 49 51 56 56 vi 8.2 Cohomology of shtuka spaces . . . . . . . . . . . . . . . . . . . . . 59 IIIThe Categorical Trace and Convolution Patterns 62 Chapter IX: More recollections on ∞-categories . . . . . . . . . . . . . . . . 63 9.1 Dual categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 9.2 Relative tensor product . . . . . . . . . . . . . . . . . . . . . . . . . 69 Chapter X: The formalism of correspondences . . . . . . . . . . . . . . . . . 71 10.1 Category of correspondences . . . . . . . . . . . . . . . . . . . . . 71 10.2 Monads and modules in the category of correspondences . . . . . . . 73 10.3 Monads and Segal objects . . . . . . . . . . . . . . . . . . . . . . . 73 10.4 The action on bimodules . . . . . . . . . . . . . . . . . . . . . . . . 75 Chapter XI: Sheaf theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 11.1 The notion of a sheaf theory . . . . . . . . . . . . . . . . . . . . . . 76 11.2 Additional base change theorems . . . . . . . . . . . . . . . . . . . 78 11.3 Category of cohomological correspondences . . . . . . . . . . . . . 80 Chapter XII: The categorical trace . . . . . . . . . . . . . . . . . . . . . . . 82 12.1 Hochschild homology . . . . . . . . . . . . . . . . . . . . . . . . . 82 12.2 Functoriality of Hochschild homology . . . . . . . . . . . . . . . . 84 12.3 Categorical traces of rigid monoidal categories . . . . . . . . . . . . 86 Chapter XIII: Geometric traces in sheaf theory . . . . . . . . . . . . . . . . . 91 13.1 Geometric Hochschild homology . . . . . . . . . . . . . . . . . . . 91 13.2 Fixed point objects and geometric traces of convolution categories . . 92 13.3 The geometric trace and relative resolutions . . . . . . . . . . . . . . 95 13.4 Comparison between geometric and ordinary traces . . . . . . . . . 102 13.5 Functoriality of categorical traces in geometric setting . . . . . . . . 105 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Part I Introduction 1 2 Chapter 1 OVERVIEW This thesis can be divided into two parts which are mathematically independent but philosophically related. In this part, we give an informal introduction to the motivations and results. The guiding question of this thesis comes from the Langlands program and specifically the geometric approach to it. Let 𝐹 be a global field, such as a number field or the field of rational functions on a projective curve over a finite field. The Langlands program (Langlands, 1970) consists of many results and conjectures establishing a deep connection between representation theory of analytic nature with arithmetic properties of the field 𝐹. Let’s recall the main protagonist of this program1, the space of automorphic functions. For every place 𝑣 of 𝐹 we denote by 𝐹𝑣 the 𝑣-adic completion of 𝐹 at 𝑣. We will denote by O𝐹 the ring of integers of 𝐹 and for every non-archimedean place 𝑣 of 𝐹, we will denote by O𝑣 the ring of integers of 𝐹𝑣 . For every finite set 𝑆 of places of 𝐹 which contains all the archimedean places, we denote Ö Ö A𝐹,𝑆 = 𝐹𝑣 × O𝑣 𝑣∈𝑆 𝑣∉𝑆 considered as a topological ring with the product topology. We denote by A𝐹 the adele ring of 𝐹. Recall that it can be described, as a topological ring, as the direct limit A𝐹 = lim A𝐹,𝑆 . −−→ 𝑆 This is sometimes referred to as the "restricted product" of the fields 𝐹𝑣 . Let 𝐺 be a connected reductive group over 𝐹, i.e. SL𝑛 , GL𝑛 , Sp2𝑛 . We can now define the space of auytomnorphic functions: A𝐺2 = 𝐿 2 (𝐺 (𝐹)\𝐺 (A𝐹 )), C) namely, it is the space of functions that are square-integrable with respect to some invariant measure on the coset space 𝐺 (𝐹)\𝐺 (A𝐹 ). This space of functions carries an action of 𝐺 (A𝐹 ) by the left action on the coset space. It is sometimes easier to 1 From the point of view of representation theorists. 3 focus on specific subspaces of the automorphic space. For any compact subgroup 𝐾 ⊆ 𝐺 (A𝐹 ) we can consider the space of automorphic functions of level 𝐾: 2 A𝐺,𝐾 = 𝐿 2 (𝐺 (𝐹)\𝐺 (A𝐹 )/𝐾), C) Î Example 1.0.1. Take 𝐹 = Q and 𝐺 = SL2 and 𝐾 = 𝑝 SL2 (Z 𝑝 ). We can relate the space of automorphic functions in the adelic description to a more familiar object. By the strong approximation theorem (Kneser, 1966), for element 𝑔 ∈ SL2 (AQ ) represented as elements (𝑔𝑣 )𝑣 where 𝑔 𝑝 ∈ SL2 (Q 𝑝 ) for prime 𝑝 and 𝑔∞ ∈ SL2 (R), there exists a matrix 𝛾 ∈ SL2 (Q) such that for 𝛾𝑔 we have (𝛾𝑔) 𝑝 ∈ SL2 (Z 𝑝 ) for all prime 𝑝. This means that we can identify the double coset spaces SL2 (Q)\SL2 (AQ )/𝐾  (SL2 (Q) ∩ 𝐾)\SL2 (R) Finally, the intersection SL2 (Q) ∩ 𝐾 consists of matrices whose entries are rational numbers that are integral for any prime 𝑝, which is the same as being integers. This gives an identification: SL2 (Q)\SL2 (AQ )/𝐾  SL2 (Z)\SL2 (R) relating the automorphic functions in the adelic description to the more familiar notion. An important way to study representations in general is to find symmetries of them in the form of commuting linear operators. An important family of such operators are the Hecke operators. For every non-archimidean place 𝑣 of 𝐹 we consider the spherical Hecke algebra: H𝑣 = 𝐶𝑐 (𝐺 (O𝑣 )\𝐺 (𝐹𝑣 )/𝐺 (O𝑣 ), C). This algebra is commutative and acts on the space of automorphic functions by convolution: ∫ ℎ ∗ 𝑓 (𝑔) = 𝑓 (𝑔𝑠−1 )ℎ(𝑠) d𝑠 𝐺 (𝐹𝑣 ) In the revolutionary seminal work (V. Lafforgue, 2018), Vincent Lafforgue showed how to expand this algebra of commuting operators in the function field case by introducing the excursion operators. In the next section, we will try to give a sense of the structures involved and the motivations for the first part in the next chapter. Finally, we will mainly focus on results that come from the algebraic-geometric 4 approach to stufying the space of automorphic functions. In that case, we focus our attention on the subset of continuous compactly supported automorphic functions: A𝐺 = 𝐶𝑐 (𝐺 (𝐹)\𝐺 (A𝐹 )), C) The second part of the thesis concerns the local side of the Langlands program. The main question considered is understanding representations of 𝐺 (𝐹𝑣 ) for all the places 𝑣. These are significant for the study of local fields but can also help understand the global picture. For example, Every irreducible continuous representation 𝜋 of the adele group 𝐺 (A𝐹 ) is equivalent to a "restricted product" 𝜋= ′ Ì 𝜋𝑣 , 𝑣 but we will not go into this more deeply here. In the second part of this overview, we will review the geometric and categorical structures surrounding Hecke, and Iwahori-Hecke algebras, and the categorical trace, which motivates the second part of this thesis. 5 Chapter 2 DRINFELD’S LEMMA AND THE GLOBAL FUNCTION FIELD LANLGANDS CORRESPONDENCE 2.1 The Langlands Correspondence over function fields Let 𝑋 be a projective curve over a finite field F𝑞 with function field 𝐹. Let 𝐺 be a split reductive group defined over 𝐹 For simplicity, we will assume in this section that the center of 𝐺 is trivial and we will focus our attention to the unramified part of the space of automorphic functions. Namely, we consider the subspace of functions 𝐶𝑐 (𝐺 (𝐹)\𝐺 (A𝐹 )/𝐺 (O), Qℓ ) ⊆ 𝐶𝑐 (𝐺 (𝐹)\𝐺 (A𝐹 , Qℓ ) Î where O = 𝑥∈𝑋 O 𝑋,𝑥 . These are the functions which invariant under the action of a maximal compact subgroup corresponding to the integral points on the function field. This double coset space is a discrete groupoid. The study of the function field version of the Langlands correspondence enjoys a rich connection to algebraic geometry thanks to the existance of an algebraic-geometric description of the double cosets 𝐺 (𝐹)\𝐺 (A𝐹 /𝐺 (O). Namely, it can be identified with the set of rational point of an algebraic stack, the stack of 𝐺-bundles on the curve. We will describe geometric objects such as schemes and stacks via their functor of points (Stacks, Tag 01J5). For convenience, for every scheme 𝑌 over F𝑞 and an 𝑏𝐹𝑞 algebra 𝑅, we will denote by 𝑌𝑅 the fiber product 𝑌 ×Spec F𝑞 Spec 𝑅. Let Bun𝐺 (𝑋) denote the moduli stack of 𝐺-bundles on the curve 𝑋. This is the stack described by the following moduli problem: for a commutative F𝑞 -algebra 𝑅, the groupoid of 𝑅-points of Bun𝐺 (𝑋) is given by the groupoid Bun𝐺 (𝑋)(𝑅) = {E a 𝐺-torsor on 𝑋 𝑅 }. The connection to the automorphic space comes from the existance of a natural isomorphism of groupoids Bun𝐺 (𝑋)(F𝑞 ) ≃ 𝐺 (𝐹)\𝐺 (A𝐹 )/𝐺 (O) (2.1) this is known as adelic uniformization. This equivalence is based on the fact that any 𝐺-torsor can be trivialized outside a finite set of points. For example, let’s fix a set 𝑆 of points 𝑥 1 , . . . , 𝑥 𝑛 𝑖𝑛|𝑋 |. By a fundamental result of Beauville and 6 Laszlo(Beauville and Laszlo, 1995), to describe a 𝐺-bundle which is trivial on 𝑋 \ 𝑆, it is enough to specify the "gluing data" consisting of modifications of the trivial bundle around each of the points 𝑥1 , . . . , 𝑥 𝑛 . More explicitly, for every point 𝑥𝑖 we can consider the "disc" and "punctured disc" D𝑖 = Spec O𝑥𝑖 , D𝑖∗ = Spec 𝐹𝑥𝑖 A modification of the trivial bundle E0 around a point 𝑥𝑖 is an automorphism of the restriction E0 | 𝐷 ∗𝑖 . These are exactly given by elements 𝑔𝑖 ∈ 𝐺 (𝐹𝑥𝑖 ), so any given choice of elements 𝑔1 , . . . , 𝑔𝑛 specifies such a bundle. However, some sets of these choices correspond to isomorphic bundles. First, two modifications that differ by an automatism of the restriction E0 | 𝐷 𝑖 to the disc give rise the same bundles. So, accounting for these means a choice of an element in the product of coset spaces: Ö 𝐺 (𝐹𝑥𝑖 )/𝐺 (O 𝑋,𝑥𝑖 ) 𝑖 Finally, two such choices can give isomorphic bundles if they differ by an automorphism of the trivial bundle restricted to 𝑋 \ 𝑆. Such automorphisms are parameterized by 𝐺 (O 𝑋 [𝑥1−1 , . . . , 𝑥 𝑛−1 ]). Thus, the groupoid of 𝐺-bundles which are trivial on 𝑋 \ 𝑆 can be identified with Ö  −1 −1 𝐺 (O 𝑋 [𝑥 1 , . . . , 𝑥 𝑛 ])\ 𝐺 (𝐹𝑥𝑖 )/𝐺 (O 𝑋,𝑥𝑖 ) 𝑖 taking the direct limit over all finite subsets of |𝑋 | results in the description (2.1). Such a description for the space of unramified automorphic functions comes to use when combined with the function-sheaf dictionary. Namely, this is a map: Dctf (Bun𝐺 (𝑋), Qℓ ) → 𝐶𝑐 (Bun𝐺 (𝑋)(F𝑞 ), Qℓ ) taking a constructible complex F ∈ Dctf (Bun𝐺 (𝑋), Qℓ ) to the function 𝑓F : Bun𝐺 (𝑋)(F𝑞 ) → Qℓ defined by sending a point 𝑥 ∈ |𝑋 | to ∑︁ 𝑓F (𝑥) = (−1) 𝑖 · Tr(Frob, H𝑖 (F𝑥 , Qℓ )), 𝑖 where F𝑥 is the stalk of F at 𝑥, considered as a chain complex of Qℓ vector spaces. Thus, studying sheaves on Bun𝐺 (𝑋) could help us understand the space of automorphic functions. 7 As a first attempt, let us try to follow the classical way automorphic function spaces are studied and try to port that into the geometric setting. An important symmetry of the space of unramified automorphic functions is the action of Hecke algebras. These can be understood in a geometric way. For every 𝑥 ∈ |𝑋 | we can parameterize the modifications of a bundle at the point 𝑥. Namely, consider the following moduli problem  Hecke𝐺,𝑥 (𝑅) = (E0 , E1 , 𝛼)|E0 , E1 ∈ Bun𝐺 (𝑋)(𝑅), 𝛼 : E0 | 𝑋𝑅 \{𝑥 𝑅 } ≃ E1 | 𝑋𝑅 \{𝑥 𝑅 } 𝑥 where 𝑥 𝑅 is the subset Spec 𝑅 → − 𝑋 𝑅 . The functor Hecke𝐺,𝑥 is ind-representable and is related to the affine Grassmannian, which parameterized modifications of the trivial bundle, see (Zhu, 2016a). Such a structure gives rise to an integral transform map. Namely, we have a correspondence: Hecke𝐺,𝑥 ℎ𝑙 ℎ𝑟 Bun𝐺 (𝑋) Bun𝐺 (𝑋) given by ℎ𝑙 (E0 , E1 , 𝛼) = E0 and ℎ𝑟 ((E0 , E1 , 𝛼) = E1 . We can use this correspondence to get an action of Dctf (Hecke𝐺,𝑥 ) on the category Dctf (Bun𝐺 (𝑋), Qℓ ). Namely, for every K ∈ Dctf (Hecke𝐺,𝑥 ) we have a functor: 𝑆K : Dctf (Bun𝐺 (𝑋), Qℓ ) → Dctf (Bun𝐺 (𝑋), Qℓ ) given by the "integral transform": 𝑆K (F ) = (ℎ𝑟 )! (ℎ∗𝑙 (F ) ⊗ K), F ∈ Dctf (Bun𝐺 (𝑋), Qℓ ) Under the function-sheaf dictionary, 𝑆K (F ) is identified with the usual Hecke operator. This presentation connects the Hecke operator at the point (or place) 𝑥 with a geometric version of it. One may ask how to make sense of the variation of the point 𝑥? how can we endode the fact that the Hecke operators depend on the choice of a point? We can consider a version of the Hecke stack which lets the point vary. Namely, consider the moduli problem:  Hecke𝐺 (𝑅) = (𝑥, E0 , E1 , 𝛼)|𝑥 ∈ 𝑋 𝑅 , 𝛼 : E0 | 𝑋𝑅 \{𝑥 𝑅 } ≃ E1 | 𝑋𝑅 \{𝑥 𝑅 } which gives rise to the correspondence: Hecke𝐺 ℎ𝑙 𝑋 × Bun𝐺 (𝑋) ℎ𝑟 Bun𝐺 (𝑋) 8 Such a structure gives us a richer theory of integral transforms. Namely, for every lisse sheaf L ∈ Dlis (𝑋, Qℓ ) and K as before, we get a functor 𝑆 L,K : Dctf (Bun𝐺 (𝑋), Qℓ ) → Dctf (Bun𝐺 (𝑋) given by
𝑆 L,K (F ) = (ℎ𝑟 )! ℎ∗𝑙 (L ⊠ F ) ⊗ K . The connection to arithmetic of the function field comes in through the equivalence Dlis (𝑋, Qℓ ) ≃ D𝑏 (Rep(𝜋1𝑒𝑡´ (𝑋), Qℓ ). The geometric Lanlgands program, which informed the work of Lafforgue suggests that it would be necessary to consider multi-point modifications. Namely, for every finite set 𝐼 we consider the moduli of modification of a bundle at 𝐼 points: n o
Hecke𝐺,𝐼 (𝑅) = (𝑥𝑖 )𝑖∈𝐼 , E0 , E1 , 𝛼 : E0 → E1 𝑥𝑖 ∈ 𝑋 𝑅 , E0 | 𝑋\{𝑥𝑖 }𝑖 ∈𝐼 ≃ E1 | 𝑋\{𝑥𝑖 }𝑖 ∈ 𝐼 which give rise to the compatible family of correspondences: Hecke 𝐼 (𝑋) 𝑝 𝑟 𝑋 𝐼 × Bun𝐺 (𝑋) Bun𝐺 (𝑋) However, unlike the case of |𝐼 | = 1, there is no way to relate lisse sheaves on 𝑋 𝐼 with the fundamental group 𝜋1𝑒𝑡´ (𝑋). This is because in general, the natural map 𝜋1𝑒𝑡´ (𝑋 𝐼 ) → 𝜋1𝑒𝑡´ (𝑋) 𝐼 is not an isomorphism. This issue was solved by Drinfeld in the seminal paper Vladimir Gershonovich Drinfeld, 1980, with the realization that the prospective presented above did not bring into account the Frobenius symmetry. Namely, note that rather than just one Frobenius morphism on 𝑋 𝐼 , we should take into account the partial Frobenius morphisms. For every 𝑖 ∈ 𝐼 we have a map 𝜑𝑖 : 𝑋 𝐼 → 𝑋 𝐼 given by the Frobenius map on the 𝑖-th component and the identity on the other components. These morphism commute and satisfy 𝜑1 ◦ · · · ◦ 𝜑𝑛 = Frob 𝑋 𝐼 9 where Frob 𝑋 is the 𝑞-Frobenius on 𝑋 𝐼 and where 𝑛 is the size of 𝑖. Drinfeld realized that we should require this symmetry on the representations as well. Namely, instead of considering all representations of 𝜋1𝑒𝑡´ (𝑋 𝐼 ), we should consider the representations 𝑉 of 𝜋1𝑒𝑡´ (𝑋 𝐼 ) equipped with pair-wise commuting morphisms: 𝜓𝑖 : 𝑉 → 𝑉 as 𝜋1𝑒𝑡´ (𝑋 𝐼 )-representations, such that the total composition 𝜓1 , . . . , 𝜓𝑛 is equivalent to the action of Frob 𝑋 𝐼 . Informally, Drinfeld established an equivalence between representations of 𝜋1𝑒𝑡´ (𝑋 𝐼 ) with a partial Frobenius equivariant structure as above and representations of 𝜋1𝑒𝑡´ (𝑋) 𝐼 . Our first main result is a generalization of Drinfeld’s lemma to the entire derived category of constructible sheaves. 2.2 Drinfeld’s lemma and the Künneth formula for Weil sheaves To state the partial Frobenius equivariance conditions and our results, it is convenient to shift our prespective and formulate the theory in terms of Weil sheaves and Weil prestacks. We give an informal account here, the full details are in Chapter 6. We will use the formalism of prestacks and constructible sheaves on them, for a review of these, see Section 7.2. Fix F = F𝑞 . Instead of working with schemes over F𝑞 and keeping track of the partial and absolute Frobenius maps, we find it preferable to work with geometric object over F. We can do this by associating to a scheme the formal quotient of the base change to the algebraic closure F by Frobenius. Namely, let Let 𝑋 be a scheme over F𝑞 and denote by 𝑋F it’s base change to F. Let 𝜙 𝑋 : 𝑋F → 𝑋F be defined by 𝜙 𝑋 = Frob 𝑋 × idSpec F . Definition 2.2.1. Let 𝑋 be a scheme over F𝑞 . The Weil prestack is defined as   𝜙𝑋 Weil 𝑋 := colim 𝑋 ×F𝑞 F ⇒ 𝑋 ×F𝑞 F ∈ PreStkF , id i.e., it is the prestack sending 𝑅 ∈ CAlgF to the colimit This description works well with products. Namely, for schemes 𝑋1 , 𝑋2 over F𝑞 we have 𝑋1 ×F𝑞 𝑋2 ×F𝑞 Spec F = 𝑋1,F ×F 𝑋2,F 10 We can consider the product of the corresponding Weil prestacks: 𝑋 Weil = 𝑋1Weil ×F 𝑋2Weil . For every prestack, we have a well defined ∞-category of constructible sheaves. The main result of the first part of the thesis can be formulated as follows: Theorem 2.2.2. Let 𝑋1 , 𝑋2 be schemes of finite type over F𝑞 . The external product functor (𝑀1 , 𝑀2 ) ↦→ 𝑀1 ⊠ 𝑀2 induces an equivalence of categories

Dcons 𝑋1Weil , Λ ⊗Λ Dcons (𝑋2Weil , Λ) → Dcons 𝑋1Weil × 𝑘 · · · ×𝑘 𝑋𝑛Weil , Λ For Λ a discrete ℓ-torsion ring, an algebraic extension of Qℓ and its ring of integers. Remark 2.2.3. For lisse sheaves, this holds if Λ is a torsion ring or an extension of Zℓ , and always holds for lisse if 𝑋1 , . . . , 𝑋𝑛 are geometrically unibranch. In particular, we have an equivalence D Repcts (Weil(𝑋), Qℓ ) 2.3
⊗𝐼 → Dlis (𝑋 Weil ) 𝐼 , Qℓ
(2.2) Application to the moduli of bundles and shtukas We can now come back to the situation of the moduli stack of bundles and review the ideas behind the construction of Lafforgue from the perspective of Weil prestacks. First, it was realized by Drinfeld, and later Lafforgue, that in order to take the Frobenius structure into account, we should not only consider bundles but the moduli of shtukas. A full account of this work is well beyond the scope of this introduction, but we will try to convey the main geometrical constructions. First, for every finite set 𝑆, we can consider the prestack of shtukas with |𝑆|-legs defined by the moduli problem whose groupoid of points for every 𝑅 ∈ CAlgF is given by ( Sht𝐺,𝑆 (𝑋)(𝑅) = ((𝑥𝑖 )𝑖∈𝑆 , E, 𝛾) |(𝑥𝑖 ) ∈ (𝑋 Weil ) |𝑆| , E ∈ Bun𝐺 (𝑋 𝑅 ), ) 𝛾 : E | 𝑋𝑅 \𝑥𝑖 ≃ Φ∗𝑋 (E)| 𝑋𝑅 \𝑥𝑖 We can then define a corresponding Hecke stack which parameterize modifications of two shtukas. These depend on three finite sets 𝑆, 𝑇, 𝐼 and given by
𝐼     ((𝑥𝑖 )𝑖∈𝐼 , S0 , S1 , 𝛼) |(𝑥𝑖 ) ∈ 𝑋 Weil ,         Hecke𝐺,𝑆,𝑇,𝐼 (𝑋)(𝑅) = S0 ∈ Sht𝐺,𝑆 (𝑋)(𝑅), S1 ∈ Sht𝐺,𝑇 (𝑋)(𝑅)         𝛼 : E | ≃ E | 0 1 𝑋 \𝑥 𝑋 \𝑥 𝑅 𝑖 𝑅 𝑖   11 where E0 , E1 are the underlying 𝐺-bundles corresponding to the shtukas S0 , S1 , respectively. We then have the Hecke correspondences Hecke𝐺,𝑆,𝑇,𝐼 (𝑋) ℎ𝑙 ℎ𝑟 (𝑋 Weil ) 𝐼 × Sht𝐺,𝑆 (𝑋) Sht𝐺,𝑇 (𝑋) Using the equivalence Dlis ((𝑋 Weil ) 𝐼 ) ≃ D𝑏 (Repcts (Weil(𝑋) 𝐼 , Qℓ )) we can get a family of integral transform functors which generalize the family used in Lafforgue’s work. Namely, for every representation 𝑉 ∈ D𝑏 (Repcts (Weil(𝑋) 𝐼 , Qℓ )) and a complex K ∈ Dcons (Hecke𝐺,𝑆,𝑇,𝐼 (𝑋, Qℓ ))) we have the functor 𝑆 𝑆,𝑇,𝐼,K,𝑉 : Dcons (Sht𝐺,𝑆 (𝑋), Qℓ )) → Dcons (Sht𝐺,𝑇 (𝑋), Qℓ )) given by
𝑆 𝑆,𝑇,𝐼,K,𝑉 (F ) = (ℎ𝑟 )! ℎ∗𝑙 (L ⊠ F ) ⊗ K . These functors are the key component in constructing the "spectral action", see (Zhu, 2021). 12 Chapter 3 THE CATEGORICAL TRACE IN THE CATEGORICAL LOCAL LANLGANDS PROGRAM 3.1 Loop groups We consider the situation where 𝐹 a non-archimedean local field, e.g. Q 𝑝 , F𝑞 ((𝑡)), and 𝑊𝐹 ⊆ Γ𝐹 its Weil group and Galois group. The goal of the local Langlands program is to understand the smooth representations of the group 𝐺 (𝐹). Fix a prime ℓ not equal to 𝑝. We denote by Rep(𝐺 (𝐹), Qℓ ) the category of smooth representations of 𝐺 (𝐹) with coefficients in Qℓ . We denote by D(Rep(𝐺 (𝐹), Qℓ )) the unbounded derived category of Rep(𝐺 (𝐹), Qℓ ). In the categorical variant of the local Langlands program, one wishes to understand the category D(Rep(𝐺 (𝐹), Qℓ )). In this chapter, we will review some of the ways to use algebraic geometry to achieve better understanding of this category. As with the global correspondence, the rich connection to algebraic geometry arises from the realization of some of the structures involved as sets of rational points of an algebraic-geometric object. We begin by describing the corresponding structures for 𝐺 (𝐹) itself. For the sake of simplicity, we will assume 𝐹 is of equal characteristic. Namely, of the form F𝑞 ((𝑡)) for a finite field F𝑞 . However, the definitions and results can be modified to include the mixed-characteristic case as well. We recall the arc-group and loop group associated to 𝐺, these are given by the group-functors sending a F𝑞 algebra 𝑅 to the groups: 𝐿 + 𝐺 (𝑅) = 𝐺 (𝑅[[𝑡]]), 𝐿𝐺 (𝑅) = 𝐺 (𝑅((𝑡))). The functor 𝐿 + 𝐺 is represented by a perfect affine scheme. In particular, 𝐺 (F𝑞 ) = 𝐺 (F𝑞 [[𝑡]]), 𝐿𝐺 (F𝑞 ) = 𝐺 (F𝑞 ((𝑡))). Let 𝐿𝐺/𝐿 + 𝐺 be the associated affine Grassmannian. It is represented by an indprojective perfect ind-scheme. We will denote by 𝐿𝐺 F and 𝐿 + 𝐺 F the base change of the loop and arc-group to the fixed algebraic closure F of F𝑞 . In the global case, the connection from algebraic object (i.e. sheaves) into the automorphic objects was facilitated by the function-sheaf dictionary. The function 13 sheaf dictionary can be seen as a way to take the "trace of the Frobenius" of an object. However, in the local theory, our target object is not a function space, but a category. This suggests the need to have a categorical way of taking the trace. This is exactly what is achieved by the categorical trace constructions. These ideas originated by Drinfeld, Gaitsgory, and others. See (Dennis Gaitsgory, 2016) for a review. We will see how these ideas work in the setting of unipotent representations. 3.2 Convolution Patterns and the Iwahori-Hecke category The abstract results of the second part of the thesis are motivated by the understanding of unipotent representations. It is the subcategory of smooth representations of 𝐺 generated by the compact induction from an Iwahori subgroup. Fix a Borel subgroup 𝐵 ⊆ 𝐺 and recall that the corresponding Iwahori subgroup 𝐼 ⊆ 𝐺 (F𝑞 [[𝑡]]) is the inverse image of 𝐵(F𝑞 ) under the surjection 𝐺 (F𝑞 [[𝑡]]) → 𝐺 (F𝑞 ). We can then consider the compactly induced representation 𝑐-ind𝐺𝐼 (𝐹) (Qℓ ). The endomorphism algebra: H 𝐼 = EndRep(𝐺 (𝐹)) (𝑐-ind𝐺𝐼 (𝐹) , 𝑐-ind𝐺𝐼 (𝐹) ), is known as the Iwahori-Hecke algebra. Up to a choice of Haar measure on 𝐺 (𝐹), there is a canonical isomorphism: H 𝐼 = 𝐼\𝐺 (F𝑞 ((𝑡)))/𝐼. The composition action on 𝐻 𝐼 corresponds to the convolution operation on the double coset space. A complex 𝑉 ∈ D(Rep(𝐺), Qℓ ) is called unipoitent if it is in the full subcategory generated by 𝑐-ind𝐺𝐼 (𝐹) by colimits and retracts. Equivalently, 𝑉 is unipotent if the cohomologies 𝐻 𝑖 (𝑉) are unipotent representations in the sense of (Lusztig, 1995). The full subcategory of unipotent representations is equivalent to the derived category of left-𝐻 𝐼 modules. This equivalence is obtained via: 𝑉 ↦→ HomD(Rep(𝐺),Qℓ ) (𝑐-ind𝐺𝐼 (𝐹) , 𝑉). We can use algebraic geometry to study the category of unipotent representations. We denote by 𝜋+ : 𝐿 + 𝐺 → 𝐺 the map corresponding to sending 𝑡 to zero. This 14 subgroup can also be lifted to an algebraic geometric object. Namely, we define the functor Iw by Iw(𝑅) = {𝑔 ∈ 𝐺 (𝑅[[𝑡]])| 𝜋+ (𝑔) ∈ 𝐵(𝑅)}. Then the Iwahori-Hecke algebra 𝐻 𝐼 identifies with the F𝑞 -points of the IwahoriHecke stack. This stack can be described by the the fpqc-quotient IwHecke ≃ [Iw\𝐿𝐺/Iw]. The category of constructible sheaves Dcons (IwHecke 𝐼,F , Qℓ ) on the base change to the algebraic closure carries a monoidal structure given by convolution. In order to get a sense of how this works, it would be useful to consider the general situation of convolution monoidal structures. In general, these arise from a "niceenough" morsphism 𝑓 : (𝑋, 𝜙 𝑋 ) → (𝑌 , 𝜙𝑌 ). In such a situation, Dcons (𝑋 ×𝑌 𝑋) has a convolution monoidal structure, given by 𝑋 ×𝑌 𝑋 ×𝑌 𝑋 Δ (𝑋 ×𝑌 𝑋) × (𝑋 ×𝑌 𝑋) 𝑚 𝑋 ×𝑌 𝑋 Namely, the monoidal operation is given by: F ★ G := 𝑚 ∗ Δ! (F ⊠ G), F , G ∈ D (𝑋 ×𝑌 𝑋) In our case, we will take 𝑋 to be the classifying stack 𝐵(IwF ) and 𝑌 to be the classifying stack 𝐵(𝐿𝐺 F ) of the loop group. For many purposes, the morphism 𝐵(IwF ) → 𝐵(𝐿𝐺 F ) behaves like a proper morphism. For more details, see (Zhu, 2021) and (Bouthier, Kazhdan, and Varshavsky, 2020). Under the identifications 𝑋 ×𝑌 𝑋 ≃ Iw\𝐿𝐺/Iw, 𝑋 ×𝑌 𝑋 ×𝑌 𝑋 ≃ Iw\𝐿𝐺 ×Iw 𝐿𝐺/Iw the general convolution monoidal structure identifies with the familiar description of convolution on Iwahori-Hecke algebras. As we did with the function sheaf dictionary, we can relate this monoidal category to the category of unipotent representations by taking the "trace of the Frobenius". Namely, the Frobenius map on Iw and 𝐿𝐺 give corresponding maps on IwF and 𝐿𝐺 F and a corresponding map on the classifying spaces. 15 3.3 The Categorical trace construction Given a Λ-linear monoidal ∞-category A together with a monoidal endomorphism 𝜙 : A → A, its categorical trace is given by the Hochschild homology Tr(A, 𝜙) := HH(A, 𝜙 A) = A ⊗A⊗A rev 𝜙 A Roughly speaking, it is determined by the following universal property: a functor Tr(A, 𝜙) → C is equivalent to a functor F : A → C equipped with equivalences 𝐹 (𝑎 ⊗ 𝑏) ≃ 𝐹 (𝑏 ⊗ 𝜙(𝑎)), 𝑎, 𝑏 ∈ Ob(A) together with all the necessary higher coherence data. In particular, there is a tautological functor Tr𝜙 : A → Tr(A, 𝜙) sending an object 𝑎 to its universal 𝜙-twisted trace. The trace Tr(A, 𝜙) always exists for general reasons, and it can be computed as the using the cyclic bar resolution. Namely, as the homotopy colimit of the simplicial object HH𝑛 (A, 𝜙 A) = A ⊗(𝑛+1) = A ⊗ · · · ⊗ A, 𝑛 ≥ 0 which has boundary maps 𝑑𝑖 (𝑎 0 ⊗ . . . , 𝑎𝑖−1 ⊗ 𝑎𝑖 ⊗ · · · ⊗ 𝑎 𝑛 ) =    𝑎 0 ⊗ · · · ⊗ 𝑎𝑖−1 𝑎𝑖 ⊗ · · · ⊗ 𝑎 𝑛  𝑖≠0   𝑎 1 ⊗ 𝑎 2 ⊗ · · · ⊗ 𝑎 𝑛 𝜙(𝑎 0 ) 𝑖=0  In general, it is difficult to compute such homotopy colimits, however, under certain conditions it is possible to say more in the case of convolution patterns as in the previous section. For monoidial structures arising from convooution patters, it is sometimes possible to realize the 𝜙-twisted trace as a full subcategory of the category of sheaves of a certain geometric object. Namely, consider the "fixed point object" L 𝜙 (𝑌 ) 𝑌 Δ𝑌 𝑌 id𝑌 ×𝜙𝑌 𝑌 We have a correspondence L 𝜙 (𝑌 ) ×𝑌 𝑋 L 𝜙 (𝑌 ) 𝑋 ×𝑌 𝑋 16 Using the identification 𝑋 ×𝑌 L 𝜙 (𝑌 ) ≃ 𝑋 ×𝑌 ×𝑌 𝑌 . The main result of the second part of the thesis can be informally formulated as follows: Theorem 3.3.1. Proposition (Ben Zvi-Nadler, H.-Zhu) Under suitable assumptions, there is a canonical commutative diagram D (𝑋 ×𝑌 𝑋) D (𝑋 ×𝑌 L 𝜙 (𝑌 )) Tr(D (𝑋 ×𝑌 𝑋), 𝜙) D (L 𝜙 (𝑌 )) with the bottom horizontal arrow fully faithful Let’s try to apply this paradigm to our case of 𝐵(IwF ) → 𝐵(𝐿𝐺 F ) and 𝜙 given by the Frobenius which we denote by 𝜎. In this case we have the fixed point object 𝐿𝐺/Ad𝜎 (𝐿𝐺) 𝐵𝐿𝐺 Δ 𝐵𝐿𝐺 id×𝜎 𝐵𝐿𝐺 and we can also identify: 𝑋 ×𝑌 𝑋 ≃ 𝐿𝐺 F /Ad𝜎 (IwF ) This means, given the correct definitions of categories of sheaves on the corresponding geometric objects, that our main result gives a fully faithful functor 𝑡𝑟 (Dcons (Iw\𝐿𝐺/Iw, Qℓ ), 𝜙) → D(𝐿𝐺/Ad𝜎 (𝐿𝐺), Qℓ ). These definitions and its implication to the study of the category of unipotent representations are the subject of a joint project with Xinwen Zhu and are beyond the scope of this thesis. Part II A Categorical Künneth Formula For Weil Sheaves 17 18 Chapter 4 RECOLLECTIONS ON INFINITY CATEGORIES Throughout this part, Λ denotes a unital, commutative ring. We briefly collect some notation pertaining to ∞-categories from (Lurie, 2017; Lurie, 2009). As in (Lurie, 2009, Section 5.5.3), PrL denotes the ∞-category of presentable ∞-categories with colimit-preserving functors. It contains the subcategory PrSt ⊂ PrL consisting of stable ∞-categories. 4.1 Monoidal aspects The category PrL carries the Lurie tensor product (Lurie, 2017, Section 4.8.1). This tensor product induces one on the full subcategory PrSt ⊂ PrL consisting of stable ∞-categories (Lurie, 2017, Proposition 4.8.2.18). For our commutative ring Λ, the ∞-category ModΛ of chain complexes of Λ-modules, up to quasi-isomorphism, is a commutative monoid in PrSt with respect to this tensor product. This structure includes, in particular, the existence of a functor ModΛ × ModΛ → ModΛ which, after passing to the homotopy categories is the classical derived tensor product on the unbounded derived category of Λ-modules. St We define PrSt Λ to be the category of modules, in Pr , over ModΛ . Noting that modules over ModΛ are in particular modules over Sp, the ∞-category of spectra, PrSt Λ can be described as the ∞-category consisting of stable presentable ∞-categories together with a Λ-linear structure, such that functors are continuous and Λ-linear. Therefore PrSt Λ carries a symmetric monoidal structure, whose unit is ModΛ . We will also denote by PrSt 𝜔 the category of compactly generated presentable with functors that send compact objects to compact objects (equivalently, those whose right adjoint is continuous). In order to express monoidal properties of ∞-categories consisting, say, of bounded complexes, recall from (Lurie, 2017, Corollary 4.8.1.4 joint with Lemma 5.3.2.11) or (Ben-Zvi, Francis, and Nadler, 2010, Proposition 4.4) the symmetric monoidal structure on the ∞-category CatEx ∞ (Idem) of idempotent complete stable ∞-categories 19 and exact functors: it is characterized by 𝐷 1 ⊗ 𝐷 2 := Ind(𝐷 1 ) ⊗ Ind(𝐷 2 )
𝜔 (4.1) that is, the compact objects in the Lurie tensor product of the Ind-completions. With respect to these monoidal structures, the Ind-completion functor (taking values in compactly generated presentable ∞-categories with the Lurie tensor product) and the functor forgetting the compact generated-ness: Ind St CatEx −→ PrSt ∞ (Idem) − 𝜔 ↩→ Pr (4.2) are both symmetric monoidal (Lurie, 2017, Lemmas 5.3.2.9, 5.3.2.11). The subcategory of compact objects in ModΛ is given by perfect complexes of Λ-modules (Lurie, 2017, Proposition 7.2.4.2.). It is denoted Perf Λ . Under the equivalence in Equation (4.2), the category Perf Λ ⊆ CatEx ∞ (Idem) corresponds to Ex ModΛ . Moreover, Perf Λ is a commutative monoid in Cat∞ (Idem), so that we can consider its category of modules, denoted as CatEx ∞,Λ (Idem). This category inherits a symmetric monoidal structure denoted by 𝐷 1 ⊗Perf Λ 𝐷 2 . Any stable ∞-category 𝐷 is canonically enriched over the category of spectra Sp. We write Hom𝐷 (−, −) for the mapping spectrum. Any category in PrSt Λ is canonically enriched over ModΛ , so that we refer to Hom𝐷 (−, −) ∈ ModΛ as the mapping complex. For example, for 𝑀, 𝑁 ∈ ModΛ , then HomModΛ (𝑀, 𝑁) is commonly also denoted by Hom(𝑀, 𝑁). Its 𝑛-th cohomology is the Hom-group Hom(𝑀, 𝑁 [𝑛]) in the classical derived category. 4.2 Fixed points of infinity categories A basic structure in Drinfeld’s lemma is the equivariance datum for the partial Frobenii. In this section, we assemble some abstract results where such ∞categorical constructions are carried out. Definition 4.2.1. Let 𝜙 : 𝐷 → 𝐷 be an endofunctor in CatEx ∞ (Idem). The category of 𝜙-fixed points is   𝜙 𝜙=id 𝐷 := Fix(𝐷, 𝜙) := lim 𝐷 ⇒ 𝐷 . id𝐷 Recall that for a symmetric monoidal ∞-category 𝐷, a commutative monoid object Λ ∈ CAlg(𝐷), the forgetful functors CAlg(𝐷) → 𝐷 and ModΛ (𝐷) → 𝐷 preserve limits (Lurie, 2017, Corollary 3.2.2.5, Corollary 4.2.3.3). In particular, if 𝐷 is in 20 addition Λ-linear, that is, an object in CatEx ∞,Λ (Idem), and 𝜙 is also Λ-linear, then Fix(𝐷, 𝜙) admits a natural Λ-linear structure as well. Because of these facts, we will usually not specify where the limit above is formed. Note that all functors Ind St (∗) St L š CatEx ∞ (Idem) −→ Pr𝜔 −→ Pr −→ Pr −→ Cat∞  (4.3) except for the forgetful functor marked (∗) preserve limits, see (Lurie, 2017, Corollary 4.2.3.3) and (Lurie, 2009, Proposition 5.5.3.13) for the rightmost two functors. To give a concrete example of that failure in our situation, note that Fix(𝐷, id𝐷 ) = Fun(𝐵Z, 𝐷), that is, objects are pairs (𝑀, 𝛼) consisting of some 𝑀 ∈ 𝐷 and some automorphism 𝛼 : 𝑀  𝑀. Now consider 𝐷 = Vectf.d Λ , the (abelian) category of finite-dimensional vector spaces over a field Λ. The natural functor  



f.d f.d f.d Ind lim Vectf.d ⇒ Vect → lim Ind Vect ⇒ Ind Vect = lim VectΛ ⇒ VectΛ Λ Λ Λ Λ is fully faithful, but not essentially surjective: given an automorphism 𝛼 of an Ð infinite-dimensional vector space 𝑀, there need not be a filtration 𝑀 = 𝑀𝑖 by finite-dimensional subspaces 𝑀𝑖 that is compatible with 𝛼. Fixed point categories inherit t-structures as follows: Lemma 4.2.2. Let 𝜙 : 𝐷 → 𝐷 be a functor in CatEx ∞ (Idem). Suppose 𝐷 carries a t-structure such that 𝜙 is t-exact. Then Fix(𝐷, 𝜙) carries a unique t-structure such that the evaluation functor is t-exact. There is a natural equivalence  Fix(𝐷 ♥ , 𝜙) −→ Fix(𝐷, 𝜙) ♥ . e := Fix(𝐷, 𝜙). For • being either “≤ 0” or “≥ 0”, we Proof. Let us abbreviate 𝐷 e• := Fix(𝐷 • , 𝜙), which is a (non-stable) ∞-category. This is clearly the only put 𝐷 choice for a t-structure making ev a t-exact functor. It satisfies the claim about the hearts of the t-structure by definition. We need to show that it is a t-structure. Being a limit of full subcategories, the e• are full subcategories of 𝐷. e Since 𝜙, being t-exact, commutes with categories 𝐷 e For 𝑀 ∈ 𝐷 e≤0 , 𝜏𝐷≤0 and 𝜏𝐷≥0 , these two functors also yield truncation functors for 𝐷. e≥1 (we use cohomological conventions), we have 𝑁∈𝐷 Hom𝐷e 𝑀, 𝑁

= lim Hom𝐷 (𝑀, 𝑁) ⇒ Hom𝐷 (𝑀, 𝑁) , 21 where on the right hand side 𝑀, 𝑁 denote the underlying objects in 𝐷. Since 𝑀 ∈ 𝐷 ≤0 , 𝑁 ∈ 𝐷 ≥1 , we have H𝑖 Hom𝐷 (𝑀, 𝑁) = 0 for 𝑖 = −1, 0. Thus, H0 Hom𝐷e (𝑀, 𝑁) = 0 as well. □ 4.2.1 can be generalized as follows: Let 𝜑 : 𝐵Z𝑛 → CatEx ∞ (Idem) be a diagram. For example, for 𝑛 = 1, this amounts to giving 𝐷 = 𝜑(∗) ∈ CatEx ∞ (Idem) and an equivalence 𝜙 = 𝜑(1) : 𝐷 → 𝐷. For 𝑛 = 2, such a datum corresponds to giving 𝐷,   equivalences 𝜙1 , 𝜙2 : 𝐷 → 𝐷 together with an equivalence 𝜙1 ◦ 𝜙2 → 𝜙2 ◦ 𝜙1 . So we define the ∞-category of simultaneous fixed points as Fix(𝐷, 𝜙1 , . . . , 𝜙𝑛 ) := lim 𝜑 ∈ CatEx ∞ (Idem). Remark 4.2.3. The statement of 4.2.2 carries over verbatim assuming that 𝐷 has a t-structure and all 𝜙𝑖 are t-exact, noting that 𝐵Z𝑛 = (𝑆 1 ) 𝑛 is a finite simplicial set. Lemma 4.2.4. Let 𝜑 : 𝐵Z𝑛 → CatEx ∞ (Idem) be a diagram. Denote 𝐷 = 𝜑(∗) and 𝜙𝑖 = 𝜑(𝑒𝑖 ) for the 𝑖-th standard vector 𝑒𝑖 ∈ Z𝑛 . The functor

Fix 𝐷, 𝜙1 , . . . , 𝜙𝑛 → Fix Ind(𝐷), 𝜙1 , . . . , 𝜙𝑛 induced from the inclusion 𝐷 ⊂ Ind(𝐷) is fully faithful and takes values in compact objects. In particular, it yields a fully faithful functor

Ind Fix(𝐷, 𝜙1 , . . . , 𝜙𝑛 ) → Fix Ind(𝐷), 𝜙1 , . . . , 𝜙𝑛 . Proof. Let 𝑀 ∈ Fix(𝐷, 𝜙1 , . . . , 𝜙𝑛 ) and denote its underlying object in 𝐷 by the same symbol. For every 𝑁 ∈ Fix Ind(𝐷), 𝜙1 , . . . , 𝜙𝑛 ), we have a limit diagram of mapping complexes
HomFix(Ind(𝐷)) (𝑀, 𝑁)  Fix HomInd(𝐷) (𝑀, 𝑁), 𝜙1 , . . . , 𝜙𝑛 . Since filtered colimits commute with finite limits in the ∞-category of anima (a.k.a. spaces) (Lurie, 2009, Proposition 5.3.3.3.), we see that 𝑀 is compact in Fix(Ind(𝐷)) because 𝑀 is so in Ind(𝐷). □ Lemma 4.2.5. Let 𝜑𝑖 : 𝐵Z → CatEx ∞,Λ (Idem), 𝑖 = 1, . . . , 𝑛 be given. Denote 𝐷 𝑖 = e𝑖 = Ind(𝐷 𝑖 ). Then there is a canonical equivalence 𝜑𝑖 (∗), 𝜙𝑖 = 𝜑𝑖 (1) and 𝐷  

 e1 , 𝜙1 ⊗ModΛ . . . ⊗ModΛ Fix 𝐷 e𝑛 , 𝜙𝑛 → e1 ⊗ModΛ . . . ⊗ModΛ 𝐷 e𝑛 , 𝜙1 , . . . , 𝜙𝑛 . Fix 𝐷 Fix 𝐷 22 e𝑖 , 𝜙𝑖 ) are compactly generated: the forgetful functor Proof. The categories Fix( 𝐷 e𝑖 , 𝜙𝑖 ) → 𝐷 e𝑖 = Ind(𝐷 𝑖 ) preserves colimits, so its left adjoint 𝐿 preserves 𝑈 : Fix( 𝐷 compact objects. Moreover, 𝑈 is conservative, so that the objects 𝐿(𝑑𝑖 ), for 𝑑𝑖 ∈ 𝐷 𝑖 , form a family of compact generators. Then, we use that any compactly generated category in PrSt Λ is dualizable (Lurie, 2018, Remark D.7.7.6 (1)) so that tensoring with it preserves limits. □ 23 Chapter 5 LISSE AND CONSTRUCTIBLE SHEAVES In order to state and prove the categorical Künneth formula for Weil sheaves, we use the framework for lisse and constructible sheaves provided by (Hemo, Richarz, and Scholbach, 2021). For the convenience of the reader, we collect here some basics of the formalism. Throughout, Λ denotes a condensed ring, for example any T1-topological ring such as discrete rings, algebraic extensions 𝐸/Qℓ or their ring of integers O𝐸 . In the synopsis below, we refer to the latter choices of Λ as the standard coefficient rings. We write Λ∗ for the underlying ring. Let D(𝑋, Λ) be the derived category of sheaves of Λ-modules on the proétale site 𝑋proét . Definition 5.0.1. For every scheme 𝑋 and every condensed ring Λ, there are the full subcategories Dlis (𝑋, Λ) ⊂ Dcons (𝑋, Λ) ⊂ D(𝑋, Λ) (5.1) By definition, the left hand category of lisse sheaves consists of the dualizable objects in the right-most category. An object (henceforth referred to as a sheaf) 𝑀 in the right hand category is constructible, if on any affine 𝑈 ⊂ 𝑋 there is a finite stratification into constructible locally closed subschemes 𝑈𝑖 ⊂ 𝑈 such that 𝑀 |𝑈𝑖 is lisse, that is, dualizable. Finally, an ind-lisse (respectively, ind-constructible) sheaf is a filtered colimit, in the category D(𝑋, Λ), of lisse (respectively, constructible) sheaves. The corresponding full subcategories of D(𝑋, Λ) are denoted by Dindlis (𝑋, Λ) ⊂ Dindcons (𝑋, Λ) ⊂ D(𝑋, Λ). For the standard coefficient rings Λ above and quasi-compact quasi-separated (qcqs) schemes 𝑋, that definition of lisse and constructible sheaves agrees with the classical ones, see loc. cit. for details. (i) Via the natural functor ModΛ∗ → D(𝑋, Λ), 𝑀 ↦→ 𝑀 ⊗Λ∗ Λ 𝑋 , the category D(𝑋, Λ) is an object in PrSt Λ∗ . The functor restricts to a functor Perf Λ∗ → Dlis (𝑋, Λ), and the categories Dlis (𝑋, Λ) ⊂ Dcons (𝑋, Λ) are objects of the category CatEx ∞,Λ (Idem). In particular, all categories listed in Equation (5.1) are stable idempotent complete Λ∗ -linear ∞-categories. 24 (ii) The extension-by-zero functor along any constructible locally closed immersion and quasi-compact étale morphisms preserves constructibility. (iii) The functors 𝑋 ↦→ Dcons (𝑋, Λ) and 𝑋 ↦→ Dlis (𝑋, Λ) satisfy proétale hyperdescent, and the functor 𝑋 ↦→ Dindcons (𝑋, Λ), resp. 𝑋 ↦→ Dindlis (𝑋, Λ) satisfies hyperdescent for quasi-compact étale, resp. finite étale covers. (iv) If Λ = colim Λ𝑖 is a filtered colimit of condensed rings and 𝑋 is qcqs, then the natural functors   colim Dlis (𝑋, Λ𝑖 ) −→ Dlis (𝑋, Λ), colim Dcons (𝑋, Λ𝑖 ) −→ Dcons (𝑋, Λ) are equivalences. The categories enjoy the following properties: (v) If 𝑋 is qcqs, then any constructible sheaf is bounded with respect to the t-structure on D(𝑋, Λ). (vi) For 𝑋 locally Noetherian (and much more generally), the t-structure on D(𝑋, Λ) restricts to one on Dlis (𝑋, Λ) and Dcons (𝑋, Λ) provided that Λ is t-admissible in the sense of (Hemo, Richarz, and Scholbach, 2021). The topological condition on the condensed structure of Λ is satisfied for all the standard coefficient rings listed above. (vii) For 𝑋 locally Noetherian (and again more generally), a sheaf is lisse if and only if it is proétale locally the constant sheaf associated to a perfect complex of Λ∗ -modules. (viii) Let 𝑋 be a qcqs scheme. If the Λ-cohomological dimension is uniformly bounded for all proétale affines 𝑈 = lim𝑖 𝑈𝑖 over 𝑋, then Ind(Dcons (𝑋, Λ)) = Dindcons (𝑋, Λ) and likewise for ind-lisse sheaves. If 𝑋 is of finite type over F𝑞 or a separably closed field, this condition holds for any of the above standard rings. For discrete 𝑝-torsion rings, algebraic extensions 𝐸/Q 𝑝 and their ring of integers O𝐸 , this holds for arbitrary qcqs schemes in characteristic 𝑝. For schemes 𝑋1 , . . . , 𝑋𝑛 over a fixed base scheme 𝑆 (for example, the spectrum of a field) and a condensed ring Λ, we denote the external product in the usual way: ⊠ : D(𝑋1 , Λ) × . . . × D(𝑋𝑛 , Λ) −→ D (𝑋1 ×𝑆 . . . ×𝑆 𝑋𝑛 , Λ) , (𝑀1 , . . . , 𝑀𝑛 ) ↦−→ 𝑀1 ⊠ . . . ⊠ 𝑀𝑛 := 𝑝 ∗1 (𝑀1 ) ⊗Λ𝑋 . . . ⊗Λ𝑋 𝑝 ∗𝑛 (𝑀𝑛 ). 25 Here 𝑝𝑖 : 𝑋 := 𝑋1 ×𝑆 . . . ×𝑆 𝑋𝑛 → 𝑋𝑖 are the projections. This functor induces the functor ⊠ : D(𝑋1 , Λ) ⊗ModΛ∗ . . . ⊗ModΛ∗ D(𝑋𝑛 , Λ) → D (𝑋1 ×𝑆 . . . ×𝑆 𝑋𝑛 , Λ) , (5.2) St in PrSt Λ∗ . Here we regard D(𝑋𝑖 , Λ) as objects in PrΛ∗ , like in Item i in the synopsis above. The external tensor product of constructible sheaves is again constructible, and hence induces a functor ⊠ : Dcons (𝑋1 , Λ) ⊗Perf Λ∗ . . . ⊗Perf Λ∗ Dcons (𝑋𝑛 , Λ) → Dcons (𝑋1 ×𝑆 . . . ×𝑆 𝑋𝑛 , Λ) , (5.3) Ex in Cat∞,Λ∗ (Idem) and likewise for the categories of ind-constructible, resp. (ind)lisse sheaves. 26 Chapter 6 CONSTRUCTIBLE WEIL SHEAVES In this section, we introduce the categories


Dlis 𝑋 Weil , Λ ⊆ Dcons 𝑋 Weil , Λ ⊆ D 𝑋 Weil , Λ consisting of lisse, resp. constructible, resp. all Weil sheaves. These are the categories featuring in the categorical Künneth formula (2.2.2). Throughout this section, 𝑋 is a scheme over a finite field F𝑞 of characteristic 𝑝 > 0. Unless the contrary is mentioned, we impose no conditions on 𝑋. Moreover, Λ is a condensed ring. We fix an algebraic closure F of F𝑞 , and denote by 𝑋F := 𝑋 ×F𝑞 Spec F the base change. Denote by 𝜙 𝑋 (resp. 𝜙F ) the endomorphism of 𝑋F that is the 𝑞-Frobenius on 𝑋 (resp. Spec F) and the identity on the other factor. Let Dlis (𝑋F , Λ) ⊂ Dcons (𝑋F , Λ) ⊂ D(𝑋F , Λ) be the categories of lisse, resp. constructible, resp. all proétale sheaves of Λ-modules on 𝑋F (5.0.1). These categories are objects in CatEx ∞,Λ∗ (Idem), that is, Λ∗ -linear stable idempotent complete symmetric monoidal ∞-categories where Λ∗ = Γ(∗, Λ) is the underlying ring. 6.1 The Weil-proétale site The Weil-étale topology for schemes over finite field is introduced in (Lichtenbaum, 2005) see also (Geisser, 2004). Our approach for the proétale topology is slightly different: Weil , is the following Definition 6.1.1. The Weil-proétale site of 𝑋, denoted by 𝑋proét Weil are pairs (𝑈, 𝜑) consisting of 𝑈 ∈ (𝑋 ) site: Objects in 𝑋proét F proét equipped with an endomorphism 𝜑 : 𝑈 → 𝑈 of F-schemes such that the map 𝑈 → 𝑋F intertwines Weil are given by equivariant maps, and a family 𝜑 and 𝜙 𝑋 . Morphisms in 𝑋proét {(𝑈𝑖 , 𝜑𝑖 ) → (𝑈, 𝜑)} of morphisms is a cover if the family {𝑈𝑖 → 𝑈} is a cover in (𝑋F )proét . Weil admits small limits formed componentwise as Note that 𝑋proét lim(𝑈𝑖 , 𝜑𝑖 ) = (lim 𝑈𝑖 , lim 𝜑𝑖 ). 27 In particular, there are limit-preserving maps of sites Weil (𝑋F )proét → 𝑋proét → 𝑋proét (6.1) given by the functors (in the opposite direction) 𝑈 ← (𝑈, 𝜑) and (𝑈F , 𝜙𝑈 ) ← 𝑈. We denote by D(𝑋 Weil , Λ) the unbounded derived category of sheaves of Λ 𝑋 -modules Weil . The maps of sites (6.1) induce functors on 𝑋proét D(𝑋, Λ) → D(𝑋 Weil , Λ) → D(𝑋F , Λ), (6.2) whose composition is the usual pullback functor along 𝑋F → 𝑋. Remark 6.1.2. The functor D(𝑋, Λ) → D(𝑋 Weil , Λ) is not an equivalence in general. This relates to the difference between continuous representations Galois versus Weil groups. See, however, Proposition 6.6.1 for filtered colimits of finite discrete rings Λ. We have the following basic functoriality: Let 𝑗 : 𝑈 → 𝑋 be a weakly étale morWeil . Then the slice phism and consider the corresponding object (𝑈F , 𝜙𝑈 ) of 𝑋proét Weil ) Weil op → PrSt , site (𝑋proét /(𝑈F ,𝜙𝑈 ) is equivalent to 𝑈proét . This gives a functor (𝑋proét ) Λ 𝑈 ↦→ D(𝑈 Weil , Λ) which is a hypercomplete sheaf of Λ∗ -linear presentable stable categories. Also, we obtain an adjunction 𝑗 ! : D(𝑈 Weil , Λ) ⇄ D(𝑋 Weil , Λ) : 𝑗 ∗ that is compatible with the (( 𝑗F )! , ( 𝑗 F ) ∗ )-adunction under (6.2). The category D(𝑋 Weil , Λ) is equivalent to the category of 𝜙 𝑋 -equivariant sheaves on 𝑋F , as we will now explain. Weil with 𝑋 = Z𝑖+1 × 𝑋 the For each 𝑖 ≥ 0, consider the object (𝑋𝑖 , Φ𝑖 ) ∈ 𝑋proét 𝑖 F countably disjoint union of 𝑋F , the map 𝑋𝑖 → 𝑋F given by projection and the endomorphism Φ𝑖 : 𝑋𝑖 → 𝑋𝑖 given by (𝑛, 𝑥) ↦→ (𝑛 − (1, . . . , 1), 𝜙 𝑋 (𝑥)) on sections. The inclusion Z𝑖 → Z𝑖+1 , 𝑛 ↦→ (0, 𝑛) induces a map of schemes 𝑋𝑖−1 → 𝑋𝑖 where 𝑋−1 := 𝑋F . By pullback, we get a limit-preserving map of sites   Weil (𝑋𝑖−1 )proét → 𝑋proét (6.3) /(𝑋𝑖 ,Φ𝑖 ) Lemma 6.1.3. For each 𝑖 ≥ 0, the map (6.3) induces an equivalence on the associated 1-topoi. 28 Proof. As universal homeomorphisms induce equivalences on proétale 1-topoi (Bhatt and Scholze, 2015, Lemma 5.4.2), we may assume that 𝑋 is perfect. In this case, the sites (6.3) are equivalent because 𝜙 𝑋 is an isomorphism. Explicitly, an à inverse is given by sending an object 𝑈 ∈ (𝑋𝑖−1 )proét to the object 𝑉 = 𝑛∈Z𝑖+1 𝑉𝑛 , 𝑉𝑛 → {𝑛} × 𝑋F defined by 𝑉𝑛 = 𝑈 (𝑛2 −𝑛1 ,...,𝑛𝑖+1 −𝑛1 ) × 𝑋F ,𝜙𝑛1 𝑋F , 𝑋 and with endomorphism 𝜑 : 𝑉 → 𝑉 defined by the maps 𝑉𝑛 = 𝑉𝑛−(1,...,1) × 𝑋F ,𝜙 𝑋 𝑋F → □ 𝑉𝑛−(1,...,1) . Weil sheaves admit the following presentation as the 𝜙∗𝑋 -fixed points of D(𝑋F , Λ), see 4.2.1: Proposition 6.1.4. The last functor in (6.2) induces an equivalence ! ∗ 𝜙𝑋 D(𝑋 Weil , Λ)  lim D(𝑋F , Λ) ⇒ D(𝑋F , Λ) . (6.4) id Remark 6.1.5. Objects in Equation (6.4) are pairs (𝑀, 𝛼) where 𝑀 ∈ D(𝑋F , Λ) and 𝛼 is an isomorphism 𝑀  𝜙∗𝑋 𝑀. Note that the composition 𝜙 𝑋 ◦ 𝜙F is the absolute 𝑞-Frobenius of 𝑋F . In particular, it induces the identity on proétale topoi, see (Bhatt and Scholze, 2015, Lemma 5.4.2). Therefore, replacing 𝜙∗𝑋 by 𝜙∗F in Equation (6.4) yields an equivalent category. Proof of Proposition 6.1.4. The structural morphism (𝑋0 , Φ0 ) → (𝑋F , 𝜙 𝑋 ) is a Weil . Its Čech nerve has objects (𝑋 , Φ ) ∈ 𝑋 Weil , 𝑖 ≥ 0 as above. By cover in 𝑋proét 𝑖 𝑖 proét descent, there is an equivalence 

 Weil D 𝑋 Weil , Λ −→ Tot D (𝑋proét )/(𝑋• ,Φ• ) , Λ . (6.5) Weil ) Under Lemma 6.1.3, the cosimplicial 1-topos associated with (𝑋proét /(𝑋• ,Φ• ) is equivalent to the cosimplicial 1-topos associated with the action of 𝜙∗𝑋 on (𝑋F )proét . The equivalence Equation (6.5) then becomes
 D 𝑋 Weil , Λ −→ lim D(𝑋F , Λ), 𝐵Z
∗ of D 𝑋 , Λ . for the diagram 𝐵Z → PrSt corresponding to the endomorphism 𝜙 F 𝑋 Λ That is, D(𝑋 Weil , Λ) is equivalent to the homotopy fixed points of D(𝑋F , Λ) with respect to the action of 𝜙∗𝑋 , which is our claim. □ 29 6.2 Weil sheaves on products The discussion of the previous section generalizes to products of schemes as follows. Let 𝑋1 , . . . , 𝑋𝑛 be schemes over F𝑞 , and denote by 𝑋 := 𝑋1 ×F𝑞 . . . ×F𝑞 𝑋𝑛 their product. For every 1 ≤ 𝑖 ≤ 𝑛, we have a morphism 𝜙 𝑋𝑖 : 𝑋𝑖,F → 𝑋𝑖,F as in the previous section. We use the notation 𝜙 𝑋𝑖 to also denote the corresponding map on 𝑋F = 𝑋1,F ×F . . . ×F 𝑋𝑛,F which is 𝜙 𝑋𝑖 on the 𝑖-th factor and the identity on the other factors. We define the site (𝑋1Weil × . . . × 𝑋𝑛Weil )proét whose underlying category consists of tuples (𝑈, 𝜑1 , . . . , 𝜑𝑛 ) with 𝑈 ∈ (𝑋F )proét and pairwise commuting endomomorphisms 𝜑𝑖 : 𝑈 → 𝑈 such that the following diagram commutes 𝜑𝑖 /𝑈 𝜙 𝑋𝑖  / 𝑋F , 𝑈  𝑋F
for all 1 ≤ 𝑖 ≤ 𝑛. As before, we denote by D 𝑋1Weil ×. . .×𝑋𝑛Weil , Λ the corresponding derived category of Λ-sheaves. Using a similar reasoning as in the previous section, we can identify this category of sheaves with the homotopy fixed points

D 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ −→ Fix D(𝑋F , Λ), 𝜙∗𝑋1 , . . . , 𝜙∗𝑋𝑛 (6.6) of the commuting family of the functors 𝜙∗𝑋𝑖 , see Remark 4.2.3. Explicitly, for 𝑛 = 2, this is the homotopy limit of the diagram 𝜙∗𝑋 D 𝑋F , Λ id  
𝜙∗𝑋 D 𝑋F , Λ 2

/ / D 𝑋F , Λ 1 id 𝜙∗𝑋 1 / id   𝜙∗𝑋 2
/ D 𝑋F , Λ . id Roughly speaking, objects in the category D(𝑋1Weil × . . . × 𝑋𝑛Weil , Λ) are given by tuples (𝑀, 𝛼1 , . . . , 𝛼𝑛 ) with 𝑀 ∈ D(𝑋F , Λ) and with pairwise commuting equivalences 𝛼𝑖 : 𝑀  𝜙∗𝑋𝑖 𝑀. That is, equipped with a collection of equivalences 𝜙∗𝑋 𝑗 (𝛼𝑖 ) ◦ 𝛼 𝑗 ≃ 𝜙∗𝑋𝑖 (𝛼 𝑗 ) ◦ 𝛼𝑖 for all 𝑖, 𝑗 satisfying higher coherence conditions. 6.3 Partial-Frobenius stability For schemes 𝑋1 , . . . , 𝑋𝑛 over F𝑞 , we denote by 𝑋 := 𝑋1 ×F𝑞 · · · ×F𝑞 𝑋𝑛 their product together with the partial Frobenii Frob 𝑋𝑖 : 𝑋 → 𝑋, 1 ≤ 𝑖 ≤ 𝑛. 30 To give a reasonable definition of lisse and constructible Weil sheaves, we need to understand the relation between partial-Frobenius invariant constructible subsets in 𝑋 and constructible subsets in the single factors 𝑋𝑖 : Definition 6.3.1. A subset 𝑍 ⊂ 𝑋 is called partial-Frobenius invariant if for all 1 ≤ 𝑖 ≤ 𝑛 we have Frob 𝑋𝑖 (𝑍) = 𝑍. The composition Frob 𝑋1 ◦ · · · ◦ Frob 𝑋𝑛 is the absolute 𝑞-Frobenius on 𝑋 and thus induces the identity on the topological space underlying 𝑋. Therefore, in order to check that 𝑍 ⊂ 𝑋 is partial-Frobenius invariant, it suffices that, for any fixed 𝑖, the subset 𝑍 is Frob 𝑋 𝑗 -invariant for all 𝑗 ≠ 𝑖. This remark, which also applies to 𝑋F = 𝑋1 ×F𝑞 . . . ×F𝑞 𝑋𝑛 ×F𝑞 Spec F, will be used below without further comment. We first investigate the case of two factors with one being a separably closed field. This eventually rests on Drinfeld’s descent result (V. G. Drinfeld, 1987, Proposition 1.1) for coherent sheaves: Lemma 6.3.2. Let 𝑋 be a qcqs F𝑞 -scheme, and let 𝑘/F𝑞 be a separably closed field. Denote by 𝑝 : 𝑋 𝑘 → 𝑋 the projection. Then 𝑍 ↦→ 𝑝 −1 (𝑍) induces a bijection {constructible subsets in 𝑋 } ↔ {partial-Frobenius invariant, constructible subsets in 𝑋 𝑘 } Proof. The injectivity is clear because 𝑝 is surjective. It remains to check the surjectivity. Without loss of generality we may assume that 𝑘 is algebraically closed, and replace Frob 𝑋 by Frob𝑘 which is an automorphism. Given that 𝑍 ↦→ 𝑝 −1 (𝑍) is compatible with passing to complements, unions and localizations on 𝑋, we are reduced to proving the bijection for constructible closed subsets 𝑍 and for 𝑋 affine over F𝑞 . By Noetherian approximation (6.3.4), we reduce further to the case where 𝑋 is of finite type over F𝑞 and still affine. Now we choose a locally closed embedding 𝑋 → PF𝑛 𝑞 into projective space. A closed subset 𝑍 ′ ⊂ 𝑋 𝑘 is 𝜙 𝑘 -invariant if and only if its closure inside P𝑛𝑘 is so. Hence, it is enough to consider the case where 𝑋 = PF𝑛 𝑞 is the projective space. Let 𝑍 ′ be a closed Frob𝑘 -invariant subset of 𝑋 𝑘 . When viewed as a reduced subscheme, the isomorphism 𝜙 𝑘 restricts to an isomorphism of 𝑍 ′. In particular, O𝑍 ′ is a coherent O 𝑋𝑘 -module equipped with an isomorphism O𝑍 ′  𝜙∗𝑘 O𝑍 ′ . Hence, Drinfeld’s descent result (V. G. Drinfeld, 1987, Proposition 1.1) (see also (Kedlaya, 2019, Section 4.2) for a recent exposition) yields 𝑍 ′ = 𝑍 𝑘 for a unique closed subscheme 𝑍 ⊂ 𝑋. □ 31 The following proposition generalizes the results (Lau, 2004, Lemma 9.2.1) and (V. Lafforgue, 2018, Lemme 8.12) in the case of curves. Proposition 6.3.3. Let 𝑋1 , . . . , 𝑋𝑛 be qcqs F𝑞 -schemes, and denote 𝑋 = 𝑋1 ×F𝑞 . . . ×F𝑞 𝑋𝑛 . Then any partial-Frobenius invariant constructible closed subset 𝑍 ⊂ 𝑋 is a finite set-theoretic union of subsets of the form 𝑍1 ×F𝑞 . . . ×F𝑞 𝑍𝑛 , for appropriate constructible closed subschemes 𝑍𝑖 ⊂ 𝑋𝑖 . In particular, any partial-Frobenius invariant constructible open subscheme 𝑈 ⊂ 𝑋 is a finite union of constructible open subschemes of the form 𝑈1 ×F𝑞 . . . ×F𝑞 𝑈𝑛 , for appropriate constructible open subschemes 𝑈𝑖 ⊂ 𝑋𝑖 . Proof. By induction, we may assume 𝑛 = 2. By Noetherian approximation (Lemma 6.3.4), we reduce to the case where both 𝑋1 , 𝑋2 are of finite type over F𝑞 . In the following, all products are formed over F𝑞 , and locally closed subschemes are equipped with their reduced subscheme structure. Let 𝑍 ⊂ 𝑋1 × 𝑋2 be a partialFrobenius invariant closed subscheme. The complement 𝑈 = 𝑋1 × 𝑋2 \ 𝑍 is also partial-Frobenius invariant. In the proof, we can replace 𝑋1 (and likewise 𝑋2 ) by a stratification in the following sense: Suppose 𝑋1 = 𝐴′ ⊔ 𝐴′′ is a set-theoretic stratification into a closed subset Ð 𝐴′ with open complement 𝐴′′. Once we know 𝑍 ∩ 𝐴′ × 𝑋2 = 𝑗 𝑍1′ 𝑗 × 𝑍2′ 𝑗 and Ð 𝑍 ∩ 𝐴′′ × 𝑋2 = 𝑗 𝑍1′′𝑗 × 𝑍2′′𝑗 for appropriate closed subschemes 𝑍1′ 𝑗 ⊂ 𝐴′, 𝑍1′′𝑗 ⊂ 𝐴′′ and 𝑍2′ 𝑗 , 𝑍2′′𝑗 ⊂ 𝑋2 , we have the set-theoretic equality 𝑍= Ø 𝑗 ′ 𝑍1𝑖 × 𝑍2′ 𝑗 ∪ Ø 𝑍1′′𝑗 × 𝑍2′′𝑗 , 𝑗 where 𝑍1′′𝑗 ⊂ 𝑋1 denotes the scheme-theoretic closure. Here we note that taking scheme-theoretic closures commutes with products because the projections 𝑋1 × 𝑋2 → 𝑋𝑖 are flat, and that the topological space underlying the scheme-theoretic closure agrees with the topological closure because all schemes involved are of finite type. The proof is now by Noetherian induction on 𝑋2 , the case 𝑋2 = ∅ being clear (or, if the reader prefers the case where 𝑋2 is zero dimensional reduces to Lemma 6.3.2). In the induction step, we may assume, using the above stratification argument, that both 𝑋𝑖 are irreducible with generic point 𝜂𝑖 . We let 𝜂𝑖 be a geometric generic point over 𝜂𝑖 , and denote by 𝑝𝑖 : 𝑋1 × 𝑋2 → 𝑋𝑖 the two projections. Both 𝑝𝑖 are faithfully 32 flat of finite type and in particular open, so that 𝑝𝑖 (𝑈) is open in 𝑋𝑖 . We have a set-theoretic equality


𝑍 = (𝑋1 \ 𝑝 1 (𝑈)) × 𝑋2 ∪ 𝑋1 × (𝑋2 \ 𝑝 2 (𝑈)) ∪ 𝑍 ∩ 𝑝 1 (𝑈) × 𝑝 2 (𝑈) . Ð Once we know 𝑍∩𝑝 1 (𝑈)×𝑝 2 (𝑈) = 𝑗 𝑍1 𝑗 ×𝑍2 𝑗 for appropriate closed 𝑍𝑖 𝑗 ⊂ 𝑝𝑖 (𝑈), we are done. We can therefore replace 𝑋𝑖 by 𝑝𝑖 (𝑈) and assume that both 𝑝𝑖 : 𝑈 → 𝑋𝑖 are surjective. The base change 𝑈 × 𝑋2 𝜂2 is a 𝜙𝜂2 -invariant subset of 𝑋1 × 𝜂2 . By Lemma 6.3.2, it is thus of the form 𝑈1 × 𝜂2 for some open subset 𝑈1 ⊂ 𝑋1 . There is an inclusion (of open subschemes of 𝑋1 ×𝜂2 ): 𝑈 × 𝑋2 𝜂2 ⊂ 𝑈1 ×𝜂2 . It becomes a set-theoretic equality, and therefore an isomorphism of schemes, after base change along 𝜂2 → 𝜂2 . By faithfully flat descent, this implies that the two mentioned subsets of 𝑋1 × 𝜂2 agree. We claim 𝑈1 = 𝑋1 . Since the projection 𝑈 → 𝑋2 is surjective, in particular its image contains 𝜂2 , so that 𝑈1 is a non-empty subset, and therefore open dense in the irreducible scheme 𝑋1 . Let 𝑥 1 ∈ 𝑋1 be a point. Since the projection 𝑈 → 𝑋1 is surjective, 𝑈 ∩ ({𝑥 1 } × 𝑋2 ) is a non-empty open subscheme of {𝑥 1 } × 𝑋2 . So it contains a point lying over (𝑥 1 , 𝜂2 ). We conclude 𝑋1 × 𝜂2 ⊂ 𝑈. We claim that there is a non-empty open subset 𝐴2 ⊂ 𝑋2 such that 𝑋1 × 𝐴2 ⊂ 𝑈 or, equivalently, 𝑋1 × (𝑋2 \ 𝐴2 ) ⊃ 𝑋1 × 𝑋2 \ 𝑈. The underlying topological space of 𝑉 = 𝑋1 × 𝑋2 \ 𝑈 is Noetherian and thus has finitely many irreducible components 𝑉 𝑗 . The closure of the projection 𝑝 2 (𝑉 𝑗 ) ⊂ 𝑋2 Ñ does not contain 𝜂2 , since 𝑋1 × 𝜂2 ⊂ 𝑈. Thus, 𝐴2 := 𝑗 𝑋1 \ 𝑝 2 (𝑉 𝑗 ) satisfies our requirements. Now we continue by Noetherian induction applied to the stratification 𝑋2 = 𝐴2 ⊔ (𝑋2 \ 𝐴2 ): We have 𝑍 ∩ 𝑋1 × 𝐴2 = ∅, so that we may replace 𝑋2 by the proper closed subscheme 𝑋2 \ 𝐴2 . Hence, the proposition follows by Noetherian induction. □ The following lemma on Noetherian approximation of partial Frobenius invariant subsets is needed for the reduction to finite type schemes: Lemma 6.3.4. Let 𝑋1 , . . . , 𝑋𝑛 be qcqs F𝑞 -schemes, and denote 𝑋 = 𝑋1 ×F𝑞 . . .×F𝑞 𝑋𝑛 . Let 𝑋𝑖 = lim 𝑗 𝑋𝑖 𝑗 be a cofiltered limit of finite type F𝑞 -schemes with affine transition maps, and write 𝑋 = lim 𝑗 𝑋 𝑗 , 𝑋 𝑗 := 𝑋1 𝑗 ×F𝑞 . . . ×F𝑞 𝑋𝑛 𝑗 (see Let 𝑍 ⊂ 𝑋 be a constructible closed subset. Then the intersection 𝑛 Ù Ù 𝑍′ = Frob𝑚𝑋𝑖 (𝑍) 𝑖=1 𝑚∈Z 33 is partial Frobenius invariant, constructible closed and there exists an index 𝑗 and a partial Frobenius invariant closed subset 𝑍 ′𝑗 ⊂ 𝑋 𝑗 such that 𝑍 ′ = 𝑍 ′𝑗 × 𝑋 𝑗 𝑋 as sets. We note that each Frob 𝑋𝑖 induces a homeomorphism on the underlying topological space of 𝑋 so that 𝑍 ′ is well-defined. This lemma applies, in particular, to partial Frobenius invariant constructible closed subsets 𝑍 ⊂ 𝑋 in which case we have 𝑍 = 𝑍 ′. Proof. As 𝑍 is constructible, there exists an index 𝑗 and a constructible closed 𝑛 ∩ subscheme 𝑍 𝑗 ⊂ 𝑋 𝑗 such that 𝑍 = 𝑍 𝑗 × 𝑋 𝑗 𝑋 as sets. We put 𝑍 ′𝑗 = ∩𝑖=1 𝑚∈Z 𝑚 ′ Frob 𝑋𝑖 𝑗 (𝑍 𝑗 ). As 𝑋 𝑗 is of finite type over F𝑞 , the subset 𝑍 𝑗 is still constructible closed. As partial Frobenii induce bijections on the underlying topological spaces, one checks that Frob𝑚𝑋𝑖 𝑗 (𝑍 𝑗 ) × 𝑋 𝑗 𝑋 = Frob𝑚𝑋𝑖 (𝑍) as sets for all 𝑚 ∈ Z. Thus, 𝑍 ′ = 𝑍 ′𝑗 × 𝑋 𝑗 𝑋 which, also, is constructible closed because 𝑋 → 𝑋 𝑗 is affine. □ 6.4 Lisse and constructible Weil sheaves In this subsection, we define the subcategories of lisse and constructible Weil sheaves and establish a presentation similar to (6.4). Let 𝑋1 , . . . , 𝑋𝑛 be schemes over F𝑞 , and denote 𝑋 := 𝑋1 ×F𝑞 . . . ×F𝑞 𝑋𝑛 . Let Λ be a condensed ring.
Definition 6.4.1. Let 𝑀 ∈ D 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ . 1. The Weil sheaf 𝑀 is called lisse if it is dualizable. (Here dualizability refers
to the symmetric monoidal structure on D 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ , given by the derived tensor product of Λ-sheaves on the Weil-proétale topos.) 2. The Weil sheaf 𝑀 is called constructible if for any open affine 𝑈𝑖 ⊂ 𝑋𝑖 there exists a finite subdivision into constructible locally closed subschemes
Weil , Λ 𝑈𝑖 𝑗 ⊆ 𝑈𝑖 such that each restriction 𝑀 |𝑈 Weil ×...×𝑈 Weil ∈ D 𝑈1Weil ×. . .×𝑈 𝑛𝑗 𝑗 𝑛𝑗 1𝑗 is lisse.
The full subcategories of D 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ consisting of lisse, resp. constructible Weil sheaves are denoted by

Dlis 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ ⊂ Dcons 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ . Both categories are idempotent complete stable Γ(𝑋, Λ)-linear symmetric monoidal ∞-categories. 34 From the presentation (6.6), we get that a Weil sheaf 𝑀 is lisse if and only if the underlying object 𝑀F ∈ D(𝑋F , Λ) is lisse. So (6.6) restricts to an equivalence  

Dlis 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ  Fix Dlis 𝑋F , Λ , 𝜙∗𝑋1 , . . . , 𝜙∗𝑋𝑛 . (6.7) The same is true for constructible Weil sheaves by the following proposition:
Proposition 6.4.2. A Weil sheaf 𝑀 ∈ D 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ is constructible if and only if the underlying sheaf 𝑀F ∈ D(𝑋F , Λ) is constructible. Consequently, Equation (6.6) restricts to an equivalence  

Dcons 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ  Fix Dcons 𝑋F , Λ , 𝜙∗𝑋1 , . . . , 𝜙∗𝑋𝑛 . (6.8) Proof. Clearly, if 𝑀 is constructible, so is 𝑀F by 6.4.1. Let 𝑀 ∈ D 𝑋1Weil × . . . ×
𝑋𝑛Weil , Λ such that 𝑀F is constructible. We may assume that all 𝑋𝑖 are affine. We claim that there is a finite subdivision 𝑋F = ⊔𝑋𝛼 into constructible locally closed subsets such that 𝑀F | 𝑋 𝛼 is lisse and such that each 𝑋𝛼 is partial Frobenius invariant. Assuming the claim we finish the argument as follows. By 6.3.3, any open stratum 𝑈 = 𝑋 𝑗0 ⊂ 𝑋F is a finite union of subsets of the form 𝑈1,F ×F . . . ×F 𝑈𝑛,F and the restriction of 𝑀 to each of them is lisse. In particular, the complement 𝑋F \𝑈 is defined over F𝑞 and arises as a finite union of schemes of the form 𝑋 ′ = 𝑋1′ ×F𝑞 . . . ×F𝑞 𝑋𝑛′ for suitable qcqs schemes 𝑋𝑖′ over F𝑞 . Intersecting each 𝑋F′ with the remaining strata ⊔ 𝑗≠ 𝑗0 𝑋 𝑗 , we conclude by induction on the number of strata. It remains to prove the claim. We start with any finite subdivision 𝑋F = ⊔𝑋 ′𝑗 into constructible locally closed subsets such that 𝑀F | 𝑋 ′𝑗 is lisse. Pick an open stratum 𝑋 ′𝑗0 , and set 𝑛 Ø Ø 𝑋 𝑗0 = 𝜙𝑚𝑋𝑖 (𝑋 ′𝑗0 ). (6.9) 𝑖=1 𝑚∈Z This is a constructible open subset of 𝑋F by 6.3.4 applied to its closed complement. Furthermore, 𝑀F | 𝑋 𝑗0 is lisse by its partial Frobenius equivariance, noting that 𝜙∗𝑋𝑖 induces equivalences on proétale topoi to treat the negative powers in (6.9). As before, 𝑋F \𝑋 𝑗0 is defined over F𝑞 . So replacing 𝑋 ′𝑗 , 𝑗 ≠ 𝑗0 by 𝑋 ′𝑗 ∩ (𝑋F \𝑋 𝑗0 ), the claim follows by induction on the number of strata. □ In the case of a single factor 𝑋 = 𝑋1 , the preceding discussion implies ! 𝜙∗𝑋
Weil D• 𝑋 , Λ  lim D• (𝑋F , Λ) ⇒ D• (𝑋F , Λ) , id for • ∈ {∅, lis, cons}. (6.10) 35 6.5 Relation with the Weil groupoid In this subsection, we relate lisse Weil sheaves with representations of the Weil groupoid. Throughout, we work with étale fundamental groups as opposed to their proétale variants in order to have Drinfeld’s lemma available, see Section 7.4. The two concepts differ in general, but agree for geometrically unibranch (for example, normal) Noetherian schemes, see (Bhatt and Scholze, 2015, Lemma 7.4.10). For a Noetherian scheme 𝑋, let 𝜋1 (𝑋) be the étale fundamental groupoid of 𝑋 as defined in (Revêtements étales et groupe fondamental (SGA 1) 2003, Exposé V, §7 and §9). Its objects are geometric points of 𝑋, and its morphisms are isomorphisms of fiber functors on the finite étale site of 𝑋. This is an essentially small category. The automorphism group in 𝜋1 (𝑋) at a geometric point 𝑥 → 𝑋 is profinite. It is denoted 𝜋1 (𝑋, 𝑥) and called the étale fundamental group of (𝑋, 𝑥). If 𝑋 is connected, then the natural map 𝐵𝜋1 (𝑋, 𝑥) → 𝜋1 (𝑋) is an equivalence for any 𝑥 → 𝑋. If 𝑋 is the disjoint sum of schemes 𝑋𝑖 , 𝑖 ∈ 𝐼, then 𝜋1 (𝑋) is the disjoint sum of the 𝜋1 (𝑋𝑖 ), 𝑖 ∈ 𝐼. In this case, if 𝑥 → 𝑋 factors through 𝑋𝑖 , then 𝜋1 (𝑋, 𝑥) = 𝜋1 (𝑋𝑖 , 𝑥). Definition 6.5.1. Let 𝑋1 , . . . , 𝑋𝑛 be Noetherian schemes over F𝑞 , and write 𝑋 = 𝑋1 ×F𝑞 . . . ×F𝑞 𝑋𝑛 . The Frobenius-Weil groupoid is the stacky quotient FWeil(𝑋) = 𝜋1 (𝑋F )/⟨𝜙Z𝑋1 , . . . , 𝜙Z𝑋𝑛 ⟩, (6.11) where we use that the partial Frobenii 𝜙 𝑋𝑖 induce automorphisms on the finite étale site of 𝑋F . For 𝑛 = 1, we denote FWeil(𝑋) = Weil(𝑋). Even if 𝑋 is connected, its base change 𝑋F might be disconnected in which case the action of 𝜙 𝑋 permutes some connected components. Therefore, fixing a geometric point of 𝑋F is inconvenient, and the reason for us to work with fundamental groupoids as opposed to fundamental groups. The automorphism groups in Weil(𝑋) carry the structure of locally profinite groups: indeed, if 𝑋 is connected, then Weil(𝑋) is, for any choice of a geometric point 𝑥 → 𝑋F , equivalent to the classifying space of the Weil group Weil(𝑋, 𝑥) from (Deligne, 1980, Définition 1.1.10). Recall that this group sits in an exact sequence of topological groups 1 → 𝜋1 (𝑋F , 𝑥) → Weil(𝑋, 𝑥) → Weil(F/F𝑞 ) ≃ Z, (6.12) where 𝜋1 (𝑋F , 𝑥) carries its profinite topology and Z the discrete topology. The topology on the morphism groups in Weil(𝑋) obtained in this way is independent 36 from the choice of 𝑥 → 𝑋F . The image of Weil(𝑋, 𝑥) → Z is the subgroup 𝑚Z where 𝑚 is the degree of the largest finite subfield in Γ(𝑋, O 𝑋 ). In particular, we have 𝑚 = 1 if 𝑋F is connected. Let us add that if 𝑥 → 𝑋F is fixed under 𝜙 𝑋 , then the action of 𝜙 𝑋 on 𝜋1 (𝑋F , 𝑥) corresponds by virtue of the formula 𝜙∗𝑋 = (𝜙∗F ) −1 to the action of the geometric Frobenius, that is, the inverse of the 𝑞-Frobenius in Weil(F/F𝑞 ). Likewise, for every 𝑛 ≥ 1, the stabilizers of the Frobenius-Weil groupoid are related to the partial Frobenius-Weil groups introduced in (V. G. Drinfeld, 1987, Proposition 6.1) and (V. Lafforgue, 2018, Remarque 8.18). In particular, there is an exact sequence 1 → 𝜋1 (𝑋F , 𝑥) → FWeil(𝑋, 𝑥) → Z𝑛 , for each geometric point 𝑥 → 𝑋F . This gives FWeil(𝑋) the structure of a locally profinite groupoid. Let Λ be either of the following coherent topological rings: a coherent discrete ring, an algebraic field extension 𝐸 ⊃ Qℓ for some prime ℓ, or its ring of integers O𝐸 ⊃ Zℓ . For a topological groupoid 𝑊, we will denote by RepΛ (𝑊) the category of continuous representations of 𝑊 with values in finitely presented Λ-modules and by f.p RepΛ (𝑊) ⊂ RepΛ (𝑊) its full subcategory of representations on finite projective Λmodules. Here finitely presented Λ-modules 𝑀 carry the quotient topology induced from the choice of any surjection Λ𝑛 → 𝑀, 𝑛 ≥ 0 and the product topology on Λ𝑛 . Lemma 6.5.2. In the situation above, the category RepΛ (𝑊) is Λ∗ -linear and f.p abelian. In particular, its full subcategory RepΛ (𝑊) is Λ∗ -linear and additive. Proof. Let 𝑊disc be the discrete groupoid underlying 𝑊, and denote by RepΛ (𝑊disc ) the category of 𝑊disc -representations on finitely presented Λ-modules. Evidently, this category is Λ∗ -linear. It is abelian since Λ is coherent. We claim that RepΛ (𝑊) ⊂ RepΛ (𝑊disc ) is a Λ∗ -linear full abelian subcategory. If Λ is discrete (and coherent), then every finitely presented Λ-module carries the discrete topology and the claim is immediate, see also (Stacks, Tag 0A2H). For Λ = 𝐸, O𝐸 , one checks that every map of finitely presented Λ-modules is continuous, every surjective map is a topological quotient and every injective map is a closed embedding. For the latter, we use that every finitely presented Λ-module can be written as a countable filtered colimit of compact Hausdorff spaces along injections, and that every injection of compact Hausdorff spaces is a closed embedding. This implies the claim. □ 37 We apply this for 𝑊 being either of the locally profinite groupoids 𝜋1 (𝑋), 𝜋1 (𝑋F ) or FWeil(𝑋). Note that restricting representations along 𝜋1 (𝑋F ) → FWeil(𝑋) induces an equivalence of Λ∗ -linear abelian categories


RepΛ FWeil(𝑋)  Fix RepΛ 𝜋1 (𝑋F ) , 𝜙 𝑋1 , . . . , 𝜙 𝑋𝑛 , (6.13)
f.p and similarly for the Λ∗ -linear additive category RepΛ FWeil(𝑋) . {−𝑛,𝑛} Definition 6.5.3. For an integer 𝑛 ≥ 0, we write Dlis (𝑋, Λ) for the full subcategory of Dlis (𝑋, Λ) of objects 𝑀 such that 𝑀 and its dual 𝑀 ∨ lie in degrees [−𝑛, 𝑛] with respect to the t-structure on D(𝑋, Λ). Lemma 6.5.4. In the situation above, there is a natural functor

♥ RepΛ FWeil(𝑋) → D 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ , (6.14) that is fully faithful. Moreover, the following properties hold if Λ is either finite discrete or Λ = O𝐸 for 𝐸 ⊃ Qℓ finite: 1. An object 𝑀 lies in the essential image of Equation (6.14) if and only if its underlying sheaf 𝑀F is locally on (𝑋F )proét isomorphic to 𝑁 ⊗Λ∗ Λ 𝑋F for some finitely presented Λ∗ -module 𝑁. 2. The functor (6.14) restricts to an equivalence of Λ∗ -linear additive categories

f.p {0,0} RepΛ FWeil(𝑋) −→ Dlis 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ . 3. If Λ∗ is regular (so that Λ is t-admissible, cf. Chapter 5 Item vi), then Equation (6.14) restricts to an equivalence of Λ∗ -linear abelian categories

♥ RepΛ FWeil(𝑋) −→ Dlis 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ . If all 𝑋𝑖 , 𝑖 = 1, . . . , 𝑛 are geometrically unibranch, then (1), (2) and (3) hold for general coherent topological rings Λ as above. œ proét Proof. There is a canonical equivalence of topological groupoids 𝜋1 (𝑋F )  𝜋1 (𝑋F ) with the profinite completion of the proétale fundamental groupoid, see (Bhatt and Scholze, 2015, Lemma 7.4.3). It follows from (Bhatt and Scholze, 2015, Lemmas proét 7.4.5, 7.4.7) that restricting representations along 𝜋1 (𝑋F ) → 𝜋1 (𝑋F ) induces full embeddings

proét RepΛ 𝜋1 (𝑋F ) ↩→ RepΛ 𝜋1 (𝑋F ) ↩→ D(𝑋F , Λ) ♥ , (6.15) 38 that are compatible with the action of 𝜙 𝑋𝑖 for all 𝑖 = 1, . . . , 𝑛. So we obtain the fully faithful functor (6.14) by passing to fixed points, see (6.13), (6.7) and Lemma 4.2.2 (see also 4.2.3).
proét Part Item 1 describes the essential image of RepΛ 𝜋1 (𝑋F ) ↩→ D(𝑋F , Λ) ♥ . So if Λ is finite discrete or profinite, then the first functor in (6.15) is an equivalence, and we are done. Part Item 2 is immediate from Item 1, noting that an object in the essential image of Equation (6.15) is lisse if and only if its underlying module is finite projective. Likewise, part Item 3 is immediate from Item 1, using Item vii. Here we need to exclude rings like Λ = Z/ℓ 2 in order to have a t-structure on lisse sheaves. Finally, if all 𝑋𝑖 are geometrically unibranch, so is 𝑋F which follows from the proét characterization (Stacks, Tag 0BQ4). In this case, we get 𝜋1 (𝑋F )  𝜋1 (𝑋F ) by (Bhatt and Scholze, 2015, Lemma 7.4.10). This finishes the proof. □ 6.6 Weil-étale versus étale sheaves We end this section with the following description of Weil sheaves with (ind-)finite coefficients. Note that such a simplification in terms of ordinary sheaves is not possible for Λ = Z, Zℓ , Qℓ , say. Proposition 6.6.1. Let 𝑋 be a qcqs F𝑞 -scheme. Let Λ be a finite discrete ring or a filtered colimit of such rings. Then the natural functors

Dlis (𝑋, Λ) → Dlis 𝑋 Weil , Λ , Dcons (𝑋, Λ) → Dcons 𝑋 Weil , Λ , are equivalences. Proof. Throughout, we repeatedly use that filtered colimits commute with finite limits in Cat∞ . Using compatibility of Dcons with filtered colimits in Λ (Chapter 5 Item iv), we may assume that Λ is finite discrete. By the comparison result with the classical bounded derived category of constructible sheaves, we can identify the categories D• (𝑋, Λ), resp. D• (𝑋F , Λ) for • ∈ {lis, cons} with full subcategories of the derived category of étale Λ-sheaves D(𝑋𝑒𝑡´ , Λ), resp. D(𝑋F,𝑒𝑡´ , Λ). Write 𝑋 = lim 𝑋𝑖 as a cofiltered limit of finite type F𝑞 -schemes 𝑋𝑖 with affine transition maps (Stacks, Tag 01ZA). Using the continuity of étale sites (Stacks, Tag 03Q4), there are natural equivalences  colim D• (𝑋𝑖 , Λ) −→ D• (𝑋, Λ),  colim D• (𝑋𝑖Weil , Λ) −→ D• (𝑋 Weil , Λ) (6.16) 39 for • ∈ {lis, cons}. Hence, we can assume that 𝑋 is of finite type over F𝑞 . To show full faithfulness, we claim more generally that the natural map ! ∗ 𝜙𝑋 D(𝑋𝑒𝑡´ , Λ) → lim D(𝑋F,𝑒𝑡´ , Λ) ⇒ D(𝑋F,𝑒𝑡´ , Λ) =: D 𝑋𝑒𝑡Weil ,Λ ´
id is fully faithful. As Λ is torsion, this is immediate from (Geisser, 2004, Corollary 5.2) applied to the inner homomorphisms between sheaves. Let us add that this induces fully faithful functors

D+ (𝑋𝑒𝑡´ , Λ) → D+ 𝑋𝑒𝑡Weil , Λ → D 𝑋 Weil , Λ ´ (6.17) on bounded below objects, see (Bhatt and Scholze, 2015, Proposition 5.2.6 (1)). It remains to prove essential surjectivity. Using a stratification as in 6.4.1, it is enough to consider the lisse case. Pick 𝑀 ∈ Dlis (𝑋 Weil , Λ). It is enough to show that 𝑀 lies is in the essential image of Equation (6.17), noting that the functor detects dualizability. As 𝑀 is bounded, this will follow from showing that for every 𝑗 ∈ Z, the cohomology sheaf H 𝑗 (𝑀) ∈ D(𝑋 Weil , Λ) ♥ is in the essential image of Equation (6.17). Fix 𝑗 ∈ Z. As 𝑀 is lisse, the underlying sheaf H 𝑗 (𝑀)F ∈ D(𝑋F , Λ) ♥ is proétalelocally constant (Chapter 5 Item vii) and valued in finitely presented Λ-modules. By Lemma 6.5.4 Item 1, it comes from a representation of Weil(𝑋). Restriction of representations along Weil(𝑋) → 𝜋1 (𝑋) fits into a commutative diagram RepΛ 𝜋1 (𝑋)  D(𝑋𝑒𝑡´ , Λ) ♥
 / Rep Λ Weil(𝑋)

/ D 𝑋 Weil , Λ ♥ , where the upper horizontal arrow is an equivalence since Λ is finite. In particular, the object H 𝑗 (𝑀) is in the essential image of the fully faithful functor Equation (6.17). □ 40 Chapter 7 THE CATEGORICAL KÜNNETH FORMULA 7.1 The main result We continue with the notation of Chapter 6. In particular, F𝑞 denotes a finite field of characteristic 𝑝 > 0. Recall from Chapter 4 the tensor product of Λ∗ -linear idempotent complete stable ∞-categories. The external tensor product of sheaves (𝑀1 , . . . , 𝑀𝑛 ) ↦→ 𝑀1 ⊠ . . . ⊠ 𝑀𝑛 as in Equation (5.2) induces a functor


D• 𝑋1Weil , Λ ⊗Perf Λ∗ . . . ⊗Perf Λ∗ D• 𝑋𝑛Weil , Λ → D• 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ , (7.1) for • ∈ {lis, cons}. Throughout, we consider the following situation. In Remark 7.1.3 we explain the compatibility of Equation (7.1) with certain (co-)limits in the schemes 𝑋𝑖 and coefficients Λ, which allows to relax these assumptions on 𝑋 and Λ somewhat. Situation 7.1.1. The schemes 𝑋1 , . . . , 𝑋𝑛 are of finite type over F𝑞 , and Λ is the condensed ring associated with one of the following topological rings: (a) a finite discrete ring of prime-to-𝑝-torsion; (b) the ring of integers O𝐸 of an algebraic field extension 𝐸 ⊃ Qℓ for ℓ ≠ 𝑝 (for example Z̄ℓ ); (c) an algebraic field extension 𝐸 ⊃ Qℓ for ℓ ≠ 𝑝 (for example Q̄ℓ ); (d) a finite discrete 𝑝-torsion ring that is flat over Z/𝑝 𝑚 for some 𝑚 ≥ 1. Theorem 7.1.2. In Situation 7.1.1, the functor Equation (7.1) is an equivalence in each of the following cases: 1. • = cons and Λ is as in Item a, Item b or Item c; 2. • = lis and Λ is as in Item a, Item b, Item d or as in Item c if all 𝑋𝑖 , 𝑖 = 1, . . . , 𝑛 are geometrically unibranch (for example, normal). In the 𝑝-torsion free cases Item a, Item b and Item c, the full faithfulness is a direct consequence of the Künneth formula applied to the 𝑋𝑖,F . In the 𝑝-torsion case Item d, 41 we use Artin–Schreier theory instead. It would be interesting to see whether this part can be extended to constructible sheaves using the mod-𝑝-Riemann–Hilbert correspondence as in, say, (Bhatt and Lurie, 2019). In all cases, the essential surjectivity relies on a variant of Drinfeld’s lemma for Weil group representations. Before turning to the proof of Theorem 7.1.2, we record the following compatibility of the functor Equation (7.1) with (co-)limits. This can be used to reduce the case of an (infinite) algebraic extension 𝐸 ⊃ Qℓ in cases Item b and Item c above to the case where 𝐸 ⊃ Qℓ is finite. In the sequel we will therefore assume 𝐸 is finite in these cases. Remark 7.1.3 can further be used to extend Theorem 7.1.2 to qcqs F𝑞 -schemes 𝑋𝑖 and finite discrete rings like Z/𝑚 for any integer 𝑚 ≥ 1 in cases Item a and Item d. Remark 7.1.3 (Compatibility of (7.1) with certain (co-)limits). Throughout, we repeatedly use that filtered colimits commute with finite limits in CatEx ∞,Λ∗ (Idem): the Ex forgetful functors CatEx ∞,Λ∗ (Idem) → Cat∞ (Idem) → Cat∞ create these (co)limits (Lurie, 2017, Theorem 1.1.4.4), (Lurie, 2009, Corollary 4.4.5.21), and the statement holds in any compactly generated ∞-category, such as Cat∞ (Bhatt and Mathew, 2021, Example 3.6(3)). We will also throughout use that in all the stable ∞categories encountered below the tensor product preserves colimits and in particular finite limits. 1. Filtered colimits in Λ. First off, extension of scalars along any map of condensed rings Λ → Λ′ induces a commutative diagram in CatEx ∞,Λ∗ (Idem):

D• 𝑋1Weil , Λ ⊗Perf Λ∗ . . . ⊗Perf Λ∗ D• 𝑋𝑛Weil , Λ / D• (𝑋 Weil . . . × 𝑋 Weil , Λ) 

D• 𝑋1Weil , Λ′ ⊗Perf Λ′ . . . ⊗Perf Λ′ D• 𝑋𝑛Weil , Λ′  / D• (𝑋 Weil . . . × 𝑋 Weil , Λ′) ∗ ∗ 1 1 𝑛 𝑛 It follows from the compatibility of Dcons with filtered colimits in Λ (Chapter 5 Item iv) that both sides of Equation (7.1) are compatible with filtered colimits in Λ. Î 2. Finite products in Λ. Let Λ = Λ𝑖 be a finite product of condensed rings. For Î any scheme 𝑋, the natural map D• (𝑋, Λ) → D• (𝑋, Λ𝑖 ) is an equivalence for • ∈ {∅, lis, cons}, and likewise for Weil sheaves if 𝑋 is defined over F𝑞 . As Î Λ∗ = Λ𝑖,∗ , we see that (7.1) is compatible finite products in the coefficients. 42 3. Limits in 𝑋𝑖 for discrete Λ. Assume that Λ is finite discrete, see Situation 7.1.1 Item a, Item d. Let 𝑋1 , . . . , 𝑋𝑛 be qcqs F𝑞 -schemes. Write each 𝑋𝑖 as a cofiltered limit 𝑋𝑖 = lim 𝑋𝑖 𝑗 of finite type F𝑞 -schemes 𝑋𝑖 𝑗 with affine transition maps (Stacks, Tag 01ZA). As Λ is finite discrete, we can use the continuity of étale sites as in Equation (6.16) to show that the natural map

Weil Weil colim D• 𝑋1Weil . . . × 𝑋𝑛Weil , Λ , 𝑗 × . . . × 𝑋𝑛 𝑗 , Λ −→ D• 𝑋1 𝑗 is an equivalence for • ∈ {lis, cons}. Thus, Equation (7.1) is compatible with cofiltered limits of finite type F𝑞 -schemes with affine transition maps. 7.2 A formulation in terms of prestacks Before turning to the proof, we point out a formulation of the results of the previous subsection in terms of symmetric monoidality of a certain sheaf theory. This formulation makes the connection with constructions in the geometric approaches to the Langlands program (Dennis Gaitsgory, Kazhdan, et al., 2022; Zhu, 2021; V. Lafforgue and Zhu, 2019) more manifest. Readers not familiar with prestacks and formulations of sheaf theories on them can safely skip this section. The categories of constructible, resp. lisse Λ-sheaves assemble into a lax symmetric monoidal functor D•,Λ : (SchF ) op → CatEx ∞,Λ (Idem) (• = lis or cons). (7.2) Namely, as a functor it sends a scheme 𝑋 to the category of constructible, resp. lisse Λ-sheaves on 𝑋, and a morphism 𝑓 : 𝑋 → 𝑌 to the functor 𝑓 ∗ : D• (𝑌 , Λ) → D• (𝑋, Λ). These are objects, resp. maps in the ∞-category CatEx ∞,Λ (Idem) := Ex ModPerf Λ (Cat∞ (Idem)), cf. Section 4.1 for notation. The lax monoidal structure is given by the external tensor product of sheaves: ⊠ : D• (𝑋proét , Λ) ⊗Perf Λ D• (𝑌proét , Λ) → D• ((𝑋 ×F 𝑌 )proét , Λ). That is, we consider the category of schemes as symmetric monoidal with respect to the fiber product over F, and the external tensor product is natural on 𝑋 and 𝑌 in the appropriate sense, see (Dennis Gaitsgory and Lurie, 2019, Section 3.1), (Dennis Gaitsgory and Rozenblyum, 2017b, Section III.2) for details and precise statements. This functor ⊠ often fails to be an equivalence, so D•,Λ is not symmetric monoidal. The assertion of Theorem 7.1.2 is that this issue is resolved by replacing sheaves with Weil sheaves. In order to formulate 7.1.2 as the monoidality of a certain functor, we need to replace the category of schemes by a category of objects that model Weil sheaves. We will 43 represent these by taking the appropriate formal quotient by the partial Frobenius automorphism. Such formal quotients can be taken in the category of prestacks. We denote by PreStkF the category of (accessible) functors from the category CAlgF of commutative algebras over F to the ∞-category Ani of Anima. The functor of taking points embeds the category of schemes fully faithfully into PreStkF . We denote by D•,Λ : (PreStkF ) op → CatEx ∞,Λ (Idem) (7.3) the functor obtained by right Kan extension (Lurie, 2009, §4.3.2) along the inclusion fp (SchF ) op ⊂ (PreStkF ) op . Concretely, (Lurie, 2018, Proposition 6.2.1.9, Proposition 6.2.3.1), given a prestack 𝑌 which can be written as a colimit of schemes 𝑌𝛼 over some indexing category 𝐴 we have a canonical equivalence D• (𝑌 , Λ)  lim D• (𝑌𝛼 , Λ). 𝛼 (7.4) This limit is formed in CatEx ∞,Λ (Idem); recall from around (4.3) that the IndEx completion functor to Cat∞,Λ (Idem) → PrSt Λ does not preserve (even finite) limits. With this general sheaf theory in place, we can restrict our attention to the class of prestacks that is relevant to the derived Drinfeld lemma. Definition 7.2.1. Let 𝑋 be a scheme over F𝑞 . The Weil prestack is defined as   𝜙𝑋 Weil 𝑋 := colim 𝑋 ×F𝑞 F ⇒ 𝑋 ×F𝑞 F ∈ PreStkF , id i.e., it is the prestack sending 𝑅 ∈ CAlgF to the colimit   𝜙𝑋 Weil 𝑋 (𝑅) = colim 𝑋 (𝑅) ⇒ 𝑋 (𝑅) . (7.5) id fp We denote by SchWeil the smallest full monoidal subcategory of PreStkF containing the Weil prestacks of finite type schemes 𝑋/F𝑞 . Equivalently, this is the full subcategory consisting of finite products of the form 𝑋1Weil × · · · × 𝑋𝑛Weil . Lemma 7.2.2. Let 𝑋1 , . . . , 𝑋𝑛 be schemes over F𝑞 . There is a canonical equivalence
 D• (𝑋1Weil ×F · · · ×F 𝑋𝑛Weil ) −→ Fix D• (𝑋F , Λ), 𝜙∗𝑋1 , . . . , 𝜙∗𝑋𝑛 (7.6) Proof. Let Φ : BZ𝑛 → PreStkF be the functor corresponding to the commuting automorphisms 𝜙 𝑋𝑖 . Then the claim follows immediately from the identification of 𝑋1Weil ×F · · · ×F 𝑋𝑛Weil with the colimit of Φ (as an object in PreStkF ). □ 44 Theorem 7.2.3. Suppose • and Λ are as in 7.1.2. Then the restriction of D•,Λ to Weil prestacks, i.e., the following composite fp D•,Λ : (SchWeil ) op ⊂ PreStkF → CatEx ∞,Λ (Idem), (7.7) is symmetric monoidal. Proof. As was noted above, the functor in Equation (7.2) is lax symmetric monoidal. By (Torii, 2022, Proposition 2.7), the Kan extension in Equation (7.3) is still lax symmetric monoidal. To check its restriction to the (symmetric monoidal) subcatefp gory SchWeil is symmetric monoidal it suffices to show that the lax monoidal maps are in fact isomorphisms. This is precisely the content of Theorem 7.1.2. □ 7.3 Full faithfulness In this section, we prove that the functor Equation (7.1) is fully faithful under the conditions of Theorem 7.1.2. We first consider the 𝑝-torsion free cases: Proposition 7.3.1. Let 𝑋1 , . . . , 𝑋𝑛 and Λ be as in Situation 7.1.1 Item a, Item b or Item c. Then the functor Equation (7.1) is fully faithful for • ∈ {lis, cons}. Proof. In a nutshell, this is an instance of the Künneth formula: for constructible sheaves on 𝑋𝑖,F (as opposed to 𝑋𝑖Weil ), this interpretation of the Künneth formula appears already in (Dennis Gaitsgory, Kazhdan, et al., 2022, Section A.2). Throughout, we drop Λ from the notation. It is enough to verify that for all 𝑀𝑖 , 𝑁𝑖 ∈ Dcons (𝑋𝑖Weil ) the natural map 𝑛 Ì HomD(𝑋 Weil ) (𝑀𝑖 , 𝑁𝑖 ) → HomD(𝑋 Weil ×...×𝑋𝑛Weil ) (𝑀1 ⊠ . . . ⊠ 𝑀𝑛 , 𝑁1 ⊠ . . . ⊠ 𝑁𝑛 ) 𝑖 1 𝑖=1 (7.8) is an equivalence. As Equation (7.8) is functorial in the objects and compatible with shifts, it suffices, by Definition 6.4.1, to consider the case where 𝑀𝑖 , 𝑖 = 1, . . . , 𝑛 is the extension by zero of a lisse Weil Λ-sheaf on some locally closed subscheme 𝑍𝑖 ⊂ 𝑋𝑖 . Using the adjunction (𝜄𝑖 )! : Dcons (𝑍𝑖Weil ) ⇄ Dcons (𝑋𝑖Weil ) : (𝜄𝑖 ) ! , and the dualizability of lisse sheaves, we reduce to the case 𝑀𝑖 = Λ 𝑋𝑖 , 𝑖 = 1, . . . , 𝑛. That is, Equation (7.8) becomes a map of cohomology complexes. 45 By Proposition 6.1.4, we have
RΓ 𝑋𝑖Weil , 𝑁𝑖 = Fib 𝜙∗𝑋 −id  𝑖  RΓ(𝑋𝑖,F , 𝑁𝑖 ) −→ RΓ(𝑋𝑖,F , 𝑁𝑖 ) . A similar computation holds for the mapping complexes in D(𝑋1Weil × . . . × 𝑋𝑛Weil ), see Equation (6.6). Such finite limits commute with the tensor product in ModΛ . Thus, Section 7.3 reduces to the Künneth formula


RΓ 𝑋1,F , 𝑁1 ⊗ . . . ⊗ RΓ 𝑋𝑛,F , 𝑁𝑛 −→ RΓ 𝑋1,F ×F . . . ×F 𝑋𝑛,F , 𝑁1 ⊠ . . . ⊠ 𝑁𝑛 , where we use that the 𝑋𝑖 are of finite type and the coprimality assumptions on Λ, see (Stacks, Tag 0F1P). □ Next, we consider the 𝑝-torsion case: Proposition 7.3.2. Let 𝑋1 , . . . , 𝑋𝑛 and Λ be as in Situation 7.1.1 Item d. Then the functor (7.1) is fully faithful for • = lis. Proof. As in the proof of Proposition 7.3.1, we need to show that the map 𝑛 Ì

RΓ 𝑋𝑖Weil , 𝑁𝑖 → RΓ 𝑋1Weil × . . . × 𝑋𝑛Weil , 𝑁1 ⊠ . . . ⊠ 𝑁𝑛 (7.9) 𝑖=1 is an equivalence for any 𝑁𝑖 ∈ Dlis (𝑋𝑖Weil ). Using Zariski descent for both sides, we may assume that each 𝑋𝑖 is affine. As Λ is finite discrete (see also the discussion around (6.16)), the invariance of the étale site under perfection reduces us to the case where each 𝑋𝑖 is perfect. The proof now proceeds by several reduction steps: 1) reduce to 𝑁𝑖 = Λ 𝑋𝑖 ; 2) reduce to Λ = Z/𝑝; 3) reduce to 𝑞 = 𝑝 being a prime. The last step 4) is then an easy computation. Step 1): We may assume 𝑁𝑖 = Λ 𝑋𝑖 . In order to show Equation (7.9) is a quasiisomorphism, it suffices to show this after applying 𝜏 ≤𝑟 for arbitrary 𝑟. The complexes 𝑁𝑖 are bounded (Chapter 5 Item v). By shifting them appropriately, we may assume 𝑟 = 0. Note that RΓ(𝑋𝑖Weil , 𝑁𝑖 )  RΓ(𝑋𝑖 , 𝑁𝑖 ), see Proposition 6.6.1. By right Ë
Ë ≤0 exactness of the tensor product, we have 𝜏 ≤0 𝑖 RΓ(𝑋𝑖 , 𝑁𝑖 ) = 𝑖 𝜏 RΓ(𝑋𝑖 , 𝑁𝑖 ). Using the compatibility with the classical notion of constructible sheaves, there is an étale covering 𝑈𝑖 → 𝑋𝑖 such that 𝑁𝑖 |𝑈𝑖 is perfect-constant. Let 𝑈𝑖,• be the Čech nerve of this covering. By étale descent, we have RΓ(𝑋𝑖 , 𝑁𝑖 ) = lim RΓ(𝑈𝑖, 𝑗 , 𝑁𝑖 ). [ 𝑗]∈Δ 46 For each 𝑟 ∈ Z, there is some 𝑗𝑟 such that 𝜏 ≤𝑟 lim RΓ(𝑈𝑖, 𝑗 , 𝑁𝑖 ) = [ 𝑗]∈Δ lim [ 𝑗]∈Δ, 𝑗 ≤ 𝑗𝑟 𝜏 ≤𝑟 RΓ(𝑈𝑖, 𝑗 , 𝑁𝑖 ). This can be seen from the spectral sequence (note that it is concentrated in degrees 𝑗 ≥ 0 and degrees 𝑗 ′ ≥ 𝑟 for some 𝑟, since the complexes 𝑁𝑖 are bounded from below) ′ ′ ′ H 𝑗 (𝑈𝑖, 𝑗 , 𝑁𝑖 ) ⇒ H 𝑗 + 𝑗 lim RΓ(𝑈𝑖, 𝑗 , 𝑁𝑖 ) = H 𝑗 + 𝑗 (𝑋𝑖 , 𝑁𝑖 ). 𝑗 ∈Δ As the tensor product in Equation (7.9) commutes with finite limits, we may thus assume that each 𝑁𝑖 is perfect-constant. Another dévissage reduces us to the case 𝑁𝑖 = Λ 𝑋𝑖 , the constant sheaf itself. Step 2): We may assume Λ = Z/𝑝. By assumption, Λ is flat over Z/𝑝 𝑚 for some 𝑚 ≥ 1. We immediately reduce to Λ = Z/𝑝 𝑚 . For any perfect affine scheme 𝑋 = Spec 𝑅 in characteristic 𝑝 > 0, we claim that RΓ(𝑋, Z/𝑝 𝑚 ) ⊗Z/𝑝 𝑚 Z/𝑝𝑟  RΓ(𝑋, Z/𝑝𝑟 ). Assuming the claim, we finish the reduction step by tensoring Equation (7.9) with the short exact sequence of Z/𝑝 𝑚 -modules 0 → Z/𝑝 𝑚−1 → Z/𝑝 𝑚 → Z/𝑝 → 0, using that finite limits commutes with tensor products. It remains to prove the claim. The Artin–Schreier–Witt exact sequence of sheaves on 𝑋𝑒𝑡´ yields 𝐹−id RΓ(𝑋, Z/𝑝 𝑚 ) = [𝑊𝑚 (𝑅) → 𝑊𝑚 (𝑅)].  Now we use that 𝑊𝑚 (𝑅) ⊗Z/𝑝 𝑚 Z/𝑝𝑟 → 𝑊𝑟 (𝑅) compatibly with 𝐹, which holds since 𝑅 is perfect. This shows the claim, and we have accomplished Step 2). Step 3): We may assume 𝑞 is prime. Recall that 𝑞 = 𝑝𝑟 is a prime power. In order to reduce to the case 𝑟 = 1, let 𝑋𝑖′ := 𝑋𝑖 , but now regarded as a scheme over ′ = Ã𝑟 𝑋 . The Galois group Gal(F /F ) is generated by the F 𝑝 . We have 𝑋𝑖,F 𝑞 𝑝 𝑖=1 𝑖,F 𝑝-Frobenius, which acts by permuting the components in this disjoint union. Thus, we have D((𝑋𝑖′) Weil ) = D(𝑋𝑖Weil ). The same reasoning also applies to several factors 𝑋𝑖Weil , so we may assume our ground field to be F 𝑝 . Ë Step 4): Set 𝑅 := 𝑖,F 𝑝 𝑅𝑖 , 𝑅F := 𝑅 ⊗F 𝑝 F. We write 𝜙𝑖 for the 𝑝-Frobenius on 𝑅𝑖 and also for any map on a tensor product involving 𝑅𝑖 , by taking the identity on the remaining tensor factors. By Artin–Schreier theory, we have 6.6.1 𝜙𝑖 −id RΓ(𝑋𝑖Weil , Z/𝑝) = RΓ(𝑋𝑖 , Z/𝑝) = [𝑅𝑖 → 𝑅𝑖 ], 𝜙−id RΓ(𝑋1,F ×F . . . ×F 𝑋𝑛,F , Z/𝑝) = [𝑅F → 𝑅F ], 47 where 𝜙 is the absolute 𝑝-Frobenius of 𝑅F . Thus, the right hand side in (7.9) is the homotopy orbits of the action of Z𝑛+1 on 𝑅F , whose basis vectors act as 𝜙1 , . . . , 𝜙𝑛 and 𝜙. Note that 𝜙 is the composite 𝜙F ◦ 𝜙1 ◦ · · · ◦ 𝜙𝑛 , where 𝜙F is the Frobenius on F. Thus, the previously mentioned Z𝑛+1 -action on 𝑅F is equivalent to the one where the basis vectors act as 𝜙1 , . . . , 𝜙𝑛 and 𝜙F . We conclude our claim by using id−𝜙F that [𝑅F → 𝑅F ] is quasi-isomorphic to 𝑅[0]. 7.4 □ Drinfeld’s lemma The essential surjectivity in Theorem 7.1.2 is based on the following variant of Drinfeld’s lemma (V. G. Drinfeld, 1980, Theorem 2.1) (see also (L. Lafforgue, 1997, IV.2, Theorem 4), (Lau, 2004, Theorem 8.1.4), (V. Lafforgue, 2018, Lemme 8.11), and (Kedlaya, 2019, Theorem 4.2.12), (Heinloth, 2018, Lemma 6.3), (Scholze and Weinstein, 2020, Theorem 16.2.4) for expositions). Its formulation is close to (Lau, 2004, Theorem 8.1.4), and in this form is a slight extension of (V. Lafforgue, 2018, Lemme 8.2) for Zℓ -coefficients and (Xue, 2020b, Lemma 3.3.2) for Qℓ -coefficients. We will drop the coefficient ring Λ from the notation whenever convenient. Let 𝑋1 , . . . , 𝑋𝑛 be Noetherian schemes over F𝑞 , and denote 𝑋 = 𝑋1 ×F𝑞 . . . ×F𝑞 𝑋𝑛 . Recall the Frobenius–Weil groupoid FWeil(𝑋), see Definition 6.5.1. The projections 𝑋F → 𝑋𝑖,F onto the single factors induce a continuous map of locally profinite groupoids 𝜇 : FWeil(𝑋) → Weil(𝑋1 ) × . . . × Weil(𝑋𝑛 ). (7.10) Theorem 7.4.1 (Drinfelds’s lemma). Let Λ be as in Situation 7.1.1. Restriction along the map Equation (7.10) induces an equivalence

RepΛ Weil(𝑋1 ) × . . . × Weil(𝑋𝑛 ) → RepΛ FWeil(𝑋) , (7.11) between the abelian categories of continuous representations on finitely presented Λ-modules. Proof. For all objects 𝑥 ∈ FWeil(𝑋), that is, all geometric points 𝑥 → 𝑋F , passing to the automorphism groups induces a commutative diagram of locally profinite groups / FWeil(𝑋, 𝑥) / Z𝑛 / 𝜋1 (𝑋F , 𝑥) 1 𝜇𝑥 1 / Î𝑛  𝑖=1 𝜋1 (𝑋𝑖,F , 𝑥) / Î𝑛  𝑖=1 Weil(𝑋𝑖 , 𝑥) / Z𝑛 . 48 The left vertical arrow is surjective (Stacks, Tag 0BN6)0385. Thus 𝜇𝑥 is surjective as well and hence (7.11) is fully faithful. For essential surjectivity, it remains to show that any continuous representation FWeil(𝑋, 𝑥) → GL(𝑀) on a finitely presented Λmodule 𝑀 factors through 𝜇𝑥 . The key input is Drinfeld’s lemma: it implies that 𝜇𝑥 induces an isomorphism on profinite completions. Therefore, it is enough to apply Lemma 7.4.2 below with 𝐻 := FWeil(𝑋, 𝑥) → Weil(𝑋1 ) × . . . × Weil(𝑋𝑛 ) =: 𝐺 and 𝐾 := 𝜋1 (𝑋F , 𝑥). This completes the proof of (7.11). □ The following lemma formalizes a few arguments from (Xue, 2020b, §3.2.3), and we reproduce the proof for the convenience of the reader: Lemma 7.4.2 (Drinfeld, Xue). Let Λ be as in 7.1.1. Let 𝜇 : 𝐻 → 𝐺 be a continuous surjection of locally profinite groups that induces an isomorphism on profinite completions. Assume that there exists a compact open normal subgroup 𝐾 ⊂ 𝐻 containing ker 𝜇 such that 𝐻/𝐾 is finitely generated and injects into its profinite completion. Then 𝜇 induces an equivalence RepΛ (𝐺)  RepΛ (𝐻) between their categories of continuous representations on finitely presented Λmodules. Proof. The case where Λ is finite discrete is obvious, and hence so is the case Λ = O𝐸 for some finite field extension 𝐸 ⊃ Qℓ . The case Λ = 𝐸 is reduced to Λ = Qℓ . As 𝜇 is surjective, it remains to show that every continuous representation 𝜌 : 𝐻 → GL(𝑀) on a finite-dimensional Qℓ -vector space factors through 𝐺, that is, ker 𝜇 ⊂ ker 𝜌. One shows the following properties: 1. The group ker 𝜇 is the intersection over all open subgroups in 𝐾 which are normal in 𝐻. 2. The group ker 𝜌 ∩𝐾 is a closed normal subgroup in 𝐻 such that 𝐾/ker 𝜌 ∩𝐾  𝜌(𝐾) is topologically finitely generated. These properties imply ker 𝜇 ⊂ ker 𝜌 ∩ 𝐾 as follows: For a finite group 𝐿, let 𝑈 𝐿 := ∩ ker(𝐾 → 𝐿) where the intersection is over all continuous morphisms 𝐾 → 𝐿 that are trivial on ker 𝜌 ∩ 𝐾. Because of the topologically finitely generatedness in (2), this is a finite intersection so that 𝑈 𝐿 is open in 𝐾. Also, it is normal in 𝐻, and 49 hence ker 𝜇 ⊂ 𝑈 𝐿 by (1). On the other hand, it is evident that ker 𝜌 ∩ 𝐾 = ∩ 𝐿 𝑈 𝐿 because 𝐾 is profinite. For the proof of (1) observe that ker 𝜇 agrees with the kernel of 𝐻 → 𝐻 ∧  𝐺 ∧ by our assumption on the profinite completions. Using ker 𝜇 ⊂ 𝐾 and the injection 𝐻/𝐾 → (𝐻/𝐾) ∧ implies (1). For (2) it is evident that ker 𝜌 ∩ 𝐾 is a closed normal subgroup in 𝐻. Since 𝐾 is compact, its image 𝜌(𝐾) is a closed subgroup of the ℓ-adic Lie group GL(𝑀), hence an ℓ-adic Lie group itself. The final assertion follows from (Serre, 1964, théorème 2). □ For the overall goal of proving essential surjectivity in Theorem 7.1.2, we need to investigate how representations of product groups factorize into external tensor products of representations. In view of Lemma 6.5.2 and its proof, it is enough to consider representations of abstract groups, disregarding the topology. This is done in the next section. 7.5 Factorizing representations In this subsection, let Λ be a Dedekind domain (Stacks, Tag 034X). Thus, since Λ is Noetherian and of projective dimension ≤ 1, any submodule 𝑁 of a finite projective Λ-module 𝑀 is again finite projective. f.p Given any group 𝑊, we write RepΛ (𝑊) for the category of 𝑊-representations on finite projective Λ-modules. As in (Curtis and Reiner, 2006, Sections 73.8, 75), we say that such a 𝑊-representation 𝑀 is fp-simple if any subrepresentation 0 ≠ 𝑁 ⊂ 𝑀 has maximal rank. By induction f.p on the rank, every non-zero representation in RepΛ (𝑊) admits a non-zero fp-simple subrepresentation. The proof of the following lemma is left to the reader. It parallels (Curtis and Reiner, 2006, Theorem 75.6). f.p Lemma 7.5.1. A representation 𝑀 ∈ RepΛ (𝑊) is fp-simple if and only if 𝑀 ⊗Λ Frac(Λ) is fp-simple (hence, simple). The following proposition will serve in the proof of Theorem 7.1.2 using Theorem 7.4.1, where we will need to decompose representations of a product of Weil groups into decompositions of the individual Weil groups. 50 f.p Proposition 7.5.2. Let 𝑊 = 𝑊1 ×𝑊2 be a product of two groups. Let 𝑀 ∈ RepΛ (𝑊) be fp-simple. Fix a 𝑊1 -subrepresentation 𝑀1 ⊂ 𝑀 that is fp-simple. Consider the 𝑊2 -representation 𝑀2 := Hom𝑊1 (𝑀1 , 𝑀) and the associated evaluation map ev : 𝑀1 ⊠ 𝑀2 → 𝑀. 1. If Λ is an algebraically closed field, then ev is an isomorphism and 𝑀2 is simple. 2. If Λ is a perfect field, then ev is a split surjection and 𝑀2 is semi-simple. 3. If Λ is a Dedekind domain of Krull dimension 1 with perfect fraction field, then there is a short exact sequence 0 → 𝑀 ⊕ ker(ev) → 𝑀1 ⊠ 𝑀2 → 𝑇 → 0, (7.12) where 𝑇 is Λ-torsion. f.p Proof. Note that ev is a map in RepΛ (𝑊). Its image has maximal rank by the fp-simplicity of 𝑀. Thus, if Λ is a field, then it is surjective. In case Item 1, we claim that ev is an isomorphism. The following argument was explained to us by Jean-François Dat: for injectivity, observe that 𝑀1 ⊠ 𝑀2 = 𝑀1⊕ dim 𝑀2 as 𝑊1 -representations. Hence, if the kernel of ev is non-trivial, then it contains 𝑀1 as an irreducible constituent. Therefore, it suffices to prove that Hom𝑊1 (𝑀1 , ev) is injective. Since Λ is algebraically closed, we have End𝑊1 (𝑀1 ) = Λ by Schur’s lemma. Hence, the composition 𝑀2 = Hom𝑊1 (End𝑊1 (𝑀1 ), 𝑀2 )  Hom𝑊1 (𝑀1 , 𝑀1 ⊠ 𝑀2 ) → Hom𝑊1 (𝑀1 , 𝑀) = 𝑀2 is the identity. This shows that Hom𝑊1 (𝑀1 , ev) is an isomorphism. In case Item 2, we claim that 𝑀1 ⊠ 𝑀2 is semi-simple, and hence that 𝑀 appears as a direct summand. Using (Bourbaki, 2012, Section 13.4 Corollaire) applied to the group algebras it is enough to show that 𝑀1 and 𝑀2 are absolutely semi-simple. Since Λ is perfect, any finite-dimensional representation is semi-simple if and only if it is absolutely semi-simple, see (Bourbaki, 2012, Section 13.1). Hence, it remains to check that 𝑀2,Λ̄ = 𝑀2 ⊗Λ Λ̄ is semi-simple where Λ̄/Λ is an algebraic closure. The module 𝑀2,Λ̄ = Hom𝑊1 (𝑀1,Λ̄ , 𝑀Λ̄ ) splits as a direct sum according to the simple ¯ is either constituents 𝑀¯ 1 ⊂ 𝑀1,Λ̄ and 𝑀¯ ⊂ 𝑀Λ̄ . Finally, each 𝑀¯ 2 = Hom𝑊1 ( 𝑀¯ 1 , 𝑀) ¯ then it simple or vanishes: if there exists a non-zero 𝑊1 -equivariant map 𝑀¯ 1 → 𝑀, 51 must be injective by the simplicity of 𝑀¯ 1 . As Λ̄ is algebraically closed, the proof of Item 1 shows that 𝑀¯  𝑀¯ 1 ⊠ 𝑀¯ 2 so that 𝑀¯ 2 must be simple because 𝑀¯ is so. This shows that 𝑀2 is absolutely semi-simple as well. In case Item 3, abbreviate Λ′ := FracΛ, 𝑀 ′ := 𝑀 ⊗Λ Λ′ and so on. We will repeatedly use that (−) ⊗Λ Λ′ preserves and detects fp-simplicity of representations, see Lemma 7.5.1. By Item 2, the evaluation map ev′ := ev ⊗ Λ′ admits a Λ′-linear section 𝑖˜ : 𝑀 ′ → (𝑀1 ⊠ 𝑀2 ) ′. As 𝑀 ′ is finitely presented, there is some 0 ≠ 𝜆 ∈ Λ such that 𝜆𝑖˜ arises by scalar extension of a map 𝑖 : 𝑀 → 𝑀1 ⊠ 𝑀2 . By construction, the map 𝑖 ⊕ incl : 𝑀 ⊕ ker(ev) → 𝑀1 ⊠ 𝑀2 is an isomorphism after tensoring with Λ′. So its cokernel is Λ-torsion, and it is injective as both modules at the left are projective (hence Λ-torsion free). This finishes the proof of the proposition. □ 7.6 Essential surjectivity In this section, we prove the essential surjectivity asserted in Theorem 7.1.2. Throughout, we freely use the full faithfulness proven in Proposition 7.3.1 and Proposition 7.3.2. Recall that 𝑋1 , . . . , 𝑋𝑛 are finite type F𝑞 -schemes, and write 𝑋 := 𝑋1 ×F𝑞 . . . ×F𝑞 𝑋𝑛 . Let Λ be either a finite discrete ring, a finite field extension 𝐸 ⊃ Qℓ for ℓ ≠ 𝑝 or its ring of integers O𝐸 . Note that this covers all cases from Situation 7.1.1. First, we show that it suffices to prove containment in the essential image étale locally: Lemma 7.6.1. Let 𝑈𝑖 → 𝑋𝑖 be quasi-compact étale surjections for 𝑖 = 1, . . . , 𝑛. Then the following properties hold:
1. An object 𝑀 ∈ D 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ belongs to the full subcategory

Dcons 𝑋1Weil , Λ ⊗Perf Λ∗ . . . ⊗Perf Λ∗ Dcons 𝑋𝑛Weil , Λ if and only if its restriction 𝑀 |𝑈 Weil ×...×𝑈𝑛Weil belongs to the full subcategory 1


Dcons 𝑈1Weil , Λ ⊗Perf Λ∗ . . . ⊗Perf Λ∗ Dcons 𝑈𝑛Weil , Λ ⊂ D 𝑈1Weil × . . . ×𝑈𝑛Weil , Λ . 2. Assume that all 𝑈𝑖 → 𝑋𝑖 are finite étale. Then Item 1 holds for the categories of lisse sheaves. Proof. The only if direction in part Item 1 is clear. Conversely, assume that 𝑀 |𝑈 Weil ×...×𝑈𝑛Weil lies in the essential image of the external tensor product. By étale 1 52 descent, we have an equivalence:
 D 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ −→ Tot 
Weil Weil D 𝑈1,• × . . . × 𝑈𝑛,• ,Λ  . ∼ In particular, we get an equivalence |( 𝑗• )! ◦ 𝑗•∗ 𝑀 | − → 𝑀 where 𝑗• := 𝑗1,• × . . . × 𝑗 𝑛,• with 𝑗𝑖,• : 𝑈𝑖,• → 𝑋𝑖 for 𝑖 = 1, . . . , 𝑛. For each 𝑚 ≥ 0, the object 𝑗 𝑚∗ 𝑀 lies in

Weil Weil Dcons 𝑈1,𝑚 , Λ ⊗Perf Λ∗ . . . ⊗Perf Λ∗ Dcons 𝑈𝑛,𝑚 ,Λ . It follows from Chapter 5 Item ii that these subcategories are preserved under ( 𝑗 𝑚 )! . So we see

( 𝑗 𝑚 )! 𝑗 𝑚∗ (𝑀) ∈ Dcons 𝑋1Weil , Λ ⊗Perf Λ∗ . . . ⊗Perf Λ∗ Dcons 𝑋𝑛Weil , Λ for all 𝑚 ≥ 0. For every 𝑚 ≥ 0, let 𝑀𝑚 denote the realization of the 𝑚-th skeleton of  the simplicial object ( 𝑗 • )! ◦ 𝑗•∗ 𝑀 so that we have a natural equivalence colim 𝑀𝑚 − → 𝑀 in D(𝑋1Weil × . . . × 𝑋𝑛Weil , Λ). We claim that 𝑀 is a retract of some 𝑀𝑚 , and hence

lies in Dcons 𝑋1Weil , Λ ⊗Perf Λ∗ . . .⊗Perf Λ∗ Dcons 𝑋𝑛Weil , Λ by idempotent completeness. To prove the claim, note that the sheaf 𝑀F ∈ Dcons (𝑋F , Λ) underlying 𝑀 is compact in the category of ind-constructible sheaves Dindcons (𝑋F , Λ), see Chapter 5 Item viii. As taking partial Frobenius fixed points is a finite limit, so commutes with filtered colimits, we see that the natural map of mapping complexes  colim HomD(𝑋 Weil ×...×𝑋𝑛Weil ,Λ) (𝑀, 𝑀𝑚 ) −→ HomD(𝑋 Weil ×...×𝑋𝑛Weil ,Λ) (𝑀, colim 𝑀𝑚 ) 1 1  is an equivalence. In particular, the inverse equivalence 𝑀 − → colim 𝑀𝑚 factors through some 𝑀𝑚 , presenting 𝑀 as a retract of 𝑀𝑚 . This proves the claim, and hence Item 1. For Item 2, note that if 𝑈𝑖 → 𝑋𝑖 are finite étale, then the functors ( 𝑗 𝑚 )! preserve the lisse categories, see Chapter 5 Item ii. In particular, for every 𝑚 ≥ 0 the object ( 𝑗 𝑚 )! 𝑗 𝑚∗ (𝑀) is lisse and so is 𝑀𝑚 . We conclude using compactness as before. □ Using Lemma 4.2.4 and Chapter 5 Item viii, the fully faithful functor Equation (7.1) uniquely extends to a fully faithful functor

Ind D• (𝑋1Weil , Λ) ⊗ModΛ∗ . . . ⊗ModΛ∗ Ind D• (𝑋𝑛Weil , Λ) → D(𝑋1Weil ×. . .× 𝑋𝑛Weil , Λ) (7.13) for • ∈ {lis, cons}. We use this in the following variant of Lemma 7.6.1. 53 Lemma 7.6.2. The statements Item 1 and Item 2 of Lemma 7.6.1 hold for the functor Equation (7.13) with • ∈ {lis, cons}. Namely, to check that an object lies in the essential image of Equation (7.13), one can pass to a quasi-compact étale cover if • = cons, and to a finite étale cover if • = lis. Proof. This is immediate from the proof of Lemma 7.6.1: Arguing as above and using étale descent for ind-constructible, resp. ind-lisse sheaves (Chapter 5 Item iii), we see that 𝑀  colim 𝑀𝑚 with

𝑀𝑚 ∈ Ind D• (𝑋1Weil , Λ) ⊗ModΛ∗ . . . ⊗ModΛ∗ Ind D• (𝑋𝑛Weil , Λ) for all 𝑚 ≥ 0 and • = cons, resp. • = lis. As the essential image of Equation (7.13) is closed under colimits, 𝑀 lies in the corresponding subcategory as well. □ Now we have enough tools to prove the categorical Künneth formula alias derived Drinfeld’s lemma: Proof of 7.1.2. In view of Proposition 7.3.1 and Proposition 7.3.2, it remains to show the essential surjectivity of the external tensor product functor on Weil sheaves Equation (7.1) under the assumptions in Theorem 7.1.2. Part Item 1, the case of constructible sheaves, is reduced to part Item 2, the case of lisse sheaves, by taking a stratification as in Definition 6.4.1 Item 2 and using the full faithfulness already proven. Here we note that by refining the stratification witnessing the constructibility if necessary, we can even assume all strata to be smooth, so in particular geometrically unibranch. Hence, it remains to prove part Item 2, that is, the essential surjectivity of the fully faithful functor


⊠ : Dlis 𝑋1Weil , Λ ⊗Perf Λ∗ . . . ⊗Perf Λ∗ Dlis 𝑋𝑛Weil , Λ → Dlis 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ , (7.14) when either Λ is finite discrete as in cases Item a, Item d in Theorem 7.1.2 Item 2, or Λ = O𝐸 for a finite field extension 𝐸 ⊃ Qℓ , ℓ ≠ 𝑝 as in case Item b, or Λ = 𝐸 and the 𝑋𝑖 are geometrically unibranch as in the remaining case Item c. In fact, the latter two cases are easier to handle due to the presence of natural t-structures on the categories of lisse sheaves (Chapter 5 Item vi). So we will distinguish two cases below: 1) Λ = O𝐸 , or Λ = 𝐸 and all 𝑋𝑖 geometrically unibranch; 2) Λ is finite discrete. Now pick 𝑀 ∈ Dlis (𝑋1Weil × . . . × 𝑋𝑛Weil , Λ). By Chapter 5 Item v, 𝑀 is bounded in the standard t-structure on D(𝑋1Weil × . . . × 𝑋𝑛Weil , Λ). So 𝑀 is a successive extension 54 of its cohomology sheaves H 𝑗 (𝑀), 𝑗 ∈ Z. As 𝑀 is lisse, Lemma 6.5.4 Item 1 shows in both cases 1) and 2) that each H 𝑗 (𝑀) comes from a continuous representation on a finitely presented Λ-module in RepΛ FWeil(𝑋)
7.4.1  RepΛ (𝑊), (7.15) where we denote 𝑊 := 𝑊1 × . . . × 𝑊𝑛 with 𝑊𝑖 := Weil(𝑋𝑖 ). Throughout, we repeatedly use that the functor (7.14) is fully faithful, commutes with finite (co-)limits and shifts, and that its essential image is closed under retracts (as the source category is idempotent complete, by definition) and contains all perfect-constant sheaves. Case 1): Assume Λ = O𝐸 , or Λ = 𝐸 and all 𝑋𝑖 geometrically unibranch. In this case, we have a t-structure on lisse Weil sheaves so that each H 𝑗 (𝑀) belongs to Dlis (𝑋1Weil × . . . × 𝑋𝑛Weil , Λ) ♥ . By induction on the length of 𝑀, using the full faithfulness of Equation (7.14), we reduce to the case where 𝑀 is abelian, that is, a continuous 𝑊-representation on a finitely presented Λ-module. The external tensor product induces a commutative diagram  / Rep ⊠ RepΛ (𝑊1 ) × . . . × RepΛ (𝑊𝑛 )  Dlis (𝑋1Weil , Λ) ♥ × . . . × Dlis (𝑋𝑛Weil , Λ) ♥ ⊠ Λ (𝑊)   / Dlis (𝑋 Weil × . . . × 𝑋 Weil , Λ) ♥ , 1 𝑛 where the vertical equivalences are given by Lemma 6.5.4. Note that 𝑀 splits into a direct sum 𝑀tor ⊕ 𝑀f.p where the finitely presented Λ-module underlying 𝑀tor is Λ-torsion and 𝑀f.p is projective. So we can treat either case separately. Using that the essential image of Equation (7.14) is closed under extensions (by full faithfulness) and retracts, the finite projective case is reduced to the fp-simple case and, by Proposition 7.5.2, to the finite torsion case. Note that the 𝑊𝑖 -representations constructed in, say Lemma 7.5.1 (7.12), are obtained from 𝑀f.p by taking subquotients and tensor products, so are automatically continuous. Next, as the Λ-module underlying 𝑀tor is finite torsion, the Λ-sheaf 𝑀tor is perfect-constant along some finite étale cover. So we conclude by 7.6.1 Item 2. Case 2): Assume Λ is finite discrete as above. In a nutshell, the argument is similar to the last step in case 1), but a little more involved due to the absence of natural t-structures on the categories of lisse sheaves in general, see Chapter 5 Item vi. More precisely, in the special case, where Λ is a finite field, the argument of case 55 1) applies, but not so if Λ = Z/ℓ 2 , say. So, instead, we extend Equation (7.14) by passing to Ind-completions to a commutative diagram

Dlis 𝑋1Weil , Λ ⊗Perf Λ∗ . . . ⊗Perf Λ∗ Dlis 𝑋𝑛Weil , Λ ⊠ / Dlis 𝑋 Weil × . . . × 𝑋 Weil , Λ
𝑛 1  

Ind(⊠)
/ Ind Dlis (𝑋 Weil × . . . × 𝑋 Weil , Λ) , Ind Dlis (𝑋1Weil , Λ) ⊗ModΛ∗ . . . ⊗ModΛ∗ Ind Dlis (𝑋𝑛Weil , Λ) 𝑛 1 of full subcategories of D(𝑋1Weil × . . . × 𝑋𝑛Weil , Λ), see the discussion around Equation (7.13). Note that the fully faithful embedding Equation (7.13) factors through Ind(⊠). Both vertical arrows are the inclusion of the subcategories of compact objects by idempotent completeness of the involved categories and Equation (4.1). Thus, if 𝑀 lies in the essential image of Ind(⊠), then it is a retract of a finite colimit of objects in the essential image of ⊠, so lies itself in this essential image. As 𝑀 is a successive extension of its cohomology sheaves H 𝑗 (𝑀), it suffices to show

H 𝑗 (𝑀) ∈ Ind Dlis (𝑋1Weil , Λ) ⊗ModΛ∗ . . . ⊗ModΛ∗ Ind Dlis (𝑋𝑛Weil , Λ) , for all 𝑗 ∈ Z. So fix 𝑗 and denote 𝑁 := H 𝑗 (𝑀) viewed as a continuous 𝑊representation on a finitely presented Λ-module. As Λ is finite, 𝑁 comes from a continuous representation of 𝜋1 (𝑋1 ) ×. . . 𝜋1 (𝑋𝑛 ) on which some open subgroup acts trivially. Hence, there exist finite étale surjections 𝑈𝑖 → 𝑋𝑖 such that the subgroup 𝜋1 (𝑈1 ) × . . . × 𝜋1 (𝑈𝑛 ) acts trivially on 𝑁. In particular, 𝑁 |𝑈 Weil ×...×𝑈𝑛Weil is constant, 1 and hence lies in the essential image of the functor

Mod 𝑅  Ind Perf 𝑅 → Ind Dlis (𝑈1Weil × . . . × 𝑈𝑛Weil , Λ) , where 𝑅 := Γ(𝜋0 (𝑈1 ) × . . . × 𝜋0 (𝑈𝑛 ), Λ). As the sets 𝜋0 (𝑈𝑖 ) are finite discrete, each 𝑅𝑖 := Γ(𝜋0 (𝑈𝑖 ), Λ) is a finite free Λ∗ -algebra, and we have 𝑅  𝑅1 ⊗Λ∗ . . . ⊗Λ∗ 𝑅𝑛 . Thus, the external tensor product induces a commutative diagram Mod 𝑅1 ⊗ModΛ∗ . . . ⊗ModΛ∗ Mod 𝑅𝑛  / Mod 𝑅  

Ind(⊠)
/ Ind Dlis (𝑈 Weil × . . . × 𝑈 Weil , Λ) , Ind Dlis (𝑈1Weil , Λ) ⊗ModΛ∗ . . . ⊗ModΛ∗ Ind Dlis (𝑈𝑛Weil , Λ) 𝑛 1 where the upper horizontal arrow is an equivalence. So 𝑁 |𝑈 Weil ×...×𝑈𝑛Weil lies in the 1 essential image of Ind(⊠), and we conclude by Lemma 7.6.2 applied to the finite étale covers 𝑈𝑖 → 𝑋𝑖 and • = lis. □ 56 Chapter 8 IND-CONSTRUCTIBLE WEIL SHEAVES In this section, we introduce the full subcategories

Dindlis 𝑋 Weil , Λ ⊂ Dindcons 𝑋 Weil , Λ
of D 𝑋 Weil , Λ consisting of ind-objects of lisse, resp. constructible sheaves equipped with partial Frobenius action. That is, the partial Frobenius only preserves the indsystem of objects, but not necessarily each member. We will define analogous categories for a product of schemes. Similarly to the lisse, resp. constructible case, there is a fully faithful functor


Dindcons 𝑋1Weil , Λ ⊗ModΛ∗ . . .⊗ModΛ∗ Dindcons 𝑋𝑛Weil , Λ → Dindcons 𝑋1Weil ×. . .×𝑋𝑛Weil , Λ , which, however, will not be an equivalence in general, see Remark 8.1.6. Nevertheless, we can identify a class of objects that lie in the essential image and that include many cases of interest such as the shtuka cohomology studied in (V. Lafforgue, 2018; V. Lafforgue and Zhu, 2019; Xue, 2020b; Xue, 2020c). 8.1 Ind-constructible Weil sheaves Let F𝑞 be a finite field of characteristic 𝑝 > 0, and fix an algebraic closure F. Let 𝑋1 , . . . , 𝑋𝑛 be schemes of finite type over F𝑞 . Let Λ be a condensed ring associated with the one of the following topological rings: a discrete coherent torsion ring (for example, a discrete finite ring), an algebraic field extension 𝐸 ⊃ Qℓ , or its ring of integers O𝐸 . We write 𝑋 := 𝑋1 ×F𝑞 . . . ×F𝑞 𝑋𝑛 , and denote by 𝑋𝑖,F := 𝑋𝑖 ×F𝑞 Spec F and 𝑋F := 𝑋 ×F𝑞 Spec F the base change. Recall that under these assumptions, by Chapter 5 Item viii, we have a fully faithful embedding
 Ind Dcons (𝑋F , Λ) −→ Dindcons (𝑋F , Λ) ⊂ D(𝑋F , Λ), (8.1) and likewise for (ind-)lisse sheaves. Definition 8.1.1. An object 𝑀 ∈ D(𝑋1Weil × . . . × 𝑋𝑛Weil , Λ) is called ind-lisse, resp. ind-constructible if the underlying sheaf 𝑀F ∈ D(𝑋F , Λ) is ind-lisse, resp. indconstructible in the sense of Definition 5.0.1. 57 We denote by

Dindlis 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ ⊂ Dindcons 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ the resulting full subcategories of D(𝑋1Weil × . . . × 𝑋𝑛Weil , Λ) consisting of ind-lisse, resp. ind-constructible objects. Both categories are naturally commutative algebra objects in PrSt Λ∗ (see the notation from Chapter 4), that is, presentable stable Λ∗ -linear symmetric monoidal ∞-categories where Λ∗ := Γ(∗, Λ) is the ring underlying Λ. It is immediate from Definition 8.1.1 that the equivalence (6.6) restricts to an equivalence  
D• 𝑋1Weil × . . . × 𝑋𝑛Weil , Λ  Fix D• (𝑋F , Λ), 𝜙∗𝑋1 , . . . , 𝜙∗𝑋𝑛 for • ∈ {indlis, indcons}. Remark 8.1.2. Note that that we have a fully faithful embedding of 𝐷 cons (𝑋 Weil ) into Dindcons (𝑋 Weil ) whose image consists of compact objects. However, the latter category is not generated by this image. Indeed, even in the case of a point, the ind-cons category consists of Λ-modules with an action of an endomorphism, whereas the image of the embedding consists of Λ-modules with an action of an automorphism. This automorphism does not have to fix any finitely generated submodule, which would be the case for any objects generated by the image of the constructible Weil complexes. Our goal in this chapter is to obtain a categorical Künneth formula for the categories of ind-lisse, resp. ind-constructible Weil sheaves. In order to state the result, we need the following terminology. Under our assumptions on Λ, each cohomology
sheaf H 𝑗 (𝑀), 𝑗 ∈ Z for 𝑀 ∈ Dlis (𝑋F , Λ is naturally a continuous representation of proét the proétale fundamental groupoid 𝜋1 (𝑋F ) on a finitely presented Λ-module, see 6.5.4. Further, the projections 𝑋F → 𝑋𝑖,F induce a full surjective map of topological groupoids proét proét proét 𝜋1 (𝑋F ) → 𝜋1 (𝑋1,F ) × . . . × 𝜋1 (𝑋𝑛,F ). (8.2) Definition 8.1.3. Let 𝑀 ∈ D(𝑋F , Λ). proét 1. The sheaf 𝑀 is called split lisse if it is lisse and the action of 𝜋1 H 𝑗 (𝑀) factors through (8.2) for all 𝑗 ∈ Z. (𝑋F ) on 58 2. The sheaf 𝑀 is called split constructible if it is constructible and there exists a finite subdivision into locally closed subschemes 𝑋𝑖,𝛼 ⊆ 𝑋𝑖 such that for each Î 𝑋𝛼 = 𝑖 𝑋𝑖,𝛼 ⊆ 𝑋, each restriction 𝑀 | 𝑋 𝛼 is split lisse. Definition 8.1.4. An object 𝑀 ∈ D(𝑋1Weil × . . . × 𝑋𝑛Weil , Λ) is called ind-(split lisse), resp. ind-(split constructible) if the underlying object 𝑀F ∈ D(𝑋F , Λ) is a colimit of split lisse, resp. split constructible objects. As the category D• (𝑋F , Λ), • ∈ {indlis, indcons} is cocomplete, every ind-(split lisse) object is ind-lisse, and likewise, every ind-(split constructible) object is indconstructible. Theorem 8.1.5. Assume that Λ is either a finite discrete ring of prime-to-𝑝 torsion, an algebraic field extension 𝐸 ⊃ Qℓ for ℓ ≠ 𝑝, or its ring of integers O𝐸 . Then the functor induced by the external tensor product D• (𝑋1Weil , Λ) ⊗ModΛ∗ . . . ⊗ModΛ∗ D• (𝑋𝑛Weil , Λ) → D• (𝑋1Weil × . . . × 𝑋𝑛Weil , Λ) (8.3) is fully faithful for • ∈ {indlis, indcons}. For • = indlis, resp. • = indcons the essential image contains the ind-(split lisse), resp. ind-(split constructible) objects. Proof. For full faithfulness, it is enough to consider the case • = indcons. Using Lemma 4.2.5, it remains to show that the functor ! Ì Ì Dindcons 𝑋𝑖,F , Λ)  Ind Dcons (𝑋𝑖,F , Λ) → D• (𝑋F , Λ) (8.4) 𝑖 𝑖 is fully faithful. In view of Equation (8.1), this is immediate from the Künneth formula for constructible Λ-sheaves as explained in Section 7.3. To identify objects in the essential image, we note that the fully faithful functors Equation (8.3) and Equation (8.4) induce a Cartesian diagram (see Lemma 4.2.5): Ë Weil , Λ) 𝑖 D• 𝑋𝑖 Ë  𝑖 D• (𝑋𝑖,F , Λ) / D• (𝑋 Weil × . . . × 𝑋 Weil , Λ) 𝑛 1 (8.5)  / D• (𝑋F , Λ), for • ∈ {indlis, indcons}. Thus, it is enough to show that the object 𝑀F underlying an ind-split object 𝑀 lies in the image of the lower horizontal arrow. Since this essential image is closed under colimits, it remains to show it contains the split lisse objects for • = indlis, resp. the split constructible objects for • = indcons. 59 By the full faithfulness of Equation (8.4), the split constructible case reduces to the split lisse case, see also the proof of 7.1.2 in Section 7.6. So assume • = indlis and let 𝑀F ∈ D(𝑋F , Λ) be split lisse. As each cohomology sheaf H 𝑗 (𝑀F ), 𝑗 ∈ Z is at least ind-lisse, an induction on the cohomological length of 𝑀F reduces us to show that H 𝑗 (𝑀F ) lies in the essential image. By definition, being split lisse implies that proét proét proét the action of 𝜋1 (𝑋F ) on H 𝑗 (𝑀F ) factors through 𝜋1 (𝑋1,F ) × . . . × 𝜋1 (𝑋𝑛,F ). Then the arguments of Section 7.6 show that H 𝑗 (𝑀F ) lies is in the essential image of the lower horizontal arrow in Equation (8.5). We leave the details to the reader. □ Remark 8.1.6. The functor Equation (8.3) is not essentially surjective in general. To see this, note that the functor Dindcons (𝑋 Weil , Λ) → Dindcons (𝑋F , Λ) admits a left adjoint 𝐹 that adds a free partial Frobenius action. Explicitly, for an object 𝑀 ∈ Dindcons (𝑋F , Λ) the object 𝐹 (𝑀) has underlying sheaf 𝐹 (𝑀)F given by a countable direct sum of copies of 𝑀. If 𝑀 was not originally in the image of the external tensor product, then 𝐹 (𝑀) will not be either. This is, however, the only obstacle for essential surjectivity: as noted in the proof of Theorem 8.1.5, the diagram Equation (8.5) is Cartesian. 8.2 Cohomology of shtuka spaces Finally, let us mention a key application of Theorem 8.1.5. Let 𝑋 be a smooth projective geometrically connected curve over F𝑞 . Let 𝑁 ⊂ 𝑋 be a finite subscheme, and denote its complement by 𝑌 = 𝑋\𝑁. Let 𝐸 ⊃ Qℓ , ℓ ≠ 𝑝 be an algebraic field extension containing a fixed square root of 𝑞. Let O𝐸 be its ring of integers and denote by 𝑘 𝐸 the residue field. Let Λ be any of the topological rings 𝐸, O𝐸 , 𝑘 𝐸 . Let b the Langlands 𝐺 be a split (for simplicity) reductive group over F𝑞 . We denote by 𝐺 dual group of 𝐺 considered as a split reductive group over Λ. In the seminal works (V. G. Drinfeld, 1980; L. Lafforgue, 2002) (𝐺 = GL𝑛 ) and (V. Lafforgue, 2018; V. Lafforgue and Zhu, 2019) (general reductive 𝐺) on the Langlands correspondence over global function fields, the construction of the Weil(𝑌 )action on automorphic forms of level 𝑁 is realized using the cohomology sheaves of moduli stacks of shtukas, defined in (Varshavsky, 2004) and (V. Lafforgue, 2018, Section 2). As explained in (V. Lafforgue and Zhu, 2019; Dennis Gaitsgory, Kazhdan, et al., 2022; Zhu, 2021), the output of the geometric construction of Lafforgue can be encoded as a natural transformation f.p b𝐼
𝐼
H𝑁,𝐼 : RepΛ 𝐺 → Repcts Λ Weil(𝑌 ) , 𝐼 ∈ FinSet (8.6) 60 of functors FinSet → Cat from the category of finite sets to the category of 1f.p b•
assigns to a finite set 𝐼 the category of categories. Here the functor RepΛ 𝐺
𝐼 b on finite free Λ-modules, and Repcts Weil(𝑌 ) • the algebraic representations of 𝐺 Λ category of continuous representations of Weil(𝑌 ) 𝐼 in Λ-modules. In both cases, the transition maps are given by restriction of representations. Let us recall some elements of its construction. For a finite set 𝐼, (Varshavsky, 2004) and (V. Lafforgue, 2018, Section 2) define the ind-algebraic stack Cht𝑁,𝐼 classifying 𝐼-legged 𝐺-shtukas on 𝑋 with full level-𝑁-structure. The morphism sending a 𝐺-shtuka to its legs 𝔭 𝑁,𝐼 : Cht𝑁,𝐼 → 𝑌 𝐼 (8.7)
f.p b𝐼 is locally of finite presentation. For every 𝑊 ∈ RepΛ 𝐺 , there is the normalized Satake sheaf F𝑁,𝐼,𝑊 on Cht𝑁,𝐼 , see (V. Lafforgue, 2018, Définition 2.14). Base changing to F and taking compactly supported cohomology, we obtain the object H𝑁,𝐼 (𝑊) := (𝔭 𝑁,𝐼,F )! (F𝑁,𝐼,𝑊,F ) ∈ Dindcons 𝑌F𝐼 , Λ
see (V. Lafforgue, 2018, Définition 4.7) and (Xue, 2020a, Definition 2.5.1). Under the normalization of the Satake sheaves, the degree 0 cohomology sheaf H𝑁,𝐼 (𝑊) := H0 (H𝐼 (𝑊)) ∈ Dindcons 𝑌F𝐼 , Λ
♥ corresponds to the middle degree compactly supported intersection cohomology of Cht𝑁,𝐼 . Using the symmetries of the moduli stacks of shtukas, the sheaf H𝑁,𝐼 (𝑊) is endowed with a partial Frobenius equivariant structure (L. Lafforgue, 2002, §6). So we obtain objects H𝑁,𝐼 (𝑊) ∈ Dindcons (𝑌 Weil ) 𝐼 , Λ
♥ (8.8) Next, using the finiteness (Xue, 2020b) and smoothness (Xue, 2020c, Theorem 4.2.3) results, the classical Drinfeld’s lemma (Theorem 7.4.1) applies to give objects
𝐼 H𝑁,𝐼 (𝑊) ∈ Repcts Λ Weil(𝑌 ) . The construction of the natural transformation (8.6) encodes the functoriality and fusion satisfied by the objects {H𝑁,𝐼 (𝑊)} for varying 𝐼 and 𝑊. However, in order to analyze construction Equation (8.6) further, it is desirable to upgrade the natural transformation of functors Equation (8.6) to the derived level. Namely, to have construction for the complexes {H𝐼 (𝑊)} 𝐼,𝑊 and not just for their cohomology sheaves, compare with (Zhu, 2021). Such an upgrade is possible using 61 the derived version of Drinfeld’s lemma, as given in the following proposition. A further study of this construction will appear in future work of the first named author (T. H.). b𝐼 ), the shtuka Proposition 8.2.1. For Λ ∈ {𝐸, O𝐸 , 𝑘 𝐸 } and any 𝑊 ∈ RepΛ ( 𝐺 cohomology (8.8) lies in the essential image of the fully faithful functor Dindlis (𝑌 Weil , Λ) ⊗𝐼 → Dindcons (𝑌 Weil ) 𝐼 , Λ
(8.9) Proof. By (Xue, 2020c, Theorem 4.2.3), the ind-constructible sheaf H𝑁,𝐼 (𝑊) is ind-lisse. By (Xue, 2020b, Proposition 3.2.15), the action of FWeil(𝑌 𝐼 ) on H𝑁,𝐼 (𝑊) factors through the product Weil(𝑌 ) 𝐼 . In particular, the action of 𝜋1 (𝑋F𝐼 ) on H𝑁,𝐼 (𝑊) factors through the product 𝜋1 (𝑋F ) 𝐼 . So it is ind-(split lisse) in the sense of Definition 8.1.4, and we are done by Theorem 8.1.5. □ Remark 8.2.2. One can upgrade the above construction in a homotopy coherent way
to show that the whole complex H𝑁,𝐼 (𝑊) lies in Dindcons (𝑌 Weil ) 𝐼 , Λ . If 𝑁 ≠ ∅ so that H𝑁,𝐼 (𝑊) is known to be bounded, then Proposition 8.2.1 implies that H𝑁,𝐼 (𝑊) lies in the essential image of Equation (8.9). Part III The Categorical Trace and Convolution Patterns 62 63 Chapter 9 MORE RECOLLECTIONS ON ∞-CATEGORIES We review some more required categorical preliminaries mainly following (Lurie, 2009) (Lurie, 2017) and (Dennis Gaitsgory and Rozenblyum, 2017a), in the (simplified) set-up suitable for our purpose. Let Cat denote the (∞, 1)-category of small categories, and Cat the category of all (not necessarily small) (∞, 1)-categories. For a(n ordinary) commutative ring Λ with unit, let Lincatsm Λ denote the ∞-category of Λ-linear small idempotently complete stable categories with functors being Λ-linear exact functors, and let LincatΛ denote the (∞, 1)-category of all presentable Λ-linear stable categories with functors being continuous functors. For C ∈ LincatΛ , and 𝑋, 𝑌 ∈ C, we write HomC (𝑋, 𝑌 ) ∈ ModΛ representing MapModΛ (𝑀, HomC (𝑋, 𝑌 )) = MapC (𝑀 ⊗ 𝑋, 𝑌 ). Then MapC (𝑋, 𝑌 ) = 𝜏 ≤0 HomC (𝑋, 𝑌 ). Definition 9.0.1. Let C 𝑓 𝑢 𝑣 D C′ 𝑔 D′ be a commutative square in LincatΛ . That is, we are given a specified equivalence 𝑢 ◦ 𝑔 ≃ 𝑔 ◦ 𝑉. Then we say that the square above is left adjointable if 𝑓 and 𝑔 admit left adjoints 𝑓 𝐿 and 𝑔 𝐿 , and the Beck-Chevalley map 𝛽 : 𝑔 𝐿 ◦ 𝑢 → 𝑣 ◦ 𝑓 𝐿 given by 𝑔𝐿 ◦ 𝑢 → 𝑔𝐿 ◦ 𝑢 ◦ 𝑓 ◦ 𝑓 𝐿 ≃ 𝑔𝐿 ◦ 𝑔 ◦ 𝑣 ◦ 𝑓 𝐿 → 𝑣 ◦ 𝑓 𝐿 is an equivalence. If 𝑓 and 𝑔 has right adjoints 𝑓 𝑅 and 𝑔 𝑅 then we say that the square is right adjointable is the map 𝛾 : 𝑣 ◦ 𝑓 𝑅 → 𝑔 𝑅 ◦ 𝑢, obtained by a dual construction to the one above, is an equivalence. Let 𝑆 be a small infinity category. We denote by FunLAd (𝑆, LincatΛ ) (resp. FunRAd (𝑆, LincatΛ )) the subcategory of Fun(𝑆, LincatΛ ) consisting of functors 𝐹 : 𝑆 → Lincat so that for all arrows 𝑠 → 𝑠′ in 𝑆 the functor 𝐹 (𝑠) → 𝐹 (𝑠′) admits a continuous right adjoint (resp. a left adjoint) and morphisms 𝛼 : 𝐹 → 𝐹 ′ 64 such that for every 𝑠 → 𝑠′ the square 𝐹 (𝑠) 𝐹 (𝑠′) 𝐹 ′ (𝑠) 𝐹 ′ (𝑠′) is left (resp. right) adjointable. We recall (Lurie, 2017, Corollary 4.7.4.18.): Proposition 9.0.2. The ∞-categories FunLAd (𝑆, LincatΛ ), FunRAd (𝑆, LincatΛ ) are presentable and the inclusion functors FunLAd (𝑆, LincatΛ ) ⊆ Fun(𝑆, LincatΛ ) and FunRAd (𝑆, LincatΛ ) ⊆ Fun(𝑆, LincatΛ ) preserve small limits. In particular, if we have an indexing category I and diagrams C, D : I → LincatΛ with a natural transformation 𝜙 : C → D, which is equivalent to having a functor 𝐹 : I → Fun(Δ1 , LincatΛ ), then 𝐹 factors through FunLAd (Δ1 , LincatΛ ) (resp. FunRAd (Δ1 , LincatΛ )) if for every vertex 𝑖 of I, the functor 𝜙𝑖 : C𝑖 → D𝑖 has a left (right) adjoint 𝜓𝑖 : D𝑖 → C𝑖 and for all arrows 𝑖 → 𝑗 in I the commutative square C𝑖 D𝑖 C𝑗 D𝑗 is left (resp. right) adjointable. Denote C = lim𝑖 C𝑖 and D = lim𝑖 D𝑖 . Then the proposition says that under the conditions above the functor 𝜙 : C → D induced on the limits has a left (resp. right) adjoint 𝜓 and for all 𝑖 ∈ I the composition 𝜓 𝜓𝑖 D− → C → C𝑖 is equivalent to the composition D → D𝑖 −→ C𝑖 . Let 𝐹 : 𝑆 → LincatΛ be a diagram. For an arrow 𝜑 : 𝑠 → 𝑠′, we have that the corresponding functor 𝐹 (𝜑) : 𝐹 (𝑠) → 𝐹 (𝑠′) preserves colimits morphism and therefore by (Lurie, 2009, Corollary 5.5.2.9) admits a right adjoint 𝐹 𝑅 (𝜑). By passing to right adjoints we get a diagram 𝐹 𝑅 : 𝑆 𝑜 𝑝 → Cat. By (Lurie, 2009, §5.5.3) the morphism colim 𝐹 (𝑠) → lim𝑜 𝑝 𝐹 𝑅 (𝑠) 𝑠∈𝑆 𝑠∈𝑆 (9.1) determined by the maps right adjoint to ins𝑠 : 𝐹 (𝑠) → colim𝑆 𝐹 is an equivalence, where the left is computed in LincatΛ and then is mapped to Cat and the right is computed in Cat. In addition, if all 𝐹 𝑅 (𝜑) are continuous, then the r.h.s. can also be computed in LincatΛ . Denote by ev𝑠 the right adjoint of ins𝑠 . It follows from adjunction that for every object 𝑐 ∈ lim𝑆 𝐹, the natural map colim (ins𝑠 ◦ ev𝑠 (𝑐)) → 𝑐 𝑠∈𝑆 is an equivalence in colim𝑆 𝐹. (9.2) 65 Remark 9.0.3. 1. Assume that for each 𝜑 : 𝑠 → 𝑠′ the functor 𝐹 (𝜑) : 𝐹 (𝑠) → 𝐹 (𝑠′) preserves compact objects (equivaletnly, 𝐹 (𝜑) preserves compact objects, the right adjoint 𝐹 𝑅 (𝜑) is continuous). Then the functors ins𝑠 : 𝐹 (𝑠) → colim𝑆 𝐹 also preserve compact objects. 2. Assume in addition that each of the categories 𝐹 (𝑠) is compactly generated, that is, 𝐹 (𝑠) ≃ Ind(𝐹 0 (𝑠)). Then the colimit colim𝑆 𝐹 is also compactly generated, with compact objects given by retracts of objects coming from 𝐹 0 (𝑠) for 𝑠 ∈ 𝑆. In particular, if 𝑆 is filtered, we have colim 𝐹 ≃ Ind(colim 𝐹 0 (𝑠)) 𝑆 𝑠∈𝑆 with the colimit taken in the category of small infinity categories. 3. If 𝑆 is filtered and the morphisms in the image of 𝐹 have continuous right adjoints, then for an object 𝑠 in 𝑆 the composition ev𝑠 ◦ins𝑠 : 𝐹 (𝑠) → colim𝑆 𝐹 ≃ lim𝑆 𝑜 𝑝 𝐹 𝑅 → 𝐹 (𝑠) is equivalent to the colimit ev𝑠 ◦ ins𝑠 ≃ colim′ 𝐹 𝑅 (𝜑) ◦ 𝐹 (𝜑). 𝜑 : 𝑠→𝑠 Using the equivalence (9.1) and Proposition 9.0.2 it is also possible to get that adjointability preserved under taking colimits (Lurie, 2017, Proposition 4.7.4.19). Proposition 9.0.4 (Lurie). Let 𝑆, 𝑇 be small ∞-categories and let 𝐹 : 𝑆 × 𝑇 → LincatΛ be a functor. Assume that for all 𝑠 → 𝑠′ in 𝑆 and 𝑡 → 𝑡 ′ in 𝑇 the square 𝐹 (𝑠, 𝑡) 𝐹 (𝑠′, 𝑡) 𝐹 (𝑠, 𝑡 ′) 𝐹 (𝑠′, 𝑡 ′) is right adjointable. Then there is an extension 𝐹 : 𝑆 ⊲ × 𝑇 → LincatΛ of 𝐹 such that: 1. For each 𝑡 ∈ 𝑇, the diagram 𝐹 : 𝑆 ⊲ × {𝑡} → LincatΛ is a colimit diagram in LincatΛ . 66 2. For all 𝑠 → 𝑠′ in 𝑆 ⊲ and 𝑡 → 𝑡 ′ in 𝑇 the square 𝐹 (𝑠, 𝑡) 𝐹 (𝑠′, 𝑡) 𝐹 (𝑠, 𝑡 ′) 𝐹 (𝑠′, 𝑡 ′) is right adjointable. We will also need (cohomological) descent. In this setting we are usually dealing with a functor C • : Δ → Cat, also called a cosimplicial ∞-category, and we would like to say something about the totalization Tot(C • ) = lim C 𝑛 [𝑛]∈Δ of C • . Usually the totalization is difficult to compute. However, under certain left adjointability conditions it is possible to deduce that the evaluation functor Tot(C • ) → C 0 is monadic. Recall that for an ∞-category D a monad on D is an associative algebra object 𝑇 in the monoidal category Fun(D, D). Informally, this means that we are gives a multiplication map 𝑇 ◦ 𝑇 → 𝑇 and a unit map idD → 𝑇 which satisfy associativity and unit conditions up to coherent homotopy. For example, if 𝐺 : E → D is a functor between ∞-categories which admits a left adjoint 𝐹, the composition 𝑇 = 𝐺 ◦ 𝐹 has the structure of a monad on D with identity given by the unit map idD → 𝐺 ◦ 𝐹 of the adjunction and composition map induced by the co-unit 𝐹 ◦ 𝐺 → idE via 𝑇 ◦ 𝑇 = (𝐺 ◦ 𝐹) ◦ (𝐺 ◦ 𝐹) ≃ 𝐺 ◦ (𝐹 ◦ 𝐺) ◦ 𝐹 → 𝐺 ◦ 𝐹. Given a monad 𝑇 on D one can consider the category LMod𝑇 (D) of left modules over 𝑇. Informally, this category consists of objects 𝐴 in D equipped with a map 𝑇 ( 𝐴) → 𝐴 giving an action of the algebra 𝑇 on 𝐴 and morphisms giving by morphisms preserving that structure. The forgetful functor 𝐺 : LMod𝑇 (D) → D has a left adjoint given by the free construction 𝐴 ↦→ 𝑇 ( 𝐴). An adjunction 𝐹 : D ⇄ E : 𝐺 is called monadic if E is equivalent to LMod𝑇 (D) for 𝑇 = 𝐺 ◦ 𝐹 and 𝐺 given by the forgetful functor. For that purpose, we recall (Lurie, 2017, Theorem 4.7.5.2, Corollary 4.7.5.3). 67 Theorem 9.0.5 (Lurie). Let C • : Δ → Cat be a cosimplicial ∞-category. Assume that for any 𝛼 : [𝑚] → [𝑛] in Δ, the induced diagram C𝑚 𝑑0 C 𝑚+1 C𝑛 𝑑0 C 𝑛+1 is left adjointable. We denote the left adjoint of 𝑑 0 : C 𝑛 → C 𝑛+1 by 𝐹 (𝑛). Let C = Tot(C • ). Then 1. The functor 𝐺 : C → C 0 admits a left adjoint 𝐹. 2. The diagram C 𝐺 𝑑1 𝐺 C0 C0 𝑑0 C1 is left adjointable. That is, the canonical map 𝐹 (0) ◦ 𝑑 1 → 𝐺 ◦ 𝐹 is an equivalence. 3. The adjunction 𝐹 : C 0 ⇄ C : 𝐺 is monadic. That is, C is equivalent to the category of left modules LMod𝑇 (C 0 ) with 𝑇 = 𝐹 (0) ◦ 𝑑 1 ≃ 𝐺 ◦ 𝐹. Corollary 9.0.6 (Lurie). Let C • : Δ+ → Cat be an augmented cosimplicial ∞category. Denote 𝐺 : C −1 → C 0 . Assume that 1. The category C −1 admits geometric realizations of 𝐺-split simplicial objects that are preserved by 𝐺. 2. for any 𝛼 : [𝑚] → [𝑛] in Δ+ the diagram C𝑚 𝑑0 C 𝑚+1 C𝑛 𝑑0 C 𝑛+1 is left adjointable. Then the canonical map 𝜙 : C −1 → Tot(C • ) admits a fully faithful left adjoint. If, in addition C −1 → C 0 is conservative, 𝜙 is an equivalence. Remark 9.0.7. There is also a dual (co-monadic) version replacing "left adjoint" with "right adjoint", and "realizations of 𝐺-split simplicial objects" with "totalizations of 𝐺-split cosimplicial objects" in the statement above. 68 9.1 Dual categories Recall (e.g. (Lurie, 2018, §D.7)) that every object C ∈ LincatΛ which is compactly generated is dualizable. If C = Ind(C0 ) for some C0 ∈ CatΛ we can identify the dual C ∨ of C with Ind(C 𝑜 𝑝 ). Explicitly, the evaluation map C ∨ ⊗ C → ModΛ is given by the unique continuous extension of the functor given by the unique continuous extension of the functor Ind(C 𝑜 𝑝 ) ⊗ C → ModΛ (F , G) → HomC (F , G). Let A = Ind(A0 ) and B = Ind(B0 ) be objects of LincatΛ which are compactly generated and let 𝐹 : A → B be a continuous functor that preserves compact objects. We have a tautological functor 𝑜𝑝 𝑜𝑝 𝑜𝑝 𝐹0 : A0 → B0 . Taking its ind-extension, we get a functor 𝐹 𝑜 : A∨ → B∨ (9.3) which is called the conjugate functor to 𝐹. The construction A ↦→ A ∨ , 𝐹 ↦→ 𝐹 𝑜 can be upgraded to an endofunctor on the subcategory of LincatΛ consisting of compactly generated Λ-linear categories and continuous functors which preserve compact objects. Remark 9.1.1. In terms of the duality on LincatΛ , the functor 𝐹 𝑜 is equivalent to the left adjoint of 𝐹 ∨ . Equivalently, the functor 𝐹 𝑜 is the dual of the right adjoint to 𝐹, see (Dennis Gaitsgory, 2012, §1.5). Let I → LincatΛ be a cofiltered diagram of Λ-linear categories such that of each 𝑖 the category 𝑖 ↦→ A𝑖 compactly generated and all transition functors A𝑖 → A 𝑗 preserve compact objects. Denote A = colim𝑖∈I A𝑖 . It follows from (Dennis Gaitsgory and Rozenblyum, 2017a, Proposition 6.3.4) that the natural map colim A𝑖∨ → A ∨ 𝑖∈I obtained by passing to conjugate functors, is an equivalence. (9.4) 69 9.2 Relative tensor product Let us first review the general formalism of relative tensor products. Let R be a monoidal category with 1R its unit. Let Alg(R) denote the category of associative algebra objects in R. Let LMod(R) (resp. RMod(R)) the category of left (resp. right) module objects in R. I.e., objects in LMod(R) (resp. RMod(R)) consist of pairs ( 𝐴, 𝑀) with 𝐴 ∈ Alg(R) and 𝑀 a left (resp. right) 𝐴-module. For 𝐴 ∈ Alg(R) we denote by LMod 𝐴 (R) = LMod(R) ×Alg(R) {𝐴} (resp. RMod 𝐴 (R) = {𝐴} ×Alg(R) LMod(R)). Similarly, let BMod(R) denote the category of bimodule objects in R. For 𝐴, 𝐵 ∈ Alg(R) we denote by 𝐴 BMod𝐵 = {𝐴} ×Alg(R) BMod(R) ×Alg(R) {𝐵} the category of 𝐴-𝐵-bimodules. An 𝐴-𝐴-bimodule is also called as an 𝐴-bimodule. For example, 𝐴 itself can be regarded as 𝐴-bimodule via the left and the right multiplication. See (Lurie, 2017, §4.3) for detailed discussions. Given associative algebra objects 𝐴, 𝐵, 𝐶 ∈ Alg(𝑅) and bimodules 𝑀 ∈ 𝐴 BMod𝐵 and 𝑁 ∈ 𝐵 BMod𝐶 , the relative tensor product 𝑀 ⊗𝐵 𝑁, if exists, is the unique object (up to equivalence) in 𝐴 BMod𝐶 , corepresenting the functor sending 𝑋 ∈ 𝐴 BMod𝐶 to the space of 𝐵-bilinear 𝐴-𝐶bimodule maps 𝑀 ⊗ 𝑁 → 𝑋 (in appropriate homotopy sense, see (Lurie, 2017, Definition 4.4.2.3)). On the other hand, there is the two-sided bar construction op LMod(R) ×Alg(R) RMod(R) → ( 𝐴 BMod𝐶 ) Δ , (𝑀, 𝑁) ↦→ Bar 𝐵 (𝑀, 𝑁)• , where Bar 𝐵 (𝑀, 𝑁)• is a simplicial object in the category of 𝐴-𝐶-bimodules, given informally as Bar 𝐵 (𝑀, 𝑁)𝑛 = 𝑀 ⊗ 𝐵𝑛 ⊗ 𝑁 with boundary maps induced by the multiplication on 𝐵 and actions on 𝑀 and 𝑁, and degeneracy maps given by insertions of the unit of 𝐵. See (Lurie, 2017, Notation 4.4.2.4, Construction 4.4.2.7). If 𝐴 = 𝐵 = 𝐶 and 𝑀 = 𝑁 = 𝐴, we simply denote Bar 𝐴 ( 𝐴, 𝐴)• by Bar( 𝐴)• , called the bar construction of the bimodule 𝐴. We do not know in general whether 𝑀 ⊗𝐵 𝑁 is always given by the geometric realization of Bar 𝐵 (𝑀, 𝑁)• as soon as the latter exists. This is the case if R admits geometric realizations and such that the monoidal product ⊗ : R × R → R preserves geometric realizations in each variable, by (Lurie, 2017, Theorem 4.4.2.8). It is also the case in the following two examples. Example 9.2.1. Assume that 𝑀 = 𝑀0 ⊗𝐵 with 𝑀0 a left 𝐴-module (resp. 𝑁 = 𝐵⊗𝑁0 with 𝑁0 a right 𝐶-module). Then 𝑀 ⊗𝐵 𝑁 exists and is represented by 𝑀0 ⊗ 𝑁 70 (resp. 𝑀 ⊗ 𝑁0 ). To see this, we follow the argument of (Lurie, 2017, Proposition 5.2.2.6). If R admits geometric realizations and such that the monoidal product ⊗ : R × R → R preserves geometric realizations in each variable, then 𝑀 ⊗𝐵 𝑁 exists and is isomorphic to 𝑀0 ⊗ 𝑁 (resp. 𝑀 ⊗ 𝑁0 ) by (Lurie, 2017, Proposition 4.4.3.14, 4.4.3.16). The general situation reduces this case via Yoneda embedding. Example 9.2.2. Using a similar argument, one can also prove the relative tensor product exists in the following situation. Let C be a (small) category admitting finite limits. Let pt denote the final object. Then it C op has the coCartesian symmetric monoidal structure: for 𝑋, 𝑌 ∈ C, the tensor product 𝑋 ⊗ 𝑌 in C op is the finite product 𝑋 × 𝑌 in C. We note that every object 𝑋 is a commutative algebra object in C op , with the multiplication given by the diagonal map Δ 𝑋 : 𝑋 → 𝑋 × 𝑋 in C and the unit given by the structural map 𝑝 𝑋 : 𝑋 → pt. In addition, every morphism 𝑓 : 𝑋 → 𝑌 in C gives a commutative algebra homomorphism in C op . Now given two morphisms 𝑎 : 𝑀 → 𝑋, 𝑏 : 𝑁 → 𝑋 in C, we regard 𝑀, 𝑁 as 𝑋-modules in C op . We claim that 𝑀 ⊗ 𝑋 𝑁 exists and is representable by 𝑀 × 𝑋 𝑁. Namely, the two-sided bar complex as the cosimplicial object in C is Δ→C:𝑀×𝑁 id×𝑎×id −→ 𝑀 × 𝑋 × 𝑁 −→ −→ 𝑀 × 𝑋 × 𝑋 × 𝑁 · · · −→ −→ id×𝑏×id We will consider the embedding C op → Ind(C op ), where Ind(C op ) denotes the indcompletion of C op , equipped with the induced symmetric monoidal structure so that the tensor product preserves colimits in each variable. Again, by the argument of (Lurie, 2017, Proposition 5.2.2.6), it is enough to show that the geometric realization of this simplicial object Δop → Ind(C op ) is represented by 𝑀 × 𝑋 𝑁. By (Lurie, 2009, Lemma 6.1.4.7), its geometric realization can be computed as the colimit of the truncated colimit diagram (Δ≤1 ) op → Ind(C op ), which in turn is the limit of 𝑀 × 𝑁 −→ −→ 𝑀 × 𝑋 × 𝑁 in C. But this is exactly 𝑀 × 𝑋 𝑁. 71 C h a p t e r 10 THE FORMALISM OF CORRESPONDENCES In this subsection, we review the formalism of correspondences, as first appeared in (Liu and Zheng, 2012, §6.1) and (Dennis Gaitsgory and Rozenblyum, 2017a, Chapter 7). There are mainly two (closely related) usages of this formalism in the paper. First, it provides a convenient framework to discuss convolution pattern arising from algebraic geometry and representation theory and therefore is useful for our study of (geometric) trace. Second, it encodes various sheaf theories in algebraic geometry in a concise way, as first observed by Lurie. 10.1 Category of correspondences Let C be an ∞-category that admits finite limits and coproducts. The category C will play the role of the category of geometric objects. We denote by pt the final object in C. Let 𝑣𝑒𝑟𝑡, ℎ𝑜𝑟𝑖𝑧 be two classes of morphisms in C, each of which contains all equivalences in C and is stable under composition and base change. Note that the classes 𝑣𝑒𝑟𝑡, ℎ𝑜𝑟𝑖𝑧 are stable under products. That is, if 𝑓𝑖 : 𝑋𝑖 → 𝑌𝑖 , 𝑖 = 1, 2 are in 𝑣𝑒𝑟𝑡 (resp. ℎ𝑜𝑟𝑖𝑧) then the product 𝑓1 × 𝑓2 : 𝑋1 × 𝑋2 → 𝑌1 × 𝑌2 is a member of 𝑣𝑒𝑟𝑡 (resp. ℎ𝑜𝑟𝑖𝑧). We denote by C𝑣𝑒𝑟𝑡 (resp. Cℎ𝑜𝑟𝑖𝑧 ) the 1-full subcategory of C consisting of morphisms in 𝑣𝑒𝑟𝑡 (resp. ℎ𝑜𝑟𝑖𝑧). Let Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 denote the category of correspondences. Informally, objects of Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 are the same as those of C and morphisms from 𝑋 to 𝑌 are given by diagrams 𝑔 𝑍 𝑋 (10.1) 𝑓 𝑌 𝑓 𝑔 with 𝑔 ∈ ℎ𝑜𝑟𝑖𝑧 and 𝑓 ∈ 𝑣𝑒𝑟𝑡. We sometimes just write such diagram by𝑌 ← −𝑍→ − 𝑋 for short. In addition, to emphasize that such a morphism in Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 is 𝑓 ◦𝑔 −1 a correspondence rather than an actual map, we sometimes write it as 𝑋 d 𝑌 or simply as 𝑋 d 𝑌 . The composition of the correspondences 𝑌 ← 𝑆 → 𝑋 and 𝑍 ← 𝑇 → 𝑌 is given by the correspondence 𝑍 ← 𝑊 → 𝑋 which is formed by 72 taking the pullback: 𝑊 𝑆 𝑇 𝑌 𝑋 . 𝑍 Given 𝑔 : 𝑌 → 𝑋 in ℎ𝑜𝑟𝑖𝑧 we will sometimes identify it with the correspondence 𝑔 id 𝑌 ← −𝑌 → − 𝑋 and we refer to such morphisms as horizontal. Similarly, given a 𝑓 morphism 𝑓 : 𝑋 → 𝑌 in 𝑣𝑒𝑟𝑡 we will identify it with the correspondence 𝑌 ← − id 𝑋− → 𝑋 and refer to such morphisms of Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 as vertical. If 𝑣𝑒𝑟𝑡, ℎ𝑜𝑟𝑖𝑧 are all morphisms, we simply write Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 by Corr(C). We refer to (Dennis Gaitsgory and Rozenblyum, 2017a, §7.1) and (Liu and Zheng, 2012, §6.1) for precise definition of it as an ∞-category. Remark 10.1.1. The category Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 admits an (∞, 2)-categorical en𝑎𝑑𝑚 hancement Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 of the category of correspondences, depending on a 𝑎𝑑𝑚 certain class 𝑎𝑑𝑚 of morphisms of C. A 2-morphism in Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 is given by a diagram 𝑆′ 𝑟 𝑆 𝑋 𝑌 with 𝑟 ∈ 𝑎𝑑𝑚. See (Dennis Gaitsgory and Rozenblyum, 2017a, §7.1.1.2) for details. As C admits finite limits it is a symmetric monoidal category under the Cartesian monoidal structure. This induces a symmetric monoidal structure on Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 constructed in (Dennis Gaitsgory and Rozenblyum, 2017a, Chapter 9) (see also (Liu and Zheng, 2014, §6.1)). Namely, the tensor product of objects 𝑋, 𝑌 in Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 is their product 𝑋 × 𝑌 as objects of C. Note that in general 𝑋 × 𝑌 is not the product of 𝑋 and 𝑌 in Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . For this reason, sometimes we write 𝑋 ⊗ 𝑌 to emphasize we regard 𝑋 × 𝑌 as the tensor product of 𝑋 and 𝑌 in Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . We note that the 1-full subcategories C𝑣𝑒𝑟𝑡 and (Cℎ𝑜𝑟𝑖𝑧 ) op are symmetric monoidal subcategories. In particular, it makes sense to talk about associative and commutative algebra objects in Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . 73 Example 10.1.2. Every object 𝑋 ∈ C with the diagonal map Δ 𝑋 : 𝑋 → 𝑋 × 𝑋 and the structural map 𝜋 𝑋 : 𝑋 → pt belonging to ℎ𝑜𝑟𝑖𝑧 has a natural commutative algebra structure in (Cℎ𝑜𝑟𝑖𝑧 ) op with the multiplication given by Δ 𝑋 and the unit given by 𝜋 𝑋 . In addition, if 𝑋 and 𝑌 are two objects satisfying the above properties and 𝑓 : 𝑋 → 𝑌 is a morphism belonging to ℎ𝑜𝑟𝑖𝑧, then we have a commutative algebra homomorphism in (Cℎ𝑜𝑟𝑖𝑧 ) op from 𝑌 to 𝑋 induced by 𝑓 . In particular, 𝑋 is a (left) 𝑌 -module. As (Cℎ𝑜𝑟𝑖𝑧 ) op → Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 is a symmetric monoidal subcategory, we obtain the corresponding (maps between) commutative algebra objects in Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . If in addition 𝑓 ∈ C𝑣𝑒𝑟𝑡 , then 𝑓 : 𝑋 → 𝑌 is naturally a morphism of 𝑌 -modules from 𝑋 to 𝑌 on Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . 10.2 Monads and modules in the category of correspondences We will be interested in a particular class of algebra objects and their bimodules arising from monads in Corr(C). We review the description of algebras in terms of Segal objects and note how these constructions generalize to describe bimodules. Consider the (∞, 2)-category Corr(C) 𝑎𝑙𝑙 . For every 𝑋, 𝑋 ′ ∈ C we have a category HomCorr(C) (𝑋 ′, 𝑋). If 𝑋 = 𝑋 ′ this category is denoted by EndCorr(C) (𝑋). It carries a canonical monoidal structure induced by composition. In general, the category HomCorr(C) (𝑋 ′, 𝑋) is equipped with a left action of EndCorr(C) (𝑋) and a right action of EndCorr(C) (𝑋 ′). 10.3 Monads and Segal objects First recall the definition of Segal objects (also known as category objects) in an ∞-category. Definition 10.3.1. A simplicial object 𝑋• : Δ𝑜 𝑝 → C is called a Segal object if for every 𝑛 ≥ 1, the map 𝑋𝑛 → 𝑋1 × 𝑋0 𝑋1 × 𝑋0 · · · × 𝑋0 𝑋1 induced by the maps 𝛿𝑖 : [1]  {𝑖, 𝑖 + 1} ⊂ [𝑛] for 𝑖 = 0, 1, . . . , 𝑛 − 1, is an equivalence. Remark 10.3.2. If C is an ordinary category, a Segal object is fully determined by the objects 𝑋0 , 𝑋1 , the boundary maps 𝑑1 , 𝑑0 : 𝑋1 → 𝑋0 , 𝑑1 : 𝑋2 → 𝑋1 and the degeneracy map 𝑠 : 𝑋0 → 𝑋1 . These define a category object of C in the usual sense. Namely, 𝑋0 is the class of objects, 𝑋1 the morphism objects, the morphisms 74 𝑑1 , 𝑑0 : 𝑋1 → 𝑋0 as source and target. The composition is given by the morphism 𝑑1 : 𝑋2 → 𝑋1 and unit by 𝑑1 : 𝑋2 → 𝑋1 . Example 10.3.3. Let 𝑓 : 𝑋 → 𝑌 be a morphism in C. The Čech nerve 𝑋• → 𝑌 of 𝑓 (see (Lurie, 2009, §6.1.2)), where 𝑛+1 z }| { 𝑋𝑛 = 𝑋 ×𝑌 𝑋 × · · · ×𝑌 𝑋, is easily seen to be a Segal object of C. Indeed, it is even a groupoid object, in the sense of (Lurie, 2009, Definition 6.1.2.7)). This will be our main example. Example 10.3.4. A Segal object 𝑋• with 𝑋0 = pt is a monoid object (in the sense of (Lurie, 2009, p. 4.1.2)). Giving such a monoid object is equivalent to giving an associative algebra object in C by (Lurie, 2009, Proposition 4.1.2.10, 4.1.2.11). This last example admits the following generalization. Let 𝑋 ∈ C and consider the monoidal category EndCorr(C) (𝑋). Then (Dennis Gaitsgory and Rozenblyum, 2017a, Proposition 9.4.1.5, Theorem 9.4.4.2) gives: Theorem 10.3.5. 1. There is a natural lax monoidal functor Sp : EndCorr(C) (𝑋) → Corr(C). 2. There is a canonical equivalence between Segal objects 𝑋• in C with 𝑋0 = 𝑋 and associative algebra objects in EndCorr(C) (𝑋). Roughly speaking, the functor Sp from (1) sends 𝑋 ← 𝑍 → 𝑋 to the object 𝑍 ∈ C with the lax structure given by horizontal arrow 𝑍 × 𝑋 𝑍 → 𝑍 × 𝑍. To construct the equivalence from (2), let 𝐴 ∈ EndCorr(C) (𝑋). Then 𝑋 itself is an 𝐴-𝐴-bimodule of 𝐴 in EndCorr(C) (𝑋). Then we have the two-sided Bar complex Bar 𝐴 (𝑋, 𝑋)• as a simplicial object in EndCorr(C) (𝑋). Its image under Sp gives a simplicial object in C. which is the desired Segal object in C. Conversely, a segal object 𝑋• in C gives rise to an algebra object in 𝑋1 ∈ Corr(C) with multiplication given by the correspondence 𝑋1 × 𝑋0 𝑋1 ≃ 𝑋2 𝑑1 𝑋1 𝑑0 ×𝑑2 𝑋1 × 𝑋1 75 and the unit given by the correspondence 𝑋0 𝜋 𝑋0 pt 𝑠 𝑋1 Remark 10.3.6. Assume that the Segal object 𝑋• is such that: • All morphisms of the simplicial object 𝑋• are in C𝑣𝑒𝑟𝑡 ; • The diagonal map Δ 𝑋0 : 𝑋0 → 𝑋0 × 𝑋0 and the structural map 𝜋 𝑋0 : 𝑋0 → pt are in Cℎ𝑜𝑟𝑖𝑧 . Then the associative algebra object 𝑋1 can be realized as an associative algebra object of the monoidal category Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . 10.4 The action on bimodules Given 𝑋0 , 𝑋0′ ∈ C and an associative algebra objects 𝑋1 ∈ Alg(EndCorr(C) (𝑋0 )), 𝑋1′ ∈ Alg(EndCorr(C) (𝑋0′ )) the lax-monoidal functor from Theorem 10.3.5 (1) induces a functor Sp : 𝑋1 BMod 𝑋1′ (HomCorr(C) (𝑋0′ , 𝑋0 )) → 𝑋1 BMod 𝑋1′ (Corr(C)). Letting 𝑋0 = 𝑋0′ and 𝑋1 = 𝑋1′ , we have a lax monoidal functor Sp 𝑋1 : 𝑋1 BMod 𝑋1 (EndCorr(C) (𝑋0 )) → 𝑋1 BMod 𝑋1 (Corr(C)). The unit of the monoidal category BMod(EndCorr(C) (𝑋0 )) is sent by Sp 𝑋1 to the algebra 𝑋0 with multiplication given by the horizontal arrow 𝑋0 → 𝑋0 × 𝑋0 (it corresponds to the Segal object given by the constant simplicial object 𝑋0 ). Moreover, it is clear that as an algebra, 𝑋0rev is isomorphic to 𝑋0 . Then Theorem 10.3.5 (1) upgrades to a lax monoidal functor Sp 𝑋0 : EndCorr(C) (𝑋0 ) → 𝑋0 BMod 𝑋0 (Corr(C)) (10.2) Explicitly, an object 𝑄 ∈ EndCorr(C) (𝑋0 ) has left and right actions given by the two maps 𝑋0 ← 𝑄 → 𝑋0 . The simple structure of such modules gives a simple description of the relative tensor product over such bimodules in the category of correspondences. Remark 10.4.1. Let 𝑄 ∈ 𝑋1 BMod 𝑋1 (EndCorr(C) (𝑋0 )) with 𝑋1 , 𝑋0 as in Remark 10.3.6. If the action maps 𝑋1 × 𝑋0 𝑄 → 𝑄 and 𝑄 × 𝑋0 𝑋1 → 𝑄 are in C𝑣𝑒𝑟𝑡 , then the corresponding module under (10.2) is an object of Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . 76 C h a p t e r 11 SHEAF THEORIES 11.1 The notion of a sheaf theory We fix C as before. By definition, a sheaf theory of C is a lax symmetric monoidal functor D : Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 → LincatΛ . (11.1) id 𝑔 For a horizontal morphism 𝑌 ← −𝑌 → − 𝑋 we will denote the corresponding functor 𝑓 id by 𝑔★ : D (𝑋) → D (𝑌 ). For a vertical morphism 𝑌 ← − 𝑋 − → 𝑋 we denote the corresponding functor by 𝑓† : D (𝑋) → D (𝑌 ). Then for a general correspondence 𝑓 𝑔 𝑌← −𝑍→ − 𝑋 the associated functor is equivalent to 𝑓† ◦ 𝑔★. Let us spread out some structures encoded by such a functor. 1. We can pass to right adjoints. For (𝑔 : 𝑋 → 𝑌 ) ∈ Cℎ𝑜𝑟𝑖𝑧 , let 𝑔★ be the (not necessarily continuous) right adjoint of 𝑔★, and for ( 𝑓 : 𝑋 → 𝑌 ) ∈ C𝑣𝑒𝑟𝑡 , let 𝑓 † be the (not necessarily continuous) right adjoint of 𝑓† . 2. The functoriality of D encodes a "base change theorem". Namely, let 𝑋′ 𝑔′ 𝑋 𝑓′ 𝑌′ (11.2) 𝑓 𝑔 𝑌 be a pullback square in C. Then if 𝑓 ∈ 𝑣𝑒𝑟𝑡 and 𝑔 ∈ ℎ𝑜𝑟𝑖𝑧, we have that 𝑓 ′ ∈ 𝑣𝑒𝑟𝑡, 𝑔′ ∈ ℎ𝑜𝑟𝑖𝑧 and part of the data of the functor D is to give an equivalence ( 𝑓 ′)† ◦ (𝑔′)★ ≃ 𝑔★ ◦ 𝑓† . (11.3) 3. The lax symmetric monoidal structure of D provides a functor ⊠ : D (𝑋) ⊗ D (𝑌 ) → D (𝑋 × 𝑌 ), 𝑋, 𝑌 ∈ C (11.4) together with all necessary higher coherence conditions. Let 𝑋 be as in Example 10.1.2. This induces a symmetric monoidal structure on the category D (𝑋). Informally, the symmetric monoidal structure is given by the 77 composition ⊠ Δ★𝑋 F , G ↦→ F ⊗ G := Δ★𝑋 (F ⊠ G). (11.5) We let Λ 𝑋 ∈ D (𝑋) denote the unit object with respect to this symmetric monoidal structure, which corresponds to the functor D (𝑋) ⊗ D (𝑋) − → D (𝑋 × 𝑋) −−→ D (𝑋), 𝜋★𝑋 ModΛ → D (pt) −−→ D (𝑋). (11.6) This symmetric monoidal structure is closed. That is, for every pair of objects F1 , F2 ∈ D (𝑋) there is an object Hom(F1 , F2 ) such that for every G ∈ D (𝑋) there is a canonical equivalence

MapD (𝑋) G, Hom(F1 , F2 ) ≃ MapD (𝑋) G ⊗ F1 , F2 . (11.7) Note we have Hom(F1 ⊗ F2 , F3 ) = Hom(F1 , Hom(F2 , F3 ). (11.8) 4. Let 𝑓 : 𝑋 → 𝑌 in Cℎ𝑜𝑟𝑖𝑧 as in Example 10.1.2. Then 𝑓 ★ : D (𝑌 ) → D (𝑋) is a symmetric monoidal functor. In particular, it endows D (𝑋) with a structure of a D (𝑌 ) monoidial category. If in addition 𝑓 ∈ C𝑣𝑒𝑟𝑡 as well, then 𝑓† : D (𝑋) → D (𝑌 ) is a morphism of D (𝑌 )-modules. In particular, for F ∈ D (𝑋), G ∈ D (𝑌 ) we have a canonical equivalence 𝑓† (F ) ⊗ G ≃ 𝑓† (F ⊗ 𝑓 ★ (G)). (11.9) which encodes a "projection formula" for 𝑓† and 𝑓 ★. Along with the co-unit of the adjunction ( 𝑓† , 𝑓 † ), it gives, for every F , G ∈ D (𝑌 ), a natural map 𝑓 † (G) ⊗ 𝑓 ★ (F ) → 𝑓 † (G ⊗ F ) (11.10) adjoint to 𝑓† ( 𝑓 † (G) ⊗ 𝑓 ★ (F )) ≃ 𝑓† 𝑓 † (G) ⊗ F → G ⊗ F , In particular, one has the natural transformation of functors 𝑓 † (Λ𝑌 ) ⊗ 𝑓 ★ → 𝑓 † (11.11) In addition, we have 𝑓 † Hom(F , G) ≃ Hom( 𝑓 ★ F , 𝑓 † G), 𝑓★Hom(F , 𝑓 † G) ≃ Hom( 𝑓† F , G). (11.12) 78 5. Assume that 𝑝 : 𝑋 → pt belongs to 𝑣𝑒𝑟𝑡. Let D 𝑋 : D (𝑋) op → D (𝑋), D 𝑋 F := Hom(F , 𝑝 †𝑋 Λpt ). (11.13) Then for 𝑓 : 𝑋 → 𝑌 in 𝑣𝑒𝑟𝑡 ∩ ℎ𝑜𝑟𝑖𝑧, the isomorphisms in (11.12) specialize to isomorphisms D 𝑋 𝑓 ★ = 𝑓 † D𝑌 and 𝑓★D 𝑋 = D𝑌 𝑓† . Remark 11.1.1. In many examples of a sheaf theory, the functor 11.4 is fully faithful and admits a continuous right adjoint. Indeed, fully faithfulness is equivalent to a Künneth type formula. Remark 11.1.2. Let 𝑆 ∈ C be as in Example 10.1.2. Then there is the (non-full) embedding Corr(C/𝑆 )𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 → Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 which is a lax symmetric monoidal functor. Therefore, one can restrict D along this embedding to obtain a sheaf theory on C/𝑆 , denoted by D/𝑆 . The lax symmetric monoidal structure is provided by Δ★𝑆 D (𝑋) ⊗ D (𝑌 ) → D (𝑋 × 𝑌 ) −−→ D (𝑋 ×𝑆 𝑌 ) Remark 11.1.3. Let 𝑋 ∈ C such that both 𝜋 𝑋 and Δ 𝑋 are in C𝑣𝑒𝑟𝑡 ∩ Cℎ𝑜𝑟𝑖𝑧 , then 𝑋 is self-dual in Corr(C). Then the evaluation and the unit maps are given by 𝜋𝑋 Δ𝑋 pt ←−− 𝑋 −−→ 𝑋 × 𝑋, Δ𝑋 𝜋𝑋 𝑋 × 𝑋 ←−− 𝑋 −−→ pt It follows that if the sheaf theory D is symmetric monoidal (rather than just lax symmetric monoidal), then D (𝑋) is self-dual. 11.2 Additional base change theorems The sheaf theory D in practice often satisfies certain additional base change theorems for certain class of morphisms. The general setting is as follows. Let C𝑠𝑚 ⊆ Cℎ𝑜𝑟𝑖𝑧 and C𝑝𝑟𝑜 𝑝 ⊆ C𝑣𝑒𝑟𝑡 be another two classes of morphisms of C stable under composition and base change and containing equivalences. We make the following assumptions on these classes: (a). For 𝑓 ∈ C𝑝𝑟𝑜 𝑝 , the functor 𝑓 † is continuous. (b). For 𝑔 ∈ C𝑠𝑚 , the functor 𝑔★ is continuous. 79 We make the following assumptions on various compatibilities between these adjoints. Consider the Cartesian square (11.2). Then consider the following assumptions. Assumptions 11.2.1. The adjoint pairs ((−)★, (−)★), ((−)† , (−) † ) satisfy the following conditions: (1) If 𝑓 ∈ C𝑝𝑟𝑜 𝑝 and 𝑔 ∈ C𝑣𝑒𝑟𝑡 the natural map 𝑓†′ ◦ 𝑔′† → 𝑔 † ◦ 𝑓† is an equivalence. (2) If 𝑓 ∈ C𝑠𝑚 and 𝑔 ∈ Cℎ𝑜𝑟𝑖𝑧 then the natural map 𝑓 ★ ◦ 𝑔★ → 𝑔★′ ◦ 𝑓 ′★ is an equivalence. (3) If 𝑓 ∈ C𝑝𝑟𝑜 𝑝 and 𝑔 ∈ C𝑠𝑚 the natural map 𝑔′★ ◦ 𝑓 † → 𝑓 ′† ◦ 𝑔★ is an equivalence. (4) If 𝑓 ∈ C𝑝𝑟𝑜 𝑝 and 𝑔 ∈ Cℎ𝑜𝑟𝑖𝑧 the natural map 𝑓† ◦ 𝑔★′ → 𝑔★ ◦ 𝑓†′ is an equivalence. (5) For 𝑓𝑖 : 𝑋𝑖 → 𝑌𝑖 in C𝑝𝑟𝑜 𝑝 with 𝑖 = 1, 2, and F𝑖 ∈ D (𝑌𝑖 ) the natural map 𝑓1† (F1 ) ⊠ 𝑓2† (F2 ) → ( 𝑓1 × 𝑓2 ) † (F1 ⊠ F2 ) is an equivalence. (6) For 𝑓𝑖 : 𝑋𝑖 → 𝑌𝑖 in C𝑠𝑚 with 𝑖 = 1, 2, and F𝑖 ∈ D (𝑋𝑖 ) the natural map ( 𝑓1 )★ (F1 ) ⊠ ( 𝑓2 )★ (F2 ) → ( 𝑓1 × 𝑓2 )★ (F1 ⊠ F2 ) is an equivalence. We make some additional assumptions. They are satisfied in certain sheaf theories, but not always. 1. Let 𝑓 ∈ 𝐶𝑣𝑒𝑟𝑡 . For 𝑔 ∈ C𝑝𝑟𝑜 𝑝 , the natural map 𝑓† ◦ 𝑔 † → 𝑔′† ◦ 𝑓†′ is an equivalence. For 𝑔 ∈ C𝑠𝑚 , the natural map 𝑓† ◦ 𝑔★′ → 𝑔★ ◦ 𝑓†′ is an equivalence. 2. Let 𝑓 ∈ 𝐶ℎ𝑜𝑟𝑖𝑧 . For 𝑔 ∈ C𝑝𝑟𝑜 𝑝 , the natural map 𝑓 ′★ ◦ 𝑔 † → 𝑔′† ◦ 𝑓 ★ is an equivalence. For 𝑔 ∈ C𝑠𝑚 , the natural map 𝑓 ★ ◦ 𝑔★′ → 𝑔★ ◦ 𝑓 ★ is an equivalence. (IX) For 𝑓 ∈ C𝑝𝑟𝑜 𝑝 ∩ Cℎ𝑜𝑟𝑖𝑧 , 𝑓 †  𝑓 ★; (X) For 𝑓 ∈ C𝑣𝑒𝑟𝑡 ∩ C𝑠𝑚 , 𝑓★  𝑓† [𝑛] for some 𝑛 𝑓 ∈ Z. 80 Example 11.2.2. We have the following basic example of a sheaf theory. Let D : C op → LincatΛ be a lax symmetric monoidal functor. Then let ℎ𝑜𝑟𝑖𝑧 = 𝑎𝑙𝑙 and let 𝑣𝑒𝑟𝑡 consist of the following class of morphisms: 𝑓 : 𝑋 → 𝑌 belongs to 𝑣𝑒𝑟𝑡 if the functor D ( 𝑓 ) : D (𝑌 ) → D (𝑋) admits a continuous left adjoint D ( 𝑓 ) 𝐿 and for every diagram as (11.2), the induced Beck-Chevalley map is an equivalence D ( 𝑓 ′) 𝐿 ◦ D (𝑔)  D (𝑔′) ◦ D ( 𝑓 ) Then D extends to a sheaf theory D : Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 → LincatΛ , such that for 𝑔 ∈ ℎ𝑜𝑟𝑖𝑧, 𝑔★ = D (𝑔) and 𝑓 ∈ 𝑣𝑒𝑟𝑡, 𝑓† = D ( 𝑓 ) 𝐿 . In this case, we can let 𝑝𝑟𝑜 𝑝 = 𝑣𝑒𝑟𝑡. Then Assumptions 11.2.1 1 4 5 automatically hold. 11.3 Category of cohomological correspondences The sheaf theory D can also be (largely) encoded as a symmetric monoidal category CorrD (C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 , usually called the category of cohomological correspondences. Namely, let D be a sheaf theory on C as above. Applying the symmetric monoidal Grothendieck construction ((Lurie, 2017, Proposition 2.4.3.16)) to the lax symmetric monoidal functor Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 → LincatΛ → Cat, one obtains a coCartesian fibration CorrD (C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 → Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . Objects of CorrD (C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 consist of pairs (𝑋, F ) where 𝑋 ∈ C and F ∈ D (𝑋), and morphisms been (𝑋, F ) and (𝑌 , G) consist of pairs (𝐶, 𝑢), where 𝐶 a correspondence ← → −𝑐 𝑐− −𝑐 in ℎ𝑜𝑟𝑖𝑧, and 𝑢 : → −𝑐 ★ F → ← 𝑌 ←− 𝐶 −→ 𝑋, with ← 𝑐− in 𝑣𝑒𝑟𝑡 and → 𝑐−† G. There is a natural symmetric monoidal structure on CorrD (C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 , with the symmetric monoidal structure given by (𝑋, F ) ⊗ (𝑌 , G) = (𝑋 × 𝑌 , F ⊠ G). The unit is given by (pt, Λpt ). We refer to (Xiao and Zhu, 2017, Appendix A) and (Lu and Zheng, 2020) for more detailed discussions of constructions in this category. We recall some results about dualizable objects in CorrD (C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . We make the following assumption throughout the rest of the section. Assumptions 11.3.1. Assume that ℎ𝑜𝑟𝑖𝑧 consist of all morphisms in C and the class 𝑣𝑒𝑟𝑡 has the following property: for a sequence of morphisms 𝑍1 → 𝑍2 → 𝑍3 in C, if 𝑍1 → 𝑍3 belongs to 𝑣𝑒𝑟𝑡, so is 𝑍1 → 𝑍2 . Assume that the structural map 𝜋 𝑋 : 𝑋 → pt belongs to 𝑣𝑒𝑟𝑡. Then as argued in (Lu and Zheng, 2020, Lemma 2.8), for (𝑋, F ), (𝑌 , G) and (𝑍, H ) ∈ CorrD (C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 , 81 we have

Map (𝑍, H ) ⊗ (𝑋, F ), (𝑌 , G) = Map (𝑍, H ), (𝑋 × 𝑌 , Hom(( 𝑝 𝑋 )★ F , ( 𝑝𝑌 ) † G)) , where Hom is the bifunctor from (11.7) (for 𝑋 × 𝑌 ), and 𝑝 𝑋 : 𝑋 × 𝑌 → 𝑋, 𝑝𝑌 : 𝑋 ×𝑌 → 𝑌 are the two natural projections. That is, the internal hom between (𝑋, F ) and (𝑌 , G) exists and is represented by (𝑋 × 𝑌 , Hom( 𝑝★𝑋 F , 𝑝𝑌† G)). The following lemma then follows by general facts about symmetric monoidal categories. Lemma 11.3.2. 1. An object (𝑋, F ) ∈ CorrD (C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 is dualizable if and only if the natural map F ⊠ D 𝑋 F → Hom(( 𝑝 1 )★ F , ( 𝑝 2 ) † F ) is an isomorphism, where 𝑝𝑖 : 𝑋 2 → 𝑋, 𝑖 = 1, 2 are two projections. 2. If (𝑋, F ) is dualizable, then its dual is (𝑋, D 𝑋 F ), with the evaluation map given by the tautological map 𝑒 (𝑋,F ) : (Δ 𝑋 )★ (F ⊠ D 𝑋 F ) = F ⊗ D 𝑋 F → (𝜋 𝑋 ) † Λpt , (11.14) and with the unit map is given by the dual of 𝑒 (𝑋,F ) . (Note that (𝑋 × 𝑋, F ⊠ D 𝑋 F ) is self-dual in CorrD (C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 .) Lemma 11.3.3. Let (𝑋, F ) be a dualizable object in CorrD (C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . If (Δ 𝑋 )† sends compact objects to compact objects, and if Λ 𝑋 is compact in D (𝑋), then F ∈ D (𝑋) is compact. Proof. Note that (Δ 𝑋 ) † and HomD (𝑋) (Λ 𝑋 , −) commute with colimits. Therefore, HomD (𝑋) (F , colim G𝑖 ) = HomD (𝑋) (Λ 𝑋 , (Δ 𝑋 ) † (D 𝑋 F ⊠ (colim G𝑖 ))) 𝑖 𝑖 = colim HomD (𝑋) (Λ 𝑋 , (Δ 𝑋 ) † (D 𝑋 F ⊠ G𝑖 )) = colim HomD (𝑋) (F , G𝑖 ) 𝑖 𝑖 □ 82 C h a p t e r 12 THE CATEGORICAL TRACE 12.1 Hochschild homology Let R be a symmetric monoidal category. The trace of an endomorphism of a dualizable object in R is a classical notion. Namely, if 𝑋 is a dualizable object in R equipped with an endomorphism 𝑓 : 𝑋 → 𝑋, its trace tr(𝑋, 𝑓 ) ∈ End(1R ) is given by the composition 𝑢 id𝑋∨ ⊗ 𝑓 𝑒 1R → − 𝑋 ∨ ⊗ 𝑋 −−−−−−→ 𝑋 ∨ ⊗ 𝑋  𝑋 ⊗ 𝑋 ∨ → − 1R . (12.1) On the other hand, in this article, we will also need a difference type of trace construction, known as the categorical trace. Let us first review the general formalism. We refer to Section 9.2 for a review of bimodules and relative tensor products. Let 𝐴 and 𝐵 be two associative algebras in R. By (Lurie, 2017, Proposition 4.6.3.11)1, an 𝐴-𝐵-bimodule can also be regarded as a left ( 𝐴 ⊗ 𝐵rev )-module or a right (𝐵 ⊗ 𝐴rev )-module, where 𝐴rev (resp. 𝐵rev ) is the algebra 𝐴 (resp. 𝐵) with the multiplication reversed. For an associative algebra 𝐴, and an 𝐴-𝐴-bimodule 𝐹, the Hochschild homology of 𝐹, if exists, is defined as Tr( 𝐴, 𝐹) = 𝐴 ⊗ 𝐴⊗ 𝐴rev 𝐹 ∈ R. (12.2) [−] 𝐹 : 𝐹 → Tr( 𝐴, 𝐹) (12.3) We write for the natural morphism, sometimes called the universal trace morphism. On the other hand, there always exists the Hochschild complex of 𝐹 defined as HH( 𝐴, 𝐹)• = Bar( 𝐴)• ⊗ 𝐴⊗ 𝐴rev 𝐹 = 𝐴 ⊗• ⊗ 𝐹, (12.4) regarded as a simplicial object Δop → R. Explicitly, on the level of simplicies and morphisms, for every 𝑛 ≥ 0 we have an equivalence HH( 𝐴, 𝐹)𝑛 ≃ 𝐴 ⊗𝑛 ⊗ 𝐹. (12.5) 1 Note that assumption (★) of loc. cit. is not essential, as explained before (Lurie, 2017, Notation 4.6.3.3). 83 Under this identification, for 0 < 𝑖 < 𝑛 the face map 𝑑𝑖HH is given by the multiplication map applied to the 𝑖-th and the (𝑖 + 1)-th factors in 𝐴 ⊗𝑛 , and the face map 𝑑0HH is given by multiplying the first factor in 𝐴 ⊗𝑛 to 𝑄 from the right and the face map 𝑑𝑛HH is given by multiplying the 𝑛-th factor in 𝐴 ⊗𝑛 to 𝑄 from the left. If R admits geometric realizations and the tensor product preserves geometric realizations in each variable, then the Hochschild homology of 𝐹 exists and can be computed as the geometric realization of the Hochschild complex. Remark 12.1.1. Associated to R, there is a symmetric monoidal 2-category Morita(R) whose objects are associative algebras in R and whose morphism categories are given by categories of bimodules: MapMorita(R) ( 𝐴, 𝐵) = 𝐵 BMod 𝐴 and compositions are given by the relative tensor products (assuming relative tensor products exist in R). Then every 𝐴-bimodule 𝐹 gives an endomorphism of 𝐴 in Morita(R). In addition, every algebra 𝐴 is a dualizable object in Morita(R). Under the equivalence 𝐴 BMod 𝐴  1 R BMod 𝐴rev ⊗ 𝐴  𝐴⊗ 𝐴rev BMod1 R , the natural 𝐴-bimodule structure on 𝐴 itself gives unit and evaluation maps 𝐴𝑒 ∈ 𝐴⊗ 𝐴rev BMod1 R , 𝐴𝑐 ∈ 1 R BMod 𝐴rev ⊗ 𝐴 , (12.6) which identify the dual of 𝐴 (in Morita(R)) as 𝐴rev . Then the Hochschild homology of ( 𝐴, 𝐹) is nothing but the trace 𝐹 in the sense of (12.1), regarded as an endomorphism of 𝐴 in Morita(R). This justifies our choice of notations. However, we will not systematically explore this approach in this article. On the one hand, we do not want to systematically review the formalism of (∞, 2)-categories. On the other hand, in our applications where R will mainly be LincatΛ , we implicitly need some 3-category structure on Morita(LincatΛ ). Example 12.1.2. If the 𝐴-bimodule 𝐹 = 𝑀 ⊗ 𝑁, where 𝑀 is a left 𝐴-module and 𝑁 a right 𝐴-module. Then Tr( 𝐴, 𝐹) = 𝐴 ⊗ 𝐴⊗ 𝐴rev (𝑀 ⊗ 𝑁)  𝑁 ⊗ 𝐴 𝑀. In fact, HH( 𝐴, 𝐹)•  Bar 𝐴 (𝑁, 𝑀)• . Example 12.1.3. Of particular importance in this paper is the following type of bimodules. Let 𝜙 be an endomorphism of the algebra 𝐴. For an 𝐴-bimodule 𝐹 we will denote by 𝜙 𝐹 the bimodule obtained by the same action on the right but with 84 a pre-composition with 𝜙 for the left action. In this case we will also denote the Hochschild homology of the bimodule 𝜙 𝐴 by Tr( 𝐴, 𝜙). That is, Tr( 𝐴, 𝜙) ≃ 𝐴 ⊗ 𝐴⊗ 𝐴rev 𝜙 𝐴. (12.7) In addition, 𝜙 = id, we simply write Tr( 𝐴, id) by Tr( 𝐴). Now let R = LincatΛ . Then an algebra object 𝐴 in LincatΛ is a presentable Λ-linear monoidal category with monoidal product commutes with colimits separately in each variable. Let 𝐹 be an 𝐴-bimodule category. In this case, Tr( 𝐴, 𝐹) always exists, and is sometimes called the categorical trace of ( 𝐴, 𝐹). Note that it is a presentable Λ-linear category. 12.2 Functoriality of Hochschild homology Let 𝐴, 𝐵 ∈ Alg(R) be associative algebras and let 𝑀 ∈ 𝐴 BMod𝐵 . Recall (Lurie, 2017, §4.6.2) that a left dual of 𝑀 is given by an object 𝑁 ∈ 𝐵 BMod 𝐴 together with a unit (or co-evaluation) 𝑢 : 𝐵 → 𝑁 ⊗𝐴 𝑀 (12.8) which is a morphism in 𝐵 BMod𝐵 , and a co-unit (or evaluation) 𝑒 : 𝑀 ⊗𝐵 𝑁 → 𝐴 (12.9) which is a morphism in 𝐴 BMod 𝐴 , such that the compositions id⊗𝑢 𝑒⊗id 𝑢⊗id id⊗𝑒 𝑀 ≃ 𝑀 ⊗𝐵 𝐵 −−−→ 𝑀 ⊗𝐵 𝑁 ⊗ 𝐴 𝑀 −−−→ 𝐴 ⊗ 𝐴 𝑀 ≃ 𝑀 𝑁 ≃ 𝐵 ⊗𝐵 𝑁 −−−→ 𝑁 ⊗ 𝐴 𝑀 ⊗𝐵 𝑁 −−−→ 𝑁 ⊗ 𝐴 𝐴 ≃ 𝑁. (12.10) (12.11) are equivalent to the identities on 𝑀 and 𝑁. Specializing to 𝐵 = 1R , we obtain the notion of a left dual of a left 𝐴-module. Remark 12.2.1. By (Lurie, 2017, Proposition 4.6.2.1) such data are equivalent to giving a right adjoint to the functor 𝑀 ⊗𝐵 (−) : LMod𝐵 (R) → LMod 𝐴 (R). Namely, right adjoint is given by 𝑁 ⊗ 𝐴 (−). Now let 𝐹𝐴 ∈ 𝐴 BMod 𝐴 and 𝐹𝐵 ∈ 𝐵 BMod𝐵 be two bimodules. Assume that we are given left dualizable 𝑀 ∈ 𝐴 BMod𝐵 together with a morphism of bimodules 𝛼 : 𝑀 ⊗𝐵 𝐹𝐵 → 𝐹𝐴 ⊗ 𝐴 𝑀. (12.12) 85 Then we can associate to (𝑀, 𝛼) a morphism in R Tr(𝑀, 𝛼) : Tr(𝐵, 𝐹𝐵 ) → Tr( 𝐴, 𝐹𝐴 ), (12.13) given by 𝑢⊗id Tr(𝐵, 𝐹𝐵 ) = 𝐵 ⊗𝐵⊗𝐵rev 𝐹𝐵 −−−→ (𝑁 ⊗ 𝐴 𝑀) ⊗𝐵⊗𝐵rev 𝐹𝐵 ≃ 𝐴 ⊗ 𝐴⊗ 𝐴rev (𝑀 ⊗𝐵 𝐹𝐵 ⊗𝐵 𝑁) id⊗𝛼⊗id id⊗id⊗𝑒 −−−−−−→ 𝐴 ⊗ 𝐴⊗ 𝐴rev (𝐹𝐴 ⊗ 𝐴 𝑀 ⊗𝐵 𝑁) −−−−−−→ 𝐴 ⊗ 𝐴⊗ 𝐴rev 𝐹𝐴 = Tr( 𝐴, 𝐹𝐴 ), where the isomorphism (𝑁 ⊗ 𝐴 𝑀) ⊗𝐵⊗𝐵rev 𝐹𝐵 ≃ 𝐴 ⊗ 𝐴⊗ 𝐴rev (𝑀 ⊗𝐵 𝐹𝐵 ⊗𝐵 𝑁) can be established by the same way as in Example 12.1.2. In the particular case when 𝐵 = 𝐹𝐵 = 1R is the unit object of R, 𝑀 is just a left 𝐴-module, and 𝛼 is a map 𝑀 → 𝐹𝐴 ⊗ 𝐴 𝑀. Then the above definition of Tr(𝑀, 𝛼) is simplified as 𝑢 1R → − 𝑁 ⊗ 𝐴 𝑀 ≃ 𝐴 ⊗ 𝐴⊗ 𝐴rev (𝑀 ⊗ 𝑁) id⊗(𝛼⊗id) id⊗𝑒 −−−−−−−−→ 𝐴 ⊗ 𝐴⊗ 𝐴rev (𝐹𝐴 ⊗ 𝐴 𝑀 ⊗ 𝑁) −−−→ 𝐴 ⊗ 𝐴⊗ 𝐴rev 𝐹𝐴 = Tr( 𝐴, 𝐹𝐴 ). (12.14) In this case, we also denote Tr(𝑀, 𝛼) as [𝑀, 𝛼] 𝐹 , thought as a point in the space Map(1R , Tr( 𝐴, 𝐹𝐴 )). Example 12.2.2. Let 𝑀 = 𝐴, regarded as a left 𝐴-module. Then 𝑀 admits a left dual given by 𝑁 = 𝐴 regarded as the right 𝐴-module. The unit and evaluation maps are 1𝐴 𝑚 1R −−→ 𝐴  𝐴 ⊗ 𝐴 𝐴, 𝐴 ⊗ 𝐴 − → 𝐴. Giving a left 𝐴-module morphism 𝛼 : 𝑀 → 𝐹𝐴 ⊗ 𝐴 𝑀 is equivalent to giving a map 𝛼0 : 1R → 𝐹𝐴 . Then [ 𝐴, 𝛼] 𝐹𝐴 : 1R → Tr( 𝐴, 𝐹𝐴 ) is canonically equivalent to [−] 𝐹𝐴 ◦ 𝛼0 : 1R → 𝐴 → 𝐹𝐴 . In the particular case R = LincatΛ , giving 𝛼0 : 1R → 𝐹𝐴 is equivalent to giving an object 𝑋 ∈ 𝐹𝐴 . We denote the corresponding left 𝐴-module morphism 𝑀 → 𝐹𝐴 ⊗ 𝐴 𝑀 by 𝛼 𝑋 . Then we have a canonical isomorphism in Tr( 𝐴, 𝐹𝐴 ) [ 𝐴, 𝛼 𝑋 ] 𝐹𝐴  [𝑋] 𝐹𝐴 . 86 Now suppose R = LincatΛ and 𝐴 and 𝐵 two algebras in LincatΛ . Then we say an 𝐴-𝐵-bimodule 𝑀 is fully dualizable if the maps (12.8) and (12.9) admit continuous right adjoint 𝑢 𝑅 and 𝑒 𝑅 respectively, and 𝑢 𝑅 (resp. 𝑒 𝑅 ) is a 𝐵-bimodule (resp. 𝐴-bimodule) homomorphism. Suppose 𝑀 is fully dualizable, and suppose 𝛼 also admits continuous right adjoint 𝛼 𝑅 , and suppose 𝛼 𝑅 is also an 𝐴-𝐵-bimodule homomorphism, then Tr(𝑀, 𝛼) admits continuous right adjoint Tr(𝑀, 𝛼) 𝑅 . Example 12.2.3. We consider the situation as in Example 12.2.2 with R = LincatΛ . We call 𝐴 a rigid monoidal category if 𝑀 = 𝐴 as a left 𝐴-module is fully dualizable. This is equivalent to the following conditions: 1. The unit object 1 𝐴 is compact; 2. The multiplication functor 𝑚 : 𝐴 ⊗ 𝐴 → 𝐴 admits a continuous right adjoint 𝑚𝑅; 3. The functor 𝑚 𝑅 : 𝐴 → 𝐴 ⊗ 𝐴 is a functor of 𝐴-bimodules. So the above definition of rigidity is exactly the same as the one from (Dennis Gaitsgory and Rozenblyum, 2017a, §1.9.1). If 𝐴 is also compactly generated this agrees with the more familiar defintion requiring that compact objects of 𝐴 admit both left and right duals, see (Lurie, 2018, Definition D.7.4.1)) and (Dennis Gaitsgory and Rozenblyum, 2017a, Lemma 1.9.1.5). Now, let 𝑋 ∈ 𝐹𝐴 , giving 𝛼 𝑋 : 𝑀 → 𝐹𝐴 ⊗ 𝐴 𝑀. Then 𝛼 𝑋 admits a continuous right adjoint (𝛼 𝑋 ) 𝑅 if and only if 𝑋 is a compact object in 𝐹𝐴 , in which case (𝛼 𝑋 ) 𝑅 is automatically a left 𝐴-module homomorphism as it is a right-lax functor, see (Dennis Gaitsgory and Rozenblyum, 2017a, Lemma 9.3.6.). It follows that [𝑋] 𝐹𝐴 = [ 𝐴, 𝛼 𝑋 ] 𝐹𝐴 , regarded as a functor ModΛ → Tr( 𝐴, 𝐹𝐴 ) admits a continuous right adjoint. That is, [𝑋] 𝐹𝐴 is a compact object in Tr( 𝐴, 𝐹𝐴 ). This can also be deduced from Corollary 12.3.3 below. 12.3 Categorical traces of rigid monoidal categories We let R = LincatΛ . Categorical traces of rigid monoidal categories have particularly nice formal properties. Namely, the rigidity implies (in fact is equivalent to) the monadicity of Hochschild complex. 87 Lemma 12.3.1. Assume that 𝐴 is rigid and 𝐹 is an 𝐴-bimodule. Then for every 𝛼 : [𝑛] → [𝑚], the diagram HH( 𝐴, 𝐹)𝑚+1 𝑑0HH  HH( 𝐴, 𝐹)𝑚 / HH( 𝐴, 𝐹)𝑛+1  (12.15) 𝑑0HH / HH( 𝐴, 𝐹)𝑛 is right-adjointable. I.e., the cosimplicial object HH( 𝐴, 𝐹) • obtained from HH( 𝐴, 𝐹)• by passing to the right adjoints satisfies the Beck-Chevalley conditions. In particular, |HH( 𝐴, 𝐹)• |  Tot(HH( 𝐴, 𝐹) • )  LMod𝑇 (𝐹), with 𝑇 the monad given to 𝑑0HH ◦ (𝑑1HH ) 𝑅 . Proof. To prove the first statement, we may assume that 𝛼 is either a coboundary map or a coface map. In the case that 𝛼 is a coboundary map, the induced map HH( 𝐴, 𝐹)𝑚 = 𝐴 ⊗𝑚 ⊗ 𝐹 → HH( 𝐴, 𝐹)𝑛 = 𝐴 ⊗𝑛 ⊗ 𝐹 is a right 𝐴-module morphism with respect to the right 𝐴-action on the last factor 𝐹. As such morphism is induced by inserting the unit at an appropriate position, it admits a continuous right adjoint, which by (Dennis Gaitsgory and Rozenblyum, 2017a, Lemma 1.9.3.6) is an 𝐴module morphism. This exactly means that the commutative diagram in the lemma is right-adjointable when 𝛼 is a coboundary map. Next, assume that 𝛼 = 𝑑𝑖 is a coface map. As explained in (Dennis Gaitsgory and Rozenblyum, 2017a, Lemma 1.9.3.2), for a left 𝐴-module 𝑀, the action map 𝑎 : 𝐴 ⊗ 𝑀 → 𝑀 admits a continuous right adjoint 𝑎 𝑅 given by (𝑚 𝑅 ◦1 𝐴)⊗id 𝑀 id 𝐴 ⊗𝑎 𝑀  ModΛ ⊗ 𝑀 −−−−−−−−−−−→ 𝐴 ⊗ 𝐴 ⊗ 𝑀 −−−−→ 𝐴 ⊗ 𝑀 and moreover the functor 𝑎 𝑅 is a morphism of 𝐴-modules (a-priori it is only lax). Similar statements hold for right 𝐴-modules. From these facts, one sees that the commutative diagram is right-adjointable for 𝛼 = 𝑑𝑖 . Indeed, since each of the functors only these claim reduce to the diagrams with [𝑛] ≤ 1. and the requirement of right adjointability is exactly the fact that 𝐹 → 𝐴 ⊗ 𝐹 and 𝐴 → 𝐴 ⊗ 𝐴 are morphisms of 𝐴-modules. That |HH( 𝐴, 𝐹)• |  Tot(HH( 𝐴, 𝐹) • ) follows from (9.1) and the last equivalence follows from the first assertion and Theorem 9.0.5. □ The following statements directly follow from the proof. 88 Corollary 12.3.2. Assume that 𝐴 is rigid. Let 𝐹1 → 𝐹2 be a fully faithful functor of 𝐴-bimodules. Then the induced functor Tr( 𝐴, 𝐹1 ) → Tr( 𝐴, 𝐹2 ) is fully faithful. Proof. By Lemma 12.3.1 the functor between the trace categories can be realized as a limit of functors between Hochschild complexes, so to get fully faithfulness it is enough to have level-wise fully faithfulness of the functors 𝐴 ⊗𝑛 ⊗Λ 𝐹1 → 𝐴 ⊗𝑛 ⊗Λ 𝐹2 for all 𝑛 ≥ 0. Each of these can be realized via a bar construction with ModΛ so it is enough to show that As 𝐴 is rigid, it is canonically self-dual as an object of R, it is enough to show that the functor LFunΛ ( 𝐴, 𝐹1 ) → LFunΛ ( 𝐴, 𝐹2 ) between categories of Λ-linear functors, is fully faithful. Fully faithfulness is then known (see for example (Gepner, Haugseng, and Nikolaus, n.d., Lemma 5.2.) □ Corollary 12.3.3. An 𝐴-bimodule functor 𝐹1 → 𝐹2 induces a functor between cosimplicial objects HH( 𝐴, 𝐹1 ) • → HH( 𝐴, 𝐹2 ) • . In addition, the following diagram is right adjointable 𝐹1  𝐹2 [−] 𝐹1 [−] 𝐹2 / Tr( 𝐴, 𝐹1 )  / Tr( 𝐴, 𝐹2 ) If 𝐹1 → 𝐹2 admits a continuous right adjoint (as plain Λ-linear presentable stable categories), then the following diagram is right adjointable 𝐹1 [−] 𝐹1  Tr( 𝐴, 𝐹1 ) / 𝐹2  [−] 𝐹2 / Tr( 𝐴, 𝐹2 ) Remark 12.3.4. Assume that 𝐴 is rigid. Let 𝑀 be a left 𝐴-module. Let ModΛ → 𝑁 ⊗ 𝐴 𝑀 and 𝑀 ⊗ 𝑁 → 𝐴 be the duality datum for 𝑀 as a left 𝐴-module. Then it follows easily from the above corollary that 𝑁 is the dual of 𝑀 in LincatΛ with the duality datum given by [−] 𝑅 ModΛ → 𝑁 ⊗ 𝐴 𝑀 −−−→ 𝑁 ⊗ 𝑀, Hom(1 𝐴,−) 𝑀 ⊗ 𝑁 → 𝐴 −−−−−−−−→ ModΛ , where [−] 𝑅 denotes the continuous right adjoint of [−] 𝑀 ⊗𝑁 . 89 One can also easily deduce the following corollary, which has appeared in (D. Gaitsgory et al., 2019, Theorem 3.8.5). Corollary 12.3.5. Assume that 𝐴 is rigid, equipped with a monoidal endomorphism 𝜙 : 𝐴 → 𝐴. Then tr( 𝐴, 𝜙)  EndTr( 𝐴,𝜙) ([1 𝐴 ] 𝜙 𝐴 ) as Λ-algebras. Let 𝑀 be a left 𝐴-module equipped with an 𝐴-module homomorphism 𝛼 : 𝑀 → 𝜙 𝐴 ⊗ 𝐴 𝑀 = 𝜙 𝑀. Assume that 𝑀 is dualizable in LincatΛ . Then under the above isomorphism, HomTr( 𝐴,𝜙) ([1 𝐴 ] 𝜙 𝐴 , [𝑀, 𝛼] 𝜙 𝐴 ) is isomorphic to tr(𝑀, 𝜙) as modules. Proof. Applying the monad from Lemma 12.3.1 to 1 𝐴 , we see that EndTr( 𝐴,𝜙) ([1 𝐴 ] 𝜙 𝐴 ) is isomorphic to Hom 𝜙 𝐴 (1 𝐴 , 𝑇 (1 𝐴 )) as algebras, which is given by 1𝐴 𝑚𝑅 sw 𝑚 1𝑅𝐴 ModΛ −−→ 𝐴 −−→ 𝜙 𝐴 ⊗ 𝐴  𝐴 ⊗ 𝜙 𝐴 − → 𝐴 −−→ ModΛ . 𝑚 𝑅 ◦1 𝐴 By (Dennis Gaitsgory and Rozenblyum, 2017a, §9.2.1), the pair ModΛ −−−−−→ 𝐴⊗ 𝐴 1 𝑅𝐴 ◦𝑚 and 𝐴 ⊗ 𝐴 −−−−→ ModΛ form a duality datum of 𝐴 (regarded as object in LincatΛ ). It follows that Hom 𝜙 𝐴 (1 𝐴 , 𝑇 (1 𝐴 ))  tr( 𝐴, 𝜙). The case of module is similar. □ The following description of hom spaces between certain objects in Tr( 𝐴, 𝜙) is useful in practice. Corollary 12.3.6. Assume that 𝐴 is rigid and is compactly generated, with {𝑐𝑖 } being a set of compact generators. Then for 𝑋, 𝑌 ∈ 𝐴 with 𝑋 compact in 𝐴, HomTr( 𝐴,𝜙) ([𝑋] 𝜙 𝐴 , [𝑌 ] 𝜙 𝐴 )  colim Hom 𝐴 (𝑋, 𝑐 𝑗 ⊗ 𝜙(𝑐𝑖 )), 𝐶⊗𝐶/𝑚 𝑅 (𝑌 ) where 𝐶 ⊗ 𝐶 ⊂ 𝐴 ⊗ 𝐴 denotes the full subcategory spanned by {𝑐𝑖 ⊠ 𝑐 𝑗 }𝑖, 𝑗 . Note that a morphism 𝑐𝑖 ⊠𝑐 𝑗 → 𝑚 𝑅 (𝑌 ) in 𝐴⊗ 𝐴 is equivalent to a morphism 𝑐𝑖 ⊗𝑐 𝑗 → 𝑌 in 𝐴. So informally, this corollary says that every morphism [𝑋] 𝜙 𝐴 → [𝑌 ] 𝜙 𝐴 in Tr( 𝐴, 𝜙) can be represented as a pair of morphisms (𝑋 → 𝑐 𝑗 ⊗ 𝜙(𝑐𝑖 ), 𝑐𝑖 ⊗ 𝑐 𝑗 → 𝑌 ) in 𝐴 (compare with (Zhu, 2016b, §3.1)). Proof. We have HomTr( 𝐴,𝜙) ([𝑋] 𝜙 𝐴 , [𝑌 ] 𝜙 𝐴 )  Hom 𝜙 𝐴 (𝑋, 𝑚 ◦ sw ◦ 𝑚 𝑅 (𝑌 )). 90 As 𝐴 is compactly generated, so is 𝐴 ⊗ 𝐴 with a set of compact generators given by {𝑐𝑖 ⊠ 𝑐 𝑗 }𝑖, 𝑗 . Then 𝑚 𝑅 (𝑌 ) = colim𝑐𝑖 ⊠𝑐 𝑗 →𝑌 𝑐𝑖 ⊠ 𝑐 𝑗 . As 𝑋 is compact, the corollary follows. □ Remark 12.3.7. In fact, the above corollary admits a more economic form. Namely, suppose we write 𝑚 𝑅 (1 𝐴 )  colim𝑖 (𝑐𝑖,1 ⊠ 𝑐𝑖,2 ) as a filtered colimit of compact objects in 𝐴 ⊗ 𝐴. Then as 𝑚 𝑅 is a right 𝐴-module homomorphism, we have 𝑚 𝑅 (𝑌 )  colim𝑖 (𝑐𝑖,1 ⊠ (𝑐𝑖,2 ⊗ 𝑌 )). Therefore, HomTr( 𝐴,𝜙) ([𝑋] 𝜙 𝐴 , [𝑌 ] 𝜙 𝐴 )  colim Hom 𝐴 (𝑋, 𝑐𝑖,2 ⊗ 𝑌 ⊗ 𝜙(𝑐𝑖,1 )). 𝑖 As mentioned above, if 𝐴 is rigid, then it is dualizable as object in LincatΛ . It follows from (Dennis Gaitsgory and Rozenblyum, 2017a, Proposition 9.5.3) that 𝐴𝑐 and 𝐴𝑒 from (12.6) admit left duals, denoted by 𝑆 𝐴 and 𝑇𝐴 respectively, which we can identify with 𝐴-bimodules under the equivalence 𝐴rev ⊗ 𝐴 BMod1  𝐴 BMod 𝐴 (resp. 1 BMod 𝐴⊗ 𝐴rev  𝐴 BMod 𝐴 ). By (Lurie, 2017, Remark 4.6.5.4), there is an automorphism 𝜎𝐴 of 𝐴 (as a monoidal category), usually called the Serre functor of 𝐴, such that 𝑆 𝐴 = 𝜎𝐴 𝐴 as 𝐴-bimodules. Example 12.3.8. We recall that a pivotal category is a rigid monoidal category equipped with an isomorphism id 𝐴  𝜑 𝐴 . If 𝐴 is compactly generated, this means that for 𝑎 ∈ 𝐴 compact, 𝑎 ∨,𝐿 and 𝑎 ∨,𝑅 are functorially isomorphic. Assume that 𝐴 is pivotal. Assume that 𝑄 is an 𝐴-bimodule, dualizable as a Λ-linear category, with ∨ 𝑄 its left dual. Then Tr( 𝐴, 𝑄) is dualizable with Tr( 𝐴, 𝑄 ∨ ) is dual. The unit is given by 𝑢 [−] 𝐴 ModΛ → − 𝐴 −−−→ Tr( 𝐴, 𝐴) → Tr( 𝐴, ∨ 𝑄 ⊗ 𝐴 𝑄) → Tr( 𝐴, 𝑄) ⊗ Tr( 𝐴, ∨ 𝑄). 91 C h a p t e r 13 GEOMETRIC TRACES IN SHEAF THEORY Following ideas of (Ben-Zvi and Nadler, 2009) (Ben-Zvi, Nadler, and Preygel, 2017) to develop a method to calculate the (twisted) categorical trace of monoidal categories arising from convolution pattern in algebraic geometry. As mentioned before, Compared with the work of loc. cit., we will first calculate a geometric version of categorical trace. Then we will compare the geometric version with the usual version in favorable cases. Our approach allows us to bypass integral transform of sheaf theories, which usually do not hold in the ℓ-adic setting. 13.1 Geometric Hochschild homology We use the formalism of category of correspondences and sheaf theory as in Chapter 10. Let 𝐴 be an associative algebra object in Corr(C)𝑣𝑒𝑟𝑡,ℎ𝑜𝑟𝑖𝑧 , and let 𝑀 be a left 𝐴module object in Corr(C)𝑣𝑒𝑟𝑡,ℎ𝑜𝑟𝑖𝑧 . As D is a lax symmetric monoidal functor, D ( 𝐴) is an algebra object in LincatΛ and D (𝑀) is a D ( 𝐴)-module object in LincatΛ , with multiplication and action maps given by ⊠ D ( 𝐴) ⊗ D ( 𝐴) − → D ( 𝐴 × 𝐴) → D ( 𝐴) ⊠ D ( 𝐴) ⊗ D (𝑀) − → D ( 𝐴 × 𝑀) → D (𝑀) Similarly, if 𝐹 is an 𝐴-bimodule, then D (𝐹) is an D ( 𝐴)-bimodule. Then one can form its Hochschild homology (a.k.a categorical trace) of (D ( 𝐴), D (𝐹)) Tr(D ( 𝐴), D (𝐹)) = D ( 𝐴) ⊗D ( 𝐴)⊗D ( 𝐴) rev D (𝐹) ∈ LincatΛ . In practice, however, we need to consider a variant Trgeo (D ( 𝐴), D (𝐹)), which we call the geometric trace of D (𝐹). Namely, we consider the Yoneda embedding Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 → P (Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 ), where P (Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 ) is the category of presheaves on Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 equipped with the induced symmetric monoidal structure, which by definition preserves colimits in each variable (see (Lurie, 2017, Corollary 4.8.1.12). Then we have the Hochschild homology of the 𝐴-bimodule 𝐹 in P (Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 ) Tr( 𝐴, 𝐹) := |HH( 𝐴, 𝐹)• | ∈ P (Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 ). 92 By the universal property of P (Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 ), the functor D : Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 → LincatΛ extends to a continuous functor D : P (Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 ) → LincatΛ . Then we define the geometric trace of D (𝐹) as Trgeo (D ( 𝐴), D (𝐹)) := D (Tr( 𝐴, 𝐹)). Explicitly, Trgeo (D (𝐹), D ( 𝐴)) can be computed in the following way. We first apply the functor D to the standard Hochschild complex (12.4) (which now is a simplicial object in Corr(C)𝑣𝑒𝑟𝑡,ℎ𝑜𝑟𝑖𝑧 ) to obtain a simplicial object D (HH• ( 𝐴, 𝐹)) in LincatΛ . Then the geometric trace Trgeo (D ( 𝐴), D (𝐹)) is the geometric realization of this simplicial object in LincatΛ Trgeo (D ( 𝐴), D (𝐹))  |D (HH( 𝐴, 𝐹)• )|. (13.1) We emphasize that Trgeo (D ( 𝐴), D (𝐹)) depends not only on D (𝐹), but on the 𝐴-bimodule 𝐹 itself (and of course the functor D). In particular, for 𝐴 equipped with an algebra endomorphism 𝜙 : 𝐴 → 𝐴 we have the 𝐴-bimodule 𝐹 = 𝜙 𝐴 in Corr(C)𝑣𝑒𝑟𝑡,ℎ𝑜𝑟𝑖𝑧 as before. We write Trgeo (D ( 𝐴), 𝜙) = Trgeo (D ( 𝐴), D ( 𝜙 𝐴)). Remark 13.1.1. As D is equipped with a lax monoidal structure we get a natural comparison functor Tr(D ( 𝐴), D (𝐹)) ≃ |D ( 𝐴) ⊗• ⊗ D (𝐹)| → |D ( 𝐴 ⊗• ⊗ 𝐹)| = Trgeo (D ( 𝐴), D (𝐹)) (13.2) from the usual trace of D (𝐹) to the geometric trace. This functor is not an equivalence in general. Of course, if for each 𝑛, the functor D ( 𝐴) ⊗𝑛 ⊗D (𝑄) → D ( 𝐴𝑛 ×𝑄) is an equivalence, then the comparison map (13.2) is an equivalence. We will see later that this functor is an equivalence in many more cases of interest. 13.2 Fixed point objects and geometric traces of convolution categories We specialize the previous constructions to the situation appearing in our applications. Let 𝑋 ∈ C satisfying conditions as in Example 10.1.2 (that is, Δ 𝑋 : 𝑋 → 𝑋 ×𝑋 and 𝜋 𝑋 : 𝑋 → pt belong to 𝐶ℎ𝑜𝑟𝑖𝑧 ). Let 𝑓 : 𝑋 → 𝑌 in C𝑣𝑒𝑟𝑡 and assume that the relative diagonal map Δ 𝑋/𝑌 : 𝑋 → 𝑋 ×𝑌 𝑋 belongs to 𝑣𝑒𝑟𝑡. Let 𝑋• → 𝑌 denote the Čech nerve of 𝑓 . From Example 10.3.3 and Remark 10.3.6, it gives rise to an object Alg(EndCorr(C) (𝑋)) which induces 𝑋1 := 𝑋 ×𝑌 𝑋 93 with the structure of an associative algebra object in Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . The multiplication and unit maps are given by 𝑋 ×𝑌 𝑋 ×𝑌 𝑋 id×Δ𝑋 ×id (𝑋 ×𝑌 𝑋) × (𝑋 ×𝑌 𝑋) 𝑋 Δ𝑋/𝑌 , id× 𝑓 ×id pt , 𝑋 ×𝑌 𝑋 𝑋 ×𝑌 𝑋 Let 𝑍 ∈ C equipped with two morphisms 𝑔𝑖 : 𝑍 → 𝑌 , 𝑖 = 1, 2 in C and let 𝑄 = 𝑋 ×𝑌 𝑍 ×𝑌 𝑋 = 𝑍 ×𝑌 ×𝑌 (𝑋 × 𝑋). Then via the construction of Section 10.4 the object 𝑄 admits the structure of an (𝑋 ×𝑌 𝑋)-bimodule in Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . In particular, the left action is given by the diagram 𝑋 ×𝑌 𝑋 ×𝑌 𝑍 ×𝑌 𝑋 (𝑋 ×𝑌 𝑋) × (𝑋 ×𝑌 𝑍 ×𝑌 𝑋) id× 𝑓 ×id×id 𝑋 ×𝑌 𝑍 ×𝑌 𝑋 and we have a similar diagram for the right action. Consider the following diagram 𝑋 ×𝑌 ×𝑌 𝑍 𝑞=( 𝑓 ×id 𝑍 ) 𝛿0 =(Δ𝑋 ×id 𝑍 ) / (𝑋 × 𝑋) ×𝑌 ×𝑌 𝑍 (13.3)  𝑌 ×𝑌 ×𝑌 𝑍, which induces a functor 𝑞 † ◦ (𝛿0 )★ : D (𝑋 ×𝑌 𝑍 ×𝑌 𝑋) → D (𝑌 ×𝑌 ×𝑌 𝑍). Proposition 13.2.1. The following diagram is commutative D (𝑋 ×𝑌 𝑍 ×𝑌 𝑋) (𝛿0 )★ / D (𝑋 ×𝑌 ×𝑌 𝑍)  Tr(D (𝑋 ×𝑌 𝑋), D (𝑋 ×𝑌 𝑍 ×𝑌 𝑋))  Trgeo (D (𝑋 ×𝑌 𝑋), D (𝑋 ×𝑌 𝑍 ×𝑌 𝑋)) 𝑞†  / D (𝑌 ×𝑌 ×𝑌 𝑍). Assume that the sheaf theory D satisfies Assumptions 11.2.1 1-3, and assume that: 1. Δ 𝑋 : 𝑋 → 𝑋 × 𝑋 ∈ C𝑠𝑚 , 2. 𝑓 : 𝑋 → 𝑌 ∈ C𝑝𝑟𝑜 𝑝 , 94 3. Δ 𝑋/𝑌 : 𝑋 → 𝑋 ×𝑌 𝑋 is in C𝑝𝑟𝑜 𝑝 . Then the bottom horizontal functor of the above diagram is fully faithful, with the essential image is generated under colimits by the image of 𝑞 † ◦ 𝛿★0 . The proof of the proposition will be given at the end of Section 13.3. We note that there is no assumption on (𝑔1 , 𝑔2 ) : 𝑍 → 𝑌 × 𝑌 . We specialize Proposition 13.2.1 to the following two cases. First, assume we are given morphisms 𝜙 𝑋 : 𝑋 → 𝑋 and 𝜙𝑌 : 𝑌 → 𝑌 in C intertwined by 𝑓 , that is, equipped with an equivalence 𝑓 ◦ 𝜙 𝑋 ≃ 𝜙𝑌 ◦ 𝑓 . We will usually abuse notation and denote both maps by 𝜙 if it is clear from context. We let 𝑍 = 𝑌 with the map 𝑔1 = id and 𝑔2 = 𝜙. In this case, 𝑍 ×𝑌 ×𝑌 𝑌 is nothing but the 𝜙-fixed point object L 𝜙 (𝑌 ), defined by the pullback L 𝜙 (𝑌 ) 𝑌 (13.4) Δ𝑌 𝑝𝜙 id×𝜙 𝑌 𝑌 × 𝑌. We assume in addition that 𝜙 𝑋 is an equivalence. In this case the (𝑋 ×𝑌 𝑋)-module 𝑋 ×𝑌 𝑍 ×𝑌 𝑋 is isomorphic to the 𝜙-twisted module 𝜙 (𝑋 ×𝑌 𝑋), with the isomorphism sending (𝑥, 𝑧, 𝑥 ′) ∈ 𝑋 ×𝑌 𝑍 ×𝑌 𝑋 to (𝜙(𝑥), 𝑥 ′) ∈ 𝜙 (𝑋 ×𝑌 𝑋). Then (13.3) becomes 𝑋 ×𝑌 L 𝜙 (𝑌 ) 𝛿0 / 𝑋 ×𝑌 𝑋 𝑞  L 𝜙 (𝑌 ) Corollary 13.2.2. Under the same assumption as in Proposition 13.2.1 and given 𝜙 𝑋 , 𝜙𝑌 as above, there is a canonical factorization D (𝑋 ×𝑌 𝑋) (𝛿0 )★ D (𝑋 × 𝑋×𝑋 (𝑋 ×𝑌 𝑋)) 𝑞† Trgeo (D (𝑋 ×𝑌 𝑋), 𝜙) D (L 𝜙𝑌 ) with the lower horizontal arrow is fully faithful. The essential image is generated under colimits by the image of 𝑞 † ◦ 𝛿★0 . Another case we need to consider is 𝑍 = 𝑊1 × 𝑊2 with 𝑔𝑖 : 𝑊𝑖 → 𝑌 two maps in C. In this case, 𝑍 ×𝑌 ×𝑌 𝑌 = 𝑊1 ×𝑌 𝑊2 , 𝑍 ×𝑌 ×𝑌 (𝑋 × 𝑋) = (𝑊1 ×𝑌 𝑋) × (𝑋 ×𝑌 𝑊2 ), 95 We denote: geo D (𝑊1 ×𝑌 𝑋) ⊗D (𝑋× 𝑋) D (𝑋 ×𝑌 𝑊2 ) := Trgeo (D (𝑋 ×𝑌 𝑋), D (𝑍 ×𝑌 ×𝑌 (𝑋 × 𝑋)), 𝑌 (13.5) which is the geometric analogue of the relative tensor product. Corollary 13.2.3. Under the same assumption as in Proposition 13.2.1, we have a canonical square D ((𝑊1 ×𝑌 𝑋) × (𝑋 ×𝑌 𝑊2 )) geo (id𝑊1 ×Δ𝑋 ×id𝑊2 )★ / D (𝑊1 ×𝑌 𝑋 ×𝑌 𝑊2 )   (id𝑊1 × 𝑓 ×id𝑊2 )† / D (𝑊1 ×𝑌 𝑊2 ) D (𝑊1 ×𝑌 𝑋) ⊗D (𝑋× 𝑋) D (𝑋 ×𝑌 𝑊2 ) 𝑌 with the bottom functor fully faithful. The essential image is generated under colimits by the image of (id𝑊1 × 𝑓 × id𝑊2 )† ◦ (id𝑊1 × Δ 𝑋 × id𝑊2 )★. Again, there is no assumption on 𝑔1 and 𝑔2 . 13.3 The geometric trace and relative resolutions Now we prove Proposition 13.2.1. In fact, (to save notations) we will prove a slightly general statement. We consider the geometric trace for pair BMod(Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 )
arising from an associative algebra 𝑋1 ∈ Alg EndCorr(C) (𝑋0 ) and a bi-module object 𝑄 ∈ 𝑋1 BMod 𝑋1 (EndCorr(C) (𝑋0 )). Roughly speaking, these are pairs (𝑋1 , 𝑄) seen as objects in the category BMod(Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 ) consisting of an algebra 𝑋1 ∈ Alg(Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 ) whose multiplication and unit maps are of the form 𝑋1 × 𝑋0 𝑋1 𝜂 𝑋1 × 𝑋1 , 𝑚 𝜋 𝑋0 𝑋0 𝑢 pt (13.6) , 𝑋1 𝑋1 and an 𝑋1 -bimodule 𝑄 ∈ 𝑋1 BMod 𝑋1 (Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 ) whose action maps are of the form 𝑋1 × 𝑋0 𝑄 𝑎𝑙 𝑄 𝜉𝑙 𝑋1 × 𝑄 𝑄 × 𝑋0 𝑋1 , 𝑎𝑟 𝜉𝑟 𝑄 × 𝑋1 . (13.7) 𝑄 Here we require Δ 𝑋0 : 𝑋0 → 𝑋0 × 𝑋0 and 𝜋 𝑋0 : 𝑋0 → pt belong to Cℎ𝑜𝑟𝑖𝑧 (so 𝜂, 𝜉𝑙 , 𝜉𝑟 ∈ Cℎ𝑜𝑟𝑖𝑧 as well) and 𝑚, 𝑢, 𝑎 𝑙 , 𝑎𝑟 ∈ C𝑣𝑒𝑟𝑡 . See Remark 10.4.1 for more details and references. 96 We can then consider the geometric trace Trgeo (D (𝑋1 ), D (𝑄)) = D (Tr(𝑋1 , 𝑄))  |D (HH(𝑋1 , 𝑄)• )| defined in the previous section. On the other hand, the extra structure on the algebra and module allows one to construct a variant of the geometric trace. In the monoidal category EndCorr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 (𝑋0 ) we consider the Bar complex of the algebra object 𝑋1 , which we denoted by Bar 𝑋0 (𝑋1 )• . Under the lax monoidal functor EndCorr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 (𝑋0 ) → Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 , it gives a simplicial object in Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 (in fact in C𝑣𝑒𝑟𝑡 ), denoted by the same notation, which can be written as Bar 𝑋0 (𝑋1 )•  𝑋• × 𝑋0 ×𝑋0 (𝑋1 × 𝑋1 ), where the two maps 𝑋𝑛 → 𝑋0 corresponds to {0} ⊂ {0, 1, . . . , 𝑛} and {𝑛} ⊂ {0, 1, . . . , 𝑛} respectively and 𝑋1 × 𝑋1 → 𝑋0 × 𝑋0 is given by (𝑑1 , 𝑑0 ). The action of (𝑋1 × 𝑋1 ) × 𝑄 → 𝑄 by right and left multiplication gives Bar 𝑋0 (𝑋1 )• ⊗ 𝑄 = 𝑋• × 𝑋0 ×𝑋0 (𝑋1 × 𝑋1 ) × 𝑄 → 𝑋• × 𝑋0 ×𝑋0 𝑄 =: HH 𝑋0 (𝑋1 , 𝑄)• which is (𝑋1 ⊗ 𝑋1 )-bilinear and therefore induces Bar 𝑋0 (𝑋1 )• ⊗ 𝑋1 ⊗𝑋1 𝑄 → HH 𝑋0 (𝑋1 , 𝑄)• The lax monoidal functor EndCorr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 (𝑋0 ) → Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 also induces a natural map of simplicial objects Bar(𝑋1 )• → Bar 𝑋0 (𝑋1 )• in Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 It follows that we obtain a map of simplicial objects in 𝛿• : HH(𝑋1 , 𝑄)• = Bar(𝑋1 )• ⊗ 𝑋1 ⊗𝑋1 𝑄 → Bar 𝑋0 (𝑋1 )• ⊗ 𝑋1 ⊗𝑋1 𝑄 → HH 𝑋0 (𝑋1 , 𝑄)• , (13.8) which is given on each level 𝑛 ≥ 0 by the horizontal arrow id 𝛿𝑛 𝑋𝑛 × 𝑋0 ×𝑋0 𝑄 ← − 𝑋𝑛 × 𝑋0 ×𝑋0 𝑄 −−→ 𝑋1𝑛 × 𝑄. Now we define the 𝑋0 -relative Hochschild homology of 𝑄 as Tr 𝑋0 (𝑋1 , 𝑄) = |HH 𝑋0 (𝑋1 , 𝑄)• | ∈ P (Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 ), 97 and define the 𝑋0 -relative geometric trace of D (𝑄) as the geometric realization in LincatΛ 𝑋0 Trgeo (D (𝑋1 ), D (𝑄)) := D (Tr 𝑋0 (𝑋1 , 𝑄))  |D (HH 𝑋0 (𝑋1 , 𝑄)• )|. Then (13.8) gives a functor 𝑋0 𝛿★ : Trgeo (D (𝑋1 ), D (𝑄)) → Trgeo (D (𝑋1 ), D (𝑄)), which fits into a commutative diagram (𝛿0 )★ D (𝑄) D (𝑋0 × 𝑋0 ×𝑋0 𝑄) Tr(D (𝑋1 ), D (𝑄)) Trgeo (D (𝑋1 ), D (𝑄)) (13.9) 𝛿★ 𝑋0 Trgeo (D (𝑋1 ), D (𝑄)). Proposition 13.3.1. Assume that the sheaf theory D satisfies Assumptions 11.2.1 1-3. In addition, in the notations of (13.6) and (13.7) assume that 1. 𝑚 : 𝑋1 × 𝑋0 𝑋1 → 𝑋1 and 𝑢 : 𝑋0 → 𝑋1 are in C𝑝𝑟𝑜 𝑝 , 2. 𝑎 𝑙 : 𝑋1 × 𝑋0 𝑄 → 𝑄 and 𝑎𝑟 : 𝑄 × 𝑋0 𝑋1 → 𝑄 are in C𝑝𝑟𝑜 𝑝 , 3. the diagonal Δ 𝑋0 : 𝑋0 → 𝑋0 × 𝑋0 is in C𝑠𝑚 , 4. 𝜋 𝑋0 : 𝑋0 → pt is in Cℎ𝑜𝑟𝑖𝑧 . Then the functor 𝛿★ from (13.9) is fully faithful. The essential image is generated un(𝛿0 )★ 𝑋0 der colimits by the image of D (𝑄) −−−−→ D (𝑋0 × 𝑋0 ×𝑋0 𝑄) → Trgeo (D (𝑋1 ), D (𝑄)). Proof. Passing to right adjoints gives a natural transformation (𝛿• )★ : D (HH 𝑋0 (𝑋1 , 𝑄)• ) → D (HH(𝑋1 , 𝑄)• ) of cosimplicial categories. To prove the left adjoint 𝛿★ is fully faithful we will use Theorem 9.0.5 and Corollary 9.0.6. The first step is to verify each of the cosimplicial categories satisfies the Beck-Chevalley conditions. Lemma 13.3.2. Under the assumptions of Proposition 13.3.1, the cosimplicial categories obtained from the simplicial categories D (HH(𝑋1 , 𝑄)• ) and D (HH 𝑋0 (𝑋1 , 𝑄)• ) by passing to right adjoints satisfy the Beck-Chevalley conditions. 98 Proof. For the cosimplicial category D (HH 𝑋0 (𝑋1 , 𝑄)• ), the face maps corresponding to 0 ↦→ 0 ∈ [𝑛] are given by the right adjoints (𝑑0 ) † of (𝑑0 )† . All the morphisms 𝑚, 𝑎 𝑙 , 𝑎𝑟 are in C𝑝𝑟𝑜 𝑝 by assumption and therefore all maps of the simplicial object D (HH 𝑋0 (𝑋1 , 𝑄)• ) are in C𝑝𝑟𝑜 𝑝 as well. Then from Assumptions 11.2.11 all the necessary Beck-Chevalley maps are equivalences. It is left to deal with D (HH• (𝑋1 , 𝑄)). For every map 𝛼 : [𝑚] → [𝑛], we have the diagram 𝑋1𝑚 × 𝑄 0 𝑑𝑚 𝑋1𝑚+1 × 𝑄 (13.10) 𝑋1𝑛 × 𝑄 𝑑 𝑛0 𝑋1𝑛+1 × 𝑄 in Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 (see (10.1) and after for notations). We need to show that the induced diagram D (𝑋1𝑚 × 𝑄) 0 𝑑𝑚 D (𝑋1𝑚+1 × 𝑄) (13.11) D (𝑋1𝑛 × 𝑄) 𝑑 𝑛0 D (𝑋1𝑛+1 × 𝑄) is left adjointable. Clearly, it is enough to consider the case when 𝛼 is either a co-face or a co-degeneracy map. In the case of co-face map, we may assume that 𝛼 = 𝑑𝑛 : [𝑛] → [𝑛 + 1], that is, 𝛼(𝑖) = 𝑖 for all 𝑖. (The proof is similar in all other cases.) Then the diagram (13.10) is explicitly given by 𝑋1𝑛 × 𝑄 𝑎𝑟 ,𝑛 𝑎 𝑙,𝑛 𝑋1𝑛 × 𝑋1 × 𝑋0 𝑄 𝜉𝑟 ,𝑛 𝑎˜ 𝑟 𝑋1𝑛 × 𝑋1 × 𝑋0 𝑄 × 𝑋0 𝑋1 𝜉˜𝑟 𝜉˜𝑙 𝑎𝑟 ,𝑛+1 𝑋1𝑛+1 × 𝑄 × 𝑋0 𝑋1 𝑋1𝑛+1 × 𝑄 𝑎 𝑙,𝑛+1 𝑎˜ 𝑙 𝜉𝑙,𝑛 𝑋1𝑛+1 × 𝑄 𝑋1𝑛 × 𝑄 × 𝑋0 𝑋1 𝑋1𝑛+1 × 𝑋1 × 𝑋0 𝑄 𝜉𝑙,𝑛+1 𝜉𝑟 ,𝑛+1 𝑋1𝑛+2 × 𝑄, as a diagram in C, where 𝑎 𝑙,𝑛 = id 𝑋1𝑛 × 𝑎 𝑙 , etc. Note that all squares are Cartesian in C. ★ , We have 𝑑 0𝑘 = (𝜉𝑙,𝑘 )★ ◦ (𝑎 𝑙,𝑘 ) † (for 𝑘 = 𝑛, 𝑛 + 1) with the left adjoint (𝑎 𝑙,𝑘 )† ◦ 𝜉𝑙,𝑘 and the vertical arrows in (13.11) are given by (𝜉𝑙,𝑘 )★ ◦ (𝑎 𝑙,𝑘 ) † . Left adjointability of 13.11 then means that the natural map (𝑎 𝑙,𝑛+1 )† ◦ (𝜉𝑟,𝑛+1 )★ ◦ (𝜉𝑙,𝑛+1 )★ ◦ (𝑎 𝑙,𝑛+1 ) † → (𝜉𝑙,𝑛 )★ ◦ (𝑎 𝑙,𝑛 ) † ◦ (𝑎𝑟,𝑛 )† ◦ (𝜉𝑟,𝑛 )★ 99 is an equivalence. It is enough to show that the Beck-Chevalley maps ( 𝑎˜𝑟 )† ◦ ( 𝑎˜ 𝑙 ) † → (𝑎 𝑙,𝑛 ) † ◦ (𝑎𝑟,𝑛 )† , (𝜉𝑟,𝑛+1 )★ ◦ (𝜉𝑙,𝑛+1 )★ → ( 𝜉˜𝑙 )★ ◦ ( 𝜉˜𝑟 )★ (13.12) ( 𝜉˜𝑟 )★ ◦ (𝑎 𝑙,𝑛+1 ) † → ( 𝑎˜ 𝑙 ) † ◦ (𝜉𝑟,𝑛 )★, (𝑎𝑟,𝑛+1 )† ◦ ( 𝜉˜𝑙 )★ → (𝜉𝑙,𝑛 )★ ◦ ( 𝑎˜𝑟 )† (13.13) are equivalences. Note that the maps 𝑎 𝑙,𝑘 , 𝑎𝑟,𝑘 , 𝑎˜ 𝑙 , 𝑎˜𝑟 are in C𝑝𝑟𝑜 𝑝 . So the left equivalence of (13.12) holds by Assumptions 11.2.11. As the maps 𝜉𝑙,𝑘 , 𝜉𝑟,𝑘 , 𝜉˜𝑙 , 𝜉˜𝑟 belong to C𝑠𝑚 , the right equivalence of (13.12) holds by Assumptions 11.2.12 and the equivalences (13.13) hold by Assumptions 11.2.1 3 and 4. In the case of co-degeneracy map, we can assume 𝛼 = 𝑠0 : [𝑛 + 1] → [𝑛] given by 𝑠(0) = 0 and 𝑠(𝑖) = 𝑖 − 1 for 1 ≤ 𝑖 ≤ 𝑛. As in the previous case, the diagram (13.10) is explicitly given as 𝑋1𝑛+1 × 𝑄 𝑎𝑟 ,𝑛+1 𝑋1𝑛+1 × 𝑄 × 𝑋0 𝑋1 𝑢𝑛 𝑋1𝑛 × 𝑋0 × 𝑄 𝑋1𝑛+2 × 𝑄 𝑢 𝑛+1 𝑢˜ 𝑎˜ 𝑟 𝑋1𝑛 × 𝑋0 × 𝑄 × 𝑋0 𝑋1 𝜋𝑛 𝑋1𝑛 × 𝑄 𝜉𝑟 ,𝑛+1 𝜉˜𝑟 𝑋1𝑛+1 × 𝑋0 × 𝑄 𝜋 𝑛+1 𝜋˜ 𝑎𝑟 ,𝑛 𝑋1𝑛 × 𝑄 × 𝑋0 𝑋1 𝜉𝑟 ,𝑛 𝑋1𝑛+1 × 𝑄, where all squares are Cartesian. As before, in order to show the corresponding adjointability equivalence it is enough to show the corresponding commutativity in each square. As the map 𝑢 : 𝑋0 → 𝑋1 is in C𝑝𝑟𝑜 𝑝 , the upper two squares of the diagram can be handled as in the co-face map case. For the remaining squares, we need the maps (𝜉𝑟,𝑛 )★ ◦ (𝜋𝑛+1 )★ → ( 𝜋) ˜ ★ ◦ ( 𝜉˜𝑟 )★, (𝑎𝑟,𝑛 )† ◦ ( 𝜋) ˜ ★ → (𝜋𝑛 )★ ◦ ( 𝑎˜𝑟 )† to be equivalences. As 𝜋 𝑋0 : 𝑋0 → pt is in Cℎ𝑜𝑟𝑖𝑧 the same is true for 𝜋 𝑘 , 𝜋˜ so the equivalences hold by Assumptions 11.2.1, 2 and 4. □ We continue to prove Proposition 13.3.1. Passing to the right adjoint of (13.9) gives D (𝑄) (𝛿0 )★ Trgeo (D (𝑋1 ), D (𝑄)) 𝛿★ D (𝑋0 × 𝑋0 ×𝑋0 𝑄) 𝑋0 Trgeo (D (𝑋1 ), D (𝑄)). 100 with horizontal arrows monadic (by Lemma 13.3.2). Let 𝑇 denote the monad corresponding to the cosimplicial category D (HH 𝑋0 (𝑋1 , 𝑄)• ) and by 𝑉 the monad corresponding to D (HH(𝑋1 , 𝑄)• ). Then to show that 𝛿★ is fully faithful it is enough to show that the natural map 𝑉 → (𝛿0 )★ ◦ 𝑇 ◦ (𝛿0 )★, is an equivalence. The monad 𝑇 is given by (𝑑0 )† ◦ (𝑑1 ) † with 𝑑1 , 𝑑0 : 𝑋1 × 𝑋0 ×𝑋0 𝑄 → 𝑋0 × 𝑋0 ×𝑋0 𝑄. Recall that the map 𝑑1 is induced by the left action of 𝑋1 on 𝑄 in EndCorr(C) (𝑋0 ) by 𝑋1 × 𝑋0 ×𝑋0 𝑄 ≃ 𝑋0 × 𝑋0 ×𝑋0 (𝑋1 × 𝑋0 𝑄) → 𝑋0 × 𝑋0 ×𝑋0 𝑄 Likewise, the map 𝑑0 is induced by the right action via 𝑋1 × 𝑋0 ×𝑋0 𝑄 ≃ 𝑋0 × 𝑋0 ×𝑋0 (𝑄 × 𝑋0 𝑋1 ) → 𝑋0 × 𝑋0 ×𝑋0 𝑄. The monad 𝑉 is given by 𝑉 ≃ (𝑎𝑟 )† ◦ (𝜉𝑟 )★ ◦ (𝜉𝑙 )★ ◦ (𝑎 𝑙 ) † . These maps fit into a commutative diagram diagram in C: 𝜉𝑙 𝑋1 × 𝑄 𝜉𝑟 𝑄 𝛿0 𝜁 𝜒 𝑄 × 𝑋0 𝑋1 𝑎𝑟 𝑄 𝑎𝑙 𝑋1 × 𝑋0 𝑄 𝑋1 × 𝑋0 ×𝑋0 𝑄 𝑑1 𝑋0 × 𝑋0 ×𝑋0 𝑄 (13.14) 𝑑0 𝛿0 𝑋0 × 𝑋0 ×𝑋0 𝑄 such that the two upper squares are Cartesian. Then it is enough to show that the natural maps (𝜉𝑟 )★ ◦ (𝜉𝑙 )★ → 𝜒★ ◦ 𝜁 ★, 𝜁 ★ ◦ (𝑎 𝑙 ) † → (𝑑1 ) † ◦ (𝛿0 )★, (𝑎𝑟 )† ◦ 𝜒★ → (𝛿0 )★ ◦ (𝑑0 )† are equivalences, which hold by Assumptions 11.2.1, 2, 3 and 4, respectively, and the fact that 𝜉𝑙 , 𝜉𝑟 , 𝜁, 𝜒 ∈ C𝑠𝑚 and 𝑎 𝑙 , 𝑎𝑟 , 𝑑0 , 𝑑1 ∈ C𝑝𝑟𝑜 𝑝 . □ 101 Now we specialize the above discussions to the case 𝑋1 = 𝑋×𝑌 𝑋 and 𝑄 = 𝑋×𝑌 𝑍×𝑌 𝑍 as in Proposition 13.2.1. In this case, the relative Hochschild complex has a simple interpretation. Consider the fiber product 𝑌 ×𝑌 ×𝑌 𝑍 𝑌 Δ𝑌 𝑍 𝑔=𝑔1 ×𝑔2 𝑌 ×𝑌 in C and consider the map 𝑞 = 𝑓 × id𝑍 : 𝑋 ×𝑌 ×𝑌 𝑍 → 𝑌 ×𝑌 ×𝑌 𝑍. Lemma 13.3.3. There is a canonical equivalence of simplicial objects in 𝐶. HH 𝑋 (𝑋 ×𝑌 𝑋, 𝑋 ×𝑌 𝑍 ×𝑌 𝑋)• ≃ 𝑋• ×𝑌 ×𝑌 𝑍 where the right hand side is the Čech nerve of 𝑞 : 𝑋 ×𝑌 ×𝑌 𝑍 → 𝑌 ×𝑌 ×𝑌 𝑍. Under the identification, the map 𝛿0 from (13.8) is the horizontal map in (13.3). Proof. The construction of the left hand side is natural in 𝑋 and applying it to the identity map 𝑌 → 𝑌 gives the right hand side. Thus, 𝑓 : 𝑋 → 𝑌 induces an augmentation HH 𝑋 (𝑋 ×𝑌 𝑋, 𝑋 ×𝑌 𝑍 ×𝑌 𝑋)• of the corresponding simplicial object. In order to identify this augmented simplicial object with the Čech nerve of 𝑞, can use the characterization (Lurie, 2009, Proposition 6.1.2.11) as it is easy to check that the necessary squares are pullbacks. □ Proof of Proposition 13.2.1. Only fully faithfulness requires a proof. Consider the augmented simplicial category associated to the Čech nerve of 𝑞. As 𝑓 ∈ C𝑝𝑟𝑜 𝑝 so is the map 𝑋 ×𝑌 ×𝑌 𝑌 → 𝑌 ×𝑌 ×𝑌 𝑍. Using Lemma 13.3.3 we identify the Čech nerve of this map with the relative Hochschild complex. By passing to right adjoints and using Corollary 9.0.6 we get a fully faithful functor D (HH 𝑋 (𝑋 ×𝑌 𝑋, 𝑍 ×𝑌 ×𝑌 (𝑋 × 𝑋))• ) → D (𝑍 ×𝑌 ×𝑌 𝑌 ). (13.15) Composition with the fully faithful functor from Proposition 13.3.1 gives the desired functor. The essential image of (13.15) is generated by the image of 𝑞 † so the description of the essential image follows. □ We record the following functorality for later purpose. Let ( 𝑓 : 𝑋 → 𝑌 ) ∈ C𝑣𝑒𝑟𝑡 , with Δ 𝑋 , 𝜋 𝑋 ∈ Cℎ𝑜𝑟𝑖𝑧 and Δ 𝑋/𝑌 ∈ C𝑣𝑒𝑟𝑡 . Let 𝑍 ← 𝐶 → 𝑍 ′ be a morphism in Corr(C/𝑌 ×𝑌 )𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 , i.e. all 𝑍, 𝑍 ′, 𝐶 are equipped with morphisms to 𝑌 × 𝑌 and 𝐶 → 𝑍 and 𝐶 → 𝑍 ′ are (𝑌 × 𝑌 )-morphisms in C. 102 Let 𝑋• be as above and let 𝑄 ′ = 𝑋 ×𝑌 𝑍 ′ ×𝑌 𝑋 and 𝑄 = 𝑋 ×𝑌 𝑍 ×𝑌 𝑋. Then the following diagram is commutative D (𝑄 ′) / Tr(D (𝑋1 ), D (𝑄 ′)) / Trgeo (D (𝑋1 ), D (𝑄 ′)) / D (𝑌 ×𝑌 ×𝑌 𝑍 ′)   / Tr(D (𝑋1 ), D (𝑄))  / Trgeo (D (𝑋1 ), D (𝑄))  / D (𝑌 ×𝑌 ×𝑌 𝑍). D (𝑄) (13.16) Remark 13.3.4. When we take C to be the category of (nice) algebraic stacks over C and the sheaf theory D to be D-modules, one always has Trgeo (D(𝑋 ×𝑌 𝑋), 𝜙) = Tr(D(𝑋 ×𝑌 𝑋), 𝜙) as D(𝑋) ⊗ D(𝑌 )  D(𝑋 × 𝑌 ). Therefore, Corollary 13.2.2 recovers (Ben-Zvi and Nadler, 2009, Theorem 6.6). In loc. cit., instead of directly considering D(HH 𝑋 (𝑋 ×𝑌 𝑋, 𝑋 ×𝑌 𝑋)• ), the authors used the relative bar resolution for the monoidal category D(𝑋 ×𝑌 𝑋) and then used integral transforms to embed each level in the resulting simplicial object of this relative resolution fully faithfully into the corresponding level of D(HH 𝑋 (𝑋 ×𝑌 𝑋, 𝑋 ×𝑌 𝑋)• ). Our method bypasses using the integral transforms, which might fail in other sheaf theoretic content. See Section 13.4 for discussions. 13.4 Comparison between geometric and ordinary traces In practice, we need to compare the geometric trace defined and studied as above with the ordinary traces reviewed in Section 12.1. The easiest situation has been discussed in Remark 13.1.1. On the other hand, the monadicity of the simplicial objects in Lemma 13.3.2 can be used to compare the usual trace and the geometric trace in other situations. Recall notations in (13.6). Proposition 13.4.1. Assume that the sheaf theory D : Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 → LincatΛ satisfies the following condition: for every 𝑋, 𝑌 ∈ C, the exterior tensor product ⊠ : D (𝑋) ⊗ D (𝑌 ) → D (𝑋 × 𝑌 ) is fully faithful and admits a continuous right adjoint ⊠ 𝑅 (see Remark 11.1.1.) In addition, Assumptions 11.2.1 1-6 hold for D. Let 𝑋• , 𝑄 as in the statement of Proposition 13.3.1. Assume that 1. the unit object Λ 𝑋0 ∈ D (𝑋0 ) (for the symmetric monoidal structure of D (𝑋0 ) as in Section 11.1) is compact; 103 2. the image of Λ 𝑋0 under the functor 𝜂★ ◦ 𝑚 † ◦ 𝑢 † belongs to D (𝑋1 ) ⊗ D (𝑋1 ) ⊂ D (𝑋1 × 𝑋1 ). Then D (𝑋1 ) is rigid and the comparison map Tr(D (𝑋1 ), D (𝑄)) → Trgeo (D (𝑋1 ), D (𝑄)) (see (13.2)) is an equivalence. Proof. We first show that D (𝑋1 ) is rigid. We need to check the conditions of Example 12.2.3. The unit 1 𝑋1 is equivalent to 𝑢 † (Λ 𝑋0 ), which is compact as Λ 𝑋0 is compact. The multiplication map is given by the composition 𝑚 † ◦ 𝜂★ ◦ ⊠, with the continuous right adjoint given by ⊠ 𝑅 ◦ 𝜂★ ◦ 𝑚 † . It remains show that that this right adjoint is a D (𝑋1 )-bimodule homomorphism. To see that it is a left D (𝑋1 )module morphism (the case of right D (𝑋1 )-module structure is similar), consider the following diagram D (𝑋1 ) ⊗ D (𝑋1 ) ⊠  D (𝑋1 × 𝑋1 ) 𝑚 † ◦𝜂★  D (𝑋1 ) id⊗(𝜂★◦𝑚 † ) (id×𝜂)★ ◦(id⊗𝑚) † 𝜂★ ◦𝑚 † 𝑅 / D (𝑋1 ) ⊗ D (𝑋1 × 𝑋1 ) id⊗⊠ / D (𝑋1 ) ⊗ D (𝑋1 ) ⊗ D (𝑋1 ) ⊠   / D (𝑋1 × 𝑋1 × 𝑋1 )  D (𝑋1 × 𝑋1 ) ⊗ D (𝑋1 ) (𝑚×id)† ◦(𝜂×id)★ / D (𝑋1 × 𝑋1 ) ⊠𝑅  / D (𝑋1 ) ⊗ D (𝑋1 ). By Assumptions 11.2.1 5-6, the left upper square is commutative. By Lemma 13.3.2, the left lower square is commutative. In other words, the functor 𝜂★ ◦ 𝑚 † is a left D (𝑋1 )-module homomorphism. Then Assumption (2) of the proposition implies that the essential images of the functor ⊠ 𝑅 ◦ 𝜂★ ◦ 𝑚 † belong to D (𝑋1 ) ⊗ D (𝑋1 ) ⊂ D (𝑋1 × 𝑋1 ). Therefore, the outer square of the above diagram is commutative. (However we do not claim the right square is commutative.) Next we show that the comparison map (13.2) is an equivalence. Recall that it is induced by the morphism of simplicial objects HH(D (𝑋1 ), D (𝑄))• = D (𝑋1 ) ⊗• ⊗ D (𝑄) → D (𝑋1• × 𝑄) = D (HH(𝑋1 , 𝑄)• ). 𝑖 to denote The 0-th objects of both simplicial objects are given by D (𝑄). We use 𝑑𝑚 the 𝑖th face map 𝑑𝑖 : D (HH(𝑋1 , 𝑄)𝑚+1 ) → D (HH(𝑋1 , 𝑄)𝑚 ). By Lemma 12.3.1, the co-simplicial object D (𝑋1 ) ⊗• ⊗ D (𝑄) obtained by passing to right adjoints on HH(D (𝑋1 ), D (𝑄))• , satisfies the Beck-Chevalley conditions, and that the resulting monad on D (𝑄) is given by 𝑑0 ◦⊠◦⊠ 𝑅 ◦ (𝑑1 ) 𝑅 . By Lemma 13.3.2, D (HH(𝑋1 , 𝑄)• ) also satisfies the Beck-Chevallay conditions, and that the resulting 104 monad on D (𝑄) is given by 𝑑0 ◦ (𝑑1 ) 𝑅 . It remains to show that the two monads are identified which will imply the proposition. Note that (𝑑1 ) 𝑅 is given by the composition of the bottom functors in the following commutative diagram id⊗((𝑎 𝑙 )† ◦(𝜉𝑙 )★◦⊠) D (𝑋1 3) ⊗ D (𝑋1 ) ⊗ D (𝑄) (⊠ 𝑅 ◦𝜂★◦𝑚 † )⊗id / D (𝑋1 ) ⊗ D (𝑄) ⊠⊗id D (𝑄) 1𝑋1 ⊗id / D (𝑋1 ) ⊗ D (𝑄) (𝜂★◦𝑚 † )⊗id  / D (𝑋1 × 𝑋1 ) ⊗ D (𝑄) (𝜂×id)★◦(𝑚×id) †  / D (𝑋1 × 𝑋1 × 𝑄) ⊠ D (𝑄)  / D (𝑋1 × 𝑄) (𝑢×id)† ◦(𝜋 𝑋0 ×id)★ D (𝑋1 ) ⊗ D (𝑄) ⊠ ⊠ (id×𝑎 𝑙 )† ◦(id×𝜉𝑙 )★  / D (𝑋1 × 𝑄). We remark that the commutativity of the triangle follows from Assumption (2), and the commutativity of the square below the triangle follows from Assumptions 11.2.1 5-6. It follows that ⊠ ◦ ⊠ 𝑅 ◦ (𝑑1 ) 𝑅 = (𝑑1 ) 𝑅 . In particular, the two monads are identified. □ Corollary 13.4.2. Assume that the sheaf theory D is as in ??. Let 𝑋, 𝑌 , 𝑍 be as in Proposition 13.2.1. Assume that Λ 𝑋 is compact and its image under the functor (id × Δ 𝑋 × id)★ ◦ (id × 𝑓 × id) † ◦ (Δ 𝑋/𝑌 )† belongs to D (𝑋 ×𝑌 𝑋) ⊗ D (𝑋 ×𝑌 𝑋), then the canonical map Tr(D (𝑋 ×𝑌 ×𝑋), D (𝑋 ×𝑌 𝑍 ×𝑌 𝑋)) → D (𝑌 ×𝑌 ×𝑌 𝑍) is fully faithful. If in addition, 𝑍 = 𝑊1 × 𝑊2 as in Corollary 13.2.3, then the canonical map D (𝑊1 ×𝑌 𝑋) ⊗D (𝑋×𝑌 𝑋) D (𝑋 ×𝑌 𝑊2 ) → D (𝑊1 ×𝑌 𝑊2 ). is fully faithful. Remark 13.4.3. If the sheaf theory D satisfies ???, i.e. for 𝑓 ∈ C𝑝𝑟𝑜 𝑝 ∩ Cℎ𝑜𝑟𝑖𝑧 , 𝑓 † = 𝑓 ★, and for 𝑔 ∈ C𝑠𝑚 ∩ C𝑣𝑒𝑟𝑡 , 𝑔★ = 𝑔† then under the assumption that 𝑓 ∈ C𝑝𝑟𝑜 𝑝 ∩ Cℎ𝑜𝑟𝑖𝑧 and Δ 𝑋 ∈ C𝑠𝑚 ∩ C𝑣𝑒𝑟𝑡 , we have (id × Δ 𝑋 × id)★ ◦ (id × 𝑓 × id) † ◦ (Δ 𝑋/𝑌 )† (Λ 𝑋 )  (Δ 𝑋1 )† (𝜋 𝑋1 ) † Λ. Proof. The first statement is a combination of Proposition 13.2.1 and ??. As D (𝑋 ×𝑌 𝑊1 ) ⊗ D (𝑊2 ×𝑌 𝑋) → D (𝑋 ×𝑌 𝑊1 × 𝑊2 ×𝑌 𝑋) is a fully faithful D (𝑋1 )-bimodule homomorphism, the second statement follows from the first and Corollary 12.3.2. □ 105 Example 13.4.4. Consider the case 𝑍 = 𝑊 × 𝑋 for some 𝑔 : 𝑊 → 𝑌 . Then we have a split augmented simplicial object HH(𝑋 ×𝑌 𝑋, (𝑊 × 𝑋) ×𝑌 ×𝑌 (𝑋 × 𝑋))• → 𝑊 ×𝑌 𝑋 with the last map given by the action of 𝑋 ×𝑌 𝑋 on 𝑊 ×𝑌 𝑋. In this case, we reduce to the tautological equivalence ∼ D (𝑊 × 𝑋 𝑋) ⊗D (𝑋×𝑌 𝑋) D (𝑋 ×𝑌 𝑋) − → D (𝑊 ×𝑌 𝑋). 13.5 Functoriality of categorical traces in geometric setting Next we discuss functoriality of categorical traces arising from convolution patterns. We start with the following observation. Let 𝑓 : 𝑋 → 𝑌 be as in Section 13.2. Let 𝑍 d 𝑍 ′ be a morphism in Corr(C/𝑌 ×𝑌 )𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 , given as 𝑍 ′ ← 𝐶 → 𝑍. Lemma 13.5.1. Assumptions are as in Corollary 13.4.2 and assume that the sheaf theory D satisfies Assumptions 11.2.1 1-2. Then the following diagram is right adjointable. Tr(D (𝑋 ×𝑌 𝑋), D (𝑋 ×𝑌 𝑍 ×𝑌 𝑋)) / D (𝑌 ×𝑌 ×𝑌 𝑍)   / D (𝑌 ×𝑌 ×𝑌 𝑍 ′). Tr(D (𝑋 ×𝑌 𝑋), D (𝑋 ×𝑌 𝑍 ′ ×𝑌 𝑋)) Proof. As D (𝑋 ×𝑌 𝑋) is rigid, using Lemma 12.3.1 it is enough to show that the following diagram is right adjointable D (𝑋 ×𝑌 𝑍 ×𝑌 𝑋) / D (𝑌 ×𝑌 ×𝑌 𝑍)   / D (𝑌 ×𝑌 ×𝑌 𝑍 ′). D (𝑋 ×𝑌 𝑍 ′ ×𝑌 𝑋) But this follows from our assumption of D (in particular Assumptions 11.2.1 12). □ Next we discuss duality of modules arising from the convolution patterns. Let ( 𝑓 : 𝑋 → 𝑌 , 𝑔 : 𝑍 → 𝑌 × 𝑌 ) and let ( 𝑓 ′ : 𝑋 ′ → 𝑌 ′, 𝑔′ : 𝑍 ′ → 𝑌 ′ × 𝑌 ′) be as in Section 13.2.. Let 𝑊 → 𝑌 × 𝑌 ′ be a morphism. Let 𝑋1 = 𝑋 ×𝑌 𝑋, 𝑄 = 𝑋 ×𝑌 𝑍 ×𝑌 𝑋, 𝑋1′ = 𝑋 ′ ×𝑌 ′ 𝑋 ′, 𝑄 ′ = 𝑋 ′ ×𝑌 ′ 𝑍 ′ ×𝑌 ′ 𝑋 ′, 106 and let 𝑀 = 𝑋 ×𝑌 𝑊 ×𝑌 ′ 𝑋 ′ . We would like to know when D (𝑀) is dualizable as a D (𝑋1 )-D (𝑋1′ )-bimodule, with the dual of given by D (𝑁) where 𝑁 = 𝑋 ′ ×𝑌 ′ 𝑊 ×𝑌 𝑋. We will assume that • 𝑊 → 𝑌 ′ and 𝑊 → 𝑊 ×𝑌 ′ 𝑊 belong to Cℎ𝑜𝑟𝑖𝑧 ; • 𝑊 → 𝑌 and 𝑊 → 𝑊 ×𝑌 𝑊 belongs to C𝑣𝑒𝑟𝑡 . Then we have the morphism 𝑢 geo : 𝑌 ′ d 𝑊 ×𝑌 𝑊 in Corr(C)𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 given by 𝑊 ×𝑌 𝑊 ← 𝑊 → 𝑌 ′ and 𝑒 geo : 𝑊 ×𝑌 ′ 𝑊 d 𝑌 given by 𝑌 ← 𝑊 → 𝑊 ×𝑌 ′ 𝑊. They induce D (𝑋1′ ) D (𝑁) ⊗D (𝑋1 ) D (𝑀)  (13.17) D (id×𝑢 geo ×id) / D (𝑋 ′ ×𝑌 ′ 𝑊 ×𝑌 𝑊 ×𝑌 ′ 𝑋 ′). and D (𝑀) ⊗D (𝑋1′ ) D (𝑁) / D (𝑋 ×𝑌 𝑊 ×𝑌 ′ 𝑊 ×𝑌 𝑋) 𝑒 *  (13.18) D (id×𝑒 geo ×id) D (𝑋1 ). Here 𝑒 is defined to be the composition. Lemma 13.5.2. If the vertical morphism in (13.17) factors through a D (𝑋1′ )bimodule morphism 𝑢 : D (𝑋1′ ) → D (𝑁) ⊗D (𝑋1 ) D (𝑀) (e.g. if D (𝑁) ⊗D (𝑋1 ) D (𝑀) → D (𝑋 ′ ×𝑌 ′ 𝑊 ×𝑌 𝑊 ×𝑌 ′ 𝑋 ′) is an equivalence), then 𝑢 and 𝑒 from (13.18) give the duality datum of D (𝑀) as a D (𝑋1 )-D (𝑋1′ )-module. Proof. Write 𝑅 = 𝑋 ′ ×𝑌 ′ 𝑊 ×𝑌 𝑊 ×𝑌 ′ 𝑋 ′ and 𝑆 = 𝑋 ×𝑌 𝑊 ×𝑌 ′ 𝑊 ×𝑌 𝑋 for simplicity. 107 Note that (using (13.16)) we have the following commutative diagram D (𝑋1′ ) ⊗D (𝑋1′ ) D (𝑁) D (𝑋1′ ) ⊗D (𝑋1′ ) D (𝑁)  / D (𝑁) = D (𝑋 ′ ×𝑌 ′ 𝑌 ′ ×𝑌 ′ 𝑊 ×𝑌 𝑋)  D (𝑅) ⊗D (𝑋1′ ) D (𝑁) 𝑢⊗id D (id×𝑢 geo ×id×id) 3  +  D (𝑋 ′ ×𝑌 ′3 𝑊 ×𝑌 𝑊 ×𝑌 ′ 𝑊 ×𝑌 𝑋) D (𝑁) ⊗D (𝑋1 ) D (𝑀) ⊗D (𝑋1′ ) D (𝑁) + D (𝑁) ⊗D (𝑋1 ) D (𝑆) id⊗𝑒   D (𝑁) ⊗D (𝑋) D (𝑋) D (𝑁) ⊗D (𝑋1 ) D (𝑋1 ) D (id×id×𝑒 geo ×id)   / D (𝑁) = D (𝑋 ′ ×𝑌 ′ 𝑌 ′ ×𝑌 ′ 𝑊 ×𝑌 𝑋) The composition of functors in the right column is isomorphic to the identity functor by the base change isomorphism (11.2). It follows that (12.11) in the current setting holds. The same reasoning implies that (12.10) in the current setting also holds. □ In practice, the assumption in Lemma 13.5.2 may not hold. But under some certain technical assumptions, we can still understand the duality datum. Lemma 13.5.3. Suppose the sheaf theory D satisfies assumptions as in Corollary 13.4.2, and let 𝑓 : 𝑋 → 𝑌 and 𝑓 ′ : 𝑋 ′ → 𝑌 ′ be as in Corollary 13.4.2 (so in particular D (𝑋1 ) and D (𝑋1′ ) are rigid). Suppose D (𝑀) ⊗ D (𝑇) → D (𝑀 × 𝑇) is an equivalence for every 𝑇 ∈ C. Then 𝑢 : D (𝑋1′ ) → D (𝑋 ′ ×𝑌 ′ 𝑊 ×𝑌 𝑊 ×𝑌 ′ 𝑋 ′) → − D (𝑁) ⊗D (𝑋1 ) D (𝑀), where the last functor is the right adjoint of the vertical morphism in (13.17) and 𝑒 from (13.18) form a duality datum. Proof. As in the proof of Lemma 13.5.2, it is enough to establish the following commutative diagram D (𝑋1′ ) ⊗D (𝑋1′ ) D (𝑁) D (𝑋1′ ) ⊗D (𝑋1′ ) D (𝑁) / D (𝑁)   D (𝑅) ⊗D (𝑋1′ ) D (𝑁) 𝑢⊗id (∗∗)  s +  D (𝑋 ′ ×𝑌 ′ 𝑊 ×𝑌 𝑊 ×𝑌 ′ 𝑊 ×𝑌 𝑋) D (𝑁) ⊗D (𝑋1 ) D (𝑀) ⊗D (𝑋1′ ) D (𝑁) (∗∗) + s D (𝑁) ⊗D (𝑋1 ) D (𝑆) id⊗𝑒  D (𝑁) ⊗D (𝑋) D (𝑋)  D (𝑁) ⊗D (𝑋1 ) D (𝑋1 )   / D (𝑁), 108 where the arrows labelled by (∗∗) are right adjoint of the corresponding arrows in the diagram from the proof of Lemma 13.5.2. Only the commutativity of the middle parallelogram and the lower right trapzoid requires justification. For the middle parallelogram, first consider the commutative diagram D (𝑁 × 𝑀 × 𝑁) / D (𝑅 × 𝑁)   / D (𝑋 ′ ×𝑌 ′ 𝑊 ×𝑌 𝑊 ×𝑌 ′ 𝑊 ×𝑌 𝑋). D (𝑁 × 𝑆) with horizontal morphisms are induced by the correspondence 𝑌 ← 𝑋 → 𝑋 × 𝑋 and vertical morphisms induced by 𝑌 ′ ← 𝑋 ′ → 𝑋 ′ × 𝑋 ′. As 𝑓 , 𝑓 ′ ∈ C𝑝𝑟𝑜 𝑝 and Δ 𝑋 , Δ 𝑋 ′ ∈ C𝑠𝑚 , the above diagram is right adjointable by the same proof as in Lemma 13.3.2. Under our assumption that D (𝑀) ⊗ D (𝑇) → D (𝑀 ×𝑇) is an equivalence for every 𝑇 ∈ C, we may replace D (𝑁 × 𝑀 × 𝑁) by D (𝑁) ⊗ D (𝑀) ⊗ D (𝑁), D (𝑅 × 𝑁) by D (𝑅) ⊗ D (𝑁) and D (𝑁 × 𝑆) by D (𝑁) ⊗ D (𝑆). Then as D (𝑋1′ ) and D (𝑋1 ) are rigid, using Lemma 12.3.1, we obtain the commutativity of the parallelogram. Similarly, the lower right trapzoid is commutative by Lemma 13.5.1 and that geo □ D (𝑁) ⊗D (𝑋1 ) D (𝑆)  D (𝑁) ⊗D (𝑋 ) D (𝑆) := Trgeo (D (𝑋1 ), D (𝑁 × 𝑆)). 1 Now suppose we are given a D (𝑋1 )-D (𝑋1′ )-bimodule homomorphism 𝛼 : D (𝑀) ⊗D (𝑋1′ ) D (𝑄 ′) → D (𝑄) ⊗D (𝑋1 ) D (𝑀). Then as explained above, under certain dualizability assumption of D (𝑀), there is a functor Tr(D (𝑀), 𝛼) : Tr(D (𝑋1′ ), D (𝑄 ′)) → Tr(D (𝑋1 ), D (𝑄)). On the other hand, suppose we are given a correspondence 𝛼geo : 𝑊 ×𝑌 ′ 𝑍 ′ d 𝑍 ×𝑌 𝑊 in Corr(C/𝑌 ×𝑌 ′ )𝑣𝑒𝑟𝑡;ℎ𝑜𝑟𝑖𝑧 . One can form the correspondence 𝐶 (𝑊, 𝛼geo ) : 𝑌 ′ ×𝑌 ′ ×𝑌 ′ 𝑍 ′ d 𝑌 ×𝑌 ×𝑌 𝑍 given by the composition 𝑢 geo ×id 𝑌 ′ ×𝑌 ′ ×𝑌 ′ 𝑍 ′ d (𝑊 ×𝑌 𝑊)𝑌 ′ ×𝑌 ′ 𝑍 ′  𝑌 ×𝑌 ×𝑌 (𝑊 ×𝑌 ′ 𝑍 ′ ×𝑌 ′ 𝑊) id×𝛼geo ×id d 𝑌 ×𝑌 ×𝑌 (𝑍 ×𝑌 𝑊 ×𝑌 ′ 𝑊) id×id×𝑒 geo d 𝑌 ×𝑌 ×𝑌 𝑍. (13.19) 109 The sheaf theory D then induces a functor D (𝐶 (𝑊, 𝛼geo )) : D (𝑌 ′ ×𝑌 ′ ×𝑌 ′ 𝑍 ′) → D (𝑌 ×𝑌 ×𝑌 𝑍). We would like to relate Tr(D (𝑀), 𝛼) with the above functor under certain assumptions. (I) We assume that the following diagram is commutative Assumptions 13.5.4. D (𝑀) ⊗D (𝑋1′ ) D (𝑄 ′) / D (𝑋 ×𝑌 𝑊 ×𝑌 ′ 𝑍 ′ ×𝑌 ′ 𝑋 ′) 𝛼  D (𝑄) ⊗D (𝑋1 ) D (𝑀)  (13.20) D (id×𝛼geo ×id) / D (𝑋 ×𝑌 𝑍 ×𝑌 𝑊 ×𝑌 ′ 𝑋 ′). (II) We assume that the following diagram is commutative D (𝑀) ⊗D (𝑋1′ ) D (𝑄 ′) o D (𝑋 ×𝑌 𝑊 ×𝑌 ′ 𝑍 ′ ×𝑌 ′ 𝑋 ′) 𝛼  D (𝑄) ⊗D (𝑋1 ) D (𝑀) o  (13.21) D (id×𝛼geo ×id) D (𝑋 ×𝑌 𝑍 ×𝑌 𝑊 ×𝑌 ′ 𝑋 ′). where the horizontal arrows are right adjoint of the natural ones. Remark 13.5.5. Note that Assumptions 13.5.4 holds in the case 𝑍 ′ = 𝑌 ′ with 𝑔1′ = 𝜙𝑌 ′ : 𝑌 ′ → 𝑌 , 𝑍 = 𝑌 with 𝑔2 = 𝜙𝑌 : 𝑌 → 𝑌 and there is 𝜙𝑊 : 𝑊 → 𝑊 compatible with 𝜙𝑌 ′ and 𝜙𝑌 . Proposition 13.5.6. Under the assumption in Lemma 13.5.2 and Assumptions 13.5.4 I, then the following diagram is commutative Tr(D (𝑋1′ ), D (𝑄 ′))  D (𝑌 ′ ×𝑌 ′ ×𝑌 ′ 𝑍 ′) Tr(D (𝑀),𝛼) / Tr(D (𝑋1 ), D (𝑄)) D (𝐶 (𝑊,𝛼geo ))  / D (𝑌 ×𝑌 ×𝑌 𝑍). Under the assumption in Lemma 13.5.3 and Assumptions 13.5.4 II, the following diagram is commutative Tr(D (𝑋1′ ), D (𝑄 ′))  D (𝑌 ′ ×𝑌 ′ ×𝑌 ′ 𝑍 ′) Tr(D (𝑀),𝛼) / Tr(D (𝑋1 ), D (𝑄)) O D (𝐶 (𝑊,𝛼geo )) / D (𝑌 ×𝑌 ×𝑌 𝑍) 110 Proof. The first case follows from the following commutative diagram / D (𝑌 ′ ×𝑌 ′ ×𝑌 ′ 𝑍 ′) D (𝑋1′ ) ⊗D (𝑋1′ )⊗D (𝑋1′ ) rev D (𝑄 ′) D (𝑢 geo ×id) 𝑢⊗1   / D ((𝑊 ×𝑌 𝑊) ×𝑌 ′ ×𝑌 ′ 𝑍 ′) (D (𝑁) ⊗D (𝑋1 ) D (𝑀)) ⊗D (𝑋1′ )⊗D (𝑋1′ ) rev D (𝑄 ′)   D (𝑋1 ) ⊗D (𝑋1 )⊗D (𝑋1 ) rev (D (𝑀) ⊗D (𝑋1′ ) D (𝑄 ′) ⊗D (𝑋1′ ) D (𝑁)) 1⊗𝛼⊗1  D (𝑋1 ) ⊗D (𝑋1 )⊗D (𝑋1 ) rev (D (𝑄) ⊗D (𝑋1 ) D (𝑀) ⊗D (𝑋1′ ) D (𝑁)) 1⊗1⊗𝑒   / D (𝑌 ×𝑌 ×𝑌 (𝑊 ×𝑌 ′ 𝑍 ′ ×𝑌 ′ 𝑊)) D (id×𝛼geo ×id)  / D (𝑌 ×𝑌 ×𝑌 (𝑍 ×𝑌 𝑊 ×𝑌 ′ 𝑊))  D)id×id×𝑒 geo )  / D (𝑌 ×𝑌 ×𝑌 𝑍) D (𝑋1 ) ⊗D (𝑋1 )⊗D (𝑋1 ) rev D (𝑄) (13.22) The second case follows from a similar diagram / D (𝑌 ′ ×𝑌 ′ ×𝑌 ′ 𝑍 ′) D (𝑋1′ ) ⊗D (𝑋1′ )⊗D (𝑋1′ ) rev D (𝑄 ′) 𝑢⊗1   (D (𝑁) ⊗D (𝑋1 ) D (𝑀)) ⊗D (𝑋1′ )⊗D (𝑋1′ ) rev D (𝑄 ′) o  D ((𝑊 ×𝑌 𝑊) ×𝑌 ′ ×𝑌 ′ 𝑍 ′)  D (𝑋1 ) ⊗D (𝑋1 )⊗D (𝑋1 ) rev (D (𝑀) ⊗D (𝑋1′ ) D (𝑄 ′) ⊗D (𝑋1′ ) D (𝑁)) o 1⊗𝛼⊗1  D (𝑋1 ) ⊗D (𝑋1 )⊗D (𝑋1 ) rev (D (𝑄) ⊗D (𝑋1 ) D (𝑀) ⊗D (𝑋1′ ) D (𝑁)) o 1⊗1⊗𝑒 D (𝑢 geo ×id)   D (𝑌 ×𝑌 ×𝑌 (𝑊 ×𝑌 ′ 𝑍 ′ ×𝑌 ′ 𝑊)) D (id×𝛼geo ×id)  D (𝑌 ×𝑌 ×𝑌 (𝑍 ×𝑌 𝑊 ×𝑌 ′ 𝑊))   D (𝑋1 ) ⊗D (𝑋1 )⊗D (𝑋1 ) rev D (𝑄) o D (id×id×𝑒 geo ) D (𝑌 ×𝑌 ×𝑌 𝑍), (13.23) where the horizontal left arrows are obtained by the corresponding horizontal right arrows in (13.22) by passing to the right adjoint. We need to justify the commutativity of this diagram. First we have the commutativity of the following diagram D (𝑋1′ ) ⊗D (𝑋1′ )⊗D (𝑋1′ ) rev D (𝑄 ′) / D (𝑌 ′ ×𝑌 ′ ×𝑌 ′ 𝑍 ′)   𝑢⊗1 (D (𝑁) ⊗D (𝑋1 ) D (𝑀)) ⊗ r D (𝑋1′ )⊗D (𝑋1′ ) rev D (𝑄 ′) o D (𝑅) ⊗ D (𝑋1′ )⊗D (𝑋1′ ) rev D (𝑄 ′) o D (𝑢 geo ×id) D ((𝑊 ×𝑌 𝑊) ×𝑌 ′ ×𝑌 ′ 𝑍 ′) Indeed, the left triangle is commutative as we are in the case as in Lemma 13.5.3, and the right square is commutative as the natural functor D (𝑅) ⊗D (𝑋1′ )⊗D (𝑋1′ ) rev D (𝑄 ′) → D ((𝑊 ×𝑌 𝑊) ×𝑌 ′ ×𝑌 ′ 𝑍 ′) is fully faithful by Corollary 13.4.2. This justifies the commutativity of the top square in (13.23). For the commutativity of the third square in (13.23), by our assuption it is enough 111 to show that D (𝑋1 ) ⊗D (𝑋1 )⊗D (𝑋1 ) rev (D (𝑋 ×𝑌 𝑊 ×𝑌 ′ 𝑍 ′ ×𝑌 ′ 𝑋 ′) ⊗D (𝑋1′ ) D (𝑁)) o  D (𝑋1 ) ⊗D (𝑋1 )⊗D (𝑋1 ) rev (D (𝑋 ×𝑌 𝑍 ×𝑌 𝑊 ×𝑌 ′ 𝑋 ′) ⊗D (𝑋1′ ) D (𝑁)) o D (𝑌 ×𝑌 ×𝑌 (𝑊 ×𝑌 ′ 𝑍 ′ ×𝑌 ′ 𝑊))  D (𝑌 ×𝑌 ×𝑌 (𝑍 ×𝑌 𝑊 ×𝑌 ′ 𝑊)) is commutative. Under the assumption that D (𝑀) ⊗ D (𝑇) → D (𝑀 × 𝑇) is an equivalence for any 𝑇 ∈ C, we can use Lemma 13.5.1 twice to conclude. Similar argument also shows that the last square in (13.23) is commutative. □ Example 13.5.7. Take 𝑌 ′ = pt and 𝑊 = 𝑋. Then the naive class identifies with the element L 𝜙 ( 𝑓 )† (𝑒 𝑋 ) of D (L 𝜙 (𝑌 )) considered as a functor from ModΛ . Example 13.5.8. If 𝑌 → pt is in Cℎ𝑜𝑟𝑖𝑧 we can take 𝑍 = 𝑌 and 𝑔 = id𝑌 . Then (Δ𝑌 /𝑌 )† = id. Then the geometric class in D (L 𝜙 (𝑌 )) is equivalent to PTrgeo (𝑒 L 𝜙 (𝑌 ) ). 112 BIBLIOGRAPHY Beauville, Arnaud and Yves Laszlo (1995). “Un lemme de descente”. In: Comptes rendus de l’Académie des sciences. Série 1, Mathématique 320.3, pp. 335–340. Ben-Zvi, David, John Francis, and David Nadler (2010). “Integral transforms and Drinfeld centers in derived algebraic geometry”. In: J. Amer. Math. Soc. 23.4, pp. 909–966. issn: 0894-0347. doi: 10.1090/S0894-0347-10-00669-7. Ben-Zvi, David and David Nadler (2009). “The Character Theory of a Complex Group”. In: arXiv: 0904.1247. Ben-Zvi, David, David Nadler, and Anatoly Preygel (2017). “A spectral incarnation of affine character sheaves”. In: Compositio Mathematica 153.9, pp. 1908–1944. Bhatt, Bhargav and Jacob Lurie (2019). “A Riemann-Hilbert correspondence in positive characteristic”. In: Camb. J. Math. 7.1-2, pp. 71–217. issn: 2168-0930. doi: 10.4310/CJM.2019.v7.n1.a3. Bhatt, Bhargav and Akhil Mathew (2021). “The arc-topology”. In: Duke Math. J. 170.9, pp. 1899–1988. issn: 0012-7094. doi: 10.1215/00127094-2020-0088. Bhatt, Bhargav and Peter Scholze (2015). “The pro-étale topology for schemes”. In: Astérisque 369, pp. 99–201. issn: 0303-1179. Bourbaki, N. (2012). Éléments de mathématique. Algèbre. Chapitre 8. Modules et anneaux semi-simples. Second revised edition of the 1958 edition [MR0098114]. Springer, Berlin, pp. x+489. isbn: 978-3-540-35315-7; 978-3-540-35316-4. doi: 10.1007/978-3-540-35316-4. Bouthier, Alexis, David Kazhdan, and Yakov Varshavsky (2020). Perverse sheaves on infinite-dimensional stacks, and affine Springer theory. eprint: arXiv:2003. 01428. Curtis, Charles W. and Irving Reiner (2006). Representation theory of finite groups and associative algebras. Reprint of the 1962 original. AMS Chelsea Publishing, Providence, RI, pp. xiv+689. isbn: 0-8218-4066-5. doi: 10.1090/chel/356. Deligne, Pierre (1980). “La conjecture de Weil. II”. In: Inst. Hautes Études Sci. Publ. Math. 52, pp. 137–252. issn: 0073-8301. Drinfeld, V. G. (1980). “Langlands’ conjecture for GL(2) over functional fields”. In: Proceedings of the International Congress of Mathematicians (Helsinki, 1978). Acad. Sci. Fennica, Helsinki, pp. 565–574. – (1987). “Moduli varieties of 𝐹-sheaves”. In: Funktsional. Anal. i Prilozhen. 21.2, pp. 23–41. Drinfeld, Vladimir Gershonovich (1980). “Langlands’ conjecture for GL2 over functional fields”. In: Proceedings of the International Congress of Mathematicians (Helsinki, 1978). Vol. 2, pp. 565–574. 113 Gaitsgory, D. et al. (2019). “A toy model for the Drinfeld-Lafforgue shtuka construction”. In: arXiv: 1908.05420. Gaitsgory, Dennis (2012). “Notes on geometric langlands: Generalities on dg categories”. In: Preprint. – (2016). “From geometric to function-theoretic Langlands (or how to invent shtukas)”. In: preprint. arXiv:1606.09608. Gaitsgory, Dennis, David Kazhdan, et al. (2022). “A toy model for the Drinfeld– Lafforgue shtuka construction”. In: Indagationes Mathematicae 33.1, pp. 39–189. eprint: 1908.05420. Gaitsgory, Dennis and Jacob Lurie (2019). Weil’s conjecture for function fields. Vol. 1. Vol. 199. Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, pp. viii+311. isbn: 978-0-691-18214-8; 978-0-691-18213-1. Gaitsgory, Dennis and Nick Rozenblyum (2017a). A study in derived algebraic geometry. Vol. 1. American Mathematical Soc. – (2017b). A study in derived algebraic geometry. Vol. I. Correspondences and duality. Vol. 221. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, xl+533pp. isbn: 978-1-4704-3569-1. Geisser, Thomas (2004). “Weil-étale cohomology over finite fields”. In: Math. Ann. 330.4, pp. 665–692. issn: 0025-5831. doi: 10.1007/s00208-004-0564-8. Gepner, D, R Haugseng, and Th Nikolaus (n.d.). “Lax colimits and free fibrations in ∞-categories (2015)”. In: Preprint. arXiv 1501 (). Heinloth, Jochen (2018). “Langlands parameterization over function fields following V. Lafforgue”. In: Acta Math. Vietnam. 43.1, pp. 45–66. issn: 0251-4184. Hemo, Tamir, Timo Richarz, and Jakob Scholbach (2021). Constructible sheaves on schemes. Kedlaya, Kiran S. (2019). Sheaves, Stacks, and Shtukas. Vol. 242. in Perfectoid spaces: Mathematical Surveys and Monographs. Lectures from the 2017 Arizona Winter School, held in Tucson, AZ, March 11–17, Edited and with a preface by Bryden Cais, With an introduction by Peter Scholze. American Mathematical Society, Providence, RI, pp. xii+297. isbn: 978-1-4704-5015-1. doi: 10.1090/ surv/242. Kneser, Martin (1966). “Strong approximation”. In: Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965). Vol. 187, p. 196. Lafforgue, Laurent (1997). “Chtoucas de Drinfeld et conjecture de RamanujanPetersson”. In: Astérisque 243, pp. ii+329. issn: 0303-1179. – (2002). “Chtoucas de Drinfeld et correspondance de Langlands”. In: Invent. Math. 147.1, pp. 1–241. issn: 0020-9910. 114 Lafforgue, Vincent (2018). “Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale”. In: J. Amer. Math. Soc. 31.3, pp. 719–891. issn: 08940347. Lafforgue, Vincent and Xinwen Zhu (2019). Décomposition au-dessus des paramètres de Langlands elliptiques. arXiv: 1811.07976 [math.AG]. Langlands, Robert P (1970). “Problems in the theory of automorphic forms”. In: Lecture Notes in Math 170, pp. 18–86. Lau, Eike Sören (2004). On generalised 𝐷-shtukas. Vol. 369. Bonner Mathematische Schriften [Bonn Mathematical Publications]. Dissertation, Rheinische FriedrichWilhelms-Universität Bonn, Bonn, 2004. Universität Bonn, Mathematisches Institut, Bonn, pp. xii+110. Lichtenbaum, S. (2005). “The Weil-étale topology on schemes over finite fields”. In: Compos. Math. 141.3, pp. 689–702. issn: 0010-437X. doi: 10 . 1112 / S0010437X04001150. Liu, Yifeng and Weizhe Zheng (2012). “Enhanced six operations and base change theorem for higher Artin stacks”. In: arXiv: 1211.5948. – (2014). “Enhanced adic formalism and perverse t-structures for higher Artin stacks”. In: arXiv: 1404.1128. Lu, Qing and Weizhe Zheng (2020). Categorical traces and a relative LefschetzVerdier formula. arXiv: 2005.08522. Lurie, Jacob (2009). Higher topos theory. Vol. 170. Annals of Mathematics Studies. Princeton, NJ: Princeton University Press, pp. xviii+925. isbn: 978-0-691-140490. – (2017). Higher Algebra. url: http://www.math.harvard.edu/~lurie/. – (2018). Spectral Algebraic Geometry. url: http://www.math.harvard.edu/ ~lurie/. Lusztig, George (1995). “ Classification of unipotent representations of simple p -adic groups ”. In: International Mathematics Research Notices 1995. Revêtements étales et groupe fondamental (SGA 1) (2003). Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3. Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61], Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]. Paris: Société Mathématique de France, pp. xviii+327. isbn: 2-85629-141-4. Scholze, Peter and Jared Weinstein (2020). Berkeley lectures on 𝑝-adic geometry. English. Vol. 207. Ann. Math. Stud. Princeton, NJ: Princeton University Press. isbn: 978-0-691-20209-9; 978-0-691-20208-2; 978-0-691-20215-0. doi: 10.1515/9780691202150. 115 Serre, Jean-Pierre (1964). “Groupes analytiques 𝑝-adiques”. In: Séminaire Bourbaki, 1963/64, Fasc. 2, Exposé 270. Secrétariat mathématique, Paris, p. 10. Stacks Project Authors, The (n.d.). Stacks Project. https : / / stacks . math . columbia.edu. Torii, Takeshi (2022). A perfect pairing for monoidal adjunctions. eprint: arXiv: 2202.02493. Varshavsky, Yakov (2004). “Moduli spaces of principal 𝐹-bundles.” English. In: Sel. Math., New Ser. 10.1, pp. 131–166. issn: 1022-1824; 1420-9020/e. Xiao, Liang and Xinwen Zhu (2017). “Cycles on Shimura varieties via geometric Satake”. In: arXiv:1707.05700. Xue, Cong (2020a). “Cuspidal cohomology of stacks of shtukas”. In: Compos. Math. 156.6, pp. 1079–1151. issn: 0010-437X. doi: 10.1112/s0010437x20007058. – (2020b). “Finiteness of cohomology groups of stacks of shtukas as modules over Hecke algebras, and applications”. In: Épijournal Géom. Algébrique 4, Art. 6, 42 pp.–41. – (2020c). Smoothness of cohomology sheaves of stacks of shtukas. arXiv: 2012. 12833 [math.AG]. Zhu, Xinwen (2016a). “An introduction to affine Grassmannians and the geometric Satake equivalence”. In: arXiv:1603.05593. – (2016b). “Geometric Satake, categorical traces, and arithmetic of Shimura varieties”. In: Current Developments in Mathematics, pp. 145–206. – (2021). Coherent sheaves on the stack of Langlands parameters. arXiv: 2008. 02998 [math.AG].