Two Categorifications of the Local Langlands Correspondence for the Torus Citation Fu, Ruide (2025) Two Categorifications of the Local Langlands Correspondence for the Torus. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/3fcs-9c06. https://resolver.caltech.edu/CaltechTHESIS:06022025-080334841 Abstract The stack of local Langlands parameters is a Picard stack when the relevant reductive group is a torus. We explicitly determine its Picard dual and show that the Fourier-Mukai transform gives rise to the integral categorical local Langlands correspondence for the torus. This is the categorification of the local Langlands correspondence and answers a conjecture of X. Zhu. Moreover, we establish a geometric version of this correspondence. This second categorification relates to the previous correspondence in the sense that taking the categorical trace construction allows one to reproduce the previous result. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Mathematics, Representation Theory, Number Theory Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Zhu, Xinwen Thesis Committee: Flach, Matthias Mantovan, Elena (chair) Fu, Yu Zhu, Xinwen Defense Date: 14 May 2025 Record Number: CaltechTHESIS:06022025-080334841 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06022025-080334841 DOI: 10.7907/3fcs-9c06 ORCID: Author ORCID Fu, Ruide 0009-0005-8025-1110 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17359 Collection: CaltechTHESIS Deposited By: Frid Fu Deposited On: 05 Jun 2025 21:08 Last Modified: 14 Nov 2025 21:10 Thesis Files PDF - Final Version See Usage Policy. 432kB Repository Staff Only: item control page

Two Categorifications of the Local Langlands Correspondence for the Torus Thesis by Ruide Fu In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2025 Defended May 14th 2025 ii © 2025 Ruide Fu ORCID: 0009-0005-8025-1110 All rights reserved iii ACKNOWLEDGEMENTS I am in debt to my advisor Zhu, Xinwen for his generous guidance and academic support. I am especially grateful for his financial assistance, without which my visit to Stanford would not have been possible, and for his understanding and encouragement during times of difficulty. I want to thank Tamir Hemo, Justin Campbell and Longke Tang for stimulating discussions and helpful advice contributing directly or indirectly to my thesis. I want to thank Tamir Hemo, Justin Campbell, Aaron Mazel-Gee for organizing seminars which introduced me to the world of goemetric representation theory. I would like to thank my friends and family. I thank my parents for their constant help and support behind every decision that I made during the years, and showed me what unconditioned love is. I want to thank my cousin Mengyun Fu, and my uncle’s famliy as a whole, whose love and care have meant a great deal to me. I want to thank Li, Heyun and Sean Bai to be my best friends. We shared memories of so many restaurant adventures, of so many cooking trial and errors, of so many national parks and hikes, of numerous visits to the Hungtington library, and of the scent of roses and the singing of birds. They were there for me at my most difficult times. I wouldn’t finish my degree without them. I love them deeply. Cheers to Alexander Froebel and Michael Wolman, for the harmony of heart and mind, echoing over drinks late into the night. I thank Forte Shinko for sharing his passion for music, and for carring me through all the sight reading which was so difficult at the beginning. I want to thank Yu, Jize, Yi, Lingfei, and Yang, Liyang, for their encouragement and companionship. I thank Leo Wu for making a big difference in my life with his constant witty comments and his philosophy of “win by presence”. I thank Ashemeet Singh and Jagannadh Kumar for the great tennis games and an unforgettable summer. I want to thank Qiao, Zhuoran and Zhang, Yuanjin for being great roommates and spent the few unforgettable months with me at the beginning of the Covid crisis. I want to thank Lin, Junxiong for the memorable meals and TV shows we shared in the bay area. I want to thank Shi, Zheng for letting me stay in his beautiful house in times of need. I want to thank Li, Zhen for being my big brother and for having been generously helping me with my career. I want to thank Sameera Vemulapalli and Chen, Xiangning for being great climbing buddies. iv I hold all of my friends very dear to my heart. I would like to thank many Caltech members who make me feel truly loved and cared for. I want to thank Michelle Vine, our “mom” at the Linde hall, who provides us with everything we ever need for teaching and doing maths. She takes real interest in our progress and in our lives, and shares every joy with us no less than if we were her kids. She even manages two math department’s guest apartments, where I had the privilege to stay when I was staying at Stanford and visited Caltech. I completely felt at home with shampoo, body wash and toilet rolls all ready in the bathroom and with a full pantry and a fully functional kitchen all at my disposal. I want to thank our choir director, Nancy Sulahian, for creating such an open and welcoming space where people of all ages and backgrounds come together to make music. She radiates infectious energy and happiness when she plucks streams and rings thunders, all with her bare hands. She is also incredibly supportive and patient when I brought my first chorus composition to her. She sat down with me and examined every note and every word. She believes in music and how music makes us human. She taught us faith in music, and above all, faith in ourselves. I want to thank Divina Bautista at our clinic. She explains everything rigorously and clearly, and even prints out handouts on the spot for us to “study” at home. Her presence is the chicken soup that soothes away my stress and discomfort. I just want to give my biggest hug to them. I want to thank our International Student Program (ISP), for navigating us through years filled with uncertainty. I want to give special thanks to Laura Flower Kim, the superhero who sent me the I-20 with travel signatures within hours on the weekend, right before I board the plane to the US. I want to thank Gretchen Lantz, who helped me sort out the bureaucracy at Stanford. I want to thank Caltech alumnus and trustee Ronald Linde (MS ’62, PhD ’64) and his wife, Maxine Linde, though we have never met in person, for their generous donation which made the renovation of the Linde Hall possible. I tremendously appreciate the modern interior design that facilitates collaboration and innovation. I love the printing rooms with any stationary men or women alike could ever dream of. I want to thank Michelle again, for the facility would not be in shape without her constant maintenance. v ABSTRACT The stack of local Langlands parameters is a Picard stack when the relevant reductive group is a torus. We explicitly determine its Picard dual and show that the Fourier-Mukai transform gives rise to the integral categorical local Langlands correspondence for the torus. This is the categorification of the local Langlands correspondence and answers a conjecture of X. Zhu. Moreover, we establish a geometric version of this correspondence. This second categorification relates to the previous correspondence in the sense that taking the categorical trace construction allows one to reproduce the previous result. vi TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter II: The First Categorification . . . . . . . . . . . . . . . . . . . . . . 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Quasi-Coherent Sheaves on Tor𝑇,iso𝐹 . . . . . . . . . . . . . . . . . 2.3 A short exact sequence . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Categorical LLC for the torus . . . . . . . . . . . . . . . . . . . . . Chapter III: The Second Categorification . . . . . . . . . . . . . . . . . . . . 3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Representation Stack of Inertia . . . . . . . . . . . . . . . . . . . . 3.3 Review of the Categorical Trace . . . . . . . . . . . . . . . . . . . . 3.4 Geometric LLC for the torus . . . . . . . . . . . . . . . . . . . . . . Appendix A: A Lemma in Homological Algebra . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii v vi 1 4 4 7 9 14 23 27 27 29 32 34 40 46 1 Chapter 1 INTRODUCTION Let 𝐺 be a reductive group over a non-archimedean local field 𝐹. X. Zhu (2021) provided an in-depth study of the stack of local and global Langlands parameters Loc𝑐 𝐺,𝐹 defined over Z[1/𝑝], and proposed a categorical arithmetic local Langlands correspondence (LLC). The automorphic side of this Langlands correspondence is given by the category of ℓ-adic sheaves on the stack of isocrystals 𝔅(𝐺) = 𝐿𝐺/Ad𝜎 𝐿𝐺, where 𝜎 is the Frobenius on the residue field 𝜅 𝐹 of 𝐹. A central part of the conjecture is as follows. Conjecture 1.0.1. (X. Zhu, 2021, Conjecture 4.6.4.) Let (𝐺, 𝐵, 𝑇, 𝑒) be a pinned quasi-split reductive group over 𝐹 and Λ = Zℓ , Qℓ or Fℓ . Then there is a natural equivalence of stable ∞-categories L𝐺 : Shv(𝔅(𝐺), Λ) → IndCohN̂𝑐 𝐺 (Loc𝑐 𝐺,𝐹 ) sending the Whittaker sheaf to the structural sheaf OLoc𝑐 𝐺,𝐹 . The category IndCohN̂𝑐 𝐺 (Loc𝑐 𝐺 ) ⊂ IndCoh(Loc𝑐 𝐺 ) is a subcategory of ind-coherent sheaves with certain singular support condition. In a more recent development, Zhu established the tame categorical local Langlands correspondence with Qℓ -coefficient and the unipotent correspondence with Fℓ -coefficient (Xinwen Zhu, 2025). 1.0.2. The First Categorification. In this paper, we verify a more general form of the conjecture that allows Z-coefficient for an arbitrary torus over 𝐹. When 𝐺 = 𝑇 is a torus, the stack 𝔅(𝑇) is a disjoint union of copies of the classifying stack [∗/𝑇 (𝐹)] indexed by the Kottwitz set 𝐵(𝑇). It is possible to define a characteristic 0 analogue of 𝔅(𝑇) which also consists of copies of [∗/𝑇 (𝐹)]. In fact, we can define an algebraic stack Tor𝑇,iso𝐹 over Z, such that     IndShv 𝔅(𝑇), Zℓ  QCoh Tor𝑇,iso𝐹 ⊗ Zℓ . 2 The algebraic stack Tor𝑇,iso𝐹 is essentially the stack of 𝑇-isocystals. Let 𝐹˘ be the completion of a maximal unramified extension of 𝐹, and let 𝜎 be the Frobenius action. The groupoid Tor𝑇,iso𝐹 (Z) is ˘ ˘ Tor𝑇,iso𝐹 (Z) = [𝑇 ( 𝐹)/Ad 𝜎 𝑇 ( 𝐹)]. One of the main result of this paper is as follows. Theorem 1.0.3. There is a canonical family of Poincaré line bundles L𝑇 : Tor𝑇,iso𝐹 × Loc𝑐 𝑇,𝐹 → BG𝑚 for every torus 𝑇 over 𝐹. Furthermore, these Poincaré line bundles induce isomorphism Loc∨𝑐 𝑇,𝐹  Tor𝑇,iso𝐹 for all torus 𝑇. We shall see in Section 2.4 that the isomorphism Loc∨𝑐 𝑇,𝐹 → Tor𝑇,iso𝐹 is essentially taking the cup product with the fundamental class of local class field theory. This answers a conjecture of Zhu (X. Zhu, 2021). As an application, we establish an integral local Langlands correspondence for torus. Theorem 1.0.4. The Fourier-Mukai transform via the Poincaré line bundle L𝑇 gives the equivalence of stable ∞-categories L : IndCoh(Loc𝑐 𝑇,𝐹 )  QCoh(Tor𝑇,iso𝐹 ). The Fourier-Mukai transform was first introduced to the Langlands program in the work of Laumon, where the transform is applied to construct the geometric Langlands correspondence for 𝐺 𝐿(1). Later, in the work of Braverman-Bezrukavnikov (Bezrukavnikov and Braverman, 2007) and Chen-Zhu (Chen and X. Zhu, 2014), Fourier-Mukai is used to establish a generic version of the Langlands correspondence in positive characteristic (for any reductive group 𝐺). 1.0.5. The Second Categorification. Let Λ = Fℓ , Qℓ or Zℓ and let 𝐿 +𝑇 be the positive loop group of 𝑇. There is an isomorphism between abelian groups of continuous characters of Serre’s fundamental group 𝜋1 (𝐿 +𝑇) and character sheaves on 𝐿 +𝑇: Homcts (𝜋1 (𝐿 +𝑇), Λ× ) = CS(𝐿 +𝑇, Λ). The second main result of this paper is a categorification of this isomorphism: 3 Theorem 1.0.6. There exists a fully-faithful, 𝑡-exact, monoidal functor geom Ch : IndCoh(Loc𝑐 𝑇,𝐹 ) → IndShv(𝐿𝑇), geom where Loc𝑐 𝑇,𝐹 is the representation stack of the inertia group of 𝐹 in the dual group 𝑐 𝑇. This can be viewed as a “categorification of categorification” in the the following sense. Let Shvmon (𝐿𝑇) denote the essential image of Ch. It is the thick subcategory compactly generated by all character sheaves on 𝐿𝑇. Let Chmon denote the the equivalence of categories geom Chmon : IndCoh(Loc𝑐 𝑇,𝐹 )  IndShvmon (𝐿𝑇). Note that both categories carry a Frobenius structure. Proposition 1.0.7. There is a commutative diagram geom Tr(IndShvmon (𝐿𝑇), 𝜎) Tr(IndCoh(Loc𝑐 𝑇,𝐹 ), 𝜎) ∼ ∼ IndCoh(Loc𝑐 𝑇,𝐹 ) L (1.1) IndShv(𝔅(𝑇)), where the top arrow is induced by Chmon , the bottom arrow is the functor L under the identification IndShv(𝔅(𝑇))  QCoh(Tor𝑇,iso𝐹 ). Furthermore, he two vertical arrows are canonical equivalences. Some results in this paper are independently obtained by K. Zou (Zou, 2024), sometimes by different methods. There are two main differences between this work and Zou’s. On the one hand, Zou considers integral ℓ-adic sheaves on Bun𝐺 over the Fargues-Fontaine curve, while we consider quasi-coherent sheaves on Tor𝑇,iso𝐹 which is defined over Z. On the other hand, Zou uses the spectral action on the Whittaker sheaf to establish the equivalence, while we use the Fourier-Mukai transform. The organization of the paper is as follows. Section 2.1 introduces the notations and definitions of Loc𝑐 𝑇,𝐹 and Tor𝑇,iso𝐹 . Section 2.3 establishes an important short exact sequence involving Loc𝑐 𝑇,𝐹 . Section 2.4 constructs the Poincaré line bundles and proves the main theorem. Section 2.5 discusses the Fourier-Mukai transform and the equivalence of categories. 4 Chapter 2 THE FIRST CATEGORIFICATION 2.1 Notation 2.1.1. Galois groups. We first fix our notations. Let 𝐹 be a local field, 𝜅 𝐹 its residue field, and 𝐸/𝐹 a Galois extension that splits the torus 𝑇 defined over 𝐹. Let Γ = Gal(𝐸/𝐹) and let 𝑊𝐹 and 𝑊𝐸 be the Weil groups of 𝐹 and 𝐸. Recall that the relative Weil group of 𝐸/𝐹 is defined as 𝑊𝐸/𝐹 = 𝑊𝐹 /[𝑊𝐸 , 𝑊𝐸 ]. By local class field theory, the relative Weil group is also the unique group extension 1 → 𝐸 × → 𝑊𝐸/𝐹 → Γ → 1 corresponding to the fundamental class 𝛼 ∈ 𝐻 2 (Γ, 𝐸 × ). Fix {𝑤 𝜏 }, 𝜏 ∈ Γ to be a system of representatives of (right) cosets of 𝐸 × . For any 𝑔, 𝑤 ∈ 𝑊𝐸/𝐹 , there is a unique 𝛿(𝑔, 𝑤) ∈ 𝐸 × and a unique 𝜏 ∈ Γ such that 𝑔𝑤 = 𝛿(𝑔, 𝑤)𝑤 𝜏 . The assignment (𝜏, 𝜎) ↦→ 𝛿(𝑤 𝜏 , 𝑤 𝜎 ) is a cycle and it represents the fundamental class 𝛼. Let 𝑝 : 𝑊𝐸/𝐹 → Γ be the projection as above, we extend the meaning of the notation 𝑤 𝜏 to allow elements of 𝑊𝐸/𝐹 appear in the subscript, for example, 𝑤 𝑔 = 𝑤 𝑝(𝑔) , and 𝑤 𝑔𝜏 = 𝑤 𝑝(𝑔)𝜏 , for all 𝑔 ∈ 𝑊𝐸/𝐹 , 𝜏 ∈ Γ. 2.1.2. Torus and dual torus. Let 𝐿 = 𝑋 ∗ (𝑇) and 𝐿ˆ = 𝑋∗ (𝑇) be the weight lattice and coweight lattice of 𝑇. They are both Γ-modules and therefore 𝑊𝐸/𝐹 -modules. The dual torus 𝑇ˆ = 𝐿 ⊗ G𝑚 is defined over Z. The C-group of 𝑇 is an enhancement of the Langlands dual group 𝑐 𝑇 := 𝑇ˆ ⋊ (G𝑚 × Γ), where Γ acts naturally on 𝑇 and G𝑚 acts trivially. The G𝑚 factor in the C-group is only useful for non-abelian reductive groups, but we include it here to be consistent with notations in (X. Zhu, 2021). Let the homomorphism 𝜒 : 𝑊𝐸/𝐹 → G𝑚 × Γ be trivial on the factor G𝑚 and be the canonical projection on the factor Γ. 5 2.1.3. The stack of Langlands parameters. We now define the scheme of framed Langlands parameters Loc□ 𝑐 𝑇,𝐹 and the stack of Langlands parameters Loc𝑐 𝑇,𝐹 . All (Picard) stacks in this note will live over the fpqc site of 𝑆 = Spec Z. Let 𝑀 be an affine group scheme over 𝑆, and 𝐺 an abstract group. The functor R 𝐺,𝑀 that sends every Z-algebra 𝑅 to the set of group homomorphisms from 𝐺 to 𝑀 (𝑅) is represented by an affine scheme. This is because, if 𝐼 ⊂ 𝐺 is a set of generators, R 𝐺,𝑀 is a closed subset of 𝑀 𝐼 according to the relations between the generators. The derived scheme of representations is studied in (X. Zhu, 2021). By local class field theory, there is the short exact sequence 1 → 𝐸 × → 𝑊𝐸/𝐹 → Γ → 1. Let 𝑈 (𝑛) be the 𝑛-th congruence subgroup of 𝐸 × , so 𝑈 (0) = O𝐸× and 𝑈 (𝑛) = 1 + 𝔪𝑛𝐸 , 𝑛 ≥ 1. Let 𝑊 (𝑛) = 𝑊𝐸/𝐹 /𝑈 (𝑛) . The scheme of framed Langlands parameters ˆ ˆ Loc□ 𝑐 𝑇,𝐹 classifies continuous cross homomorphisms from 𝑊 𝐸/𝐹 to 𝑇, with 𝑇 (𝑅) endowed with the discrete topology. Namely, □,(𝑛) Loc□ 𝑐 𝑇,𝐹 = lim Loc𝑐 𝑇,𝐹 , −−→ where 𝑐𝑙 Loc□,(𝑛) R𝑊 (𝑛) ,𝑐 𝑇 ×𝑐𝑙 R (𝑛) 𝑐 𝑇,𝐹 = 𝑊 ,G𝑚 ×Γ { 𝜒}. 𝑛 1 ˆ Equivalently, Loc□ 𝑐 𝑇,𝐹 is the scheme whose 𝑅-points are 𝑍 (𝑊 𝐸/𝐹 , 𝑇 (𝑅)), where 𝑍 1 denotes the set of continuous cocycles. Since 𝑇ˆ is commutative, Loc□ 𝑐 𝑇,𝐹 has a canonical Picard stack structure. The stack of Langlands parameters is defined as ˆ Loc𝑐 𝑇,𝐹 = Loc□ 𝑐 𝑇,𝐹 /𝑇. Recall that in (Deligne, n.d.), Deligne defined the functor ch : 𝐷 [−1,0] (𝑆, Z) → PS/𝑆 that sends a complex of abelian sheaves over 𝑆 whose cohomology concentrates in degree -1 and 0 to a Picard stack over 𝑆. It is convenient to think of Loc𝑐 𝑇,𝐹 as
ˆ [1]) , Loc𝑐 𝑇,𝐹 = ch 𝜏≤0 (𝐶 • (𝑊𝐸/𝐹 , 𝑇) ˆ is the complex of abelian sheaves calculating group cohomolowhere 𝐶 • (𝑊𝐸/𝐹 , 𝑇) ˆ gies 𝐻 • (𝑊𝐸/𝐹 , 𝑇). 2.1.4. The category of coherent sheaves on Loc𝑐 𝑇,𝐹 . Recall that Loc□ 𝑐 𝑇,𝐹 = □,(𝑛) □,(𝑛) ˆ ˆ lim Loc𝑐 𝑇,𝐹 and Loc𝑐 𝑇,𝐹 = Loc□ 𝑐 𝑇,𝐹 /𝑇. As the action of 𝑇 stablizes Loc𝑐 𝑇,𝐹 for −−→ large 𝑛, Loc𝑐 𝑇,𝐹 has a natural ind-scheme structure. Accordingly, we define the category Coh(Loc𝑐 𝑇,𝐹 ) to be coherent sheaves supported on finitely many connected components and IndCoh(Loc𝑐 𝑇,𝐹 ) to be the ind-completion of it. 6 2.1.5. The stack of isocrystals. We follow the setup in (Kottwitz, 2014), and let ˆ = 𝑍 1 (𝐸 × , 𝑇 (𝐸)) be the adjoint of the identity. We form 𝐿ˆ → Hom(𝐸 × , 𝐸 × ⊗ 𝐿) the set of algebraic cycles by the following fibre product: 1 (𝑊 𝑍alg 𝐸/𝐹 , 𝑇 (𝐸)) 𝑍 1 (𝑊𝐸/𝐹 , 𝑇 (𝐸)) 𝐿ˆ res 𝑍 1 (𝐸 × , 𝑇 (𝐸)). In other words, the algebraic cycles are those 1-cycles whose restriction to 𝐸 × ⊂ 𝑊𝐸/𝐹 is an algebraic character of 𝑇. Recall that 𝑈 (𝑛) are congrunce subgroups of 𝐸 × . The sets 𝑈𝑛 = 𝐿ˆ ⊗ 𝑈 (𝑛) forms a basis of open subgroups of 𝑇 (𝐸) that are 𝑊𝐸/𝐹 -invariant, and 𝑉𝑛 = 𝑈𝑛 ∩ 𝑇 (𝐹) forms a basis of open subgroups of 𝑇 (𝐹). We define the stack Tor𝑇,iso𝐹 as follows (non-zero terms of the chain complex are situated in degree -1 and 0)   1 Tor𝑇,iso𝐹 := lim ch · · · → 0 → 𝑇 (𝐸)/𝑈𝑛 → 𝑍alg (𝑊𝐸/𝐹 , 𝑇 (𝐸)/𝑈𝑛 ) → 0 → · · · . ←−− 𝑛 Since the topos of fpqc-sheaves is replete, it is shown that inverse limit of surjective homomorphisms have no higer limit (Bhatt and Scholze, 2014). Therefore the inverse limit can also be taken at each degree:   1 Tor𝑇,iso𝐹 = ch · · · → 0 → lim 𝑇 (𝐸)/𝑈𝑛 → lim 𝑍alg (𝑊𝐸/𝐹 , 𝑇 (𝐸)/𝑈𝑛 ) → 0 → · · · . ←−− ←−− Remark 2.1.6. (i) Let us consider the groupoid Tor𝑇,iso𝐹 (Z). The isomorphism 1 (𝑊 class of the groupoid is the cohomology of algebraic cycles 𝐻alg 𝐸/𝐹 , 𝑇 (𝐸)), and it is isomorphic to the Kottwitz set 𝐵(𝑇) = 𝑋 ∗ (𝑇ˆ Γ ) = 𝑋∗ (𝑇) Γ = 𝐿ˆ Γ which classifies 𝑇-isocrystals. The automorphism group of the identity object is 𝑇 (𝐹). (ii) The stack Tor𝑇,iso𝐹 is not the constant groupoid with objects 𝐿ˆ Γ and with automorphism group of the identity 𝑇 (𝐹). The fpqc site remembers the pro-finite topology of the automorphism group 𝑇 (𝐹). (iii) Let 𝐹˘ be the completion of a maximal unramified extension of 𝐹, 𝜎 the Frobenius. Tor𝑇,iso𝐹 (Z) is isomorphic to the groupoid of pairs (E, 𝜙) where E is a 𝑇-torsor over 𝐹˘ and 𝜙 : E  𝜎 ∗ E is an isomorphism of 𝑇-torsors. By a Tannakian formalism, this is the groupoid of exact tensor functors from Rep𝑇 to the monoidal category of isocrystals over 𝐹. This explains the notation Tor𝑇,iso𝐹 . 7 2.2 Quasi-Coherent Sheaves on Tor𝑇,iso𝐹 2.2.1. Recall that 𝑉𝑛 is a system of open neighbourhoods of the topological group 𝑇 (𝐹). Let 𝑇 (𝐹)0 be the maximal compact subgroup of 𝑇. We define two sheaves of abelian groups on the fpqc-site of Z T = lim 𝑇 (𝐹)/𝑉𝑛 , ←−− T0 = lim 𝑇 (𝐹)0 /𝑉𝑛 , ←−− where 𝐴 means the constant sheaf of an abelian group 𝐴. 2.2.2. We also define a sub-functor T ◦ of T so that for every unitary ring 𝐴, Ø T ◦ ( 𝐴) = 𝑡 · T0 ( 𝐴). 𝑡∈𝑇 (𝐹) In other words, T ◦ consists of sections which lie uniformly in one coset of 𝑇 (𝐹)0 . The sheafification of T ◦ is T . We take pre-stack quotients BT ◦ = ∗/T ◦ and BT0 = ∗/T0 . Their sheafification are the stacks B𝑇 (𝐹) and BT0 . In the next proposition, we will use the pre-stack quotients because they are easier to work with. However, sheafification has no effect on the category of quasi-coherent sheaves. We quickly recall the argument for this fact in the following. Consider the presheaf QCoh that associates a scheme 𝑆 with the ∞-category of quasicoherent sheaves on 𝑆 and associates a morphism with its ∗-pullback. This presheaf is in fact a sheaf by fpqc-descent. Let Y ♭ be a prestack and Y its sheafification. A quasi-coherent sheaf on Y ♭ is a morphism of pre-sheaves Y ♭ → QCoh. However, any morphism of pre-sheaf must factor through the sheafification Y ♭ → Y. This is to say that there is a canonical equivalence QCoh(Y ♭ )  QCoh(Y). Proposition 2.2.3. Let Λ be any commutative ring. Pulling back along ∗ → BT0 and ∗ → B𝑇 (𝐹) produces the following equivalences of categories: QCoh(BT0 ⊗ Λ)  Repsm (𝑇 (𝐹)0 , Λ), QCoh(BT ◦ ⊗ Λ)  Repsm (𝑇 (𝐹), Λ). Proof. Let BT0 and BT ◦ also denote their base-change over Λ. Let 𝑝 denote ∗ → BT0 . Consider the adjoint pair 𝑝 ∗ : QCoh(BT0 ) Λ-Mod : 𝑝 ∗ . 8 It is clear that 𝑝 ∗ is conservative and exact. The category QCoh(BT0 ) admits arbitrary limits. Therefore, 𝑝 ∗ exhibits QCoh(BT0 ) as co-monadic over Λ-Mod. Since T0 is pro-finite, it is represented by an affine scheme Spec( 𝐴). The co-monad endomorphism 𝑝 ∗ 𝑝 ∗ is the functor 𝑀 ↦→ 𝐴 ⊗ 𝑀. QCoh(BT0 ) is hence equivalent to the category of Λ-modules with 𝐴-co-algebra structure, i.e., the category Repsm (𝑇 (𝐹)0 , Λ). Let 𝑓 : BT0 → BT ◦ the map induced by the inclusion T0 → T ◦ . It is schematic, quasi-compact, and quasi-separated. Therefore there exists a pair of adjoint functors 𝑓 ∗ : QCoh(BT ◦ ) QCoh(BT0 ) : 𝑓∗ . The pullback functor 𝑓 ∗ can be identified with pre-composing a map BT ◦ → QCoh with 𝑓 . It is clear that 𝑓 ∗ is conservative and exact. The category QCoh(BT0 ) admits arbitrary limits. Therefore, 𝑓 ∗ exhibits QCoh(BT ◦ ) as co-monadic over QCoh(BT0 ). On the other hand, we have another co-monad by induction-restriction: Res : Repsm (𝑇 (𝐹), Λ) Repsm (𝑇 (𝐹)0 , Λ) : Ind, where Res exhibit Repsm (𝑇 (𝐹)) as monadic over Repsm (𝑇 (𝐹)0 ). Let 𝑞 : ∗ → BT ◦ . By identifying QCoh(BT0 ) with Repsm (𝑇 (𝐹)0 ), we obtain the following commutative diagram: 𝑞∗ QCoh(BT ◦ ) Repsm (𝑇 (𝐹), Λ) 𝑓∗ Res Repsm (𝑇 (𝐹)0 , Λ) This put us in the situation of the following lemma, which is a weaker version of the dual statement of Corollary 4.7.3.16 in (Lurie, 2017). Lemma 2.2.4. Suppose we have a commutative diagram of ∞-categories 𝐻′ 𝐻 D Assume that: C′ 𝑈 C 9 1. 𝐻 and 𝐻 ′ admit right adjoints 𝐺 and 𝐺 ′. 2. 𝐻 exhibit C as co-monadic over D. 3. 𝐻 ′ exhibit C ′ as co-monadic over D. 4. For each object 𝐷 ∈ D, the unit and co-unit map 𝑈𝐺 → 𝐺 ′ 𝐻 ′𝑈𝐺 → 𝐺 ′ 𝐻𝐺 → 𝐺 ′ induce equivalence 𝑈𝐺 (𝐷) → 𝐺 ′ (𝐷). Then 𝑈 is an equivalence of ∞-categories. All conditions in the lemma are readily satisfied except 4. We notice that in our case 𝑈𝐺 = 𝑞 ∗ 𝑓∗ and 𝐺 ′ = Ind𝑇𝑇 (𝐹) . Using base change in the following diagram: (𝐹) 0 Î 𝑇 (𝐹)/𝑇 (𝐹)0 ∗ BT0 𝑓 ∗ 𝑞 BT ◦ Î we compute 𝑈𝐺 (𝑀) = 𝑇 (𝐹)/𝑇 (𝐹)0 𝑀 for any 𝑀 ∈ Repsm (𝑇 (𝐹)0 ) and this is canonically isomorphic to Ind𝑇𝑇 (𝐹) 𝑀. This canonical isomorphism 𝑈𝐺  𝐺 ′ must (𝐹)0 agree with the morphism constructed in condition 4. □ 2.3 A short exact sequence The main result of this section is as follows. Proposition 2.3.1. For an arbitrary torus 𝑇 over a local field 𝐹, we have a short exact sequence of Picard stacks 1 → Hom( 𝐿ˆ Γ , BG𝑚 ) → Loc𝑐 𝑇,𝐹 → Homcts (𝑇 (𝐹), G𝑚 ) → 1. (2.1) Remark 2.3.2. (i) Note the first term has the alternative form Hom( 𝐿ˆ Γ , BG𝑚 ) = 𝐵𝑇ˆ Γ , because 𝑋 ∗ (𝑇ˆ Γ ) = 𝐿ˆ Γ . (ii) From the proposition, we see the isomorphism classes of Loc𝑐 𝑇,𝐹 is dual to the automorphism group of Tor𝑇,iso𝐹 and vice versa. This results in the duality that is treated in section 2.4. (iii) The inner-hom Homcts (𝑇 (𝐹), G𝑚 ) is defined as the colimit of inner-homs Hom(𝑇 (𝐹)/𝑈, G𝑚 ) for all open subgroups 𝑈 ⊂ 𝑇 (𝐹). 10 Our strategy is to first show the following short exact sequence for any Z-algebra 𝑅: 0 → Ext1 ( 𝐿ˆ Γ , 𝑅 × ) → 𝐻 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)) → Homcts (𝑇 (𝐹), 𝑅 × ) → 0. (2.2) Heuristically, this is our desired exact sequence (2.1) on the level of coarse moduli space. The group cohomology 𝐻 1 is interpreted as continuous cross homomorphisms up to conjugation. After (2.2) is shown, we will show (2.1) directly. We let 𝑍¯ 1 , 𝐻¯ 1 be all cross homomorphisms and all cross homomorphisms up to conjugation respectively, removing the continuous conditions. We first show a version of the short exact sequence (2.2) without the topology constraints. Lemma 2.3.3. We have the short exact sequence 0 → Ext1 ( 𝐿ˆ Γ , 𝑅 × ) → 𝐻¯ 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)) → Hom(𝑇 (𝐹), 𝑅 × ) → 0. (2.3) Proof. Let 𝐶• → Z → 0 be the bar resolution of the trivial module Z by free Z[𝑊𝐸/𝐹 ]-modules. The group cohomology 𝐻¯ 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)) is exactly the first cohomology of the following complex (we abbreviate 𝑊𝐸/𝐹 by 𝑊 at times): ˆ 𝑊 , 𝑅 × ). HomZ[𝑊] (𝐶• , 𝑇ˆ (𝑅)) = HomZ[𝑊] (𝐶• , 𝐿 ⊗ 𝑅 × ) = Hom((𝐶• ⊗ 𝐿) ˆ 𝑊 is a complex of free abelian groups, and its Because the complex (𝐶• ⊗ 𝐿) ˆ we apply the homology caculates the group homology of the 𝑊𝐸/𝐹 -module 𝐿, universal coefficient theorem and get ˆ 𝑅 × ) → 𝐻¯ 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)) → Hom(𝐻1 (𝑊𝐸/𝐹 , 𝐿), ˆ 𝑅 × ) → 0. 0 → Ext1 (𝐻0 (𝑊𝐸/𝐹 , 𝐿), ˆ = 𝐿ˆ 𝑊 = 𝐿ˆ Γ . In the third term, Langlands(Langlands, In the first term, 𝐻0 (𝑊𝐸/𝐹 , 𝐿) ˆ → 𝐻1 (𝐸 × , 𝐿) ˆ Γ is an 1997) proved that the corestriction map cores : 𝐻1 (𝑊𝐸/𝐹 , 𝐿) isomorphism, and therefore ∼ ˆ − ˆ Γ = 𝑇 (𝐹). 𝐻1 (𝑊𝐸/𝐹 , 𝐿) → 𝐻1 (𝐸 × , 𝐿) This finishes the proof of (2.3). □ ˆ 𝑊 = Z[𝑊𝐸/𝐹 ] ⊗ 𝐿, ˆ We spell out the corestriction map explicitly. We have (𝐶1 ⊗ 𝐿) Í Í ˆ 𝑊 = 𝐿ˆ and the differential is 𝑤 ⊗ 𝑥 = 𝑤 −1 𝑥 − 𝑥. The corestriction map (𝐶0 ⊗ 𝐿) ˆ → 𝐻1 (𝐸 × , 𝐿) ˆ Γ is induced by the chain map cores : 𝐻1 (𝑊𝐸/𝐹 , 𝐿) Z[𝑊𝐸/𝐹 ] ⊗ 𝐿ˆ → Z[𝐸 × ] ⊗ 𝐿ˆ ∑︁ 𝑤 ⊗ 𝑥 ↦→ 𝛿(𝑤 𝜏 , 𝑤) ⊗ 𝑤 𝜏 𝑥. 𝜏 (2.4) 11 Lemma 2.3.4. We have a short exact sequence 0 → Ext1 ( 𝐿ˆ Γ , 𝑅 × ) → 𝐻 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)) → Homcts (𝑇 (𝐹), 𝑅 × ) → 0. Proof. We first show the image of Ext1 ( 𝐿ˆ Γ , 𝑅 × ) is contained in 𝐻 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)). To show this, we work out an explicit formula for the inclusion Ext1 ( 𝐿ˆ Γ , 𝑅 × ) → ˆ so we have 𝐻¯ 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)) in (2.3). Let 𝐿ˆ 0 = {𝑔𝑥 − 𝑥|𝑔 ∈ 𝑊𝐸/𝐹 , 𝑥 ∈ 𝐿}, 0 → 𝐿ˆ 0 → 𝐿ˆ → 𝐿ˆ Γ → 0. This short exact sequence gives rise to 𝛼 Hom( 𝐿ˆ 0 , 𝑅 × ) − → Ext1 ( 𝐿ˆ Γ , 𝑅 × ) → 0. On the other hand, for each 𝜙 ∈ Hom( 𝐿ˆ 0 , 𝑅 × ), we associate a map 𝑊𝐸/𝐹 × 𝐿ˆ → 𝑅 × by (𝑔, 𝑥) ↦→ 𝜙(𝑔 −1 𝑥 − 𝑥). By adjunction this defines a cross homomorphism 𝑊𝐸/𝐹 → 𝐿 ⊗ 𝑅 × = 𝑇ˆ (𝑅). We denote this map by 𝛽 : Hom( 𝐿ˆ 0 , 𝑅 × ) → 𝐻¯ 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)). The inclusion map is then 𝛽 ◦ 𝛼−1 : Ext1 ( 𝐿ˆ Γ , 𝑅 × ) → 𝐻¯ 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)), whose result is independent of the choice of a preimage of 𝛼. It remains to show under the surjection in (2.3), an element lies in 𝐻 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)) if and only if its image is a continuous homomorphism. Recall the bar resolution 𝐵• → Z → 0 is a resolution of the trivial module Z by tr free Z[𝑊𝐸/𝐹 ]-modules, and it begins with Z[𝑊𝐸/𝐹 ] → − Z. The first homology ˆ ˆ is therefore represented by a cycle living in Z[𝑊𝐸/𝐹 ] ⊗ 𝐿. 𝐻1 (𝑊𝐸/𝐹 , 𝐿) The surjection in (2.3) is induced by 𝜋 on the level of cocycle as follows: 𝑍¯ 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)) 𝜙 : 𝑊𝐸/𝐹 → 𝐿 ⊗ 𝑅 × ˆ 𝑅× ) Hom(𝐻1 (𝑊𝐸/𝐹 , 𝐿), 𝜋 ( Í 𝑤 ⊗ 𝑥 ↦→ Í ⟨𝜙(𝑤), 𝑥⟩) , Í where 𝜙 is a cocycle, 𝑤 ⊗ 𝑥 ∈ Z[𝑊𝐸/𝐹 ] ⊗ 𝐿ˆ a chosen cycle reprensenting some ˆ and the pairing is by evaluating 𝐿ˆ on 𝐿. It remains to show element in 𝐻1 (𝑊𝐸/𝐹 , 𝐿), that 𝜙 is continuous if and only if 𝜋(𝜙) is. This is the content of the next lemma. □ 12 Lemma 2.3.5. A cocycle 𝜙 is continuous if and only if 𝜋(𝜙) is. Proof. We basically follow the argument given in (Langlands, 1997). We have ˆ → 𝐻1 (𝑊𝐸/𝐹 , 𝐿). ˆ A homomorphism 𝜓 : the corestriction map cor : 𝐻1 (𝐸 × , 𝐿) ˆ → 𝑅 × is continuous if and only if 𝜓 ◦ cor is. On the other hand, it is 𝐻1 (𝑊𝐸/𝐹 , 𝐿) clear that a cocycle is continuous if and only if its restriction to 𝐸 × is. We have the following commutative diagram: ˆ 𝑍¯ 1 (𝑊𝐸/𝐹 , 𝑇) 𝜋 ˆ 𝑅× ) Hom(𝐻1 (𝑊𝐸/𝐹 , 𝐿), ˆ 𝑍¯ 1 (𝐸, 𝑇) 𝜏 ˆ 𝑅 × ), Hom(𝐻1 (𝐸, 𝐿), ˆ → 𝑅× , where the map 𝜏 send a cocycle 𝑓 to an homomorphism 𝜏( 𝑓 ) : 𝐻1 (𝐸, 𝐿) 𝑎 ⊗ 𝑥 ↦→ ⟨ 𝑓 (𝑎), 𝑥⟩. It is then clear that 𝑓 is continuous if and only if 𝜏( 𝑓 ) is, and the lemma follows. □ 2.3.6. Proof of proposition 2.3.1. The functor ch : 𝐷 [−1,0] (𝑆, Z) → PS/𝑆 is an equivalence of category. Let (−) ♭ be an quasi-inverse. The existence of the short exact sequence (2.1) is by definition to say there exists an exact triangle +1 Hom( 𝐿ˆ Γ , BG𝑚 ) ♭ → Loc♭𝑐 𝑇,𝐹 → Homcts (𝑇 (𝐹), G𝑚 ) ♭ −−→ . We first define the homomorphisms 𝛽¯ 𝜋¯ − Loc♭𝑐 𝑇,𝐹 → − Homcts (𝑇 (𝐹), G𝑚 ) ♭ Hom( 𝐿ˆ Γ , BG𝑚 ) ♭ → as follows, where each column is a 2-term complex of abelian sheaves corresponding to the Picard stack above: deg = 0 Hom( 𝐿ˆ 0 , G𝑚 ) 𝛽′ res deg = −1 ˆ G𝑚 ) Hom( 𝐿, 𝜋′ Loc□ 𝑐 𝑇,𝐹 Homcts (𝑇 (𝐹), G𝑚 ) 𝑑 ∼ 𝑇ˆ 0. The map 𝛽′ (𝑅) is exactly 𝛽 : Hom( 𝐿ˆ 0 , 𝑅 × ) → 𝐻 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)) as defined in Lemma 2.3.4. The map 𝜋′ (𝑅) is the restriction of 𝜋 to the continuous cocycles 𝑍 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)) 𝜙 : 𝑊𝐸/𝐹 → 𝐿 ⊗ 𝑅 × ˆ 𝑅× ) Homcts (𝐻1 (𝑊𝐸/𝐹 , 𝐿), ( Í 𝑤 ⊗ 𝑥 ↦→ ⟨𝜙(𝑤), 𝑥⟩) . 13 It is immediate to check that the squares commute, so they define homomorphisms of complexes. To show the above three terms forms an exact sequence of Picard stacks amounts to checking they form an exact triangle. We do so by showing a quasi-isomorphism ¯  Hom (𝑇 (𝐹), G𝑚 ). cofib( 𝛽) cts ¯ is nothing other than the following complex: Note cofib( 𝛽)  (id,−res) 𝑑 𝛽′  ˆ G𝑚 ) −−−−−−→ 𝑇ˆ ⊕ Hom( 𝐿ˆ 0 , G𝑚 ) −−−−→ Loc□ Hom( 𝐿, 𝑐 𝑇,𝐹 . ¯ = 0. Next, because 𝑑 ◦ id = 𝛽′ ◦ res Because the first map is injective, 𝐻 −2 (cofib( 𝛽)) ¯ = 0. and 𝛽′ is injective, 𝐻 −1 (cofib( 𝛽))   □ 0 ¯ We know that 𝐻 (cofib( 𝛽)) = Loc𝑐 𝑇,𝐹 /Im 𝛽𝑑′ . To evaluate this, we first take the   ′ 𝑑′ ′ quotient Loc□ /Im 𝑐 𝑇,𝐹 𝛽 as presheaf, where Im is the image of maps of presheaves. We have ′ ′ ′ Loc□ 𝑐 𝑇,𝐹 (𝑅)/(Im (𝑑) + Im (𝛽 )) = 𝐻 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅))/(Im′ (𝛽′)/(Im′ (𝛽′) ∩ Im′ (𝑑))) = 𝐻 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅))/Ext1 ( 𝐿ˆ Γ , 𝑅 × ) = Homcts (𝑇 (𝐹), 𝑅 × ). The last line of the equation is just the 𝑅-points of Homcts (𝑇 (𝐹), G𝑚 ). □ The next proposition is crutial for the duality Tor𝑇,iso𝐹  Loc∨𝑐 𝑇,𝐹 . Proposition 2.3.7. The +1 map is zero for the following exact triangle we just show +1 Hom( 𝐿ˆ Γ , BG𝑚 ) ♭ → Loc♭𝑐 𝑇,𝐹 → Homcts (𝑇 (𝐹), G𝑚 ) ♭ −−→ . Proof. Recall that the +1 map is the inverse of the quasi-isomorphism showed above composed with a canonical projection ∼ ¯ → 𝐵𝑇ˆ Γ [1]. Homcts (𝑇 (𝐹), G𝑚 ) ← cofib( 𝛽) Due to the universal coefficient theorem, there exists a collection of maps ˆ 𝑅 × ) → 𝑍¯ 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)) 𝜌 : Hom(𝐻 1 (𝑊𝐸/𝐹 , 𝐿), 14 that is functorial in 𝑅, such that after composing with the projection maps 𝑍¯ 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)) → 𝐻¯ 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)), give splittings of the following surjections: ˆ 𝑅 × ). 𝜋 : 𝐻¯ 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)) → Hom(𝐻 1 (𝑊𝐸/𝐹 , 𝐿), ˆ → 𝑅 × , 𝜌( 𝑓 ) is automatically For a continuous homomorphism 𝑓 : 𝐻 1 (𝑊𝐸/𝐹 , 𝐿) a continuous cocycle because its orbit under action of 𝑇ˆ (𝑅) must contain one continuous cocycle, but since all coboundries are continuous, 𝜌( 𝑓 ) is continuous. Therefore we have the following maps: ˆ 𝑅 × ) → 𝑍 1 (𝑊𝐸/𝐹 , 𝑇ˆ (𝑅)). Homcts (𝐻 1 (𝑊𝐸/𝐹 , 𝐿), (2.5) ¯ → The maps (2.5) give rise to an inverse of the quasi-isomorphism cofib( 𝛽) Homcts (𝑇 (𝐹), G𝑚 ) by the composition ¯ Homcts (𝑇 (𝐹), G𝑚 ) → Loc□ 𝑐 𝑇,𝐹 → cofib( 𝛽). ¯ → 𝐵𝑇ˆ Γ [1] is nothing but the +1 map, but it Composing this inverse with cofib( 𝛽) is also zero by construction. □ 2.4 Duality The main result of this paper is the following. Theorem 2.4.1. There is a unique family of Poincaré line bundles (or a family of pairings) L𝑇 : Tor𝑇,iso𝐹 × Loc𝑐 𝑇,𝐹 → BG𝑚 for every torus 𝑇 over 𝐹 that satisfies the three conditions below. Furthermore, these Poincaré line bundles induce isomorphism Tor𝑇,iso𝐹  Loc∨𝑐 𝑇,𝐹 for every torus 𝑇. a) Functorality. Let 𝑓 : 𝑆 → 𝑇 be a map between torus. It induces 𝛼 : Loc𝑐 𝑇,𝐹 → Loc𝑐 𝑆,𝐹 and 𝛽 : Tor𝑆,iso𝐹 → Tor𝑇,iso𝐹 . The following two line bundles on Tor𝑆,iso𝐹 × Loc𝑐 𝑇,𝐹 are canonically isomorphic (𝛽 × id) ∗ L𝑇 = (id ×𝛼) ∗ L 𝑆 . b) Split case. For the split torus 𝑇 = G𝑚 , both Tor𝑇,iso𝐹 and Loc𝑐 𝑇,𝐹 canonically split as Tor𝑇,iso𝐹 = Z × B𝐹 × , Loc𝑐 𝑇,𝐹 = Homcts (𝐹 × , G𝑚 ) × BG𝑚 . 15 Then L𝑇 is the canonical line bundle Tor𝑇,iso𝐹 × Loc𝑐 𝑇,𝐹 → BG𝑚 via the tautological pairings Homcts (𝐹 × , G𝑚 ) × 𝐵𝐹 × → BG𝑚 Z × BG𝑚 → BG𝑚 . c) Induced case. For an induced torus 𝑆 = Res𝐸/𝐹 G𝑚 , let 𝑇 be the split torus over 𝐸, the Shapiro isomorphism Tor𝑆,iso𝐹 × Loc𝑐 𝑆,𝐹  Tor𝑇,iso𝐸 × Loc𝑐 𝑇,𝐸 identifies the line bundles for 𝑆 and 𝑇. Our strategy is to introduce an auxilary stack T𝑇 for every torus 𝑇 in place of Tor𝑇,iso𝐹 . When it is clear from the context, we will suppress the torus 𝑇 from the notation T𝑇 . We will first define a family of line bundles on T ×Loc𝑐 𝑇,𝐹 . Then we verify the main theorem for T instead of Tor𝑇,iso𝐹 . Finally, we will show a cononical isomorphism T  Tor𝑇,iso𝐹 . 2.4.2. Definition of T . Recall that 𝑈 (𝑛) ⊂ 𝐸 × ⊂ 𝑊𝐸/𝐹 is a basis of open neighbourhoods . We define 𝑊 (𝑛) = 𝑊𝐸/𝐹 /𝑈 (𝑛) . ˆ be the chain complex that Notice that 𝐿ˆ is a 𝑊 (𝑛) -module for all 𝑛. Let 𝐶• (𝑊 (𝑛) , 𝐿) ˆ The boundries of this chain calculates the group homology of the 𝑊 (𝑛) -module 𝐿. complex are defined by   𝑑 ˆ = Im 𝐶𝑖+1 (𝑊 (𝑛) , 𝐿) ˆ → ˆ . 𝐵𝑖 (𝑊 (𝑛) , 𝐿) − 𝐶𝑖 (𝑊 (𝑛) , 𝐿) We define  ˆ 𝑑 𝐶1 (𝑊 (𝑛) , 𝐿) (𝑛) ˆ → − 𝐶0 (𝑊 , 𝐿) → 0 → · · · , T𝑛 := ch · · · → 0 → ˆ 𝐵1 (𝑊 (𝑛) , 𝐿) 
ˆ and T := lim T𝑛 . It is convenient to think of T as ch 𝜏≥−1 (𝐶• (𝑊𝐸/𝐹 , 𝐿)) with a ←−− pro-stack structure. The groupoid T𝑛 (Z) calculates group homology. Hence, it has isomorphism classes 𝐿ˆ Γ . The following two lemma calculates its automorphism group to be 𝑇 (𝐹)/𝑉𝑛 . (𝑉𝑛 is defined in section 2.1.5). Lemma 2.4.3. The subgroups 𝑈 (𝑛) and 𝑈𝑛 ⊂ 𝑇 (𝐸) has no higer Galois cohomologies for sufficiently large 𝑛. 16 Proof. For sufficiently large 𝑛, the group 𝑈 (𝑛) is isomorphic to the additive group O𝐸 via the logarithm map. Since this is an induced module, its higher cohomology groups vanish. Moreover, 𝑈𝑛 = 𝐿ˆ ⊗ 𝑈 (𝑛) is a direct sum of copies of 𝑈 (𝑛) , so the same conclusion holds. □ Next we introduce a lemma in homological algebra. It generalizes a result of Langlands (Langlands, 1997). It also plays an important role in the second categorification which is the theme of the next chapter. Lemma 2.4.4. Let Γ be a finite group, 𝐶 be an abelian group equipped with a Γ-action. Let 𝛼 ∈ 𝐻 2 (Γ, 𝐶) represent the group extension 1 → 𝐶 → 𝐺 → Γ → 1. Let 𝑀 be any Γ-module, in other words, it is a 𝐺-module on which 𝐶 acts trivially. Then we have a commutative diagram 𝐻2 (Γ, 𝑀) 𝐻1 (𝐶, 𝑀) Γ ∪𝛼 0 𝐻1 (𝐺, 𝑀) res 𝐻ˆ −1 (Γ, 𝑀 ⊗ 𝐶) (𝑀 ⊗ 𝐶) Γ (𝑀 ⊗ 𝐶) Γ 𝐻1 (Γ, 𝑀) 0 ∪𝛼 𝐻ˆ 0 (Γ, 𝑀 ⊗ 𝐶) 0. The first row of the diagram is the long exact sequence associated to the LyndonHochschild-Serre spectral sequence. The second row is the definition of the Tate cohomology. The first and last vertical map is the cup product with 𝛼. The map res is the restriction map 𝐻1 (𝐺, 𝑀) → 𝐻1 (𝐶, 𝑀) Γ . In particular, if Γ and 𝐶 satisfies the condition of the Tate-Nakayama lemma, we would have 𝐻1 (𝐺, 𝑀)  (𝑀 ⊗ 𝐶) Γ . Proof. See appendix A. □ Corollary 2.4.5. (i) Applying the lemma to the group extension 1 → 𝐸 × → 𝑊𝐸/𝐹 → Γ → 1, we get the result of Langlands which claims ˆ  (𝐸 × ⊗ 𝐿) ˆ Γ = 𝑇 (𝐹). 𝐻1 (𝑊𝐸/𝐹 , 𝐿) 17 (ii) Similarly, with the group extension 1 → 𝐸 × /𝑈 (𝑛) → 𝑊 (𝑛) → Γ, we get  Γ ˆ  (𝐸 × /𝑈 (𝑛) ) ⊗ 𝐿ˆ = 𝑇 (𝐹)/𝑉𝑛 . 𝐻1 (𝑊 (𝑛) , 𝐿) (iii) Recall the definition of the relative inertia group rel 1 → 𝐼 𝐸ab → 𝐼 𝐸/𝐹 → 𝐼 → 1. Applying the lemma to this group extension gives us rel ˆ = (𝐼 𝐸ab ⊗ 𝐿) 𝐼 . 𝐻1 (𝐼 𝐸/𝐹 , 𝐿) We now move towards proof of the main theorem. Proposition 2.4.6. There is a canonical pairing ( · , · ) : T × Loc𝑐 𝑇,𝐹 → BG𝑚 . Proof. Recall that   𝑑 ˆ → ··· , Loc𝑐 𝑇,𝐹 = colim ch · · · → 𝑇ˆ → − 𝑍 1 (𝑊 (𝑛) , 𝑇)   ˆ 𝑑 𝐶1 (𝑊 (𝑛) , 𝐿) ˆ T = lim ch · · · → 𝐵 (𝑊 (𝑛) , 𝐿) − 𝐶0 (𝑊𝐸/𝐹 , 𝐿) → · · · . ˆ → 1 Loc𝑐 𝑇,𝐹 has an ind-stack structure, and T has a corresponding pro-stack structure, so to define a pairing between them, it suffices to define it for each Loc𝑐(𝑛) 𝑇,𝐹 and T . The pairing in the proposition is induced by two pairings: ⟨, ⟩: 𝑇ˆ × 𝐿ˆ ˆ × [ , ] : 𝑍 1 (𝑊 (𝑛) , 𝑇) → G𝑚 , ˆ 𝐶1 (𝑊 (𝑛) , 𝐿) ˆ 𝐵1 (𝑊 (𝑛) , 𝐿) → G𝑚 . ˆ The The first pairing is the natural pairing, since 𝐿ˆ is the character lattice of 𝑇. second pairing is given by ˆ × 𝐶1 (𝑊 (𝑛) , 𝐿) ˆ → G𝑚 𝑍 1 (𝑊 (𝑛) , 𝑇) ∑︁ (𝜙, 𝜓) ↦→ ⟨𝜙(𝑤), 𝜓(𝑤)⟩. 𝑤∈𝑊 (𝑛) , 𝐿) ˆ ˆ because in Section 2.3, we This pairing factors through 𝐶1 (𝑊 (𝑛) , 𝐿)/𝐵 1 (𝑊 ˆ and 𝐻1 (𝑊 (𝑛) , 𝐿) ˆ = show that the same formula defines a pairing between 𝑍 1 (𝑊 (𝑛) , 𝑇) (𝑛) , 𝐿). ˆ ˆ 𝑍1 (𝑊 (𝑛) , 𝐿)/𝐵 1 (𝑊 18 Our desired pairing T × Loc𝑐 𝑇,𝐹 → BG𝑚 is given by the following assignment. For 𝜓 : 𝑥 → 𝑥 + 𝑑𝜓 an isomorphism in T and 𝑡 : 𝜙 → 𝜙 + 𝑑𝑡 an isomorphism in Loc𝑐 𝑇,𝐹 , there is an isomorphism in the fibred groupoid Loc𝑐 𝑇,𝐹 × T (𝑡, 𝜓) : (𝜙, 𝑥) → (𝜙 + 𝑑𝑡, 𝑥 + 𝑑𝜓). We assign to this morphism a morphism in BG𝑚 which is given by [𝜙, 𝜓] · ⟨𝑡, 𝑥 + 𝑑𝜓⟩. To show this assignment defines a homomorphism of stacks, one must check this assignment respects composition of morphisms, and a simple calculation shows it boils down to check [𝑑𝑡, 𝜓] = ⟨𝑡, 𝑑𝜓⟩, ∀𝑡, 𝜓. The verification is straightforward: ∑︁ [𝑑𝑡, 𝜓] = ⟨𝑑𝑡 (𝑤), 𝜓(𝑤)⟩ 𝑤∈𝑊 = ∑︁ ⟨𝑤𝑡 − 𝑡, 𝜓(𝑤)⟩ 𝑤∈𝑊 = ⟨𝑡, ∑︁ 𝑤 −1 𝜓(𝑤) − 𝜓(𝑤)⟩ 𝑤∈𝑊 = ⟨𝑡, 𝑑𝜓⟩. □ 2.4.7. Proof of T  Loc∨𝑐 𝑇,𝐹 . We have the following short exact sequences by truncation of the 𝑡-structure: ˆ → T𝑛 → 𝐿ˆ Γ → 1. 1 → 𝐵𝐻1 (𝑊 (𝑛) , 𝐿) The +1 map of these exact sequences are zero. This is because, in the derived category of abelian groups, every complex is quasi-isomorphic to the direct sum of its cohomologies (Keller, 1996). Therefore, the +1 map in any truncation is zero. The above exact sequences are in the image of the exact functor embedding the drived category of abelian groups to the drived category of abelian sheaves on Spec(Z), and hence its +1 map is zero. Since in the fpqc-topology, there is no higher limit, taking derived limit on the short exact sequence gives us ˆ → T → 𝐿ˆ Γ → 1. 1 → lim B𝐻1 (𝑊 (𝑛) , 𝐿) ←−− 19 Note the +1 map is zero. By Lemma 2.4.4, we can rewrite the short exact sequence as 1 → lim B(𝑇 (𝐹)/𝑉𝑛 ) → T → 𝐿ˆ Γ → 1. ←−− Since Ext1 (Homcts (𝑇 (𝐹), G𝑚 ), BG𝑚 ) = 0 (for generality on the dualality for Picard stacks, see (Brochard, 2014)), by dualizing the short exact sequence (2.1) 1 → B𝑇ˆ Γ → Loc𝑐 𝑇,𝐹 → Homcts (𝑇 (𝐹), G𝑚 ) → 1 we get 1 → Homcts (𝑇 (𝐹), G𝑚 ) ∨ → Loc∨𝑐 𝑇,𝐹 → (B𝑇ˆ Γ ) ∨ → 1. The +1 map of the above exact triangle is zero because it is induced by the +1 map of (2.1), which is shown to be zero in section 2.3.7. We form the following diagram where the first and the third vertical maps are canonical isomorphisms and the second is the map induced by our pairing: lim 𝐵(𝑇 (𝐹)/𝑉𝑛 ) ←−− 1 Homcts (𝑇 (𝐹), G𝑚 ) ∨ T 𝐿ˆ Γ Loc∨𝑐 𝑇,𝐹 (𝐵𝑇ˆ Γ ) ∨ 1 ∼ ∼ 1 1. The second vertical map is an isomorphism because, first, the two visible squares commute by the construction of the pairing, and second, the square of the +1 maps commutes because the two +1 maps are shown to be zero. This concludes the proof that T  Loc∨𝑐 𝑇,𝐹 . 2.4.8. Proof of theorem 2.4.1. Now we prove the main theorem with T in stead of Tor𝑇,iso𝐹 . We established above that for each torus 𝑇 the pairing defined in proposition 2.4.6 induces an ismorphism T  Loc∨𝑐 𝑇,𝐹 . It remains to check that they are a unique family of line bundles on T × Loc𝑐 𝑇,𝐹 satisfying the three conditions in the theorem. Condition (a) is equivalent to the following proposition. Proposition 2.4.9. Let 𝑓 : 𝑆 → 𝑇 be a map between torus. It induces 𝛼 : Loc𝑐 𝑇,𝐹 → Loc𝑐 𝑆,𝐹 and 𝛽 : T𝑆 → T𝑇 . Let 𝑡 : 𝜙 → 𝜙 be an isomorphism in Loc𝑐 𝑇,𝐹 , and let 𝜓 : 𝑥 → 𝑥 be an isomorphism in T𝑆 . Under the pairing, (𝛼(𝜙), 𝜓) and (𝜙, 𝛽(𝜓)) are isomorphisms in BG𝑚 . We then have (𝛼(𝜙), 𝜓) = (𝜙, 𝛽(𝜓)). 20 ˆ and T𝑇 is Proof. Note that Loc𝑐 𝑇,𝐹 is the truncation of the complex 𝐻 • (𝑊𝐸/𝐹 , 𝑇), ˆ 𝑔 : 𝐿ˆ 𝑆 → 𝐿ˆ 𝑇 ˆ If we let 𝑓ˆ : 𝑇ˆ → 𝑆, the truncation of the complex 𝐻• (𝑊𝐸/𝐹 , 𝐿). ˆ the proposition follows ˆ and 𝑥 ∈ 𝑆, be the homomorphisms induced by 𝑓 , let 𝑡 ∈ 𝑇, from the trivial fact that ⟨ 𝑓ˆ(𝑡), 𝑥⟩ = ⟨𝑡, 𝑔(𝑥)⟩. □ Condition (b) follows from the definition of the pairing. We verify condition (c). Let 𝑆 = Res𝐸/𝐹 G𝑚 an induced torus and 𝑇 = G𝑚 a split torus over 𝐸. The cocharacters of the two torus satisfy 𝐿ˆ 𝑆 = IndGal(𝐸/𝐹) 𝐿ˆ 𝑇 . ∗ We apply Shapiro’s lemma to get isomorphisms   ˆ [1]) Loc𝑐 𝑆,𝐹 = ch 𝜏≤0 (𝐻 • (𝑊𝐸/𝐹 , 𝑆)
ˆ [1]) = Loc𝑐 𝑇,𝐸 ,  ch 𝜏≤0 (𝐻 • (𝐸 × , 𝑇)
Tor𝑆,iso𝐹 = ch 𝜏≥−1 𝐻• (𝑊𝐸/𝐹 , 𝐿ˆ 𝑆 )
 ch 𝜏≥−1 𝐻• (𝐸 × , 𝐿ˆ 𝑇 ) = Tor𝑇,iso𝐸 . The cap product that is used to define the pairings clearly interwines with the Shapiro isomorphisms, and hence condition (c) is satisfied. By condition (b) and (c), any such family of pairings is uniquely determined on all induced torus. To show such a family of pairings is unique, it remains to show the pairing for every torus 𝑇 is induced by the pairing for an induced torus 𝑆 that covers 𝑇. For every torus 𝑇, there is always an induced torus 𝑆 and a cover 𝑆 → 𝑇 that is surjective on cocharacters 𝐿ˆ 𝑆 → 𝐿ˆ 𝑇 . Let 𝑡 : 𝜙 → 𝜙 + 𝑑𝑡 be an isomorphism in Loc𝑐 𝑇,𝐹 and 𝜓 : 𝑥 → 𝑥 + 𝑑𝜓 an isomorphism in T𝑇 . By surjectivity of 𝐿ˆ 𝑆 → 𝐿ˆ 𝑇 , we can lift 𝜓 : 𝑥 → 𝑥 + 𝑑𝜓 to 𝜓˜ : 𝑥˜ → 𝑥˜ + 𝑑 𝜓˜ in T𝑆 . By functorality of the pairings, we have ˜ (𝑡, 𝜓) = (𝛼(𝑡), 𝜓). Therefore the pairing on 𝑇 is predetermined by that of 𝑆. This concludes the proof of the main theorem. We finish the section with 21 2.4.10. Proof of T  Tor𝑇,iso𝐹 . Recall that   1 Tor𝑇,iso𝐹 = lim ch · · · → 0 → 𝑇 (𝐸)/𝑈𝑛 → 𝑍alg (𝑊 (𝑛) , 𝑇 (𝐸)) → 0 → · · · . We first define a map between groupoids T (Z) → Tor𝑇,iso𝐹 (Z) by defining the two vertical maps in the following diagram such that the square commutes ˆ ˆ 𝐶1 (𝑊𝐸/𝐹 , 𝐿)/𝐵 1 (𝑊𝐸/𝐹 , 𝐿) 𝑑 𝐿ˆ 𝑐0 cores 𝑑 𝑇 (𝐸) (2.6) 1 (𝑊 𝑍alg 𝐸/𝐹 , 𝑇 (𝐸)). In the following, we make the identification 𝑇 (𝐸) = 𝐿ˆ ⊗ 𝐸 × , with 𝑊𝐸/𝐹 acting diagonally. We define the first verticle map to be the corestriction map (2.4) × ˆ ˆ ˆ −cores ˆ 𝐶1 (𝑊𝐸/𝐹 , 𝐿)/𝐵 −−→ 𝐶1 (𝐸 × , 𝐿)/𝐵  𝑇 (𝐸). 1 (𝑊𝐸/𝐹 , 𝐿) 1 (𝐸 , 𝐿) Í Following (2.4), it sends 𝑤 ⊗ 𝑥 to 𝜏 𝑤 𝜏 𝑥 ⊗ 𝛿(𝑤 𝜏 , 𝑤). The fact that this map induces isomorphism on the cohomology of degree −1 is provided by Corollary 2.4.5. 1 (𝑊 The second verticle map 𝑐 0 : 𝐿ˆ → 𝑍alg 𝐸/𝐹 , 𝑇 (𝐸)) is defined via two maps, Γ 1 1 (𝑊 𝐿ˆ → 𝐿ˆ and 𝐿ˆ → 𝑍 (𝑊𝐸/𝐹 , 𝑇 (𝐸)), since 𝑍alg 𝐸/𝐹 , 𝑇 (𝐸)) is defined as a fibre product 𝐿ˆ ×𝑍 1 (𝐸 × ,𝑇 (𝐸)) 𝑍 1 (𝑊𝐸/𝐹 , 𝑇 (𝐸)). We order the first map to be the norm map, and the second to be the composition inf 𝐿ˆ → 𝑍 1 (𝐸 × , 𝑇 (𝐸)) −−→ 𝑍 1 (𝑊𝐸/𝐹 , 𝑇 (𝐸)). The inflation map on cocyles is defined as follows. Let 𝜙 : 𝐸 × → 𝑇 (𝐸) be a cocycle. The inflation of this cocycle is usually given by ∑︁ 𝑔 ↦→ 𝑤 −1 𝜏 𝜙(𝛿(𝑤 𝜏 , 𝑔)). 𝜏 We define the inflation map by the same formula, however with a different system of representatives given by { 𝑤˜ 𝜏 = 𝑤 −1 , 𝜏 ∈ Γ}. 𝜏 −1 Explicitly, 𝑐 0 (𝑥) is the cocycle 𝑔 ↦→ = = ∑︁ ∑︁𝜏 𝜏 ∑︁ 𝜏 ˜ ˜ 𝜏 , 𝑔)) 𝑤˜ −1 𝜏 (𝑥 ⊗ 𝛿( 𝑤 ˜ ˜ 𝜏 , 𝑔) 𝑤˜ −1 ˜ −1 𝜏 𝑥 ⊗𝑤 𝜏 𝛿( 𝑤 𝑤 𝜏 −1 𝑥 ⊗ 𝛿(𝑔, 𝑤 𝑔 −1 𝜏 −1 ). 22 (For notation, see section 2.1.1.) The map 𝑐 0 agrees with the map 𝑐 0 Kottwitz assign to the Tate-Nakayama triple (Z, 𝐸 × , 𝛼) in (Kottwitz, 2014), and that it induce isomorphism on the cohomology of degree 0 is exactly Lemma 5.1 in loc.cit. The following calculation verifies the diagram (2.6) commutes. Proposition 2.4.11. We have 𝑐 0 ◦ 𝑑 = 𝑑 ◦ cores. Proof. From above, 𝑑 ◦ cores(𝑤 ⊗ 𝑥) is the cocycle ∑︁ ∑︁ 𝑔 ↦→ 𝑔𝑤 𝜏 𝑥 ⊗ 𝑔𝛿(𝑤 𝜏 , 𝑤) − 𝑤 𝜏 𝑥 ⊗ 𝛿(𝑤 𝜏 , 𝑤). 𝑤𝜏 𝑤𝜏 We change the variable 𝜏 → 𝑔 −1 𝜏 in the first summation. The right hand side becomes ∑︁ 𝑤 𝜏 𝑥 ⊗ (𝑔𝛿(𝑤 𝑔 −1 𝜏 , 𝑤) − 𝛿(𝑤 𝜏 , 𝑤)) = 𝜏 ∑︁ 𝑤 𝜏 𝑥 ⊗ (𝛿(𝑔, 𝑤 𝑔 −1 𝜏𝑤 ) − 𝛿(𝑔, 𝑤 𝑔 −1 𝜏 )). (2.7) 𝜏 On the other hand, 𝑐 0 ◦ 𝑑 (𝑤 ⊗ 𝑥) is the cocycle ∑︁ ∑︁ 𝑔 ↦→ 𝑤 𝜏 −1 𝑤 −1 𝑥 ⊗ 𝛿(𝑔, 𝑤 𝑔 −1 𝜏 −1 ) − 𝑤 𝜏 −1 𝑥 ⊗ 𝛿(𝑔, 𝑤 𝑔 −1 𝜏 −1 ). 𝜏 𝜏 We change the variable 𝜏 → 𝑤 −1 𝜏 in the first summation. The right hand side becomes ∑︁ 𝑤 𝜏 −1 𝑥 ⊗ (𝛿(𝑔, 𝑤 𝑔 −1 𝜏 −1 𝑤 ) − 𝛿(𝑔, 𝑤 𝑔 −1 𝜏 −1 )). 𝜏 This equals (2.7). □ We continue with the proof that T  Tor𝑇,iso𝐹 . Both T and Tor𝑇,iso𝐹 has a pro-stack structure. We can use the same construction for the commuting diagram (2.6) when we replace 𝑊𝐸/𝐹 by 𝑊 (𝑛) and 𝑇 (𝐸) by 𝑇 (𝐸)/𝑈𝑛 . The definition of the algebraic 1 (𝑊 1 (𝑛) , 𝑇 (𝐸)/𝑈 ). We cycles 𝑍alg 𝑛 𝐸/𝐹 , 𝑇 (𝐸)) can be easily adapted to define 𝑍 alg (𝑊 claim that the diagram is a quasi-isomorphism for sufficiently large 𝑛, and therefore together they define an isomorphism T  Tor𝑇,iso𝐹 . By Lemma 2.4.3, there is an isomorphism for large 𝑛 1 1 1 𝐻alg (𝑊𝐸/𝐹 , 𝑇 (𝐸)) → 𝐻alg (𝑊𝐸/𝐹 , 𝑇 (𝐸)/𝑈𝑛 ) = 𝐻alg (𝑊 (𝑛) , 𝑇 (𝐸)/𝑈𝑛 ). This shows the induced map on cohomology at degree 0 is an isomorphism for large 𝑛. Lemma 2.4.4 says the induced map on cohomology at degree -1 is an isomorphism for large 𝑛. This finishes the proof. 23 2.5 Categorical LLC for the torus In the previous section, we constructed a canonical family of Poincaré line bundles L : Tor𝑇,iso𝐹 × Loc𝑐 𝑇,𝐹 → BG𝑚 . They induce isomorphisms Tor𝑇,iso𝐹  Loc∨𝑐 𝑇,𝐹 . Theorem 2.5.1. Let 𝜋1 , 𝜋2 be the projects of Loc𝑐 𝑇,𝐹 × Tor𝑇,iso𝐹 to its factors. A Fourier-Mukai transform via the line bundle L establish the equivalence of ∞categories that preseres t-structures: 𝜋1! (𝜋2∗ (−) ⊗ L −1 ) : QCoh(Tor𝑇,iso𝐹 ) ⇋ IndCoh(Loc𝑐 𝑇,𝐹 ) : 𝜋2∗ (𝜋1∗ (−) ⊗ L), where the functor 𝜋1! is the right adjoint of 𝜋1∗ and 𝜋2∗ is the left adjoint of 𝜋2∗ . Before we can prove the theorem, we gather some useful facts as follows. Recall that by the system of congrunce subgroups 𝑈 (𝑛) ⊂ O𝐸× give rise to a basis of open subgroups 𝑉𝑛 of 𝑇 (𝐹). Using Corollary 2.4.5, we have the following short exact sequence: 0 → Hom( 𝐿ˆ Γ , BG𝑚 ) → Loc𝑐(𝑛) 𝑇,𝐹 → Homcts (𝑇 (𝐹)/𝑉𝑛 , G𝑚 ) → 0. We let Loc = Homcts (𝑇 (𝐹), G𝑚 ) and let Loc (𝑛) = Homcts (𝑇 (𝐹)/𝑉𝑛 , G𝑚 ). By proposition 2.3.7, Loc𝑐 𝑇,𝐹 splits as Loc𝑐 𝑇,𝐹 = Loc × B𝑇ˆ Γ and similarly Loc𝑐(𝑛) 𝑇,𝐹 = Loc (𝑛) × B𝑇ˆ Γ . We define the regular function ring OLoc to be the union of all OLoc (𝑛) , i.e., OLoc consists of functions supported on some Loc (𝑛) for some 𝑛. Let 𝑇 (𝐹)0 be the maximal compact subgroup of 𝑇 (𝐹) and let 𝜇 be the (left) Haar measure on 𝑇 (𝐹) such that 𝜇(𝑇 (𝐹)0 ) = 1. We define the Hecke algebra of 𝑇 (𝐹) to be Q-valued compactly supported smooth function on 𝑇 (𝐹) equipped with the convolution product with respect to 𝜇 and denote it by H (𝑇 (𝐹), Q). Lemma 2.5.2. There is an isomorphism OLoc ⊗ Q  H (𝑇 (𝐹), Q). Proof. We choose a splitting 𝑇 (𝐹) = 𝑇 (𝐹)0 ⊕ 𝑅, where 𝑅 is a free abelian group of finite rank. It suffice to check the claim on each subgroup. The Hecke algebra of 𝑅 is isomorphic to the group ring Z[𝑅], and this is isomorphic to the function ring of Hom(𝑅, G𝑚 ). Let 𝐺 𝑛 = 𝑇 (𝐹)0 /𝑉𝑛 be a tower of finite abelian groups with projections 𝑝 𝑛 : 𝐺 𝑛+1 → 𝐺 𝑛. 24 Consider the map Q[𝐺 𝑛 ] → H (𝐺 𝑛 , Q) that sends an element 𝑔 ∈ 𝐺 𝑛 to |𝐺 𝑛 | · 𝛿𝑔 . By direct computation, this map is an algebra isomorphism (with C-coefficient, this is the familiar discrete Fourier transform). The Hecke algebra H (𝑇 (𝐹)0 , Q) is the union of all H (𝐺 𝑛 , Q), where the inclusion map H (𝐺 𝑛 , Q) → H (𝐺 𝑛+1 , Q) is the pullback of functions along projections 𝑝 𝑛 . Let 𝑖 𝑛 : Q[𝐺 𝑛 ] → Q[𝐺 𝑛+1 ] be the unique map such that the following diagram commutes: ∼ Q[𝐺 𝑛 ] H (𝐺 𝑛 , Q) 𝑖𝑛 Q[𝐺 𝑛+1 ] ∼ H (𝐺 𝑛+1 , Q). It remains to identify the map 𝑖 𝑛 with the inclusion OLoc (𝑛) ⊂ OLoc (𝑛+1) . This is readily manifested given that the image of 1 under 𝑖 𝑛 is an idempotent ∑︁ 1 𝑖 𝑛 (1) = 𝑔, |𝐺 𝑛+1 | 𝑔∈ker( 𝑝 𝑛 ) and that 𝑖 𝑛 (1) · Q[𝐺 𝑛+1 ] is precisely the image of Q[𝐺 𝑛 ]. □ Remark 2.5.3. This proposition amounts to saying that under our Langlands correspondence, OLoc𝑐 𝑇 ,𝐹 is send to the Whittaker sheaf, i.e., the Hecke algebra with the usual 𝑇 (𝐹)-action. 2.5.4. Decomposition of categories. Under the splitting Loc𝑐 𝑇,𝐹 = Loc × B𝑇ˆ Γ , the Poincaré line bundle L on Loc𝑐 𝑇,𝐹 × Tor𝑇,iso𝐹 can be regarded as the structural sheaf over Loc× 𝐿ˆ Γ with both a 𝑇ˆ Γ and a 𝑇 (𝐹)-action, such that 𝑇ˆ Γ acts by the character in 𝑥 ∈ 𝐿ˆ Γ on the 𝑥-component, and 𝑡 ∈ 𝑇 (𝐹) acts on Loc𝑛 × 𝐿ˆ Γ by 𝑡𝑉𝑛 ∈ Z[𝑇 (𝐹)/𝑉𝑛 ] as described in the previous lemma. Since Loc𝑐 𝑇,𝐹 = Loc × B𝑇ˆ Γ is a 𝑇ˆ Γ -gerbe, ind-coherent sheaves on Loc𝑐 𝑇,𝐹 decompose as Ö IndCoh(Loc𝑐 𝑇,𝐹 ) = IndCoh𝛼 (Loc𝑐 𝑇,𝐹 ). (2.8) 𝛼∈𝑋 ∗ (𝑇ˆ Γ ) The subcategory IndCoh𝛼 (Loc𝑐 𝑇,𝐹 ) comprise ind-coherent sheaves on Loc𝑐 𝑇,𝐹 that has an action of 𝑇ˆ Γ via the character 𝛼. It is equivalent to IndCoh(Loc). Connected components of Tor𝑇,iso𝐹 are indexed by the Kottwitz set 𝐵(𝑇) = 𝑋 ∗ (𝑇ˆ Γ ). Let the corresponding component of 𝛽 ∈ 𝑋 ∗ (𝑇ˆ Γ ) be 𝐵𝑇 (𝐹) 𝛽 . We have a decomposition by connected components Ö QCoh(𝐵𝑇 (𝐹) 𝛽 ). QCoh(Tor𝑇,iso𝐹 ) = 𝛽∈𝑋 ∗ (𝑇ˆ Γ ) 25 We have an open embedding 𝐵𝑇 (𝐹) 𝛽 × Loc𝑐 𝑇,𝐹 ↩→ Tor𝑇,iso𝐹 × Loc𝑐 𝑇,𝐹 , and we denote the restriction of the Poincaré line bundle by L 𝛽 . By abuse of notation, we let 𝜋1 and 𝜋2 be the two projection of 𝐵𝑇 (𝐹) 𝛽 × Loc𝑐 𝑇,𝐹 to its factors. A simple calculation should reveal that for F ∈ Coh𝛼 (Loc𝑐 𝑇,𝐹 ) and 𝛽 ≠ −𝛼 𝜋2∗ (𝜋1∗ F ⊗ L 𝛽 ) = 0. Therefore, showing the equivalence of category QCoh(Tor𝑇,iso𝐹 )  Coh(Loc𝑐 𝑇,𝐹 ) is reduced to showing the equivalence of the subcategories by pull-push −1 𝜋1! (𝜋2∗ (−) ⊗ L−𝛼 ) : QCoh(𝐵𝑇 (𝐹)−𝛼 ) ⇋ IndCoh𝛼 (Loc𝑐 𝑇,𝐹 ) : 𝜋2∗ (𝜋1∗ (−) ⊗ L−𝛼 ). 2.5.5. The equivalence. In this section we show the following functor is an equivalence: 𝜋2∗ (𝜋1∗ (−) ⊗ L−𝛼 ) : IndCoh𝛼 (Loc𝑐 𝑇,𝐹 ) → QCoh(𝐵𝑇 (𝐹)−𝛼 ).  Loc𝑐(𝑛) 𝑇,𝐹 (2.9)  . Consider the coherent sheaf 𝜋1∗ F ⊗ L−𝛼 on Loc𝑐 𝑇,𝐹 × B𝑇 (𝐹)−𝛼 , it can be regarded as a coherent sheaf on Loc with a 𝑇ˆ Γ and a 𝑇 (𝐹)-action. By definition of L−𝛼 , the 𝑇ˆ Γ -action of 𝜋 ∗ F ⊗ L−𝛼 is trivial, and the 𝑇 (𝐹)-action Let F ∈ Coh 𝛼 1 is identified with the OLoc (𝑛) = Z[𝑇 (𝐹)/𝑉𝑛 ]-action. The pushforward by 𝜋2∗ simply forgets the 𝑇ˆ Γ -action. In summary, the pull-push factors as 𝜋2∗ (𝜋1∗ (−)⊗L − 𝛼 ) Coh𝛼 (Loc𝑐(𝑛) 𝑇,𝐹 ) QCoh(𝐵𝑇 (𝐹)−𝛼 ). ∼ n finite Z[𝑇 (𝐹)/𝑉𝑛 ]-Mod o By taking colimit, this results in the equivalence of categories 𝜋2∗ (𝜋1∗ (−)⊗L − 𝛼 ) IndCoh𝛼 (Loc) ∼ QCoh(𝐵𝑇 (𝐹)−𝛼 ). ∼ Repsm (𝑇 (𝐹)) 2.5.6. The inverse functor. The functor 𝜋1∗ preserves limits and is a right adjoint. We denote its left adjoint by 𝜋1! . We first note that both 𝜋1∗ and 𝜋1! 26 respect the decomposition of IndCoh(Loc𝑐 𝑇,𝐹 × B𝑇 (𝐹)−𝛼 ) and IndCoh(Loc𝑐 𝑇,𝐹 ) as 𝑇ˆ Γ -gerbes, we can restrict the following discussion to 𝑝 ∗1 and 𝑝 1! for the map 𝑝 1 : Loc × B𝑇 (𝐹)−𝛼 → Loc and 𝑝 2 : Loc × B𝑇 (𝐹)−𝛼 → B𝑇 (𝐹)−𝛼 . Let 𝑋 = Loc × Tor𝑇,iso𝐹 and 𝑌 = Loc. On the level of abelian categories, let 𝐺 : QCoh(𝑋) ♥ → QCoh(𝑌 ) ♥ be the functor of taking 𝑇 (𝐹)-coinvariants. Then 𝐺 and 𝑝 ∗1 is an adjoint pair. The derived functors 𝐿𝐺 and 𝑝 ∗1 (no need to derive) lift to ∞-categories and is still an adjoint pair. Therefore 𝐿𝐺 = 𝑝 1! . Let 𝑀 ∈ QCoh(B𝑇 (𝐹)−𝛼 ) such that 𝑀 is fixed by 𝑉𝑛 . The quasi-coherent sheaf (𝑛) −1 = 𝑀 ⊗ O and it has an 𝑝 ∗2 𝑀 ⊗ L−𝛼 Loc restricts to 𝑀 ⊗ Z[𝑇 (𝐹)/𝑉𝑛 ] on Loc 𝑇 (𝐹)-action that is identified with its Z[𝑇 (𝐹)/𝑉𝑛 ]-module structure. Due to the fact that Z[𝑇 (𝐹)/𝑉𝑛 ] is a projective 𝑇 (𝐹)-module, we have −1 𝑝 1! ( 𝑝 ∗2 𝑀 ⊗ L−𝛼 )| Loc (𝑛) = 𝐿𝐺 (𝑀 ⊗ Z[𝑇 (𝐹)/𝑉𝑛 ])  𝑀. We emphesize again that this isomorphism endows 𝑀 the with structure of an OLoc -module which is identified with its 𝑇 (𝐹)-module structure. In general, take 𝑀 = colim 𝑀𝑛 where 𝑀𝑛 = 𝑀 𝑉𝑛 . As 𝑝 1! commute with filtered colimits, the same statement holds. In other words, the functor 𝜋1! (𝜋2∗ (−) ⊗ L −1 ) is the inverse of 𝜋2∗ (𝜋1∗ (−) ⊗ L). 27 Chapter 3 THE SECOND CATEGORIFICATION 3.1 Notation 3.1.1. The inertia group. We let 𝐼 ⊂ Γ be the inertia group of the field extension 𝐸/𝐹. Assume Ω𝐸 and Ω𝐹 are the maximal unramified extension of 𝐸 ¯ 𝐸 ) and and 𝐹, we denote the absolute inertia group of 𝐸 and 𝐹 by 𝐼 𝐸 = Gal( 𝐸/Ω ¯ 𝐹 ). They are subgroups of 𝑊𝐸 and 𝑊𝐹 respectively. We also define 𝐼 𝐹 = Gal( 𝐹/Ω rel := 𝐼 /[𝐼 , 𝐼 ]. the relative inertia group as 𝐼 𝐸/𝐹 𝐹 𝐸 𝐸 The relative inertia group is a group extension: rel 1 → 𝐼 𝐸ab → 𝐼 𝐸/𝐹 → 𝐼 → 1. 3.1.2. The loop group. We denote by 𝑘 and O the residue field and ring of integers of 𝐹. Let 𝜛 denote a uniformizer of O. For a 𝑘-algebra 𝑅, we let 𝑊 (𝑅) denote its ring of Witt vectors, let 𝑊O (𝑅) = 𝑊 (𝑅) ⊗𝑊 (𝑘) O and 𝑊O,𝑛 (𝑅) = 𝑊 (𝑅) ⊗𝑊 (𝑘) O/𝜛 𝑛 . Let Y be a finite-type O-scheme. According to Greenberg, the following two presheaves on the category of affine 𝑘-schemes: 𝐿 +𝑝 Y (𝑅) = Y (𝑊O (𝑅)), 𝐿 𝑛𝑝 Y (𝑅) = Y (𝑊O,𝑛 (𝑅)), are represented by schemes over 𝑘. We denote their perfection by −∞ 𝐿 + Y = (𝐿 +𝑝 Y) 𝑝 , −∞ 𝐿 𝑛 Y = (𝐿 𝑛𝑝 Y) 𝑝 , and call them 𝑝-adic jet spaces. Let 𝑌 be an affine scheme of finite type over 𝐹. The 𝑝-adic loop space 𝐿𝑌 of 𝑌 is a perfect space defined by assigning a perfect 𝑘-algebra 𝑅 the set 𝐿𝑌 (𝑅) = 𝑌 (𝑊O (𝑅) [1/𝑝]). 𝐿𝑌 is represented by an ind-perfect scheme. Assume Y is an affine scheme of finite type over O and 𝑋 = Y ⊗O 𝐹, then 𝐿 + Y ⊂ 𝐿𝑌 is a closed subscheme. Take 𝑌 = 𝑇. We let 𝐿 ≥𝑛𝑇 ⊂ 𝐿 +𝑇 be the congruence subgroup such that 𝐿 ≥𝑛𝑇 (𝜅 𝐹 ) = 𝑉𝑛 . 28 3.1.3. Character sheaves and Serre’s fundamental group. Let Λ = Zℓ , Qℓ or Fℓ . Let 𝐻 be a connected pro-algebraic group over Fℓ , and let 𝑚 be the multiplication map of 𝐻. Recall that a character sheaf with coefficient in Λ on 𝐻 is a rank one Λ-local system Ch𝜉 on 𝐻 equipped with an isomorphism 𝑚 ∗ Ch𝜉  Ch𝜉 ⊠Λ Ch𝜉 , which satisfies the usual cocycle conditions. Definition 3.1.4. Let 𝐻 be a (not necessarily) connected pro-algebraic group, and let 𝐻 ◦ be its central component. We define the character sheaves on 𝐻 to consists of all the translations of character sheaves on 𝐻 ◦ . Denote by P the category of commutative pro-algebraic groups over 𝑘 = Fℓ and by P0 the category of abelian profinite groups. Serre defined the fundamental groups 𝜋1 as the first derived functor of the right exact functor 𝜋0 : P → P0 , 𝜋0 (𝐺) := 𝐺/𝐺 ◦ (Serre, 1961). Assume 𝐻 is abelian and connected. A key property of this fundamental group is that the abelian group of character sheaves on 𝐻, which we denote by CS(𝐻, Λ), is isomorphic to the abelian group of continuous rank 1 representations of 𝜋1 (𝐻): Homcts (𝜋1 (𝐻), Λ× )  CS(𝐻, Λ). The goal of the second categorification is essentially to categorify this isomorphism into a fully faithful functor (take 𝐻 = 𝐿 +𝑇)
Ch : Coh Homcts (𝜋1 (𝐿 +𝑇), G𝑚 ) → Shv(𝐿 +𝑇, Λ). The following theorem, which can be called the geometric class field theory, will play a key role in the sequel. Theorem 3.1.5 (Serre (1961); Deshpande and Wagh (2023)). Let 𝑇 be an arbitrary torus defined over 𝐹 and splits over a finite Galois extension 𝐸 of 𝐹. Let 𝐼 be the inertia group of the field extension 𝐹/𝐸 and let 𝐼 𝐸ab be the abelianization of the (absolute) inertia group of 𝐸. Then there is a canonical isomorphism  𝐼 𝜋1 (𝐿 +𝑇)  𝐿ˆ ⊗ 𝐼 𝐸ab . 29 3.1.6. The fixed point stack. Let 𝑌 be an arbitrary stack equipped with an automorphism 𝜎𝑌 : 𝑌 → 𝑌 . We define the fix-point stack of 𝜎𝑌 to be L𝜎 𝑌 = 𝑌 Γ, 𝑌 ×𝑌 , Δ 𝑌 , where Γ = (1, 𝜎𝑌 ) is the graph of 𝜎𝑌 and Δ is the diagonal embedding. Proposition 3.1.7. The fixed point stack L𝜎 B𝐿𝑇 = 𝐿𝑇 . Ad𝜎 𝐿𝑇 Proof. By definition L𝜎 B𝐿𝑇 is the fibre product L𝜎 B𝐿𝑇 B𝐿𝑇 Δ B𝐿𝑇 id×𝜎 B𝐿𝑇 × B𝐿𝑇 . Objects of L𝜎 B𝐿𝑇 are represented by a tuple (𝑥, 𝑦) ∈ 𝐿𝑇 and morphisms (𝑡, 𝑠) : (𝑥, 𝑦) → (𝑥 ′, 𝑦′) are a tuple (𝑡, 𝑠) ∈ 𝐿 +𝑇 such that ∗×∗ (𝑥,𝑦) (𝑡,𝜎𝑡) ∗×∗ ∗×∗ (𝑠,𝑠) (𝑥 ′ ,𝑦 ′ ) ∗ × ∗. In other words, (𝑡, 𝑠) : (𝑥, 𝑦) → (𝑥 + 𝑠 − 𝑡, 𝑦 + 𝑠 − 𝜎𝑡). It is straightforward from here to see that L𝜎 B𝐿𝑇 = Ad𝐿𝑇 . □ 𝜎 𝐿𝑇 3.2 Representation Stack of Inertia Definition 3.2.1. The moduli space R 𝐼𝐹 ,𝑐 𝑇 of the (strongly) continuous representaˆ For every Zℓ -algebra tions of 𝐼 𝐹 in 𝑐𝑇 is defined over Zℓ as follows. Let 𝑟 = rank 𝑇. 𝐴, R 𝐼𝐹 ,𝑐 𝑇 ( 𝐴) classifies the cross homomorphisms 𝜌 : 𝐼 𝐹 → 𝑇ˆ ( 𝐴) ⊂ 𝐴𝑟 such that for each coordinate map 𝜈 : 𝐴𝑟 → 𝐴, the Zℓ subspace 𝑀 generated by 𝜈 ◦ 𝜌(𝐼 𝐹 ) is a finitely generated Zℓ -module, and the representation of 𝐼 𝐹 on 𝑀 is continuous, with 𝑀 equipped with the usual ℓ-adic topology. geom ˆ Definition 3.2.2. The representation stack of inertia is Loc𝑐 𝑇,𝐹 := R 𝐼𝐹 ,𝑐 𝑇 /𝑇. geom ˆ Explicitly, the action The Frobenius 𝜎 acts on Loc𝑐 𝑇,𝐹 via its action on 𝐼 𝐹 and 𝑇. sends a cross homomorphism 𝑐 : 𝐼 𝐹 → 𝑇ˆ to 𝜎𝑐 such that 𝜎𝑐(𝑔) = 𝜎 𝑐(𝜎 −1 𝑔𝜎). geom The following proposition reveals the connection between Loc𝑐 𝑇,𝐹 and Loc𝑐 𝑇,𝐹 . 30 geom Proposition 3.2.3. The fix-point stack L𝜎 Loc𝑐 𝑇,𝐹 is canonically isomorphic to Loc𝑐 𝑇,𝐹 ⊗ Zℓ . geom Proof. There is a natural map Loc𝑐 𝑇,𝐹 ⊗ Zℓ → L𝜎 Loc𝑐 𝑇,𝐹 induced by restricting a ˆ The goal is to show that this map cross homomorphism 𝑐 : 𝑊𝐹 → 𝑇ˆ to 𝑐′ : 𝐼 𝐹 → 𝑇. is surjective on objects and is a bijection on the automorphism group of an object. geom ˆ and Objects of L𝜎 Loc𝑐 𝑇,𝐹 can be represented by a tuple ([𝑐], 𝑡) for 𝑐 ∈ 𝑍 1 (𝐼 𝐹 , 𝑇) ˆ if 𝑐˜ satisfies 𝑐| 𝑡 ∈ 𝑇ˆ such that 𝜎𝑐 = 𝑐 + 𝑑𝑡. It is the image of 𝑐˜ ∈ 𝑍 1 (𝑊𝐹 , 𝑇) ˜ 𝐼𝐹 = 𝑐 and 𝑐(𝜎) ˜ = 𝑡. To show that such 𝑐˜ exists, we first need to check that 𝑐 factors through a finite quotient of 𝐼 𝐹 . The restriction of 𝑐 to the wild inertia 𝑃 𝐹 clearly factors through a finite quotient. Therefore, it suffices to show the claim for the restriction of 𝑐 to 𝐼 𝐹tame . Let the action of 𝐼 𝐹tame on 𝑇ˆ factor through an open subgroup 𝐾. Then for all 𝑔 ∈ 𝐾, we have 𝑐(𝑔) = 𝜎 𝑐(𝜎 −1 𝑔𝜎). Henceforth 𝜎 𝑐(𝑔) = 𝑐(𝜎𝑔𝜎 −1 ) = 𝑐(𝑔 𝑞 ) = 𝑐(𝑔) 𝑞 . Let ˆ we have 𝑐(𝑔) = 𝑐(𝑔) 𝑞 𝑁 . That 𝑁 be a positive integer so that 𝜎 𝑁 acts trivially on 𝑇, 𝑁 is, for every 𝑔 ∈ 𝐾, 𝑐(𝑔 𝑞 −1 ) = 1. We conclude that 𝑐 factors through an open 𝑁 −1 subgroup of 𝐼 𝐹tame as {𝑔 𝑞 |𝑔 ∈ 𝐾 } is open. One define 𝑐(𝜎 ˜ 𝑛 ) recursively via 𝑐(𝜎 ˜ 𝑛 ) = 𝑡 + 𝜎 𝑐(𝜎 ˜ 𝑛−1 ) for all 𝑛 ∈ N and define −𝑛 𝑐(𝜎 ˜ −𝑛 ) = −𝜎 𝑐(𝜎 ˜ 𝑛 ), 𝑛 𝑐(𝜎 ˜ 𝑚 𝑔) = 𝑐(𝜎 ˜ 𝑚 ) + 𝜎 𝑐(𝑔), for all 𝑛 ∈ N, 𝑚 ∈ Z and for all 𝑔 ∈ 𝐼 𝐹 . The condition 𝜎𝑐 = 𝑐 + 𝑑𝑡 ensures that 𝑐˜ is a 1-cocycle. An automorphism of the object ([𝑐], 𝑡) is represented by an element 𝑤 ∈ 𝑇ˆ such that 𝑑𝑤 = 0 and the following diagram commutes: 𝑡 𝑐 𝜎𝑐 𝑤 𝜎𝑤 𝑡 𝑐 𝜎𝑐. That is, 𝑤 ∈ 𝑇ˆ 𝐼𝐹 and 𝜎𝑤 = 𝑤, hence 𝑤 ∈ 𝑇ˆ 𝑊𝐹 . This automorphism group is exactly in bijection with the automorphism group of an object of Loc𝑐 𝑇,𝐹 . □ 31 Proposition 3.2.4. We have a canonically split short exact sequence geom 1 → Hom( 𝐿ˆ 𝐼 , BG𝑚 ) → Loc𝑐 𝑇,𝐹 → Homcts (( 𝐿ˆ ⊗ 𝐼 𝐸ab ) 𝐼 , G𝑚 ) → 1. (3.1) Proof. The proof of this proposition is similar to that of proposition 2.3.1. The only ˆ  𝑇 (𝐹) with difference is that we need to replace the isomorphism 𝐻1 (𝑊𝐸/𝐹 , 𝐿) rel , 𝐿) ˆ  ( 𝐿ˆ ⊗ 𝐼 ab ) 𝐼 , which is supplied by Corollary 2.4.5. □ 𝐻1 (𝐼 𝐸/𝐹 𝐸 geom 3.2.5. Loc𝑐 𝑇,𝐹 as an ind-scheme. We want to understand the geometry of the third term in this exact sequence. We first note that the abelian group 𝐼 𝐸ab splits into its tame and wild parts as an Γ-module. By class field theory, we have 𝐼 𝐸ab = lim ←−− O𝐾× . 𝐾/𝐸 finite Galois, unramified Each term O𝐾× = 𝜇 𝐾 × 𝑈𝐾(1) splits as a Γ-module, where 𝜇 𝐾 is the group of roots of unity in 𝐾 and 𝑈𝐾(1) = 1 + 𝔪 𝐾 the principal units. This splitting clearly respects the norm map, which is the transition map in the projective limit. Therefore we have a splitting tame 𝐼 𝐸ab = lim 𝑈𝐾(1) × 𝐼 𝐸tame = 𝑃ab 𝐸 × 𝐼𝐸 . ←−− The group lim 𝑈𝐾(1) can be identified with the abelianization of the wild inertia ←−− 𝑃ab 𝐸 (Iwasawa, 1955). Although we do not rely on this fact, but we use it to simplify our notation. With the splitting of the group 𝐼 𝐸ab , we obtain a decomposition 𝐼 Homcts (( 𝐿ˆ ⊗ 𝐼 𝐸ab ) 𝐼 , G𝑚 ) = Homcts (( 𝐿ˆ ⊗ 𝐼 𝐸tame ) 𝐼 , G𝑚 ) × Homcts (( 𝐿ˆ ⊗ 𝑃ab 𝐸 ) , G𝑚 ) which is characterized by that ( 𝐿ˆ ⊗ 𝐼 𝐸tame ) 𝐼 is a prime-to-𝑝 profinite group and 𝐼 ( 𝐿ˆ ⊗ 𝑃ab 𝐸 ) is a pro-𝑝 group. This gives us Ä Ä ˆ G𝑚 ) = Homcts ( 𝐽, R 𝐽,G (3.2) Homcts (( 𝐿ˆ ⊗ 𝐼 𝐸ab ) 𝐼 , G𝑚 ) = ˆ 𝑚. 𝑥∈Ξ 𝑥∈Ξ tame 𝐼 𝐼 ˆ ˆ where Ξ = Homcts (( 𝐿ˆ ⊗ 𝑃ab 𝐸 ) , G𝑚 ), and 𝐽 = ( 𝐿 ⊗ 𝐼 𝐸 ) . The functor R 𝐽,G ˆ 𝑚 can be made explicit by embedding it into an algebraic torus. We choose 𝜏 a topological generator of 𝐼 𝐸tame , and let the finitely generated abelian 32 group ( 𝐿ˆ ⊗ ⟨𝜏⟩) 𝐼 ⊂ 𝐽ˆ be denoted by 𝐽. Denote by 𝐻ˆ the torus R 𝐽,G𝑚 . There is an ˆ by restricting a representation of 𝐽ˆ to 𝐽. inclusion R 𝐽,G ˆ 𝑚 → 𝐻 Let us denote 𝐻ˆ 𝑝 ⊂ 𝐻ˆ the subfunctor which is the union of all closed subschemes 𝑖 𝑍 : 𝑍 ⊂ 𝐻ˆ that are finite over Zℓ and 𝑍 (Fℓ ) ⊂ 𝐻ˆ (Fℓ ) 𝑝 , where 𝐻 (Fℓ ) 𝑝 consists of all elements in 𝐻ˆ (Fℓ ) of order prime to 𝑝. Note that by construction 𝐻ˆ 𝑝 has a natural structure of an ind-scheme. ˆ identifies R 𝐽,G ˆ 𝑝 . ConseProposition 3.2.6. The inclusion R 𝐽,G ˆ 𝑚 → 𝐻 ˆ 𝑚 with 𝐻 quently, R 𝐽,G ˆ 𝑚 is an ind-scheme. Proof. Let 𝐴 be an Zℓ -algebra. Every 𝜙 ∈ R 𝐽,G ˆ 𝑚 ( 𝐴) represents a continuous cross homomorphism 𝜑 : 𝐽ˆ → 𝐴× ⊂ 𝐴 such that the image of 𝜑 in 𝐴 spans a finitely generated Zℓ -module 𝑀, and the map 𝜙 : 𝐽ˆ → 𝑀 is continuous. The image of 𝜙 ∈ 𝐻ˆ gives a ring homomorphism which we also denote by 𝜙 : Zℓ [𝐽] → 𝐴. This is justified because this 𝜙 coincide with the cross homomorphism in their value in 𝐴. Let 𝜙 factor as Zℓ [𝐽] → 𝑅 ↩→ 𝐴, where Spec 𝑅 is the schematic image of Spec 𝐴. Our conditions on 𝜙 imply that 𝑅 is a finite Zℓ -algebra. Henceforth, all Fℓ -points of Spec 𝑅 can lift to Fℓ -points of Spec 𝐴, their image in 𝐻ˆ (Fℓ ) must have finite and prime-to-ℓ order, because 𝜙 : Zℓ [𝐽] → 𝐴 can be lifted ˆ → 𝐴. to a continuous map Zℓ [ 𝐽] □ 3.3 Review of the Categorical Trace 3.3.1. Hochschiled homology. Let us first review the general formalism of the Hochschild homology. We will mostly follow ((Hemo, 2023)). Let R be a symmetric monoidal category, 𝐴 and 𝐵 be two associative algebras in R. We denote by 𝐴rev (resp. 𝐵rev ) to be the algebra 𝐴 with the reversed multiplication. An 𝐴-𝐵-bimodule can be regarded as a left ( 𝐴 ⊗ 𝐵rev )-module or a right (𝐵 ⊗ 𝐴rev )module. For an 𝐴-𝐴-bimodule 𝐹, the Hochschild homology of 𝐹, if exists, is defined as Tr( 𝐴, 𝐹) = 𝐴 ⊗ 𝐴⊗ 𝐴rev 𝐹 ∈ R. However, the Hochschild complex of 𝐹 always exists. It is a simplicial object in R given by HH( 𝐴, 𝐹)• = Bar( 𝐴)• ⊗ 𝐴⊗ 𝐴rev 𝐹 = 𝐴 ⊗• ⊗ 𝐹. Example 3.3.2. Let 𝜙 be an endomorphism of the algebra 𝐴, and 𝐹 an 𝐴-bimodule. The 𝜙-twisted bimodule 𝐹, which we denote by 𝜙 𝐹, is the bimodule whose left 33 𝐴-action is pre-composed with 𝜙 and whose right action stays the same. In this case, we also denote the Hochschild homology of 𝜙 𝐴 by Tr( 𝐴, 𝜙). Roughly speaking, Tr( 𝐴, 𝜙) is determined by the universal property that a functor Tr( 𝐴, 𝜙) → 𝐶 is equivalent to a functor 𝐺 : 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. Now consider R = LincatΛ . An algebra object 𝐴 in LincatΛ is a presentable Λlinear monoidal category such that the 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 called the categorical trace of ( 𝐴, 𝐹). Note that the output of categorical trace is again a presentable Λ-linear category. We recall two key propositions which are particularly useful in the calculation of categorical traces, one in the coherent sheaf setting, and one in the Λ-sheaf setting. The gist is that the categorical trace of a sheaf theory can often be identified with, or embed in, the category of sheaves of certain fixed point object. Proposition 3.3.3. Let 𝑋 be a smooth Artin stack with an automorphism 𝜎 : 𝑋 → 𝑋, and let 𝑖 : L𝜎 𝑋 → 𝑋. There is a canonical equivalence of categories 𝐻 : Tr(IndCoh(𝑋), 𝜎)  IndCoh(L𝜎 𝑋), and the following diagram commutes: 𝑖∗ IndCoh(𝑋) IndCoh(L𝜎 𝑋) 𝐻 Tr(IndCoh(𝑋), 𝜎). Proposition 3.3.4. Let 𝑋 be a placid stack such that the diagonal Δ 𝑋 : 𝑋 → 𝑋 × 𝑋 is representable pro-smooth. Let 𝑌 be a prestack and let 𝑓 : 𝑋 → 𝑌 be a indproper morphism such that the relative diagonal 𝑋 → 𝑋 ×𝑌 𝑋 is ind-proper. Let 𝜎𝑋 : 𝑋 → 𝑋 and 𝜎𝑌 : 𝑌 → 𝑌 be endomorphisms intertwined by 𝑓 . 34 Consider the following diagram: 𝑋 ×𝑌 L𝜎𝑌 𝛿0 𝑋 × 𝑓 ,𝑌 , 𝑓 ◦𝜎𝑋 𝑋 (3.3) 𝑞 L 𝜎𝑌 , where 𝛿0 is the map Δ𝑋 𝑋 ×𝑌 (𝑌 ×Δ𝑌 ,𝑌 ×𝑌 ,1×𝜎𝑌 𝑌 )  𝑋 ×𝑌 ×𝑌 𝑌 −−→ (𝑋 × 𝑋) ×𝑌 ×𝑌 𝑌  𝑋 × 𝑓 ,𝑌 , 𝑓 ◦𝜎𝑋 𝑋 and 𝑞 is the projection. There is a canonical factorization 𝛿0∗ Shv(𝑋 ×𝑌 𝑋) Shv(𝑋 ×𝑌 L𝜎𝑌 ) 𝑞! 𝐺 Tr(Shv(𝑋 ×𝑌 𝑋), 𝜎) Shv(L𝜎𝑌 ), where the functor 𝐺 is fully faithful. 3.4 Geometric LLC for the torus In this section, let Λ = Fℓ or Qℓ . By abuse of notation, we will use Loc𝑐 𝑇,𝐹 and geom geom Loc𝑐 𝑇,𝐹 to denote the base-change Loc𝑐 𝑇,𝐹 ⊗ Λ and Loc𝑐 𝑇,𝐹 ⊗ Λ in the sequel. We will use Shv(−) to refer to the category of bounded constructable Λ-sheaves. Theorem 3.4.1. (i) Let 𝑇 be an algebraic torus over 𝐹. There exists a fully-faithful, 𝑡-exact, monoidal functor geom Ch : IndCoh(Loc𝑐 𝑇,𝐹 ) → IndShv(𝐿𝑇). Let Shvmon (𝐿𝑇) denote the essential image of Ch. It is the thick subcategory compactly generated by all character sheaves on 𝐿𝑇. Let Chmon denote the equivalence of categories geom Chmon : IndCoh(Loc𝑐 𝑇,𝐹 )  IndShvmon (𝐿𝑇). (ii) Both categories carry a Frobenius structure, and there is a commutative diagram geom Tr(IndCoh(Loc𝑐 𝑇,𝐹 ), 𝜎) T 𝐺 𝐻 IndCoh(Loc𝑐 𝑇,𝐹 ) Tr(IndShvmon (𝐿𝑇), 𝜎) L IndShv(𝔅(𝑇)), (3.4) 35 where T is induced by Chmon and the functorality of the categorical trace construction, L is the canonical equivalence (2.9) under the identification IndShv(𝔅(𝑇))  QCoh(Tor𝑇,iso𝐹 ), and two vertical arrows are the canonical maps 𝐻 and 𝐺 supplied by Proposition 3.3.3 and Proposition 3.3.4. Furthermore, 𝐻 and 𝐺 are equivalences. 3.4.2. Construction of Ch. IndShv(𝐿𝑇) as follows. geom We define the functor Ch : IndCoh(Loc𝑐 𝑇,𝐹 ) → Using the split short exact sequence (3.1), we have a decomposition Ö geom geom Coh(Loc𝑐 𝑇,𝐹 ) = Coh𝛼 (Loc𝑐 𝑇,𝐹 ). 𝛼∈ 𝐿ˆ 𝐼 On the other hand, since 1 → 𝐿 +𝑇 → 𝐿𝑇 → 𝐿ˆ 𝐼 → 1, we have another decomposition Shv(𝐿𝑇) = Ö Shv(𝐿 +𝑇). 𝛽∈ 𝐿ˆ 𝐼 The index set of the two decomposition are canonically identified. However, for our purpose, the functor sends the 𝛼 component of ind-coherent sheaves to the −𝛼 component of Λ-sheaves. Given that Coh(Homcts (𝜋1 (𝐿 +𝑇), G𝑚 )) = Coh0 (Loc𝑐 𝑇,𝐹 ), the next step is to define the functor Ch0 : Coh(Homcts (𝜋1 (𝐿 +𝑇), G𝑚 )) → Shv(𝐿 +𝑇), geom and naturally extend it to a functor between ind-completions of both sides. Notice that 𝐿 +𝑇 = 𝑇 × 𝐿 ++𝑇 with 𝐿 ++𝑇 being the first congruence subgroup. 𝐼 Recall that Ξ = Hom(( 𝐿ˆ ⊗ 𝑃ab 𝐸 ) , G𝑚 ), and each point 𝑥 ∈ Ξ gives a local system on 𝐿 ++𝑇 that we denote by L𝑥++ . By Proposition 3.2.6, with a choice of a topological generator 𝜏 ∈ 𝐼 𝐸tame , we have an ˆ and coherent sheaves on R 𝐽,G inclusion R 𝐽,G ˆ 𝑚 → 𝐻 ˆ 𝑚 are given by Coh(R 𝐽,G ˆ 𝑚 ) = colim Coh(𝑍). 𝑍 ⊂ 𝐻ˆ Therefore we can define a functor  ˆ Chtame : Coh(R 𝐽,G → Shv(𝑇). ˆ 𝑚 )  finite 𝐽-module 36 This functor is clearly independent of the choice of the topological generator and fully faithful. Now we can define the functor Ch0 : Coh(Homcts (𝜋1 (𝐿 +𝑇), G𝑚 ) → Shv(𝐿 +𝑇) = Shv(𝑇) ⊗ Shv(𝐿 ++𝑇) by sending an ind-coherent sheaf F on the 𝑥-component to Chtame (F ) ⊗ L𝑥++ . of Theorem 3.4.1 (i). It is obvious from the construction that Ch is 𝑡-exact. To show Ch is fully faithful, we can decompose both categories into blocks ÊÊ geom ˆ G𝑚 ) IndCoh(Loc𝑐 𝑇,𝐹 ) = IndCoh(Homcts ( 𝐽, 𝛼∈ 𝐿ˆ 𝐼 𝑥∈Ξ IndShv(𝐿𝑇) = ÊÊ IndShv(𝑇), 𝛽∈ 𝐿ˆ 𝐼 𝑥∈Ξ where Ch respects both direct sums (with a twist 𝛽 = −𝛼). On each block, Ch restricts to Chtame and we already know the latter is fully faithful. □ 3.4.3. Decategorification. We move on to prove part (ii) of the Theorem 3.4.1. Let 𝑟 = rank 𝑇. Consider the following diagram: geom Chmon IndShvmon (𝐿𝑇) geom T Tr(IndShvmon (𝐿𝑇), 𝜎) IndCoh(Loc𝑐 𝑇,𝐹 ) Tr(IndCoh(Loc𝑐 𝑇,𝐹 ), 𝜎) 𝑞 ∗ ◦𝛿0∗ [2𝑟] 𝑖∗ 𝐺 𝐻 IndCoh(Loc𝑐 𝑇,𝐹 ) L IndShv(𝔅(𝑇)). The functorality of the trace construction yields the upper square. The lower two vertical maps and the two curved maps are the canonical maps characterized in Theorem 3.3.3 and Theorem 3.3.4. They will be made explicit in the sequel. By Proposition 3.3.3, 𝐻 is an equivalence of categories, and 𝐺 is automatically an equivalence once we established that the lower square commutes. Instead of showing the lower square commutes directly, we first show that for a family of objects F ∈ IndCoh(𝑋) we have canonical isomorphisms L ◦ 𝑖 ∗ (F )  𝑞 ∗ ◦ 𝛿0∗ ◦ Chmon (F ) [2𝑟]. (3.5) 37 Then we show that this family is big enough to force the commutativity of the lower square. Each element 𝑥 ∈ Homcts (𝜋1 (𝐿 +𝑇), Λ× ) defines a closed point of Loc𝑐 𝑇,𝐹 . In particular, we fix an element 𝑥 ∈ Homcts (𝜋1 (𝐿 +𝑇), Λ× ) 𝜎 and consider the skyscraper sheaf Λ𝑥 at this point. We show that (3.5) holds for all such Λ𝑥 . geom Definition 3.4.4. For a character 𝜒 : 𝑇 (𝐹)0 → Λ× , we define 𝐼 𝜒 to be the augmentation ideal {𝑡 − 𝜒(𝑡)|𝑡 ∈ 𝑇 (𝐹)0 } in the group ring Λ[𝑇 (𝐹)]. Then we define 𝑅 𝜒 = Λ[𝑇 (𝐹)]/𝐼 𝜒 . geom geom Proposition 3.4.5. Let 𝑖 : Loc𝑐 𝑇,𝐹  L𝜎 Loc𝑐 𝑇,𝐹 → Loc𝑐 𝑇,𝐹 , and 𝑥 ∈ Homcts (𝜋1 (𝐿 +𝑇), Λ× ) 𝜎 . Let 𝑥 corresponds to a character 𝜒 under the isomorphism Homcts (𝜋1 (𝐿 +𝑇), Λ× ) 𝜎  Homcts (𝑇 (𝐹)0 , Λ× ). We have Û 𝑖 ∗ Λ𝑥 = 𝑅 𝜒 ⊗ (𝑋∗ (𝑇) ⊗ Λ[1]). Proof. The components of Loc𝑐 𝑇,𝐹 are indexed by the set of representations geom Homcts (𝑇 (𝐹)0 , Λ× ). Each component is sent to a closed subscheme in Loc𝑐 𝑇,𝐹 via Homcts (𝑇 (𝐹)0 ), Λ× ) = Homcts (𝐿 +𝑇 (𝜅 𝐹 ), Λ× ) = Homcts (𝜋1 (𝐿 +𝑇), Λ× ) 𝜎 ↩→ Homcts (𝜋1 (𝐿 +𝑇), Λ× ). Let 𝑈 ⊂ Loc𝑐 𝑇,𝐹 be the component that is sent to 𝑥 ∈ Homcts (𝜋1 (𝐿 +𝑇), Λ× ) 𝜎 . Explicitly, 𝑈 = Spec(𝑅 𝜒 ). We denote the character associated to 𝑈 by 𝜒 : 𝑇 (𝐹)0 → Λ× and study Λ𝑥 = 𝑥 ∗ Λ. The component in which 𝑥 lies can be embedded in some R 𝐽,G𝑚 as in equation (3.2). The component itself is precisely the formal neighbourhood of 𝑥 in that torus. Therefore, Û 𝐿 𝐿 𝑖 ∗ Λ𝑥 = 𝑅 𝜒 ⊗Λ[𝐽] Λ = 𝑅 𝜒 ⊗Λ[ Λ = 𝑅 ⊗ (𝑋∗ (𝑇) ⊗ Λ[1]). □ 𝜒 ˆ 𝐿] Now we compute 𝑞 ! ◦ 𝛿0∗ ◦ Chmon (F ). Let L = Chmon (Λ𝑥 ), and let 𝜒 : 𝑇 (𝐹)0 → Λ× be the character associated to 𝑥. There exists 𝑛 so that 𝜒 factors through 𝑇 (𝐹)0 /𝑉𝑛 . In Theorem 3.3.4, we take 𝑋 = B𝐿 ≥𝑛𝑇, 𝑌 = B𝐿𝑇, 𝑓 : 𝑋 → 𝑌 induced by the inclusion 𝐿 ≥𝑛𝑇 → 𝐿𝑇, and the Frobenius 𝜎 induce the automorphisms of 𝑋 and 𝑌 . 38 The diagram (3.3) becomes 𝐿𝑇 Ad𝜎 𝐿 ≥𝑛𝑇 𝛿0 𝐿 ≥𝑛𝑇 \ 𝐿𝑇 / ≥𝑛 𝐿 𝑇 (3.6) 𝑞 𝐿𝑇 , Ad𝜎 𝐿𝑇 where 𝛿0 is given by identity of 𝐿𝑇 on objects and 𝑡 ∈ 𝐿 ≥𝑛𝑇 ↦→ (𝑡, −𝜎𝑡) on morphisms, and the map 𝑞 is given by identity on objects and the inclusion of 𝐿 ≥𝑛𝑇 → 𝐿𝑇 on morphisms. Proposition 3.4.6. Let L = Chmon (Λ𝑥 ), and let 𝜒 be the character associated to 𝑥. Let 𝑟 = rank 𝑇, we have Û 𝑞 ∗ ◦ 𝛿0∗ L [2𝑟] = 𝑅 𝜒 ⊗ (𝑋∗ (𝑇) ⊗ Λ[1]). Proof. Let L be a character sheaf on the central component 𝐿 +𝑇/𝐿 ≥𝑛𝑇 such that 𝜎 ∗ L  L. By Lang’s theorem, this character sheaf corresponds to a character + 𝜒 : 𝐿𝐿≥𝑛𝑇𝑇 (𝜅 𝐹 ) → Λ× . We calculate 𝑞 ! ◦ 𝛿0∗ L as follows. Let 𝑠 : ∗ → Ad𝐿𝑇 be a point. We form the 𝜎 𝐿𝑇 pull-back 𝐿𝑇 𝐿𝑇 𝛿0 𝑠′ 𝐿𝑇 𝐿 ≥𝑛𝑇 \ / 𝐿 ≥𝑛𝑇 ≥𝑛 ≥𝑛 𝐿 𝑇 Ad𝜎 𝐿 𝑇 𝑞 𝑞′ ∗ 𝑠 𝐿𝑇 , Ad𝜎 𝐿𝑇 where 𝑠′ is given by 𝑡 ↦→ 𝑠 + 𝑡 − 𝜎𝑡 on objects and identity on morphisms. First, we determine on which components of Ad 𝜎𝐿𝑇𝐿 ≥𝑛𝑇 the pullback 𝑠′∗ ◦ 𝛿0∗ L is non-zero. Clearly, 𝛿0∗ L has the same support as L. The components of Ad 𝜎𝐿𝑇𝐿 ≥𝑛𝑇 are indexed by the set 𝑋∗ (𝑇) 𝐼 , and so is 𝐿𝐿𝑇 ≥𝑛 𝑇 . Let [𝑠] ∈ 𝑋∗ (𝑇) 𝐼 be the component in ′∗ ∗ which 𝑠 lies, then 𝑠 ◦ 𝛿0 L has support on all those components 𝛼 ∈ 𝑋∗ (𝑇) 𝐼 such that [𝑠] + 𝛼 − 𝜎𝛼 = 0. Assume [𝑠] = 𝜎𝛼 − 𝛼, then 𝑠′∗ ◦ 𝛿0∗ L has support on those components indexed by 𝛼 + (𝑋∗ (𝑇) 𝐼 ) 𝜎 . Next, we determine on which components of Ad𝐿𝑇 the sheaf 𝑠∗ ◦ 𝛿0∗ L is non𝜎 𝐿𝑇 zero. The components of Ad𝐿𝑇 is indexed by 𝐵(𝑇) = 𝑋∗ (𝑇) Γ . The fact that 𝜎 𝐿𝑇 39 [𝑠] = 𝜎𝛼 − 𝛼 in the previous analysis implies that 𝑠 lies in the central component of Ad𝐿𝑇 . Therefore 𝑠∗ ◦ 𝛿0∗ L is supported on the central component only. 𝜎 𝐿𝑇 To study 𝑠∗ ◦ 𝛿0∗ L on the central component of Ad𝐿𝑇 , it suffice to take 𝑠 = 1 ∈ 𝐿𝑇. 𝜎 𝐿𝑇 Since we assumed that 𝜎 ∗ L  L, the pullback 𝑠′∗ ◦ 𝛿0∗ L is the trivial local system on components indexed by (𝑋∗ (𝑇) 𝐼 ) 𝜎 . By Lang’s theorem, 𝑠′∗ ◦ 𝛿0∗ L restricted + to 𝐿𝐿≥𝑛𝑇𝑇 is the trivial local system with an (𝐿 +𝑇/𝐿 ≥𝑛𝑇)(𝜅 𝐹 )-action given by the character 𝜒. + Let 𝑟 = rank 𝑇. Since 𝐿𝐿≥𝑛𝑇𝑇  𝑇 × A𝑑 as an algebraic scheme for some integer 𝑑, we have  Ê Û 𝑠∗ ◦ 𝑞 ∗ ◦ 𝛿0∗ L [2𝑟] = (𝑋 ∗ (𝑇) ⊗ Λ[−1] (−1)) [2𝑟] (𝑋 (𝑇) 𝐼 ) 𝜎 = Û∗ (𝑋∗ (𝑇) ⊗ Λ[1] (−1)). The sheaf 𝑞 ∗ ◦ 𝛿0∗ L is the above sheaf equipped with a 𝑇 (𝐹)-action. Recall that 𝐼 𝜒 = {𝑡 − 𝜒(𝑡)|𝑡 ∈ 𝑇 (𝐹)0 } is the augmentation ideal. Then as a Λ[𝑇 (𝐹)]-module we have Û 𝑞 ∗ ◦ 𝛿0∗ L = (Λ[𝑇 (𝐹)]/𝐼 𝜒 ) ⊗ (𝑋∗ (𝑇) ⊗ Λ[1] (−1)) Û = 𝑅𝜒 ⊗ (𝑋∗ (𝑇) ⊗ Λ[1] (−1)) □ Proof of part (ii) of Theorem 3.4.1. Let 𝑥 ∈ Homcts (𝜋1 (𝐿 +𝑇), Λ× ) 𝜎 and consider the skyscraper sheaf Λ𝑥 at this point. We have shown the canonical isomorphism Û L ◦ 𝑖 ∗ (Λ𝑥 )  𝑞 ∗ ◦ 𝛿0∗ ◦ Chmon (Λ𝑥 ) [2𝑟] = 𝑅 𝜒 ⊗ (𝑋∗ (𝑇) ⊗ Λ[1]). Since Chmon is 𝑡-exact, so is T. Therefore all four functors in diagram 3.4 are 𝑡-exact.  geom Let F ∈ Tr IndCoh(Loc𝑐 𝑇,𝐹 ), 𝜎 be the image of Λ𝑥 . Since L ◦ 𝐻 (F )  𝐺 ◦ T(F ), (3.7) there is an isomorphism on each cohomology. The cohomology of (3.7) in degree 0 is 𝑅 𝜒 . Since the family {𝑅 𝜒 } generates the whole category of IndCoh(Loc𝑐 𝑇,𝐹 ), we conclude that the diagram commutes. □ 40 Appendix A A LEMMA IN HOMOLOGICAL ALGEBRA Lemma A.0.1. Let Γ be a finite group, 𝐴 be an abelian group equipped with a Γ-action. Let [𝛼] ∈ 𝐻 2 (Γ, 𝐴) represent the group extension 1 → 𝐴 → 𝐺 → Γ. Let 𝑀 be any Γ-module, in other words, it is a 𝐺-module on which 𝐴 acts trivially. Then we have a commutative diagram 𝐻2 (Γ, 𝑀) 𝑑2 𝐻1 ( 𝐴, 𝑀) Γ ∪𝛼 0 𝐻ˆ −1 (Γ, 𝑀 ⊗ 𝐴) 𝐻1 (𝐺, 𝑀) res (𝑀 ⊗ 𝐴) Γ (𝑀 ⊗ 𝐴) Γ 𝐻1 (Γ, 𝑀) ∪𝛼 𝐻ˆ 0 (Γ, 𝑀 ⊗ 𝐴) The first row of the diagram is the long exact sequence associated to the LyndonHochschild-Serre spectral sequence. The second row is the definition of the Tate cohomology. The first and last vertical map is the cup product with 𝛼. The map res is the restriction map 𝐻1 (𝐺, 𝑀) → 𝐻1 ( 𝐴, 𝑀) Γ . In particular, if Γ and 𝐴 satisfies the condition of the Tate-Nakayama lemma, we would have 𝐻1 (𝐺, 𝑀)  (𝑀 ⊗ 𝐴) Γ . Proof. The crux of the proof is the commutativity of the left-most square, which we will prove in the sequel. The commutativity of the rest of the squares are fairly standard and will be omitted. A.0.2. Notations. We follow the notation of Atiyah and Wall (Cassels, 1987). The (homogeneous) bar resolution of the 𝐺-module Z is given by Ê 𝐶𝑛 (𝐺, Z) = Z(𝑔0 , 𝑔1 , · · · , 𝑔𝑛 ), 𝐺 𝑛+1 which is the free abelian group generated by the basis 𝐺 𝑛+1 with the action of 𝑔 · (𝑔0 , · · · , 𝑔𝑛 ) = (𝑔𝑔0 , · · · , 𝑔𝑔𝑛 ). 0 0. 41 The differential is the usual 𝑑 (𝑔0 , · · · , 𝑔𝑛 ) = 𝑛 ∑︁ (−1) 𝑖 (𝑔0 , · · · , 𝑔b𝑖 , · · · , 𝑔𝑛 ). 𝑖=0 It is isomorphic to the more familiar inhomogeneous bar resolution via sending (𝑔0 , 𝑔0 𝑔1 , · · · , 𝑔0 𝑔1 · · · 𝑔𝑛 ) ↦→ (𝑔0 |𝑔1 | · · · |𝑔𝑛 ). Let 𝑃• denote a Γ-resolution of Z by finitely-generated free Γ-modules, and let 𝑃∗ = Hom(𝑃• , Z) be its dual, so that we have exact sequences · · · → 𝑃1 → 𝑃0 → Z →0 0 → Z → 𝑃0∗ → 𝑃1∗ → · · · . ∗ Let 𝑃−𝑛 = 𝑃𝑛−1 and join the two sequences together we get a doubly-infinite exact sequence 𝐿 • : · · · → 𝑃1 → 𝑃0 → 𝑃−1 → 𝑃−2 → · · · The Tate groups are then the cohomology groups of HomΓ (𝐿 • , 𝑀) for any Γ-module 𝑀, i.e. 𝐻𝑇𝑖 (Γ, 𝑀) = 𝐻 𝑖 (HomΓ (𝐿 • , 𝑀)). The cohomology in degrees 𝑛 ≤ −2 agrees with the Tate groups explicitly via isomorphisms 𝑁 𝑃• ⊗Γ 𝑀 = (𝑃• ⊗ 𝑀) Γ − → (𝑃• ⊗ 𝑀) Γ → (Hom(𝑃•∗ , 𝑀)) Γ = HomΓ (𝑃•∗ , 𝑀). A.0.3. Cup Product. Let {𝑤 𝜎 |𝜎 ∈ Γ} be a system of representatives of Γ in 𝐺. We define the cocycle 𝛿 : Γ × Γ → 𝐴 via 𝑤 𝜎 𝑤 𝜏 = 𝛿(𝜎, 𝜏)𝑤 𝜎𝜏 . Representing 𝛿 using the homogeneous bar resolution yields a cocycle 𝛼 ∈ HomΓ (𝐶2 (Γ, Z), 𝐴) which is uniquely characterized by 𝛼(1, 𝑐 1 , 𝑐 1 𝑐 2 ) = 𝛿(𝑐 1 , 𝑐 2 ). ¤ ∈ 𝐻2 (Γ, 𝑀) and 𝜙¤ ∈ HomΓ (𝐶 ∗ (Γ, Z), 𝑀). The cup product 𝜙¤ ∪ 𝛼 lives in Let [ 𝜙] 2 HomΓ (𝐶0 (Γ, Z), 𝑀 ⊗ 𝐴) and is defined by ∑︁ ¤ ∗ , 𝑐∗1 , 𝑐∗2 ) ⊗ 𝛼(𝑐 2 , 𝑐 1 , 𝑐) 𝜙¤ ∪ 𝛼(𝑐∗ ) = 𝜙(𝑐 (A.1) 𝑐 1 ,𝑐 2 = ∑︁ 𝑐 1 ,𝑐 2 ¤ ∗ , 𝑐∗1 , (𝑐 1 𝑐 2 ) ∗ ) ⊗ 𝛼(𝑐 1 𝑐 2 , 𝑐 1 , 𝑐). 𝜙(𝑐 (A.2) 42 Its image in (𝑀 ⊗ 𝐴) Γ is 𝜙¤ ∪ 𝛼(1) = ∑︁ ¤ 𝑐∗1 , (𝑐 1 𝑐 2 ) ∗ ) ⊗ 𝛼(𝑐 1 𝑐 2 , 𝑐 1 , 1) 𝜙(1, (A.3) −1 −1 𝜙(𝑐 1 , 𝑐 2 ) ⊗ 𝑐 1 𝑐 2 · 𝛼(1, 𝑐−1 2 , 𝑐2 𝑐1 ) (A.4) −1 𝜙(𝑐 1 , 𝑐 2 ) ⊗ 𝑐 1 𝑐 2 · 𝛿(𝑐−1 2 , 𝑐 1 ). (A.5) 𝑐 1 ,𝑐 2 = ∑︁ 𝑐 1 ,𝑐 2 = ∑︁ 𝑐 1 ,𝑐 2 Notice that ¤ 𝑐∗1 , (𝑐 1 𝑐 2 ) ∗ ) = 𝜙(𝑐 1 , 𝑐 2 ) 𝜙(1, is the switch from the inhomogeneous bar resolution to the homogeneous one. A.0.4. Spectral sequence. For a 𝐺-module 𝑀, we set up the chain complex that produces the Lydon-Hochschild-Serre spectral sequence. Let 𝑃• = 𝐶• (Γ, Z) and 𝑄 • = 𝐶• (𝐺, Z). The differential of the two chain complexes are denoted by 𝑑1 and 𝑑0 respectively. We define the double complex 𝐸𝑖 𝑗 = 𝑃𝑖 ⊗ 𝑄 𝑗 . The complex Tot(𝐸𝑖 𝑗 ) is a 𝐺-resolution of Z, therefore the spectral sequence associated with 𝐸𝑖 𝑗 ⊗𝐺 𝑀 calculates the group homology of 𝑀. Note that 𝑃• is also a 𝐺-module. We have
𝐸𝑖𝑀𝑗 := 𝐸𝑖 𝑗 ⊗ 𝐺 𝑀 = (𝑃𝑖 ⊗ 𝑄 𝑗 ⊗ 𝑀)𝐺 = 𝑃𝑖 ⊗ Γ (𝑄 𝑗 ⊗ 𝐴 𝑀) . Since 𝑄 𝑗 ⊗ 𝐴 (−) calculates 𝐴-homology and 𝑃𝑖 ⊗ Γ (−) calculates Γ-homology, the resulting spectral sequence of this double complex is the Lydon-Hochschild-Serre spectral sequence. We need a formula that directly link the 𝐴-homology calculated by the complex 𝑄 • with that calculated by 𝐶• ( 𝐴, Z). We choose a projection 𝑓 : 𝐺 → 𝐴 such that 𝑓 (𝑤 𝛾 ) = 1, for all 𝛾 ∈ Γ, 𝑓 (𝑎𝑔) = 𝑎 𝑓 (𝑔), for all 𝑎 ∈ 𝐴, 𝑔 ∈ 𝐺. This induce a map between 𝐴-complexes 𝐶• (𝐺, Z) → 𝐶• ( 𝐴, Z). Let   𝑍 = ker 𝐶1 (𝐺, Z) ⊗ 𝐴 𝑀 → 𝐶0 (𝐺, Z) ⊗ 𝐴 𝑀 and consider the composition 𝑍 → 𝐻1 ( 𝐴, 𝑀)  𝐴 ⊗ 𝑀. It sends a cycle (𝑔1 , 𝑔2 ) ⊗ 𝑚 ↦→ ( 𝑓 (𝑔2 ) − 𝑓 (𝑔1 )) ⊗ 𝑚. (A.6) 43 A.0.5. Differentials. Recall the definition of the map 𝑑2 : 𝐻2 (Γ, 𝑀) → 𝑀 be a cycle and assume there exists 𝜓 ∈ 𝐸 𝑀 such 𝐻1 ( 𝐴, 𝑀) Γ . Let 𝜙 ∈ 𝐸 2,0 1,1 that 𝑑0 𝜓 + 𝑑1 𝜙 = 0, then 𝑑2 ([𝜙]) = 𝑑1 ([𝜓]). From now on, we will use 𝑐, 𝑐𝑖 or Greek letters 𝜎, 𝜏 to denote an element in Γ, 𝑔, 𝑔𝑖 or ℎ an element in 𝐺. We note that in 𝐸𝑖,𝑀𝑗 , we have (𝑐 0 , 𝑐 1 , · · · , 𝑐𝑖 ) ⊗ (𝑔0 , · · · , 𝑔 𝑗 ) ⊗ 𝑚 −1 −1 −1 −1 = (1, 𝑐−1 0 𝑐 1 , · · · , 𝑐 0 𝑐𝑖 ) ⊗ (𝑤 𝑐 0 𝑔0 , · · · , 𝑤 𝑐 0 𝑔 𝑗 ) ⊗ 𝑐 0 𝑚, and therefore 𝐸𝑖,𝑀𝑗 is generated by by elements whose component in the 𝑃𝑖 factor starts with 𝑐 0 = 1. We write all cycles as a linear combination of these elements. 𝑀 , we can write For example, for any 𝜙 ∈ 𝐸 2,0 ∑︁ 𝜙= (1, 𝑐 1 , 𝑐 1 𝑐 2 ) ⊗ (𝑔) ⊗ 𝜙(𝑐 1 , 𝑐 2 , 𝑔). 𝑐 1 ,𝑐 2 ,𝑔 This expression is in general not unique. However, it is worth noting that in the 𝑀 , the expression above is unique modulo the identity special case of 𝜙 ∈ 𝐸 2,0 (1, 𝑐 1 , 𝑐 1 𝑐 2 ) ⊗ (𝑔) ⊗ 𝜙(𝑐 1 , 𝑐 2 , 𝑔) = (1, 𝑐 1 , 𝑐 1 𝑐 2 ) ⊗ (𝑎𝑔) ⊗ 𝜙(𝑐 1 , 𝑐 2 , 𝑔) for any 𝑎 ∈ 𝐴. That is to say, let ˜ 1 , 𝑐 2 , 𝜎) = 𝜙(𝑐 ∑︁ 𝜙(𝑐 1 , 𝑐 2 , 𝑔), 𝑔=𝜎 ¯ the following expression of 𝜙 is unique: ∑︁ ˜ 1 , 𝑐 2 , 𝜎). 𝜙= (1, 𝑐 1 , 𝑐 1 𝑐 2 ) ⊗ (𝑤 𝜎 ) ⊗ 𝜙(𝑐 𝑐 1 ,𝑐 2 ,𝜎 𝑀 with little change. Similar unique expression exists for elements of 𝐸 1,0 44 Now we compute 𝑑2 . The next three formulae are straightforward. 𝑑1 (1, 𝑐 1 , 𝑐 1 𝑐 2 ) ⊗ (𝑔) ⊗ 𝑚 = (𝑐 1 (1, 𝑐 2 ) − (1, 𝑐 1 𝑐 2 ) + (1, 𝑐 1 )) ⊗ (𝑔) ⊗ 𝑚 −1 = (1, 𝑐 2 ) ⊗ (𝑤 −1 𝑐 1 𝑔) ⊗ 𝑐 1 𝑚 − (1, 𝑐 1 , 𝑐 2 ) ⊗ (𝑔) ⊗ 𝑚 + (1, 𝑐 1 ) ⊗ (𝑔) ⊗ 𝑚. 𝑑0 (1, 𝑐) ⊗ (𝑔0 , 𝑔1 ) ⊗ 𝑚 = (1, 𝑐) ⊗ (𝑔0 ) ⊗ 𝑚 − (1, 𝑐) ⊗ (𝑔1 ) ⊗ 𝑚. 𝑑1 (1, 𝑐) ⊗ (𝑔1 , 𝑔2 ) ⊗ 𝑚 −1 −1 = (𝑤 −1 𝑐 𝑔1 , 𝑤 𝑐 𝑔2 ) ⊗ 𝑐 𝑚 − (𝑔1 , 𝑔2 ) ⊗ 𝑚   −1 = 𝑐−1 𝑚 ⊗ 𝑓 (𝑤 −1 𝑔 ) − 𝑓 (𝑤 𝑔 ) − 𝑚 ⊗ ( 𝑓 (𝑔2 ) − 𝑓 (𝑔1 )) 𝑐 2 𝑐 1   = 𝑐−1 𝑚 ⊗ 𝛿(𝑐−1 , 𝑔2 ) − 𝛿(𝑐−1 , 𝑔1 ) − 𝑚 ⊗ ( 𝑓 (𝑔2 ) − 𝑓 (𝑔1 )) . In the second to last equality we actually send the element to 𝑀 ⊗ 𝐴 as in (A.6). From this we deduce the boundaries of a cycle as follows. For 𝛾, 𝜎 ∈ Γ, we have ∑︁ ∑︁ ∑︁ ˜ 𝜎) = ˜ ˜ ˜ 1 , 𝑐 2 , 𝜎) 𝑑1 𝜙(𝛾, 𝑐−1 𝜙(𝑐 , 𝑐 , 𝑐 𝜎) − 𝜙(𝑐 , 𝑐 , 𝜎) + 𝜙(𝑐 1 2 1 1 2 1 𝑐 =𝛾 = 2 ∑︁ 𝑐 𝑐 =𝛾 𝑐 𝑐 ∑︁ ˜ 𝑑0 𝜓(𝛾, 𝜎) = 𝜓(𝛾, ℎ, 𝑔) − ∑︁ ∑︁ (A.7) 𝑐 𝜓(𝛾, 𝑔, ℎ). (A.8) 𝑔 ℎ=𝜎, 𝑔 𝑑1 𝜓 = 𝑐 =𝛾 ∑︁ 1 2 ∑︁ 1 −1 ˜ −1 ˜ 𝑐 𝛾, 𝜎) + ˜ 𝑐, 𝜎), 𝑐 𝜙(𝑐, 𝛾, 𝑐𝜎) − 𝜙(𝑐, 𝜙(𝛾, 𝑐−1 𝜓(𝑐, 𝑔1 , 𝑔2 ) ⊗ (𝛿(𝑐−1 , 𝑔2 ) − 𝛿(𝑐−1 , 𝑔1 )) 𝑐,𝑔1 ,𝑔2 − ∑︁ 𝜓(𝑐, 𝑔1 , 𝑔2 ) ⊗ ( 𝑓 (𝑔2 ) − 𝑓 (𝑔1 )). 𝑐,𝑔1 ,𝑔2 These is enough to determine the map 𝑑2 . (A.9) 45 To conclude, let us assume 𝑑1 𝜙 + 𝑑0 𝜓 = 0. We have 𝑑2 ( [𝜙]) = 𝑑1 𝜓 ∑︁ = 𝛾 −1 𝜓(𝛾, 𝑔1 , 𝑔2 ) ⊗ (𝛿(𝛾 −1 , 𝑔2 ) − 𝛿(𝛾 −1 , 𝑔1 )) 𝛾,𝑔1 ,𝑔2 = ∑︁
𝛾 · 𝜓(𝛾 −1 , 𝑔2 , 𝑔1 ) − 𝛾 · 𝜓(𝛾 −1 , 𝑔1 , 𝑔2 ) ⊗ 𝛿(𝛾, 𝑔1 ) 𝛾,𝑔1 ,𝑔2 = ∑︁ = ∑︁ = ∑︁ ˜ −1 , 𝜎) ⊗ 𝛿(𝛾, 𝜎) −𝛾 · 𝑑0 𝜓(𝛾 𝛾,𝜎 ˜ −1 , 𝜎) ⊗ 𝛿(𝛾, 𝜎) 𝛾 · 𝑑1 𝜙(𝛾 𝛾,𝜎
˜ 𝛾 −1 , 𝑐𝜎) − 𝜙(𝑐, ˜ 𝑐−1 𝛾 −1 , 𝜎) + 𝜙(𝛾 ˜ −1 , 𝑐, 𝜎) ⊗ 𝛿(𝛾, 𝜎) 𝑐−1 𝜙(𝑐, 𝛾,𝜎,𝑐 = ∑︁ −1 −1 −1 −1 ˜ 1 , 𝑐 2 , 𝜎) ⊗ 𝑐 1 𝑐 2 · 𝛿(𝑐−1 𝜙(𝑐 2 , 𝑐 1 𝜎) − 𝑐 1 𝑐 2 · 𝛿(𝑐 2 𝑐 1 , 𝜎) + 𝑐 1 𝛿(𝑐 1 , 𝜎) 𝑐 1 ,𝑐 2 ,𝜎 = ∑︁ −1 ˜ 1 , 𝑐 2 , 𝜎) ⊗ 𝑐 1 𝑐 2 · 𝛿(𝑐−1 𝜙(𝑐 2 , 𝑐1 ) 𝑐 1 ,𝑐 2 ,𝜎 = [𝜙] ∪ 𝛼. □
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