A Comparison of p-adic Motivic Cohomology and Rigid Cohomology Citation Lawless Hughes, Nathaniel (2019) A Comparison of p-adic Motivic Cohomology and Rigid Cohomology. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/DCJJ-E164. https://resolver.caltech.edu/CaltechTHESIS:06012019-191035765 Abstract We study two conjectures introduced by Flach and Morin in [FM18] for schemes over a perfect field of characteristic p > 0. The first conjecture relates a p -adic extension of the étale motivic cohomology with compact support on general schemes introduced by Geisser in [Gei06] to rigid cohomology with compact support, and is proved here. The second, relates a p -adic Borel-Moore motivic homology with the dual of rigid cohomology with compact support, and is proved in the smooth case. For this, we also prove a generalization of the comparison theorem from rigid cohomology to overconvergent de Rham-Witt cohomology in [DLZ11]. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Mathematics; number theory; arithmetic geometry; rigid cohomology; motivic cohomology; overconvergent de rham-whit cohomology Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Apostol Award for Excellence in Teaching in Mathematics, 2016. Thesis Availability: Public (worldwide access) Research Advisor(s): Flach, Matthias Thesis Committee: Mantovan, Elena (chair) Flach, Matthias Ramakrishnan, Dinakar Zhu, Xinwen Defense Date: 24 May 2019 Record Number: CaltechTHESIS:06012019-191035765 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06012019-191035765 DOI: 10.7907/DCJJ-E164 Related URLs: URL URL Type Description https://arxiv.org/abs/1810.10059 arXiv Paper adapted for Chapters V and VI. ORCID: Author ORCID Lawless Hughes, Nathaniel 0000-0003-2755-4065 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 11598 Collection: CaltechTHESIS Deposited By: Nathaniel Lawless Hughes Deposited On: 10 Jun 2019 22:24 Last Modified: 04 Oct 2019 00:26 Thesis Files Preview PDF (Thesis) - Final Version See Usage Policy. 513kB Repository Staff Only: item control page

A Comparison of p-adic Motivic Cohomology and Rigid Cohomology Thesis by Nathaniel Lawless Hughes In Partial Fulllment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2019 Defended 05/24/2019 ii c 2019 Nathaniel Lawless Hughes ORCID: 0000-0003-2755-4065 All rights reserved. iii ACKNOWLEDGEMENTS First and foremost, I would like to thank my advisor, Matthias Flach, for suggesting the problem, and his kind, patient and wise guidance and support over the years. He has helped me form as a mathematician, with helpful conversations, literature suggestions, and has always been there to answer my questions. I am thankful to Andreas Langer for his help and suggestions regarding the content of Chapter 5. I would also like to thank the people who have helped me reach here. To my mother, Victoria Hughes, for raising me and for her unconditional support throughout my life. To my undergraduate teachers and mentors, Cyril Rakovski and Peter Jipsen for their guidance and help. I am very thankful to my partner, Kelly Mauser, for her friendship and support during my PhD. To all the wonderful people I have met in my time at Caltech. Special thanks to my fellow student, soccer teammate and friend Andrei Frimu, and the helpful discussions and advice he has provided. To all the other great friends I made in the Mathematics department: Angad Singh, Marius Lemm, Tamir Hemo, Thomas Norton and Nathaniel Sagman. And to my fellow FC Monrovia teammates. Lastly, I would like to thank the California Institute of Technology Mathematics Department, for giving me this opportunity, and providing a great environment in which to study Mathematics. iv ABSTRACT We study two conjectures introduced by Flach and Morin in [FM18] for schemes over a perfect eld of characteristic p > 0. The rst conjecture relates a p-adic extension of the étale motivic cohomology with compact support on general schemes introduced by Geisser in [Gei06] to rigid cohomology with compact support, and is proved here. The second, relates a p-adic Borel-Moore motivic homology with the dual of rigid cohomology with compact support, and is proved in the smooth case. For this, we also prove a generalization of the comparison theorem from rigid cohomology to overconvergent de Rham-Witt cohomology in [DLZ11]. v TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter II: Background . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Cohomological Descent . . . . . . . . . . . . . . . . . . . . . . 2.2 Rigid Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Denition of rigid cohomology . . . . . . . . . . . . . . . . . . 2.4 The Tsuzuki Functor . . . . . . . . . . . . . . . . . . . . . . . 2.5 Crystalline Cohomology . . . . . . . . . . . . . . . . . . . . . . 2.6 Witt de-Rham Cohomology . . . . . . . . . . . . . . . . . . . . 2.7 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter III: p-adic motivic cohomology on singular varieties . . . . . . 3.1 The eh-topology . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Cohomology for the eh-topology . . . . . . . . . . . . . . . . . 3.3 Motivic cohomology for singular varieties . . . . . . . . . . . . Chapter IV: Conjecture A . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Proper eh-hypercoverings . . . . . . . . . . . . . . . . . . . . . 4.2 Proof of Conjecture A . . . . . . . . . . . . . . . . . . . . . . . Chapter V: Comparison of overconvergent Witt de-Rham cohomology and rigid cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The hypercovering . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The simplicial special frame . . . . . . . . . . . . . . . . . . . . 5.5 The comparison theorem . . . . . . . . . . . . . . . . . . . . . Chapter VI: Conjecture B . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Conjecture B . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iv v 1 1 1 2 3 5 5 8 13 14 17 17 18 20 20 21 22 23 23 26 39 39 41 43 46 49 57 57 60 1 Chapter 1 INTRODUCTION 1.1 Notation k will denote a perfect eld of characteristic p > 0. W (k) will be its Witt ring, and K := W (k) ⊗ Q will be the eld of fractions. We will use W = W (k) when talking of the formal scheme Spf W rather than the scheme Spec W (k). Let Schd /k denote the category of separated and nite type schemes over k of dimension ≤ d. In the case where d = ∞ we just use Sch/k . Let FSch/W be separated and nite type formal schemes over Spf W . In this thesis, we will consider all schemes and formal schemes to be separated and nite type. In the derived category D(A) for some abelian category A, and a map f : A → B , let h i f f A → B := Cone(A → B)[−1]. For a complex C of abelian groups, let CQ := C ⊗Z Q. 1.2 Motivation For a variety X over a perfect eld k of characteristic p > 0, there exist various constructions of cohomology theories with coecients in Zp or Qp , and with a suitable X satisfy the properties of Weil cohomologies (in the sense of [Kle68, 1.2]). For X smooth and proper, crystalline cohomology is a good cohomology theory (see [Ber74] and [BO78]), and can be computed as the hypercohomology of the de Rham-Witt complex W Ω•X/k by [Ill79, Proposition 2.1]. This endows it with a Frobenius action φ : σ ∗ RΓ(X/W (k)) → RΓ(X/W (k)) and a slope ltration on (H ∗ (X/W (k)) ⊗ K). For a general variety X over k , Berthelot dened rigid cohomology with coecients in the fraction eld K of W (k), by calculating a cohomology of a suitable subcomplex of the de Rham complex in a rigid analytic space over K 2 related to X . This also has a version with compact support, and has various nice properties such as existence of a Frobenius ( [Ber96]), nite dimensionality of cohomology groups ( [Ber97b]), and in the case of smooth schemes a Poncairé duality and Künneth formula ( [Ber97a]). On the other hand, we can consider construction on the motivic side: motivic cohomology, motivic cohomology with compact support, Borel-Moore motivic homology, and motivic homology. We can also consider the étale versions of these theories. These are well behaved on smooth quasi-projective schemes. In order to extend this to general varieties over k , [Gei06] used an analog method to Voevodsky's use of cdh topology in order to add abstract blowups to the Nisnevich topology, and considers an eh topology where he adds abstract blowups to the étale topology. Under strong resolution of singularities, this allows to extend the étale motivic cohomology theories to general schemes. We may consider a p-adic completion of the above étale motivic cohomology and Borel-Moore homology theories. One place where these theories arise is in the study of vanishing order for zeta functions at integers n on proper regular arithmetic schemes as explained in [FM18, Chapter 5]. Based on results on proper smooth schemes over k , we expect certain relations between the p-adic completion of the étale motivic cohomology with compact supports (resp. p-adic completion of the étale Borel-Moore homology) with rigid cohomology with compact support (resp. dual of rigid cohomology with compact support), as stated below in Conjecture 1.3.1 (resp. Conjecture 1.3.2). These relations hold in the case of proper-smooth schemes as shown in [FM18, Proposition 7.21]. 1.3 Main Results Let Z(n) be the complex of étale sheaves on Sch/k dened in [SV00], and let Zc (n) := z n (−, 2n−∗) denote the complex of étale sheaves from Bloch's higher Chow complex dened in [Blo86]. For a scheme X in Schd /k , under strong resolution of singularities R(d) (see Denition 3.1.4), let ! RΓc (Xeh , Qp (n)) := R lim RΓc (Xeh , Z(n)/pr ) ←− r Q 3 as in Denition 3.3.2. Then, we expect the following relation with rigid cohomology with compact support: Conjecture 1.3.1 (Conjecture A). Under R(d), for a separated, nite type k -scheme X , and n ∈ Z, there exists an isomorphism h i pn −φ ∼ RΓc (Xeh , Qp (n)) → RΓrig,c (X/K) → RΓrig,c (X/K) We also dene a p-adic Borel-Moore homology theory: ! RΓ(X, Qcp (n)) := R lim RΓc (Xet , Zc (n)/pr ) ←− r . Q We expect the following relationship with the dual of rigid cohomology with compact support: Conjecture 1.3.2 (Conjecture B). For a separated, nite type k-scheme X of dimension d, and n ∈ Z, there exists an isomorphism ∼ RΓ(Xet , Qcp (n)) →   n−d −φ ∗ p ∗ RΓrig,c (X/K) → RΓrig,c (X/K) We prove Conjecture A in Theorem 4.0.1, and we prove Conjecture B in the case where X is smooth over k in Theorem 6.1.1. The proof of Conjecture B in the smooth case uses a generalization of one of the main results in [DLZ11]: Theorem 1.3.3. [DLZ11, Theorem 4.40] Let X be a smooth quasi-projective scheme over k. Then we have a natural quasi-isomorphism ∼ RΓrig (X/K) → RΓ(X, W † Ω•X/k ) ⊗ Q. We generalize this result in order to drop the quasi-projectiveness condition in Theorem 5.5.5 by use of simplicial and cohomological descent methods. 1.4 Outline In Chapter 2, we introduce the necessary background. Mainly the cohomological descent and simplicial techniques from [Con03], and dierent p-adic cohomologies and their relations. In particular, we summarize some of the 4 notation and main results from the rigid cohomology version of [CT03], which will allow us to use simplicial methods on rigid cohomology. In Chapter 3, we explain the construction of the eh-site and extension of étale motivic cohomology to singular varieties done in [Gei06]. In Chapter 4, we prove Conjecture A. In order to do so, for a given scheme X , we rst form (under assumption of strong resolution of singularities) a hypercovering in the eh site by smooth schemes, which is also a proper hypercovering. This will allow cohomological descent on the motivic side, and on the rigid side. Doing this functorially, and showing independence of choices will allow to prove Conjecture A. In Chapter 5, we prove the generalization of [DLZ11, Theorem 4.40] to smooth schemes. In order to transfer their machinery, we nd a hypercovering of a given smooth scheme by ane standard smooth schemes, and use some of their results and a vanishing result to generalize the methods. In Chapter 6, we use the result from Chapter 5 and Poincaré duality on rigid cohomology to prove Conjecture B for smooth schemes. 5 Chapter 2 BACKGROUND 2.1 Cohomological Descent Simplicial Objects We summarize some of the results and notation from [Con03]. Let C be a category admitting nite inverse limits. • We denote by ∆+ the category of objects [n] = {0, ..., n} for n ≥ −1, with morphisms given by non-decreasing maps of ordered sets [n] → [m]. • We denote by ∆ the full subcategory of objects [n] with n ≥ 0. + • We denote by ∆+ ≤N the full subcategory of ∆ of objects [n] with −1 ≤ n ≤ N. • We denote by ∆≤N the full subcategory of ∆ of objects [n] with 0 ≤ n ≤ N. Then, we consider the following: • Simp(C) is the category of simplicial objects in C . That is, contravariant functors X• : ∆ → C , where Xn = X• ([n]). • Simp+ (C) is the category of augmented simplicial objects in C . That is, contravariant functors X• /S : ∆ → C , where S denotes the image of [−1]. • SimpN (C) is the category of N -truncated simplicial objects in C . That is, contravariant functors X•≤N : ∆≤N → C . • Simp+ N (C) is the category of N -truncated augmented simplicial objects in C . That is, contravariant functors X•≤N /S : ∆+ ≤N → C . + + Let skN : Simp(C) → SimpN (C) and sk+ N : Simp (C) → SimpN (C) denote the N -skeleton functor skN (X• ) = X•≤N . 6 Since C is taken to have nite inverse limits, we have the following: Theorem 2.1.1. [Con03, Theorem 3.9] For any N ≥ 0, skN admits a right adjoint coskN : SimpN (C) → Simp(C). Similarly for augmented objects and N ≥ −1. All of the above may be generalized to multisimplicial objects (see [Con03, Denition 3.13]). Hypercovers Denition 2.1.2. Let P be a class of morphisms in C which is stable under base change, preserved under composition and containing all isomorphisms. A simplicial object X• in C is said to be a P-hypercovering if, for all n ≥ 0, the natural adjunction map X• → coskn (skn (X• )) induces a map Xn+1 → coskn (skn (X• ))n+1 in degree n + 1 which is in P. Two common examples will be when C is some category of spaces (e.g. schemes), when P is the class of proper surjective maps, in which case we will talk of proper hypercoverings ; and when P is the class of étale surjective maps, in which case we will call them étale hypercoverings. In order to construct hypercoverings, we introduce the notion of split simplicial objects: Denition 2.1.3. We say that a simplicial object X• is split if there exist subobjects N Xj in each Xj such that the natural map G N X φ → Xn φ:[n][m] is an isomorphism for every n ≥ 0, where N Xφ := N Xm for a surjection φ : [n]  [m], and the natural maps are given by the composition X• (φ) N X φ ⊂ Xm → Xn . We dene truncated and augmented cases similarly. 7 We denote by N Xm,φ the image of N Xφ ⊂ Xm under this isomorphism. Note that for any epimorphism φ : [n]  [m] we have a commutative map N Xm,id[m] ⊂ Xm 9 ∼ N Xm ∼ . X• (φ) ∼ &  N Xn,φ ⊂  Xn By [Con03, Theorem 4.12], given any split n-truncated augmented simplicial scheme X•≤n /S with the splitting given by {N Xk }0≤k≤n , in order to extend it to a split (n + 1)-truncated scheme X•≤n+1 /S it suces to give an object N Xn+1 and a morphism β : N Xn+1 → coskSn (X•≤n )n+1 . Following the notation from [CT03, Section 11.2], we denote the corresponding n + 1 augmented simplicial object above by ΩSn+1 (X•≤n , N X0 , ..., N Xn+1 ) ∈ Simp+ ≤n+1 (C). This construction can be done similarly for the non-augmented case. Remark. Note that if we construct stepwise a split object X• using the above, by choosing β to be in P for every n we can form a P-hypercovering. Cohomological Descent Consider C to be some category of spaces, and X• a simplicial object in C . Denition 2.1.4. A F• sheaf of sets (resp. groups, resp. rings) on X• consists of a collection {Fn } where Fn is a sheaf of sets (resp. groups, resp. rings) on Xn satisfying some compatibility conditions. More explicitly, given φ : [n] → [m], we have a map of sheaves [φ] : X(φ)∗ (Fn ) → Fm satisfying [φ] ◦ X(φ)∗ [ψ] = [φ ◦ ψ] for composable φ, ψ . 8 Given an augmented simplicial object X• /S , let w• : X• → S denote the augmented structure. Then, we have a map of topoi w = (w•∗ , w•∗ ) : X̃• → S̃ where for a sheaf G on S , (w•∗ (G ))n := wn∗ G and for a sheaf F• on X• , (w•∗ (F• )) = ker(w0∗ F0 → w1∗ F1 ). Similarly we can do this for abelian sheaves, rings and modules over some ring. We can demonstrate that there are enough injectives and thus we obtain functors w∗ : D+ (S) → D+ (X• ), Rw∗ : D+ (X• ) → D+ (S) on the abelian level. Denition 2.1.5. • We say that w : X• → S is a morphism of cohomological descent if the natural transformation id → Rw∗ ◦ w∗ on D+ (S) is an isomorphism. • w : X• → S is said to be universally of cohomological descent if for every base change S 0 → S , the augmentation w0 : X• ×S S 0 → S 0 is of cohomological descent. 2.2 Rigid Cohomology We use [CT03] denition of rigid cohomology in order to work without assumptions of closed embeddings into a smooth formal scheme. We summarize their main notation and results below. Denition 2.2.1. • A pair of schemes (X, X) consists of an open immersion X ,→ X over k. 9 • A triple of schemes X = (X, X, X ) consists of a pair (X, X), and a closed immersion X ,→ X ×W k for a formal W -scheme X of nite type over W . We will denote triples by their corresponding fraktur letter. • Given a pair (X, X), a (X, X)-triple Y = (Y, Y , Y) is given by a commutative diagram /Y /Y Y  /X  X   k  /W k Morphisms are just pairs (and triples) of morphisms w = (ẘ, w) : (X, X) → (Y, Y ) (and w = (ẘ, w, ŵ) : X → Y) over (k, k) (and (k, k, W)) tting into the commutative diagrams X  ẘ Y /X  X w /Y  Y /X ẘ  /Y w /X  ŵ /Y Denition 2.2.2. • A morphism of pairs w : (Y, Y ) → (X, X) is strict if Y = w−1 (X). • A morphism of triples w = (ẘ, w, ŵ) : Y → X is strict if Y = ŵ−1 (X) and Y = w−1 (X). Denition 2.2.3. Let ŵ : Y → X be a morphism of formal schemes, and let Y be a subset of Y . Then, ŵ is smooth around Y if there exists an open formal subscheme U of Y such that Y ⊂ U and ŵ|U : U → X is smooth. Denition 2.2.4. For a formal scheme P over W , we have an associated rigid analytic space PK over SpmK in the sense of Raynaud [Ray74], and a specialization morphism sp : PK → P. Given a k -subscheme X in the special ber P0 := P ×W k , we let ]X[P := sp−1 (X) with the induced Grothendieck topology from PK , and call it the tube of X in PK . 10 Given a morphism of triples w : Y → X, we naturally get a morphism of rigid analytic spaces w̃ :]Y [Y →]X[X . Hypercoverings Denition 2.2.5. (1) For a simplicial pair (Y• , Y • ) → (X, X):  (Y• , Y • ) → (X, X) is an étale-proper hypercovering if Y• → X is an étale-hypercovering and Y • → X is proper (i.e. for all n, Y n+1 → coskX n (Y •≤n )n+1 is proper, possibly non-surjective).  (Y• , Y • ) → (X, X) is an étale-étale hypercovering if both Y• → X and Y • → X are étale hypercoverings, and (Yn , Y n ) → (X, X) is strict for all n.  (Y• , Y • ) → (X, X) is an proper-proper hypercovering if Y• → X is a proper-hypercovering, Y • → X is proper, and (Yn , Y n ) → (X, X) is strict for all n. (2) A simplicial triple Y• → X is a étale-proper (resp. étale-étale, resp. proper-proper) hypercovering if: i) (Y• , Y • ) → (X, X) is an étale-proper (resp. étale-étale, resp. properproper) hypercovering of pairs. X X ii) coskX n (Y•≤n )l → coskn−1 (Y•≤n−1 )l is smooth around coskn (Y•≤n )l for any n and l. • A simplicial (X, X)-triple Y• is a étale-proper (resp. étale-étale, resp. proper-proper) hypercovering if: i) (Y• , Y • ) → (X, X) is an étale-proper (resp. étale-étale, resp. properproper) hypercovering of pairs. W X ii) coskW n (Y•≤n )l → coskn−1 (Y•≤n−1 )l is smooth around coskn (Y•≤n )l for any n and l. • We dene truncated versions similarly. 11 Lemma 2.2.6. For an n-truncated étale-proper (resp. étale-étale, resp. proper- proper) hypercovering Y•≤n → X, we have that coskXn (Y•≤n ) = (coskXn (Y•≤n ), coskXn (Y •≤n ), coskXn (Y•≤n )) → X is an étale-proper (resp. étale-étale, resp. proper-proper) hypercovering. Proof. This follows from the fact that for 0 ≤ n ≥ m, by [Con03, Corollary 3.11] we have a natural isomorphism ∼ coskm (skm (−)) → coskn (skn (coskm (skm (−))). Overconvergence We introduce strict neighborhoods, to deal with overconvergence in the case of non-proper schemes: Denition 2.2.7. [Ber96, Def.1.2.1] Given a triple (X, X, X ), a subset V of ]X[X is called a strict neighborhood of ]X[X in ]X[X if {V, ]X \ X[X } is an admissible covering of ]X[X . We will simply call V a strict neighborhood if there is no possibility of confusion about ]X[X and ]X[X . For admissible open subsets V ⊂ U of ]X[X , denote by jVU : V → U the ]X[ inclusion. In the case U =]X[X , simply set jV := jV X . By [Ber96, Prop. 1.2.10.(i)], intersections of strict neighborhoods are still strict neighborhoods, so these form a ltered category. Therefore, given a sheaf of abelian groups F on a strict neighborhood U , we dene the sheaf of overconvergent sections of F on ]X[X along ]X \ X[X as jU† F := lim jV ∗ (jVU )−1 F −→ V ⊂U where V runs through strict neighborhoods contained in U . We denote † j † := j]X[ for when U =]X[X . X If F is a sheaf of rings on U (resp. O -module for O a sheaf of rings on U ), then jU† F is a sheaf of rings on ]X[X (resp. a jU† O -module). Given a morphism of triples w : Y → X, consider the natural map w̃−1 (j † O]X[X ) → j † O]Y [Y . 12 For a sheaf E of coherent j † O]X[X -modules, we can dene w† E := w̃−1 E ⊗w̃−1 (j † O]X[ ) j † O]Y [Y X which by [CT03, Prop. 2.10.1] gives a functor w† : Coh(j † O]X[X ) → Coh(j † O]Y [Y ). Given a simplicial triple X• , we get a simplicial objects of rigid spaces ]X • [X• and we may consider sheaves of rings O]X • [X• as in Denition 2.1.4. We may further apply the j † at every n to consider sheaves of rings j † O]X • [X• and sheaves of j † O]X • [X• -modules. We may generalize as follows: Denition 2.2.8. We say a sheaf E• of j † O]X • [X• -modules is coherent if • En is a sheaf of coherent j † O]X n [Xn -modules for all n. • For any φ : [n] → [m], the map j † O]X m [Xm ⊗φ̃−1 j † O ]X n [X n φ̃−1 En → Em is an isomorphism, where φ̃ :]X m [Xm →]X n [Xn is the map induced by φ. Given an augmented simplicial triple w• : Y• → X and a complex of sheaves F•• of w̃•−1 (j † O]X[X )−modules, let I•• be an injective resolution of F•• in w̃•−1 (j † O]X[X )−mod. Then, dene Rw•∗ F•• (denoted by RC † (X, Y• ; F ) in [CT03]) to be the total complex associated to ... ...O ...O ... 0 / w̃0∗ I 1 O 0 / / w̃1∗ I11 O / // ... 0 / w̃0∗ I 0 O 0 / / w̃1∗ I10 O / // ... 0 0 ... (2.1) 13 where the vertical maps come from maps in Ip• , and the horizontal come from the simplicial structure. Note that these are complexes of abelian sheaves on ]X[X . The n-truncated version w•≤n : Y•≤n → X is dened similarly, taking the total complexes of 2.1 and setting the columns larger than n to 0. We will be particularly interested in the case when X = (Spec k, Spec k, Spf W), in which case we will denote RΓ(]Y • [Y• , F•• ) = Rw•∗ F•• . Denition 2.2.9. With the same w• : Y• → X, suppose that Y• is smooth over X around Y• . We say that w• is de Rham descendable if, for any sheaf E of coherent j † O]X[X -modules, the canonical homomorphism   E → Rw•∗ w̃•† E ⊗j † O]Y [ j † Ω•]Y • [Y /]X[X • Y• • is an isomorphism in D+ (Z]X[X ). We say w• is universally de Rham descendable if, for every morhpism Z → Y of triples, the base change Y• ×X Z → Z is de Rham descendable. 2.3 Denition of rigid cohomology Denition 2.3.1. Let Y• be a simplicial (X, X) triple, such that Yn → W is smooth around Yn for all n. We say Y• is a universally de Rham descendable hypercovering of (X, X) if for any (X, X)-triple Z, the base change Y• ×(X,X,W) Z → Z is de Rham descendable. Proposition 2.3.2. Given (X, X), there always exists a universally de Rham descendable hypercovering Y• of (X, X). Furthermore, if Y• a (X, X)-triple is an étale-proper (resp. étale-étale, resp. proper-proper) hypercovering, then, Y• is a universally de Rham descendable hypercovering of (X, X) Proof. The rst part is [CT03, Corollary 10.1.5]. For the second part, see [CT03, Example 10.1.6.] for étale-étale and étale-proper cases, and [Tsu03, Proposition 2.2.2.] for the proper-proper case. 14 Denition 2.3.3. Given a k-scheme X , consider an open immersion into a proper k -scheme X ; this gives a pair (X, X). Let Y• be any universally de Rham descendable hypercovering of (X, X), then set RΓrig (X/K) := RΓ(]Y • [Y• , j † Ω•]Y • [Y ). • Such a X always exists by Nagata, and by [CT03], a universally de Rham descendable hypercovering always exists (Corollary 10.1.5), this denition is independent of the choice of universally de Rham descendable hypercovering Y• (Proposition 10.4.3.), compactication X (Corollary 10.5.4.) and agrees with Berthelot's original denition of rigid cohomology (Theorem 10.6.1). Note in the case that X is quasi-projective, we may nd some triple X = (X, X, X ) with X proper (in fact projective) and X a smooth formal W scheme. Then, we may take Y• to be the constant triple over X (that is, Yn = X), and RΓrig (X/K) ∼ = RΓ(]X[X , j † Ω•]X[X ). One important result that we will use later on, is the vanishing of rigid cohomology: Theorem 2.3.4. [Tsu04, Theorem 6.4.1] Given a scheme X over k, there i (X/K) = 0. exists an integer c such that for i > c, Hrig 2.4 The Tsuzuki Functor When constructing some simplicial triple Y• to compute rigid cohomology, we may keep control of (Y• , Y • ) using a split construction (see Denition 2.1.2). However, it proves hard to embed into some simplicial formal scheme Y• , smooth over W , or even to construct it one step at a time. Noting that for an N -truncated étale-étale (resp. étale - proper, resp. properproper) hypercovering Y•≤N of (X, X), that (X,X,W) coskN (Y•≤N ) is also a étale-étale (resp. étale - proper, resp. proper-proper) hypercovering of (X, X), then we see that we just need to do our construction at the N truncated level. In fact, the construction below shows that all we need, is a closed immersion of Y N into some smooth formal W -scheme Y . Doing this, 15 we will lose control above N , but for vanishing reasons, this will not aect computations for large enough N . We use the Tsuzuki functor introduced in [CT03, Section 11.2]. Given a category C with nite inverse limits, a non-negative integer N , and an object X , we construct a N -truncated simplicial object ΓCN (X) in Simp≤N (C) as follows: Denition 2.4.1. Set ΓCN (X)m := Y Xφ φ:[N ]→[m] where Xφ = X . To dene the simplicial maps, given α : [m0 ] → [m], we dene Γα : ΓCN (X)m → ΓCN (X)m0 by (cφ )φ:[N ]→[m] 7→ (dψ )ψ:[N ]→[m0 ] where dψ := cα◦ψ , and the product is in C . Note that this is just a product of copies of the given object X . In the case of augmented simplicial objects over some S , we take the product over S . Given any Y•≤N in Simp≤N (C), and a morphism f : YN → X in C , we construct a morphism Y•≤N → ΓCN (X) by the commutative diagram Ym / ΓC (X)m = N Q ψ:[N ]→[m] Xψ pφ Y (φ)  YN f  / X = Xφ for any m and φ : [N ] → [m], where pφ is just the projection onto the φ : [N ] → [m] factor. Letting C be the category of formal schemes over Spf(W) or of schemes over Spec(W (k)), we have the following: Lemma 2.4.2. Let C be as above. If f : YN → X is a (closed) immersion, and Y•≤N and X are separated, then the induced morphism Y•≤N → ΓCN (X)•≤N is a (closed) immersion. 16 Proof. We will do the case where C are schemes over W (k), but the case of formal schemes follows identically. For any 0 ≤ m ≤ N , consider any face morphism δ : YN → Ym (with δ = idYN if m = N ), and a corresponding degeneracy map σ : Ym → YN which is a section to δ . Then, we have / Ym δ YN  " W (k) where the vertical and diagonal maps are separated. This shows that δ is also separated. Then, by the commutative diagram Ym σ / YN  δ Ym we see that σ is a closed immersion. Finally, by the denition of the map W (k) gm : Ym → ΓN (X)m , we have a commutative diagram Q gm / ΓW (k) (X)m Ym X N φ:[N ]→[m] prσ σ  YN f  /X which shows that f ◦ σ , and thus gm is a (closed) immersion. This will be useful by the following result: Proposition 2.4.3. Suppose (Y•≤N , Y •≤N ) → (X, X) is an N -truncated étale- proper (resp. étale-étale, resp. proper-proper) hypercovering. Suppose further that there exists a closed immersion Y N ,→ Y ×k W for some smooth formal W -scheme Y . Then, X X W W coskN(X,X,W) (Y•≤N , Y •≤N , ΓW N (Y)) = (coskN (Y•≤N ), coskN (Y •≤N ), coskN (ΓN (Y)) is an étale-proper (resp. étale-étale, resp. proper-proper) hypercovering of (X,X,W) (X, X). In particular, coskN (Y•≤N , Y •≤N , ΓW N (Y)) is a universally de Rham descendable hypercovering of (X, X ). Proof. The rst part follows from the proof of [CT03, Prop. 11.4.1.] (note that using Q 0≤m≤N ΓW (Ym ) there instead of just ΓW (Y) does not seem necessary). The second part follows by Proposition 2.3.2. 17 2.5 Crystalline Cohomology The main reference is [BO78]. Given a scheme X over k , we may consider the crystalline site Cris(X/Wn ) with objects given by PD-thickenings (U ,→ T, δ) over Wn for Zariski opens U of X . Let (X/Wn )cris denote its topos. The morphism and topology is explained in Ÿ5 loc. cit. One can also take the direct limit of the sites (X/Wn )crys (see Ÿ7 loc. cit.), and obtain a site Cris(X/W), with a corresponding topos (X/W)cris . Let uX/Wn : (X/Wn )cris → XZar and uX/W : (X/W)cris → XZar denote the morphism of topoi. Then: Theorem 2.5.1. [BO78, Proposition 7.22] RΓ(X/W, OX/W ) ∼ RΓ(X/Wn , OX/Wn ) = R lim ←− n RuX/W∗ OX/W ∼ Ru O . = R lim ←− X/Wn ∗ X/Wn n 2.6 Witt de-Rham Cohomology The main reference is [Ill79]. For a given smooth scheme X , we may consider the complex of étale sheaves Wn Ω•X/k on X . We can consider the pro-complex W• Ω•X/k as a DGA with additional maps F : Wn ΩiX/k → Wn−1 ΩiX/k , V : Wn ΩiX/k → Wn+1 ΩiX/k satisfying certain compatibility conditions. We may also consider the limit W Ω•X/k := lim Wn Ω•X/k ←− n and endow it with a Frobenius endomorphism φ dened by φ = pi F on W ΩiX/k . By the proof of [Ill79, Proposition 2.1], we have that the canonical map W ΩiX/k → R lim Wn ΩiX/k ←− n is a quasi-isomorphism (even though the actual statement of the proposition also assumes properness, we do not need it for this result). For X proper and smooth, we can then consider the hypercohomology RΓ(X, W Ω•X/k ) which is a perfect complex of W (k)-modules by [Ill79, Theorem 2.7]. 18 2.7 Comparisons Theorem 2.7.1. If X is smooth, and there exists a closed embedding into a smooth formal scheme X over W , then there exists a natural quasi-isomorphism Rsp∗ Ω•]X[X ∼ = RuX/W (k) OX/W (k),Q ∼ = W Ω•X,Q on XZar . Therefore, if X is smooth and projective, this induces natural quasiisomorphisms RΓrig (X/K) ∼ = RΓ(X/W (k))Q ∼ = RΓ(X, W Ω•X )Q . Proof. From the proof of [Ber97b, Proposition 1.9] we have that ∼ Rsp∗ Ω•]X[X ← sp∗ Ω•]X[X ∼ = (sp∗ O]X[X ) ⊗ Ω•X . For the rst isomorphism, given any open ane formal scheme U = SpfA ⊂ X , we see that sp−1 (U ) ∼ = Spm(A ⊗ K) is anoid and thus quasi-Stein, thus its closed subspace sp−1 (U )∩]X[X is also quasi-Stein, and thus satises Kiehl's Theorem B. Therefore, H i (sp−1 (U )∩]X[X , Ωk]X[X ) = 0 for all k and i > 0. Since Ri sp∗ Ωk]X[X is the sheaf associated to the presheaf U ∩ X 7→ H i (sp−1 (U )∩]X[X , Ωk]X[X ) this proves the vanishing of the higher cohomologies. Then, there exists a natural morphism (sp∗ O]X[X ) ⊗ Ω•X → P̂(I) ⊗ Ω•X ,Q where I is the ideal of X in X , P(I) is the PD-envelope of X in X , and P̂(I) its p-adic completion. This is a quasi-isomorphism when X is smooth. Note that even though properness is assumed in the statement of Berthelot's proposition, we do not require it for this quasi-isomorphism. Next, by [BO78], we have natural quasi-isomorphisms R lim RuX/Wn (k) OX/Wn (k) ←− [BO78, Th.7.22.2] ∼ = RuX/W ∗OX/W [BO78, Th.7.23] ∼ = P̂(I)⊗Ω•X . n Finally, by [Ill79], we have natural quasi-isomorphisms R lim RuX/Wn (k) OX/Wn (k) ←− n [Ill79, II.Th.1.4.] ∼ = R lim Wn Ω•X ←− n [Ill79, II.Pr.2.1] ∼ = W Ω•X 19 where we again note that even though the Proposition for the last quasiisomorphism assumes properness, the quasi-isomorphism holds without properness. Tensoring with Q these last quasi-isomorphisms we complete the proof. 20 Chapter 3 P-ADIC MOTIVIC COHOMOLOGY ON SINGULAR VARIETIES We recall notation and results from [Gei06]. 3.1 The eh-topology Fix a perfect eld k . For d ∈ N ∪ ∞, let Schd /k be the category of separated schemes of nite type over k of dimension ≤ d (and drop the d in the case d = ∞), and Smd /k the full subcategory of smooth schemes over k . Denition 3.1.1. The étale h-topology (abbreviated eh-topology ) on Sch/k, is the Grothendieck topology generated by the following coverings: 1) Étale coverings. 2) Abstract blowups {Z → X, X 0 → X} coming from a cartesian square Z0 f0  Z i0 / X0 f i  /X where f is a proper morphism, i a closed embedding, and f induces an ∼ isomorphism X 0 − Z 0 → X − Z . We state the following result (cf. [SV00, Lemma 5.8]: Lemma 3.1.2. [Gei06, Lemma 2.2.a] Every proper morphism p : X 0 → X , such that for every point x ∈ X there is a point x ∈ X with p(x0 ) = x which induces an isomorphism on the residue elds, is an eh-covering. Denition 3.1.3. We call a covering as in Lemma 3.1.2 a proper eh-covering. Remark. These are called proper cdh-coverings in [SV00]. Denition 3.1.4. For d ∈ N ∪ ∞, we say the strong form of resolution of singularities holds for varieties up to dimension d if the following hold: 21 • For every integral separated scheme X ∈ Schd /k , there is a proper, birational map f : Y → X with Y ∈ Sm/k . • For every smooth scheme X ∈ Smd /k and every proper birational map f : Y → X , there is a sequence of blow-ups along smooth centers Xn → Xn−1 → ... → X1 → X such that Xn → X factors through f . If this holds, we denote it by R(d). Remark. Some known cases of resolution of singularities: • R(∞) when char(k) = 0, by [Hir64]. • R(2) in general, and R(3) for k algebraically closed of char(k) = p > 5, by [Abh56]. Note that R(d) makes all schemes in Schd /k locally smooth in the eh-topology (see Lemma 4.1.3). The inclusion of smooth schemes then induces a morphism of sites ρd : (Schd /k)eh → (Smd /k)et . This in turn induces a morphism of topoi under R(d): Lemma 3.1.5. [Gei06, Lemma 2.5.a] Assume R(d) holds. Then the functor ρd induces a morphism of topoi d ∼ ρd : (Schd /k)∼ eh → (Sm /k)et . 3.2 Cohomology for the eh-topology By usual arguments, (Schd /k)∼ eh has enough injectives, which allows to dene cohomology groups RΓ(Xeh , F ) as the right derived functor for the global section functor Γ(Xeh , −). One of the advantages of eh-topology is that we can dene the cohomology with compact support as follows: Denition 3.2.1. Let X ∈ Schd /k. Take an open embedding j : X → X with dense image into a proper scheme, and let i : Z → X be the closed complement with reduce subscheme structure. For F ∈ (Schd /k)∼ eh , take an injective resolution F → I • in (Schd /k)∼ eh , and dene   RΓc (Xeh , F ) := I • (X) → I • (Z) . 22 This is independent of the choice of X by [Gei06, Lemma 3.4], and has the expected properties, such as contravariance for proper maps, covariance for open embeddings, and long exact sequences for open-closed decompositions. 3.3 Motivic cohomology for singular varieties Using R(d), we dene motivic cohomology on any scheme X ∈ Schd /k by considering RΓ(Xeh , ρ∗d Z(n)), where Z(n) is Suslin-Voevodsky's motivic complex [SV00, Denition 3.1] for smooth schemes. However, since we will be interested in a p-adic completion of this cohomology, we will use the identication Z/pr (n) ∼ = Wr Ωnlog [−n] on Sm/k from [GL00], Theorem 8.5, where Wr Ωnlog (denoted νrn there) is the subsheaf of Wr Ωn étale locally generated by sections of the forms dlogf1 ...dlogfn , dened in [Ill79] II.5.7. Under R(d), by the same proof as [Gei06] Theorem 4.3 mod pr we get that this motivic cohomology on the eh-site coincides with the usual motivic cohomology in Smd /k . Theorem 3.3.1. Assuming R(d), for any n ∈ N and r ≥ 0 we have Z/pr (n) → Rρd∗ ρ∗d Z/pr (n) on Smd /k. ∼ In particular, for any X ∈ Smd /k, RΓ(Xet , Z/pr (n)) ∼ = RΓ(Xeh , ρ∗d Z/pr (n)). We then consider the p-adic completion of this cohomology: Denition 3.3.2. Assume R(d). For X ∈ Schd /k and n ∈ Z set RΓ(Xeh , Zp (n)) := R lim RΓ(Xeh , ρ∗d Z(n)/pr ) ←− r and RΓ(Xeh , Qp (n)) := RΓ(Xeh , Zp (n))Q , and the cohomology with compact support RΓc (Xeh , Zp (n)) := R lim RΓc (Xeh , ρ∗d Z(n)/pr ) ←− r and RΓc (Xeh , Qp (n)) := RΓc (Xeh , Zp (n))Q , 23 Chapter 4 CONJECTURE A In this chapter, we prove Conjecture A: Theorem 4.0.1 (Conjecture A). Assume R(d). Let X be in Sch/k with dim X ≤ d. Then, for any n ∈ N, h i pn −φ ∼ RΓc (Xeh , Qp (n)) → RΓrig,c (X/K) → RΓrig,c (X/K) functorially in X . In Section 4.1, we construct proper-eh hypercoverings with suitable properties for the proof of Theorem 4.0.1 in Section 4.2. 4.1 Proper eh-hypercoverings We generalize the notion of a proper hypercovering: Denition 4.1.1. For an augmented simplicial scheme a : X• → Y , we say X• is a proper eh-hypercovering if the natural maps fn+1 : Xn+1 → (coskn skn (X• ))n+1 are proper-eh coverings for all n ≥ −1. We prove some properties of proper eh-coverings which will be useful in the construction of proper-eh hypercoverings: Lemma 4.1.2. Proper eh-coverings are stable under base change, preserved under composition and contain all isomorphisms. Proof. The only thing that needs proving is the stability under base change. Given a proper eh-cover p : X → Z , and a morphism f : Y → Z , consider X ×Z Y with morphism p0 and f 0 to X and Y respectively. Then, p0 is proper. Also, given any point y ∈ Y , consider a point x ∈ X lift f (y) =: z ∈ Z with the same residue eld. Then, since k(x) ⊗k(z) k(y) ∼ = k(x), the point k(x) factors through X ×Z Y . 24 Lemma 4.1.3. Assuming R(d), for every scheme Y in Sch/k of dimension ≤ d there exists a proper eh-covering X → Y with X ∈ Sm/k . Proof. Firstly, we can assume that X is integral since the reduced subscheme and disjoint union of irreducible components are both eh-coverings. We proceed by induction on d. The base case d = 0 is thus trivial. So we assume it holds true for dimension < d. Then, by R(d) we can nd a proper birational map Y0 →X with Y 0 smooth, which is an isomorphism away from some proper closed subscheme Z of X , and thus Y 0 t Z → X is an eh-covering. But by inductive hypothesis, there is a proper eh-covering Z 0 → Z with Z 0 in Sm/k , and thus Y := Y 0 t Z 0 → X is a proper eh-covering. Denition 4.1.4. We say X• → X is a peh-resolution if it is a split proper-eh hypercovering, and for all n, Xn is smooth over k . We dene the truncated version similarly. Proposition 4.1.5. Assuming R(d), then for every scheme X in Schd /k there exists a split proper eh-hypercovering X• → X with Xm ∈ Sm/k for all m. Proof. Assuming R(d), the proof is identical to the construction of proper hypercoverings (for example in [Con03, Theorem 4.13]), replacing proper surjective maps with proper-eh coverings at every step N Xn+1 → coskX n (X•≤n )n+1 using Lemma 4.1.3. Constructing peh-resolutions in this manner, we show we can always rene two given ones, and that we can construct them functorially. Proposition 4.1.6. Assume R(d). Then, i) Given two split proper eh-hypercoverings U• , V• /X , there exists a pehresolution W• /X and morphisms f• : W• → U• , g• : W• → V• over X. 25 ii) Given a morphism f : X → Y , and a peh-resolution b• : V• /Y , then there exists a peh-resolution a : U• /X with a morphism f• : U• → V• making U•  f• a X f / V•  b /Y commute. Proof. For i), we again proceed by proceed by constructing the n-truncation of W• one step at a time. For n = 0, taking a proper eh-covering N W0 = W0 → U0 ×X V0 with N W0 smooth, then this satises the lemma at 0-truncation. Assume we have constructed W•≤n satisfying the hypothesis. Then, let (−)0 X X X 0 denote coskX n skn (−) (e.g. U• = coskn skn (U• )), let N Uk 's give the splitting, and denote the proper eh-coverings used to construct the n + 1 step by β− 0 ). (e.g. βU : N Un+1 → Un+1 Take a proper eh-covering 0 0 0 N Vn+1 , βW : N Wn+1 → (Wn+1 N Un+1 ) ×Vn+1 ×Un+1 0 0 0 are dened by functoriality of the , Vn+1 to Un+1 where the morphisms Wn+1 coskn map and N Wn+1 is some smooth scheme. Then, looking at the composition p W 0 0 0 0 0 0 N Un+1 → Wn+1 N Vn+1 → Wn+1 ×Un+1 N Un+1 ) ×Vn+1 N Wn+1 → (Wn+1 ×Un+1 we see that the two last maps are base changes by proper eh-coverings. So by Lemma 4.1.2 all three are proper eh-coverings, and thus so is the composition 0 β : N Wn+1 → Wn+1 which we can use in the construction of Wn+1 by the same method as above. This then comes with obvious maps N Wn+1 → N Un+1 , N Vn+1 , compatible with the maps on components on lower skeleta, inducing maps Wn+1 → Un+1 , Vn+1 . Part ii) follows from the proof of i) by taking a renement of the peh-resolution W• /X and (V• ×Y X)/X . Finally, since working on ane schemes will be easier later on, we introduce the following: 26 Denition 4.1.7. Given an augmented simplicial scheme X• /X , we say an augmented simplicial scheme X•0 /X is a simplicial ane covering of X• /X if there is a morphism f• : X•0 → X• over X such that for all n, Xn0 = tα∈In Xn,α for a nite open covering by anes Xn ∼ = ∪α∈I Xn,α such that the image of n each Xn,α in X is contained in some ane open subscheme of X . These will always exist under nice enough conditions, and by a proof similar to 4.1.6 we have: Lemma 4.1.8. Let X• /X be a split proper hypercovering. Then: i) [Nak12, Lemma 9.6] There exists some simplicial ane covering X•0 /X of X• /X . ii) [Nak12, Proposition 6.3.1] Given another split proper hypercovering Y• /Y , and a commutative diagram X• g• / Y• a  X f b  /Y and any simplicial ane covering Y•0 /Y of Y• /Y , then there exists a simplicial ane covering X•0 /X of X• /X and a morphism g•0 : X•0 → Y•0 tting into the commutative diagram X•0 g•0 /Y0 • a0•  X•  g• a X f  b0• / Y•  b / Y. iii) [Nak12, Proposition 6.3.2] Given a two simplicial ane covering X•0 , X 00 • /X of X• /X , there exists a third simplicial ane covering X 000 • /X of X• /X rening X•0 , X•00 . 4.2 Proof of Conjecture A The main ingredient for the proof is the following result: 27 Theorem 4.2.1. [Nak12, Theorem 11.6.3] Suppose X is a proper scheme over k. Then there exists a quasi-isomorphism ∼ RΓrig (X/K) → RΓ(X• , W Ω•X• )Q , functorial in split smooth proper hypercoverings X• → X . Proof. We x some h and some N > (h + 1)(h + 2)/2. Take some simplicial ane covering X•0 of X• , which is possible by Lemma 4.1.8. Let X•,• be the 0 0 0 l ƒech diagram of X•,0 over X• , with Xlm := coskX 0 (Xl )m = Xl ×Xl ... ×Xl Xl . Take an ane open covering X = ∪Xα , with closed embeddings Xα ,→ Xα into smooth formal schemes, and let Z = tXα , Z = tXα . Let V (Z• , Z• ) = (coskX 0 (Z), cosk0 (Z))) be its ƒech hypercovering. Then, we set lm Xlmn := coskX (Xlm ×X Z)n = (Xlm ×X Z) ×Xlm ... ×Xlm (Xlm ×X Z) ∼ = 0 ∼ = Xlm ×X (Z ×X ... ×X Z) = Xlm ×X Zn . Since the Xl are smooth, so are the Xlm and Xlmn . We then construct a closed immersion X•≤N,•,• ,→ R•≤N,•,• where R•≤N,•,• is a smooth (N, ∞, ∞)truncated trisimplicial W -scheme. To do so, since XN 0 = XN0 is a disjoint union of ane open subschemes of the smooth scheme XN , we can pick a smooth lift XN 0 over Spf(W). Then, by Lemma 2.4.2 we have a closed immersion X•≤N 0 ,→ ΓW N (XN 0 )•≤N . This in turn gives a closed immersion W X•≤N,• ,→ R•≤N,• := coskW 0 (ΓN (XN 0 )•≤N ) and ˆ X•≤N,•,• ,→ R•≤N,•,• := coskW 0 (R•≤N,• ×W Z)) given respectively by W ˆ ˆ Xlm = Xl0 ×Xl ... ×Xl Xl0 ,→ Rlm = ΓW N (XN 0 )l ×W ...×W ΓN (XN 0 )l 28 and Rlmn Xlmn (Xlm ×X Z) ×Xlm ... ×Xlm (Xlm ×X Z)   / (Rlm × ˆ W Z)× ˆ W ...× ˆ W (Rlm × ˆ W Z). The following diagram summarizes the morphisms above: / X•≤N,• X•≤N,•,•  / X•≤N  /X Z• where X•≤N,•,• , X•≤N,• and Z• have compatible closed immersions into formally smooth simplicial schemes (note that we do not require any such embeddings for X and X•≤N ). Then, (X•≤N,•,• , R•≤N,•,• ) → (Z• , Z• ) induces a map (4.1) τ≤h RΓ(]Z• [Z• , Ω•]Z• [Z• ) τ≤h RΓrig (X/K)  τ≤h RΓ(]X•≤N,•,• [R•≤N,•,• , Ω•]X•≤N,•,• [R ) •≤N,•,• where τ≤h is the canonical truncation, and we show in Lemma 4.2.3 that it is a quasiisomorphism. Now, by Theorem 2.7.1, we have a natural quasiisomorphism Rsp∗ Ω•]X•≤N,•,• [R •≤N,•,• ∼ = W Ω•X•≤N,•,• ,Q , which we use to get )∼ = (4.2) RΓ(X•≤N,•,• , W Ω•X•≤N,•,• ,Q ) ∼ = (4.3) RΓ(X•≤N,• , W Ω•X•≤N ,•,Q ) ∼ = (4.4) RΓ(]X•≤N,•,• [R•≤N,•,• , Ω•]X•≤N,•,• [R •≤N,•,• RΓ(X•≤N , W Ω•X•≤N ,Q ) ∼ = RΓ(X•≤N,et , W Ω•X•≤N ,Q ) (4.5) where we have used that X•≤N,•,• → X•≤N,• and X•≤N,• → X•≤N are Zariski hypercoverings (and thus satisfy cohomological descent), and that Zariski and étale hypercohomology agrees since the the W Ωi are quasi-coherent sheaves (see [Mil80, Remark 3.8]). 29 This gives τ≤h RΓrig (X/K) ∼ = τ≤h RΓ(X•≤N,et , W Ω•X• ≤N,Q ) ∼ = τ≤h RΓ(X•,et , W Ω•X• ,Q ) (4.6) since by the spectral sequence E1p,q = H q (Xp , W Ω•Xp ) ⇒ H p+q (X• , W Ω•X• ) the Xn with n > N don't contribute to H i (X• , W Ω•X• ) for i ≤ h. q Next, by Theorem 2.3.4 there exists an integer c such that Hrig (X/K) = 0 for q > c. Thus, since (4.6) holds for any h, we see that H q (X•,et , W Ω••,Q ) = 0 for q > c also. Taking h = c, we can drop the truncation terms and get RΓrig (X/K) ∼ = RΓ(X•,et , W Ω•X• ,Q ) (4.7) Finally, it remains to show independence of all the choices, and prove functoriality, which is Lemma 4.2.2 below. Lemma 4.2.2. With the same assumptions as Theorem 4.2.1: i) The isomorphism (4.7) in D+ (K) is independent of choices. ii) The isomorphism (4.7) in D+ (K) is functorial in split smooth proper hypercoverings X• → X . Proof. i) Independence of choices: We need to show independence of the choices of closed embeddings into smooth formal schemes XN 0 ,→ XN 0 and Z ,→ Z , independence of choice of ane Zariski coverings Z → X and X•0 → X• and independence of choice of N satisfying N > (c + 1)(c + 2)/2 with c as in the proof. Given two choices i T i := (X•0 → X• , XNi 0 ,→ XNi 0 , Z i → X, Z i ,→ Z i ) for i = 1, 2, we will set i RΓi•,• := RΓ(]X•≤N i ,• [Ri , Ω•]X i i RΓi•,•,• := RΓ(]X•≤N i ,•,• [Ri , Ω•]X i •≤N i ,• •≤N i ,•,• [ •≤N i ,• R i •≤N i ,• ), [ i •≤N i ,•,• R •≤N i ,•,• ) 30 to be the complex formed as in the proof, where if not explicitly dened, we will drop the superscript i (e.g. for independence of choice of Z , T i = (X•0 → X• , XN 0 ,→ XN 0 , Z i → X, Z i ,→ Z i ) with only Z i , Z i and Z i ,→ Z i varying). By the description of D+ (K) in terms of right roofs, to show that the two maps 8 RΓi•,•,• g ∼ RΓ(X• , W Ω•X• )Q RΓrig (X/K) i = 1, 2 are equivalent, it suces to nd some RΓ12 •,•,• tting into the commutative diagram 8 RΓ1•,•,• (4.8) g  RΓrig (X/K) RΓ(X• , W Ω•X• )Q RΓ12 •,•,• O & RΓ12 •,•,• w where all maps are quasi-isomorphisms. • Independence of choice of XN 0 : Suppose we choose two dierent closed embeddings into smooth formal schemes XN 0 ,→ XN1 0 , XN2 0 . Take the closed embedding into a smooth formal scheme ˆ W XN2 0 =: XN120 . XN 0 ,→ XN1 0 × i Letting Γi := ΓW N (XN 0 )•≤N for i = 1, 2, 12, we then have a commutative diagram of N -truncated simplicial complexes 2 > Γ1 X•≤N,0 / Γ12 , Γ2 31 which in turns gives rise to two diagrams 1 2 R•≤N,• : p1 X•≤N,• / R12 •≤N,• p2 , $ R2•≤N,• of (N, ∞), and similarly for (N, ∞, ∞)-truncated simplicial complexes, where we construct Ri•≤N,• and Ri•≤N,•,• as above for i = 1, 2, 12. Then, by Berthelot's independence of the choice of a closed immersion into a smooth formal scheme [Ber97b, Théorème 1.4] we have quasi-isomorphisms Ω•]X•≤N,•,• [ i R •≤N,•,• ∼ → Rpi∗ Ω•]X•≤N,•,• [ 12 R •≤N,•,• for i = 1, 2. This in turn gives rise to a desired diagram such as (4.8). • Independence of choice of ane covering X•0 : Given two simplicial ane 2 1 → X• , by Lemma 4.1.8.iii) we can choose a common , X•0 coverings X•0 12 tting into simplicial ane covering X•0 = 1 X•0 , / X• > 12 X•0 ! 2 X•0 12 where Xn0 is the disjoint product of some ane coverings of Xn for every n. Then, we can choose closed embeddings XNi 0 ,→ XNi 0 into smooth formal schemes, compatible with the above maps. This will give maps 1 (X•≤N,• ,→ R1•≤N,• ) 4 12 (X•≤N,• ,→ R12 •≤N,• ) * 2 (X•≤N,• ,→ R2•≤N,• ) 32 and similar for its trisimplicial counterpart. We claim that all the vertical arrows in the commutative diagram RΓ1•,•,• o ∼   o ∼ RΓ12 •,•,• RΓ12 •,• RΓ2•,•,• o ∼ RΓ2•,• O (4.9) RΓ1•,• O are quasi-isomorphisms, from which independence will follow just as before. It will suce to show this for the right vertical arrows. The fact that they are quasi-isomorphisms follows since for a xed p ≤ N , i i , Ril• ) is a univeris a Zariski covering of Xp , (Xp• and i = 1, 2, 12, as Xp0 sally de Rham descendable hypercovering of Xp by [CT03, Proposition 10.1.4]. Thus, by the independence of choice of universally de Rham descendable hypercovering ( [CT03, Lemma 10.4.1]), we have quasiisomorphisms 1 [R1p,• , Ω•]Xp,• ) RΓ(]Xp,• 1 [ 1 Rp,• ∼  12 ) RΓ(]Xp,• [R12 , Ω•]Xp,• 12 [ p,• 12 O Rp,• ∼ 2 RΓ(]Xp,• [R2p,• , Ω•]Xp,• ) 2 [ 2 Rp,• for all p ≤ N . Together with the spectral sequence i i E1p,q = H q (RΓ(]Xp,• [Rip,• , Ω•]Xp,• ) ⇒ H p+q (]X•≤N,• [Ri•≤N,• , Ω•]X i i [ i Rp,• •≤N,• [Ri •≤N,• we see that the right vertical arrows in (4.9) are quasi-isomorphisms. • Independence of choice of ane covering Z of X and closed immersion Z ,→ Z : We already know that the denition of rigid cohomology does not depend on choices of Z and Z by [CT03, Proposition 10.4.3], so we show that the comparison morphism is also independent. Given two ane coverings Z i → X (i = 1, 2), with closed immersions Z i ,→ Z i into smooth formal schemes. we can pick a simplicial ane covering Z 12 → X with a compatible closed immersion Z 12 ,→ Z 12 for ) 33 some smooth formal scheme. This will give compatible maps Z•1 ← 2 Z•12 → Z•2 and R1•≤N,•,• ← R12 •≤N,•,• → R•≤N,•,• giving the commutative diagram RΓ(]Z•1 [Z•1 , Ω•]Z•1 [ 1 ) ∼ / RΓ1 RΓ(]Z•12 [Z•12 , Ω•]Z•12 [ 12 ) ∼  / RΓ12 •,•,• O RΓ(]Z•2 [Z•2 , Ω•]Z•2 [ 2 ) ∼ / RΓ2  O (4.10) •,•,• Z• Z• •,•,• Z• By [CT03, Lemma 10.4.1], the left vertical maps in (4.10) are quasiisomorphisms, making all the vertical maps quasi-isomorphisms. • Independence of choice of N : Take N 2 ≥ N 1 satisfying N 1 > (h + 1)(h + 2)/2 for h = c as above, and choose embeddings XN i 0 ,→ XNi i 0 to construct (X•≤N i ,•,• , Ri•≤N i ,•,• ) for i = 1, 2. Firstly, by the independence of the embedding, we can replace R1•≤N 1 ,•,• with the (N 1 , ∞, ∞)-truncation R2•≤N 1 ,•,• . Then, consider the natural map τ≤c RΓ(]X•≤N 1 ,•,• [R2 •≤N 1 ,•,•  τ≤c RΓ(]X•≤N 2 ,•,• [R2 •≤N 2 ,•,• , Ω•]X ) , Ω•]X ) •≤N 1 ,•,• [R2 •≤N 1 ,•,• •≤N 2 ,•,• [R2 •≤N 2 ,•,• coming from truncation. By choice of N 1 and N 2 , the Xlmn and R2lmn with l > N 1 do not contribute to the cohomology of the bottom complex, so the above is a natural quasi-isomorphism. Similarly, we have a natural quasi-isomorphisms τ≤c RΓ(X•≤N 1 , W Ω•X•≤N 1 )Q  τ≤c RΓ(X•≤N 2 , W Ω•X•≤N 2 )Q  τ≤c RΓ(X• , W Ω•X• )Q compatible with the above. This shows independence of the choice of N . 34 Functoriality: Given a diagram of good split proper hypercoverings X• / Y• ,   /Y X and having chosen a disjoint union W of an open ane covering of Y , and a closed immersion W ,→ W into a smooth formal scheme, then we can pick Z to be a disjoint union of an ane open covering of X rening W , and a closed immersion Z ,→ Z tting into the commutative diagram Z /Z   / W. W Similarly, having chosen a simplicial ane covering Y•0 → Y• , then by Lemma 4.1.8.ii) we can choose some simplicial ane covering X•0 → X• tting into the commutative diagram X•0 / X•   / Y• . Y•0 Then, we can pick XN 0 and YN 0 tting into the commutative diagram XN 0 / XN 0 .   / YN 0 YN 0 This in turn will give morphisms of pairs and triples (X•≤N,• , R•≤N,• ) → (Y•≤N,• , S•≤N,• ), (X•≤N,•,• , R•≤N,•,• ) → (Y•≤N,•,• , S•≤N,•,• ), where S•≤N,• and S•≤N,•,• are constructed for Y the same as the R equivalents are for X . All these maps provide compatible maps at each step of the construction of the comparison morphism. 35 Lemma 4.2.3. The map (4.1) is a quasi-isomorphism. Proof. We give an outline of the proof, by drawing directly from the proof of [Nak12, Theorem 11.6], which provides a detailed explanation. We introduce some intermediate multisimplicial pairs to prove the quasi-isomorphism. Consider the proper hypercovering (X• ×X Z)/Z , and take a projective renement V• /Z using [Tsu03, Lemma 4.2.2.(1)]. That is, V• /Z is a proper hypercovering, and we have a Z -morphism V• → X• ×X Z with the natural morphisms Vn → coskZn−1 (skn−1 (V• ))n ×coskZn−1 (skn−1 (X• ×X Z))n (X• ×X Z)n are proper surjective for any n, and each Vn is projective over Z . Note that we do not require any smoothness conditions on V• , and that we get an X morphism V• → X• . Thus, we can choose a closed embedding into a smooth formal scheme VN ,→ PN , and setting Q• := ΓW N (PN ) we get a closed embedding V•≤N ,→ Q•≤N . We can then form a (N, ∞)-truncated bisimplicial complex (V•≤N,,• ), Q•≤N,,• ), where  stands for an empty spot, by setting W l ˆ Vln = coskX 0 (Vl )n , Qln = cosk0 (Ql ×W Z)n , 0 ≤ l ≤ N, n ∈ N. This gives a diagram V•≤N,,• / X•≤N   / X. Z l ∼ Also set Xln := coskX 0 (Xl ×X Z)n = Xl ×X Zn to dene X•≤N,,• . Thus, for any n ∈ N, (V•≤N,,n , Q•≤N,,n ) → (Zn , Zn ) is a N -truncated proper hypercovering. We consider the induced morphism RΓ(]Z• [Z• , Ω•]Z• [Z• ) → RΓ(]V•≤N,,• [Q•≤N,,• , Ω•]V•≤N,,• [Q •≤N,,• ). 36 We claim that applying τ≤h (−) to this morphism gives a quasi-isomorphism. To see this, consider the spectral sequences E1pq = H q (]Zp [Zp , Ω•]Zp [Zp ) ⇒ H p+q (]Z• [Z• , Ω•]Z• [Z• ), E1pq = H q (]Vp,,• [Qp,,• , Ω•]Vp,,• [Q p,,• ) ⇒ H p+q (]V•≤N,,• [Q•≤N,,• , Ω•]V•≤N,,• [Q •≤N,,• Since the maps for each p are proper hypercoverings, using [Tsu03, Theorem 2.1.3] we get that the maps on E1pq for 0 ≤ p ≤ N are isomorphisms, and by our choice of N relative to h, we see that we get the desired isomorphism H i (]Z• [Z• , Ω•]Z• [Z• ) → H i (]V•≤N,,• [Q•≤N,,• , Ω•]V•≤N,,• [Q •≤N,,• ) is an isomorphism for i ≤ h. Dene V V•≤N,•,• = cosk0 •≤N,,• (V•≤N,,• ×X•≤N X•≤N,0 ) so Vlmn ∼ = Vln ×Xl Xlm ∼ = Vln ×Xl ×X Zn (Xlm ×X Zn ) ∼ = Vln ×Xln Xlmn giving a cartesian diagram V•≤N,•,• / V•≤N,,•   / X•≤N,,• . X•≤N,•,• We can then form a closed embedding V•≤N,•,• ,→ S•≤N,•,• into a smooth formal scheme, with a pair of morphisms into X•≤N,•,• ,→ R•≤N,•,• . We claim that the induced morphism τ≤h RΓ(]X•≤N,•,• [R•≤N,•,• , Ω•]X•≤N,•,• [R ) •≤N,•,•  τ≤h RΓ(]V•≤N,•,• [S•≤N,•,• , Ω•]V•≤N,•,• [S ) •≤N,•,• is a quasi-isomorphism. Similar to above, it suces to check that the induced morphism H q (]Xp,•,• [Rp,•,• , Ω•]Xp,•,• [Rp,•,• ) → H q (]Vp,•,• [Sp,•,• , Ω•]Vp,•,• [Sp,•,• ) ). 37 is an isomorphism for p ≤ N . This in turn follows from the commutative diagram ∼ RΓ(]Xp• [Rp• , Ω•]Xp• [Rp• ) RΓrig (Xp /K) ∼ RΓrig (Xp /K) / RΓ(]Vp• [S , Ω•]Vp• [S p• / RΓ(]Xp•• [R p•• ) ∼ p• / RΓ(]Vp•• [S  p•• , Ω•]Xp•• [Rp•• ) , Ω•]Vp•• [Sp•• ) where the horizontal maps are quasi-isomorphisms since they come from Zariski hypercoverings. Then, with some additional checking and using the spectral sequences p+q (X/K), E1pq = H q (]Zp [Zp , Ω•]Zp [Zp ) ⇒ H p+q (]Z• [Z• , Ω•]Z• [Z• ) = Hrig E1pq = H q (]Xp,•,• [Rp,•,• , Ω•]Xp,•,• [Rp,•,• ) ⇒ H p+q (]X•≤N,•,• [R•≤N,•,• , Ω•]X•≤N,•,• [R •≤N,•,• the above isomorphism proves the lemma. This allows us to prove conjecture A under R(d): Proof of Theorem 4.0.1. For general X , take a compactication X ,→ X where X is proper over k and has X as a dense open subscheme. Let Z ⊂ X be the complement of X with reduced closed subscheme structure. Since in D+ (K),   RΓc (Xeh , Qp (n)) = RΓ(X eh , Qp (n)) → RΓ(Zeh , Qp (n)) and   RΓrig,c (X/K) = RΓrig (X/K) → RΓrig (Z/K) it suces to prove the theorem for X proper, along with functoriality. Take a peh-resolution X• → X using Proposition 4.1.6. Since this is a hypercovering in eh-topology (i.e. every map Xn+1 → (coskn skn (X• ))n+1 is a covering in the eh-topology as they are proper eh-coverings), we get by cohomological descent that RΓc (Xeh , Qp (n)) = R lim RΓ(Xeh , ρ∗ Wr ΩnX,log )Q [−n] ←− r ∼ RΓ(X•,eh , ρ∗ Wr ΩnX• ,log )Q [−n] = R lim ←− r ) 38 ∼ Now, by Theorem 3.3.1, we have that Z(n)/pr → Rρ∗ ρ∗ Z(n)/pr on (Sm/k)et , so (using the identication Z(n)/pr ∼ = Wr Ωn [−n] on Sm/k ), we have that log RΓc (Xeh , Qp (n)) ∼ RΓ(X•,et , Wr ΩnX• ,log )Q [−n]. = R lim ←− r Then, by [Ill79], [Ill79, I.Th.5.7.2] ∼ = " RΓc (Xeh , Qp (n)) ∼ RΓ(X•,et , Wr ΩnX• ,log )Q [−n] = R lim ←− r # 1−F R lim RΓ(X•,et , Wr ΩnX• )Q → R lim RΓ(X•,et , Wr ΩnX• )Q [−n] ←− ←− r r i h [Ill79, II.Prop.2.1.(a)] 1−F ∼ RΓ(X•,et , W ΩnX• )Q → RΓ(X•,et , W ΩnX• )Q [−n] = i [Ill79, II.Cor.3.5.] h pn −φ ∼ RΓ(X•,et , W Ω•X• )Q → RΓ(X•,et , W Ω•X• )Q = i Th.4.2.1 h pn −φ ∼ RΓrig,c (X/K) → RΓrig,c (X/K) = where we have used that RΓ(X• , W Ω• )[i,i+1) = RΓ(X• , W Ωi )[−i] from the slope decomposition for the second to last equality. This quasiisomorphism is independent of the choice of peh-resolutions, as by Proposition 4.1.6,i), for any two such peh-resolutions, we can nd a pehresolution which is a common renement of the two. As for functoriality, given f : X → Y of proper schemes over k , we may pick any peh-resolution Y• → Y , and by Proposition 4.1.6.ii) we may choose another peh-resolution X• → X such that we get a commutative diagram / Y• X•  X f  /Y so using the functoriality of Theorem 4.2.1 we see that every step is functorial. 39 Chapter 5 COMPARISON OF OVERCONVERGENT WITT DE-RHAM COHOMOLOGY AND RIGID COHOMOLOGY 5.1 Introduction Let X be a smooth scheme over a perfect eld k of characteristic p > 0, and consider its overconvergent de Rham-Witt complex of étale sheaves W † Ω•X/k , which is dened in [DLZ11] (see Denition 1.1 and Theorem 1.8). One of the main results of loc. cit. is that if X is also quasi-projective, then there exists a natural quasi-isomorphism RΓrig (X/K) ∼ = RΓ(X, W † Ω•X/k )Q , where K = W (k) ⊗ Q. The main result of this chapter is Theorem 5.5.5, where we drop the quasiprojectivity condition in the comparison. We outline the approach in [DLZ11] and the one used here. If X = Spec A is an ane smooth k -scheme, [DLZ11] consider pairs (X, F ) given by closed immersions X = Spec A ,→ F = Spec à into W (k)-schemes, called special frames. To this, the authors associate dagger spaces (in the sense of [Gro00]) ]X[†F functorially in (X, F ), which calculate RΓrig (X/K): ∼ RΓrig (X/K) → RΓ(]X[†F , Ω•]X[† ) F (5.1) (here F denotes the p-adic completion of F ). So using the specialization maps sp†∗ :]X[†F → X we have that RΓrig (X/K) ∼ = RΓ(X, Rsp†∗ Ω•]X[† ). F They also form a quasi-isomorphism of Zariski sheaves on X , sp∗ Ω•]X[† → W † Ω•X/k ⊗ Q, F̂ (5.2) 40 functorial in (X, F ). This all gives a map (5.3) RΓ(sp†∗ Ω•]X[† ) → RΓ(X, W † Ω•X/k )Q F so it suces to show that the natural map (5.4) sp†∗ Ω•]X[† → Rsp†∗ Ω•]X[† F F is a quasi-isomorphism. While vanishing of higher cohomologies in (5.4) is not known in general, we can show it in some instances. In [DLZ11], to globalize the above construction for a smooth quasi-projective X (though possibly not ane), the authors consider an open covering by a particular type of ane smooth schemes, standard smooth schemes, which may be lifted nicely over W (k), which are all coming from localizations in a common projective space provided by the quasi-projectivity of X . This gives a nice description of the intersections of such opens in coskX 0 (X0 )• , which allows them to prove the vanishing of higher cohomologies in (5.4), and then complete the proof by means of cohomological descent. For our case, when X is not quasi-projective, we do not have a common projective space in which all our open anes are open. So instead of working with the 0-coskeleton, we rene it at each level, getting an étale hypercovering X• /X so that at each level, Xn is a disjoint union of ane standard smooth schemes, which we call a special hypercovering. This is done in Section 5.3. Considering any compactication X ,→ X to a proper k -scheme X , we use W (k) the Tsuzuki functors ΓN (−) and ΓW N (−) introduced in Denition 2.4.1 to construct an N -truncated special frame (X•≤N , F•≤N ) and a N -truncated simplicial version of (5.3): RΓ(X•≤N , W † Ω•X•≤N /k )Q → RΓ(sp†∗ Ω•]X † •≤N [F •≤N ). (5.5) Then, we show the following: • Vanishing of Rspi∗ Ω•]X [† n Fn for 0 ≤ n ≤ N and i > 0: we use techniques from the proof of [DLZ11, Proposition 4.35], such as being able to replace the Fn by some Fn0 étale over Fn or equal to Fn ×W (k) ArW (k) for some r tting into a special frame (Xn , Fn0 ). 41 • Prove independence of choices and functoriality. • For large enough N , (5.5) gives a map RΓ(X, W † Ω•X/k )Q → RΓrig (X/K). This is motivated by [Nak12] and relies on the machinery of [CT03], such as vanishing of higher enough rigid cohomology groups of X , independence of the choices of rigid frames and cohomological descent methods. 5.2 Background Special Frames and Dagger Spaces The following is a summary of Section 4 of [DLZ11]. Denition 5.2.1. A special frame is a pair (X, F ) with a closed embedding X ,→ F , where X and F are smooth ane schemes over k and W (k) respectively. Given a special frame (X, F ), we can choose an embedding F ,→ AnW (k) for some n, and in turn we have an open embedding E := AnW ,→ PnW (k) =: P . Let Q = F and X be the closures of F and X respectively in P , and let F and Q be the p-adic completions of F and Q respectively. Then, X ,→ X ,→ Q is a frame for rigid cohomology in the sense of Berthelot (i.e. we have an open immersion of X into a proper scheme X over k , and a closed immersion X ,→ Q where Q is smooth around X ). So we may dene the rigid cohomology of X as RΓrig (X/K) = RΓ(]X[Q , j † Ω•]X[Q ), where j is the inclusion ]X[Q ,→]X[Q . Note also that ]X[Q =]X[F . The authors then give an explicit description of a fundamental system of strict neighborhoods of ]X[F in ]X[Q , which they use to give a dagger structure (in the sense of [Gro00]) on ]X[F , denoted by ]X[†F , along with a morphism sp†∗ :]X[†F → X which is independent of the choice of embedding of F into ane and projective spaces. Thus, we have a functorial association Special Frames / Dagger Spaces (X, F )  / ]X[† . F (5.6) 42 By [Gro00, Theorem 5.1], this gives quasi-isomorphisms ∼ RΓrig (X/K) = RΓ(]X[Q , j † Ω•]X[Q ) → RΓ(]X[†F , Ω•]X[† ). F (5.7) To such a frame (X, F ), they also form in [DLZ11, (4.32)] a map sp∗ Ω•]X[† → W † Ω•X/k ⊗ Q, (5.8) F̂ which is a quasi-isomorphism of Zariski sheaves and functorial in (X, F ). Standard Smooth Schemes Denition 5.2.2. We call a ring A a standard smooth algebra (over k) if A can be represented in the form A = k[X1 , ..., Xn ]/(f1 , ..., fm ), where m ≤ n and the determinant   ∂fi det , 1 ≤ i, j ≤ m ∂Xj is a unit in A. The scheme Spec A is then called a standard smooth scheme. Such schemes are convenient to work with, since for a standard smooth algebra represented as k[T1 , ..., Tn ]/(f1 , ..., fr ), we may choose liftings f˜1 , ..., f˜r ˜ ˜ to W [T1 , ..., Tn], and  let à be the localization of W [T1 , ..., Tn ]/(f1 , ..., fr ) with respect to det ∂ f˜i ∂Tj . Then, à is a standard smooth algebra which lifts A over W , which gives a special frame (Spec A, Spec Ã). We note that this may be done functorially in A; that is, given a homomorphism of standard smooth algebras ϕ:A→B with presentations A∼ = k[T1 , .., Tn ]/(f1 , .., fr ), B ∼ = k[S1 , ..., Sm ]/(g1 , ..., gs ), after choosing liftings f˜i to dene Ã, we may chose the representation B∼ = k[S1 , ..., Sm , T1 , ..., Tn ]/(g1 , ..., gs , f1 , ..., fr , T1 − α(T1 ), ..., Tr − α(Tr )) and then take liftings g˜j , α̃i over gj and α(Ti ) respectively to form B̃ . Note also that for any such standard smooth scheme F = Spec Ã, we have an étale map F → AnW (k) for some n. 43 5.3 The hypercovering Proposition 5.3.1. Given any étale hypercovering Z• → X , with Zn being smooth schemes over k, there exists an étale hypercovering Y• → X rening Z• → X such that for any n, Yn is the disjoint union of ane standard smooth schemes giving a nite open covering of Zn . Proof. The proof is nearly identical to [CT03, Proposition 11.3.2], with the only dierence being that when we form a nite ane Zariski covering of the smooth scheme coskX Zn+1 , n (Y•≤n )n+1 ×coskX n (Z•≤n )n+1 we require the covering to be by ane standard smooth schemes also. Denition 5.3.2. We say Y• → X is a special hypercovering if Y• is a split étale hypercovering of X , and each Yn is a disjoint union of ane standard smooth schemes which give an open covering of X . We prove the existence and some functorial property of such hypercoverings, which will be useful to work on the comparison locally. Proposition 5.3.3. Given a smooth scheme X : i) There exists a special hypercovering Y• → X . ii) Given two special hypercoverings Y• , Y•0 /X, there a third special hypercovering Y•00 /X rening them. iii) Given a morphism X → X 0 of smooth schemes, there exist special hypercoverings Y• → X and Y•0 → X 0 tting in a commutative diagram Y• /Y0 .   / X0 X • Proof. Part i) follows immediately from Proposition 5.3.1 by taking the constant simplicial scheme Z• = coskX −1 (X) (so Zn = X for all n). For part ii), we just apply Proposition 5.3.1 with Z• := Y• ×X Y•0 , 44 and for part iii) we nd some special hypercovering Y•0 → X 0 , and then again use Proposition 5.3.1 with Z• := Y•0 ×X 0 X. We will use the following version of Chow's lemma: Lemma 5.3.4. Consider a compactication over a scheme S X j /X f  g S where j is an open immersion and g is proper. Suppose that f is quasiprojective. Then, there exists a blowup Y → X at some closed subscheme Z ⊂ X disjoint from j(X) such that Y is projective over S . Proof. This follows immediately from [Ray74, Corollaire 5.7.14] since it gives us such a blowup Y → X with Y quasi-projective over S with the only requirements that X be quasi-separated and nite type over S . In this case, Y → X → S is the composition of a blowup and a proper map, so it is proper. Since Y is quasi-projective and proper over S , it is also projective. Denition 5.3.5. For a pair (X, X), we say a simplicial pair (X• , X • ) → (X, X) is a special hypercovering of pairs if: (a) Both X• and X • are split. (b) X• → X is a special hypercovering. (c) X • → X is proper. In particular, (X• , X • ) → (X, X) is an étale-proper hypercovering. (d) For all n ≥ 0, X n is projective over k . Proposition 5.3.6. Given a smooth scheme X and a compactication X ,→ X over k : i) There exists a special hypercovering of pairs (X• , X • ) → (X, X). 45 ii) Let (X, X) → (Y, Y ) be a morphism of compactications. Then, we may construct special hypercoverings of pairs (X• , X • ) and (Y• , Y • ) tting into a commutative diagram (X• , X • ) / (Y• , Y • )   / (Y, Y ) (X, X) iii) Let (X• , X • ), (X•0 , X • ) be two special hypercoverings of (X, X). Then, 00 there exists a third special hypercovering of pairs (X•00 , X • ) tting into a diagram 0 00 (X•00 , X • ) % y 0 (X•0 , X • ) (X• , X • ) & x (X, X) Proof. For i), rst x a special hypercovering X• → X by Proposition 5.3.3. Let {N Xk }k≥0 denote the splitting of X• . We construct X n+1 at step n + 1 ≥ 0, assuming we have constructed X n with a splitting {N X k }0≤k≤n (the n = −1 case is vacuous). By Nagata's compactication theorem we may take a compactication N Xn+1 ,→ N X n+1 tting into a diagram N Xn+1 / N X n+1   / coskX (X •≤n )n+1 coskX n (X•≤n )n+1 (5.9) n where the vertical arrows are proper maps. For the n + 1 = 0, note that coskX −1 (−)i = X for all i (and similar for X ), so the bottom row is just X ,→ X . Since N Xn+1 is a disjoint union of ane standard smooth schemes by construction, itself is also ane standard smooth. In particular, N Xn+1 is quasiprojective, and we may use Lemma 5.3.4 to make N X n+1 be projective over k in (5.9). This allows us to construct X •≤n+1 = ΩX n+1 (X •≤n , N X 0 , ..., N X n+1 ). In particular, X n+1 ∼ = G φ:[n+1][k] N Xφ 46 where each N X φ is projective over k . Since the disjoint union of projective schemes are still projective, it follows that X n+1 is projective over k . This completes i). For ii), we rst construct compatible X• → Y• over X → Y using Proposition 5.3.3, and we construct Y • as in i). Then, we build X • similarly, except that at each n, we take a compactication N X n+1 of N Xn+1 over coskX N Yn+1 ,→ coskX n (X •≤n )n+1 ×coskY (Y n (X•≤n )n+1 ×coskY n (Y•≤n )n+1 n •≤n )n+1 N Y n+1 , and by the same argument as i), such that N X n+1 is projective over k . This all ts into a commutative diagram / N X n+1 N Xn+1  / coskX (X •≤n )n+1 × coskX N Yn+1 n (X•≤n )n+1 ×coskY n (Y•≤n )n+1  coskY n (Y •≤n )n+1 n N Y n+1  / coskX (X •≤n )n+1  coskX n (X•≤n ))n+1 n where all horizontal morphisms are open immersions, and the vertical morphisms on the right are all proper. This gives us the desired functoriallity. 0 For iii), given (X• , X • ) and (X•0 , X • ), we may construct a special hypercovering X•00 rening X• and X•0 using Proposition 5.3.3. Then, at each step, we 00 00 over construct N X n+1 by taking a compactication of N Xn+1   00 N Xn+1 coskX n (X•≤n ) ×coskX n (X•≤n )  00 coskX N X n+1 n (X •≤n ) ×coskX n (X •≤n )   0 ×coskX N Xn+1 0 n (X•≤n ) ×coskX (X 0 n •≤n ) 0 N X n+1 00 and using the above argument so that N X n+1 is projective over k . Denition 5.3.7. We call a simplicial pair (X• , X • ) → (X, X) as in Proposition 5.3.6 a special hypercovering of pairs. 5.4 The simplicial special frame Ideally, we would like to get a simplicial special frame (X• , F• ), with an embedding F• ,→ P• for Pn being projective over W , and such that X • ,→ P• 47 is the closure of X• . Then, we could do a simplicial version of the comparison in [DLZ11] directly. This problem seems dicult to do simultaneously. However, using the Tsuzuki functor below, it will suce to do this only for one XN , rather than all. This will capture all the simplicial structure below N , and for N large enough we will be able to ignore the terms above, as they won't contribute to the cohomology. For a smooth scheme X and a compactication X ,→ X , construct a special hypercovering (X• , X • ) → (X, X) as in Proposition 5.3.6. Fix some N ≥ 0. Then, since X N is projective over k , we may nd a closed immersion X N ,→ PrkN ,→ PrWN(k) =: P for some rN ≥ 0. This gives us an immersion XN ,→ P , and thus a presentation of XN (which is ane). We may use this to lift XN to some standard smooth ane scheme F over W (k). Using the Tsuzuki functor ΓN (−) introduced in Denition 2.4.1, we construct the following: W (k) (F ). W (k) (P ). • F• := ΓN • P• := ΓN • Y• is the closure of X• in P• . • Q• is the closure of F• . • Q• is the p-adic completion of Q• . By Lemma 2.4.2, since X N ,→ P is a closed immersion, we have that X •≤N ,→ P•≤N is a closed immersion. Therefore, X•≤N → P•≤N factors through both X •≤N and Q•≤N , giving natural maps from Y•≤N to both by universal property of the closure. This all ts into a diagram (5.10) X •≤N ; O X•≤N / Y•≤N   / Q•≤N F•≤N # / P•≤N : 48 Putting all this together: Denition 5.4.1. Consider a special compactication X• / X•   /X X with X (resp. X ) being smooth (resp. proper) over k . For a given N ≥ 0, the information {F• , Q• , Q• , Y• , P• } and the diagram from (5.10) is a N -rigid special frame of (X• , X • ) → (X, X). Lemma 5.4.2. Given a commutative diagram of special hypercoverings / (X 0 , X 0 ) (X• , X • ) • •  / (X 0 , X)  (X, X) and a given N ≥ 0, we may nd N -rigid special frames of (X• , X • ) and 0 (X•0 , X • ) with maps {F• , Q• , Q• , Y• , P• } → {F•0 , Q0• , Q0• , Y•0 , P•0 } compatible with the given diagram. Proof. Consider a morphism (X• , X • ) → (X•0 , X 0 • ) over (X, X) → (X 0 , X ). 0 We may construct {F•0 , Q0• , Q0• , Y•0 , P•0 } as explained above. Then, since X N and 0 X N are projective over k , the map between them is projective, and we may nd some closed immersion of X N into P = PrW tting into the commutative diagram /P XN   / P 0. 0 XN 0 This gives representations of XN and X N in compatible ane spaces, and we thus may lift them to standard smooth ane schemes over W (k) as explained 49 in Section 5.2 in a compatible way, giving a commutative diagram XN /F /P   / F0  / P 0. XN0 Since the functor ΓW N , p-adic completion and closure are functorial, all the remaining functoriality will follow from that of P and F . This gives an N -truncated special frame (5.11) (X•≤N , F•≤N ), and an N -truncated étale-proper hypercovering of (X, X) (5.12) (X•≤N , Y•≤N , Q•≤N ). 5.5 The comparison theorem In the course of the proof, we will need some tools from [DLZ11] in order to compare special frames. In loc. cit., Proposition 4.35 proves the rst result, and the second result is Proposition 4.37. Proposition 5.5.1. i) Given a map of special frames X / F0 X  /F with the right vertical map being étale, then we get a natural isomorphism of dagger spaces † ]X[†F 0 ∼ =]X[F . ii) For some n, let (X, F × AnW (k) ) be a special frame such that the map X → AnW (k) factors through the origin. Then, the induced map gives a quasi-isomorphism ∼ Rsp∗ Ω•]X[† → Rsp∗ Ω•]X[† F F ×An W . 50 When proving the comparison, we will need to show vanishing of the higher cohomologies of Rsp∗ Ω•]Y [† m Fm for 0 ≤ m ≤ N , where (Ym , Fm ) are the special frames constructed in sections 5.3 and 5.4. The above proposition will allow us to reduce it to the following theorem, which follows from the proof of [Ber97b, Theorem 1.10]: Proposition 5.5.2. Let (X, F ) be a special frame, where F is a lifting of X over W (k). Then, Ri sp∗ Ω•]X[† = 0 for i>0. F We now prove a key ingredient of the comparison theorem: Proposition 5.5.3. Let (X•≤N , F•≤N ) be an N -truncated simplicial frame as in (5.11). Then, for 0 ≤ m ≤ N and i > 0, Ri sp∗ Ω•]Xm [† Fm = 0. Proof. Pick any 0 ≤ m ≤ N . By splitness of X• , we may write Xm = G N Xm,φ . φ:[N ][m] W (k) Fix some degeneracy map σ : [N ]  [m]. Then, by construction of ΓN (−), we have a commutative diagram / Fm = Xm Q Fα α:[N ]→[m] X• (σ)   pσ / F = Fσ YN where Fφ = F for any φ : [N ]  [m] was dened in section 5.4 as just a lift of XN in PN , and pσ is the projection, and both horizontal maps and the left vertical map are closed immersions. This gives us a closed immersion Xm ,→ Fσ . Let Fm0 := Y Fα , α:[N ]→[m],α6=σ 0 so Fm = Fm × Fσ . Then, since each of the Fα are standard smooth schemes over W (k), so is their product, and we may get an étale morphism Fm0 → AnW (k) 51 for some n. Thus, considering the commutative diagram Xm / F 0 × Fσ Xm / An Fm m  W (k) × Fσ where the right vertical morphism is étale, using Proposition 5.5.1.i) we may reduce to the case of the special frame (Xm , AnW (k) × Fσ ). Furthermore, we may assume that the map Xm → AnW (k) factors through the origin. To see this, write Xm = Spec (A) and Fσ = Spec B , so AnW (k) × Fσ = Spec B[T1 , ..., Tn ]. Then, since B  A is surjective (as Xm → Fσ is a closed immersion), we may pick b1 , ..., bn ∈ B which map to the images of T1 , ..., Tn respectively in A, and replace Ti by Ti0 := Ti − bi , giving a special frame (Xm , Spec B[T10 , ..., Tn0 ]) = (Xm , AnW (k) × Fσ ) factoring through the origin. Thus, by Proposition 5.5.1.ii), we reduce the proof to the special frame (Xm , Fσ ). Now, since G ]Xm [†Fσ =] N Xm,φ [†Fσ ∼ = G ]N Xm,φ [†Fσ φ:[m][k] φ:[m][k] we may reduce to studying the special frames (N Xm,φ , Fσ ) for any φ : [m]  [k] and 0 ≤ k ≤ m. But notice that by the construction of the frame, for any φ : [m]  [k], we have a commutative diagram N Xm,φ ⊂ Xm _ ∼ = X• (σ)  N XN,φ◦σ ⊂ XN   / Fσ F FN N FN,ψ ψ:[N ][k0 ] where ψ vary over all morphisms ψ : [N ]  [k 0 ] with 0 ≤ k 0 ≤ N , and the composite map N Xm,φ → FN is the map giving the special frame. Thus, N Xm,φ is isomorphic to N FN,φ◦σ ⊂ FN , and therefore sp−1 (N Xm,φ ) =]N Xm,φ [†Fσ =]N Xm,φ [†N FN,φ◦σ , which reduces the proof to the case of the special frame (N Xm,φ , N FN,φ◦σ ). But by construction, N FN,φ◦σ is a smooth lift of N XN,φ◦σ ∼ = N Xm,φ over W (k), and thus we can apply Proposition 5.5.2 to complete the proof. 52 We will need the following to deal with N -truncations, which basically says that for some large enough N , we only need the N -skeleton in the calculations of cohomologies on simplicial objects (such as for rigid cohomology and overconvergent Witt de-Rham). For a complex A• of K vector spaces, and any h, consider the h-truncated complex    Ai if i < h   τ≤h (A• )i = ker(Ah → Ah+1 ) if i > h    0 else. (1) For a double complex A•• , let τ≤h (A•q ) := τ≤h (A•q ), and let s : C(K) → K be the total complex map. Lemma 5.5.4. [Nak12, Lemma 2.2] Consider a double complex A•,• of K vector spaces such that Ap,q = 0 for p < 0 or q < 0. Given any N > max{i + (h − i + 1)(h − i + 2)/2 | 0 ≤ i ≤ h} = (h + 1)(h + 2)/2, (5.13) (1) the natural maps s(τ≤N (A•• ) → s(A•• ) induce a quasi-isomorphism ∼ (1) τ≤h (s(τ≤N (A•• )) → τ≤h (s(A•• )). From this, and the formation of the spectral sequence for cohomology on simplicial objects, it follows for example that for some simplicial rigid frame (Z• , Z • , Z• ), and h and N as in (5.13), we get natural quasi-isomorphisms ∼ τ≤h RΓ(]Z •≤N [Z•≤N , j † Ω•]Z •≤N [Z •≤N ) → τ≤h RΓ(]Z • [Z• , j † Ω•]Z • [Z ) • (5.14) and that for a smooth simplicial scheme X• , ∼ τ≤h RΓ(X• , W † Ω•X•/k ) → τ≤h RΓ(X•≤N , W † Ω•X•≤N/k ). (5.15) This is useful because of the vanishing of rigid cohomology from Theorem 2.3.4, which tells us that there exists a c such that ∼ τ•≤c RΓrig (X/K) → RΓrig (X/K) so letting h ≥ c and N as in (5.13), we may compute rigid chomology with an N -truncated de-Rham descendable hypercovering by (5.14). We can now prove the main comparison theorem: 53 Theorem 5.5.5. Given a smooth scheme X over k, there exists a functorial quasi-isomorphism ∼ RΓrig (X/K) → RΓ(X, W † Ω•X/k ) ⊗ Q. Proof. Choose a compactication X of X and construct {X• , X • , F• , Q• , Q• , Y• , P• } as in the previous section. Then, (X•≤N , F•≤N ) is an N -truncated special frame, so by functoriality of the construction in [DLZ11] Special Frames → Dagger Spaces (X, F ) 7→]X[†F we get an N -truncated special frame ]X•≤N [†F•≤N . Furthermore, from (5.8), we get sp∗ Ω•]X † •≤N [F •≤N → W † Ω•X•≤N /k ⊗ Q (5.16) to give us a quasi-isomorphism ∼ RΓ(X•≤N , sp∗ Ω•]X † •≤N [F •≤N ) → RΓ(X•≤N , W † Ω•X•≤N /k ) ⊗ Q. (5.17) Next, by Proposition 5.5.3, we have that sp∗ Ω•]X ∼ † •≤N [F •≤N → Rsp∗ Ω•]X † •≤N [F •≤N (5.18) so that RΓ(]X•≤N [†F•≤N , Ω•]X † •≤N [F •≤N )∼ = RΓ(X•≤N , W † Ω•X•≤N ,k ) ⊗ Q (5.19) Next, note that (X•0 , Y•0 , Q0• ) := coskN (X•≤N , Y•≤N , Q•≤N ) is an étale-proper hypercovering of (X, X) by Lemma 2.2.6, and thus a de Rham desecendable hypercovering of (X, X) by Proposition 2.3.2, so RΓrig (X/K) = RΓ(]Y•0 [Q0• , j † Ω•]Y•0 [Q0 ) • computes the rigid cohomology. (5.20) 54 Furthermore, since for any n ≤ N , Yn is the closure of Xn in Pn and Qn is the p-adic completion of the closure of Fn in Pn , we get an N -truncated simplicial version of (5.7): RΓ(]X•≤N [†F•≤N , Ω•]X † •≤N [F •≤N )∼ = RΓ(]Y•≤N [Q•≤N , j † Ω•]Y•≤N [Q •≤N ) (5.21) Putting all this together, we will get RΓ(]Y•≤N [Q•≤N , j † Ω•]Y•≤N [Q •≤N )∼ = RΓ(X•≤N , W † Ω•X•≤N /k ) ⊗ Q (5.22) for any N . We now show that the left-hand side computes RΓrig (X/K) (compare with (5.20)) and that the right hand side computes RΓ(X, W † Ω•X/k ) ⊗ Q. Using Proposition 2.3.4, pick some c such that for any h ≥ c, ∼ τ≤h RΓrig (X/K) → RΓrig (X/K), and pick N = N (h) large enough to satisfy (5.13). Then, RΓrig (X/K) ∼ =τ≤h RΓrig (X/K) ∼ = τ≤h RΓ(]Y•0 [Q0• , j † Ω•]Y•0 [Q0 ) • (5.23) ∼ = τ≤h RΓ(]Y•≤N [Q•≤N , j † Ω•]Y•≤N [Q•≤N ) so the left hand side of (5.22) will compute rigid cohomology. On the other hand, since X• → X is an étale hypercovering, and W † Ω•X/k is an étale sheaf, we have that ∼ RΓ(X, W † Ω•X/k ) → RΓ(X• , W † Ω•X• /k ) To compare to the truncated version in (5.22), note that for any h and N satisfying (5.13), we have by Lemma 5.5.3 and (5.22) that τ≤h RΓrig (X/K) ∼ = τ≤h RΓ(X•≤N , W † Ω•X•≤N /k ) ⊗ Q (5.24) ∼ = τ≤h RΓ(X, W † Ω•X/k ) = τ≤h RΓ(X• , W † Ω•X• /k ) ⊗ Q ∼ Varying h (and N ), we see that cohomology vanishes in RΓ(X, W † Ω•X/k ) above c, and thus letting h ≥ c we may drop it from (5.24) to obtain RΓrig (X/K) ∼ = RΓ(X, W † Ω•X/k ) ⊗ Q It remains to show independence and functoriality. (5.25) 55 Independence: We must show independence of the choices of X , X• , X • , F• , Q• , P• , and N . The independence of X follows from independence of X in computation of rigid cohomology ( [CT03, Corollary 10.5.4]). To show independence of {X• , X • , F• , Q• , Q• , Y• , P• }, suppose for a given pair i (X, X), we have made two choices {X•i , X • , F•i , Qi• , Qi• , Y•i , P•i } for i = 1, 2. 3 Then, by Proposition 5.3.6 it follows that we may nd (X•3 , X • ) rening them. 3 1 2 Furthermore, since X N is projective over X N and X N , we can nd some P 3 tting into the diagram 3 (XN3 , X N , P 3 ) ( v 1 (XN1 , X N , P 1 ) 2 (XN2 , X N , P 2 ) ( v (X, X, W (k)) Furthermore, we may take the standard smooth lift F 3 of XN3 over W (k) to be compatible with the standard smooth lifts F i of XNi over W (k) for i = 1, 2. W (k) All this compatibility carries over when applying ΓN , taking closures and X W completions, and applying coskX n , coskN , coskN , which gives a diagrams of N - truncated simplicial special frames 3 3 (X•≤N , F•≤N ) v ( 3 3 (X•≤N , F•≤N ) 3 3 (X•≤N , F•≤N ) and of universally de Rham descendable hypercoverings of (X, X) (X•03 , Y•03 , Q03 •) v (X•01 , Y•01 , Q01 •) ( (X•02 , Y•02 , Q02 •) where we use the notation from above. This gives a factorization of all the maps used that shows independence in D+ (K). For independence of N , we argue similarly to the proof of Lemma 4.2.2. Suppose we x (X• , X • ). Given to choices N 1 and N 2 satisfying (5.13) for h = c, 56 suppose N 2 ≥ N 1 , and construct F•i , Qi• , Qi• , Y•i , P•i } for i = 1, 2. We then have a natural map (X,X,W) (X•02 , Y•02 , Q02 • ) := coskN 2 2 2 (X•≤N 2 , Y•≤N 2 , Q•≤N 2 )  (X,X,W) 2 2 coskN 1 (X•≤N 1 , Y•≤N 1 , Q•≤N 1 ) induced by the maps coskN 2 → coskN 1 ◦ skN 1 ◦ coskN 2 ∼ = coskN 1 ◦ skN 1 . This all induces a commutative diagram in the diagrams (5.22) for N 2 and N 1 . This is compatible with the maps in (5.24). Thus, we may replace N 2 with N 1 (as long as they are both large enough), and then independence above (for N 1 xed) shows the independence of choices. Functoriality: Given a map X 1 → X 2 , we may choose compatible X and X , 1 i 2 and then pick compatible (X•i , X • ) by Proposition 5.3.6, and by Lemma 5.4.2 we may choose compatible N -rigid special frames. 57 Chapter 6 CONJECTURE B 6.1 Conjecture B We study the following conjecture: Conjecture 6.1.1 (Conjecture B). For a separated, nite type k-scheme X of dimension d, and n ∈ Z, there exists a quasi-isomorphism ∼ RΓ(Xet , Qcp (n)) →  RΓrig,c (X/K) ∗ φ−p n−d → RΓrig,c (X/K) ∗  [−2d]. Here, RΓrig,c (X/K)∗ := RHom(RΓrig,c (X/K), K). We prove this conjecture for X smooth: Theorem 6.1.2. If X is smooth, then Conjecture 6.1.1 holds. Proof. As before, since X is smooth and we have Zc (n)/pr ∼ = Wr Ωnlog [−n], we can identify RΓrig (X, Qcp (n)) ∼ RΓ(Xet , Wr ΩnX,log )Q [−n], = R lim ←− r and as in the proof of Theorem 4.2.1 we have a short exact sequence 1−F 0 → W ΩnX,log → W ΩnX → W ΩnX → 0 in Xet , and W ΩnX ∼ W Ωn , so we have = R lim ←−r r X h i 1−F RΓrig (X, Qcp (n)) ∼ = RΓ(Xet , W ΩnX ) → RΓ(Xet , W ΩnX ) [−n]. Q Next, by [Ert14, Corollary 2.4.12], we have that all logarithmic Witt de-Rham sections are overconvergent, and that 1 − F is still surjective when restricted to the overconvergent part; so we have a commutative diagram in Xet : 0 / W Ωn / W † Ωn 1−F / W † Ωn _ X _ X 0 / W Ωn  / W Ωn X,log X,log X 1−F /  W † ΩnX /0 /0 58 where the vertical arrows are given by inclusion, and both rows are short exact sequences. Thus, we get a natural quasi-isomorphism h i 1−F RΓ(Xet , W ΩnX ) → RΓ(Xet , W ΩnX ) [−n] Q ∼ = h i 1−F RΓ(Xet , W † ΩnX ) → RΓ(Xet , W † ΩnX ) [−n]. Q We consider the Frobenius on W † Ω•X by restricting that on W Ω•X . So as before, the part with slope pn must be coming from W † Ωn , thus giving h i h i pn −φ 1−F RΓ(Xet , W † ΩnX ) → RΓ(Xet , W † ΩnX ) [−n] ∼ = RΓ(Xet , W † Ω•X ) → RΓ(Xet , W † Ω•X ) . Q Q Then, we use the comparison from overconvergent Witt de-Rham cohomology to rigid cohomology for smooth schemes given by Theorem 5.5.5 to get that ∼ RΓrig (X/K) → RΓ(X, W † Ω•X/k )Q and thus h i pn −φ RΓrig (X, Qcp (n)) ∼ = RΓrig (X/K) → RΓrig (X/K) . Finally, from [Ber97a, Théorème 2.4] we can use Poincaré duality for rigid cohomology to get non-degenerate pairings ∼ 2d−i i 2d Hrig (X/K) × Hrig,c (X/K) → Hrig,c (X/K) → K(−d) compatible as F-crystals, where K(−d) is K with a Frobenius action given by multiplication by pd . Thus, we have a natural quasi-isomorphism ∼ RΓrig (X/K) → RHom(RΓrig,c (X/K)∗ [−2d] := RHom(RΓrig,c (X/K), K)[−2d] and therefore, h i pn −φ ∼ RΓrig (X, Qp (n)) = RΓrig (X/K0 ) → RΓrig (X/K0 )   d−n −φ ∗ p ∗ ∼ = RΓrig,c (X/K) → RΓrig,c (X/K) [−2d]. 59 Remark. In order to prove Conjecture B for the general case, one should try and get a map RΓ(X, Qcp (n)) →   n−d ∗ φ−p ∗ RΓrig,c (X/K) → RΓrig,c (X/K) [−2d] for the general case compatible with that used to prove the isomorphism in the smooth case. Then, using localization triangles and an induction on dimension one could prove that it is an isomorphism. 60 BIBLIOGRAPHY [Abh56] Shreeram Abhyankar. Local Uniformization on Algebraic Surfaces Over Ground Fields of Characteristic p 6= 0 1. Ann. Math., 63(3):491, May 1956. [Ber74] Pierre Berthelot. Cohomologie cristalline des schémas de caractéris- tique p > o. Number 407 in Lecture notes in mathematics. Springer, Berlin, 1974. OCLC: 1009402. [Ber96] Pierre Berthelot. 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