DOCSLIB.ORG

CRYSTALLINE COHOMOLOGY 07GI Contents 1. Introduction 1 2

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
CRYSTALLINE COHOMOLOGY 07GI Contents 1. Introduction 1 2 CRYSTALLINE COHOMOLOGY 07GI Contents 1. Introduction 1 2. Divided power envelope 2 3. Some explicit divided power thickenings 6 4. Compatibility 8 5. Affine crystalline site 8 6. Module of differentials 11 7. Divided power schemes 16 8. The big crystalline site 17 9. The crystalline site 20 10. Sheaves on the crystalline site 23 11. Crystals in modules 24 12. Sheaf of differentials 25 13. Two universal thickenings 27 14. The de Rham complex 28 15. Connections 29 16. Cosimplicial algebra 30 17. Crystals in quasi-coherent modules 31 18. General remarks on cohomology 36 19. Cosimplicial preparations 38 20. Divided power Poincaré lemma 40 21. Cohomology in the affine case 41 22. Two counter examples 44 23. Applications 46 24. Some further results 47 25. Pulling back along purely inseparable maps 53 26. Frobenius action on crystalline cohomology 58 27. Other chapters 60 References 61 1. Introduction 07GJ This chapter is based on a lecture series given by Johan de Jong held in 2012 at Columbia University. The goals of this chapter are to give a quick introduction to crystalline cohomology. A reference is the book [Ber74]. This is a chapter of the Stacks Project, version 77243390, compiled on Sep 28, 2021. 1 CRYSTALLINE COHOMOLOGY 2 We have moved the more elementary purely algebraic discussion of divided power rings to a preliminary chapter as it is also useful in discussing Tate resolutions in commutative algebra. Please see Divided Power Algebra, Section 1. 2. Divided power envelope 07H7 The construction of the following lemma will be dubbed the divided power envelope. It will play an important role later. 07H8 Lemma 2.1. Let (A, I, γ) be a divided power ring. Let A → B be a ring map. Let J ⊂ B be an ideal with IB ⊂ J. There exists a homomorphism of divided power rings (A, I, γ) −→ (D, J,¯ γ¯) such that ¯ Hom(A,I,γ)((D, J, γ¯), (C, K, δ)) = Hom(A,I)((B, J), (C, K)) functorially in the divided power algebra (C, K, δ) over (A, I, γ). Here the LHS is morphisms of divided power rings over (A, I, γ) and the RHS is morphisms of (ring, ideal) pairs over (A, I). Proof. Denote C the category of divided power rings (C, K, δ). Consider the func- tor F : C −→ Sets defined by ϕ :(A, I, γ) → (C, K, δ) homomorphism of divided power rings F (C, K, δ) = (ϕ, ψ) ψ :(B, J) → (C, K) an A-algebra homomorphism with ψ(J) ⊂ K We will show that Divided Power Algebra, Lemma 3.3 applies to this functor which will prove the lemma. Suppose that (ϕ, ψ) ∈ F (C, K, δ). Let C0 ⊂ C be the subring 0 0 generated by ϕ(A), ψ(B), and δn(ψ(f)) for all f ∈ J. Let K ⊂ K ∩ C be the 0 0 0 ideal of C generated by ϕ(I) and δn(ψ(f)) for f ∈ J. Then (C ,K , δ|K0 ) is a divided power ring and C0 has cardinality bounded by the cardinal κ = |A| ⊗ |B|ℵ0 . Moreover, ϕ factors as A → C0 → C and ψ factors as B → C0 → C. This proves assumption (1) of Divided Power Algebra, Lemma 3.3 holds. Assumption (2) is clear as limits in the category of divided power rings commute with the forgetful functor (C, K, δ) 7→ (C, K), see Divided Power Algebra, Lemma 3.2 and its proof. 07H9Definition 2.2. Let (A, I, γ) be a divided power ring. Let A → B be a ring map. Let J ⊂ B be an ideal with IB ⊂ J. The divided power algebra (D, J,¯ γ¯) constructed in Lemma 2.1 is called the divided power envelope of J in B relative to (A, I, γ) and is denoted DB(J) or DB,γ (J). Let (A, I, γ) → (C, K, δ) be a homomorphism of divided power rings. The universal property of DB,γ (J) = (D, J,¯ γ¯) is ring maps B → C divided power homomorphisms ←→ which map J into K (D, J,¯ γ¯) → (C, K, δ) and the correspondence is given by precomposing with the map B → D which corresponds to idD. Here are some properties of (D, J,¯ γ¯) which follow directly from the universal property. There are A-algebra maps (2.2.1)07HA B −→ D −→ B/J The first arrow maps J into J¯ and J¯ is the kernel of the second arrow. The elements γ¯n(x) where n > 0 and x is an element in the image of J → D generate J¯ as an ideal in D and generate D as a B-algebra. CRYSTALLINE COHOMOLOGY 3 07HB Lemma 2.3. Let (A, I, γ) be a divided power ring. Let ϕ : B0 → B be a surjection of A-algebras with kernel K. Let IB ⊂ J ⊂ B be an ideal. Let J 0 ⊂ B0 be the inverse 0 0 0 0 0 0 0 image of J. Write DB0,γ (J ) = (D , J¯ , γ¯). Then DB,γ (J) = (D /K , J¯ /K , γ¯) 0 where K is the ideal generated by the elements γ¯n(k) for n ≥ 1 and k ∈ K. 0 Proof. Write DB,γ (J) = (D, J,¯ γ¯). The universal property of D gives us a homo- morphism D0 → D of divided power algebras. As B0 → B and J 0 → J are surjec- 0 tive, we see that D → D is surjective (see remarks above). It is clear that γ¯n(k) is in the kernel for n ≥ 1 and k ∈ K, i.e., we obtain a homomorphism D0/K0 → D. Conversely, there exists a divided power structure on J¯0/K0 ⊂ D0/K0, see Divided Power Algebra, Lemma 4.3. Hence the universal property of D gives an inverse 0 0 D → D /K and we win. In the situation of Definition 2.2 we can choose a surjection P → B where P is a polynomial algebra over A and let J 0 ⊂ P be the inverse image of J. The 0 previous lemma describes DB,γ (J) in terms of DP,γ (J ). Note that γ extends to a divided power structure γ0 on IP by Divided Power Algebra, Lemma 4.2. Hence 0 0 DP,γ (J ) = DP,γ0 (J ) is an example of a special case of divided power envelopes we describe in the following lemma. 07HC Lemma 2.4. Let (B, I, γ) be a divided power algebra. Let I ⊂ J ⊂ B be an ideal. Let (D, J,¯ γ¯) be the divided power envelope of J relative to γ. Choose elements ft ∈ J, t ∈ T such that J = I + (ft). Then there exists a surjection Ψ: Bhxti −→ D of divided power rings mapping xt to the image of ft in D. The kernel of Ψ is generated by the elements xt − ft and all X δn rtxt − r0 P whenever rtft = r0 in B for some rt ∈ B, r0 ∈ I. Proof. In the statement of the lemma we think of Bhxti as a divided power ring 0 with ideal J = IBhxti + Bhxti+, see Divided Power Algebra, Remark 5.2. The existence of Ψ follows from the universal property of divided power polynomial rings. Surjectivity of Ψ follows from the fact that its image is a divided power subring of D, hence equal to D by the universal property of D. It is clear that xt − ft is in the kernel. Set M X R = {(r0, rt) ∈ I ⊕ B | rtft = r0 in B} t∈T P If (r0, rt) ∈ R then it is clear that rtxt − r0 is in the kernel. As Ψ is a homomor- P 0 P phism of divided power rings and rtxt − r0 ∈ J it follows that δn( rtxt − r0) is in the kernel as well. Let K ⊂ Bhxti be the ideal generated by xt − ft and the P elements δn( rtxt − r0) for (r0, rt) ∈ R. To show that K = Ker(Ψ) it suffices to show that δ extends to Bhxti/K. Namely, if so the universal property of D 0 gives a map D → Bhxti/K inverse to Ψ. Hence we have to show that K ∩ J is 0 preserved by δn, see Divided Power Algebra, Lemma 4.3. Let K ⊂ Bhxti be the ideal generated by the elements P (1) δm( rtxt − r0) where m > 0 and (r0, rt) ∈ R, [m] 0 (2) xt0 (xt − ft) where m > 0 and t , t ∈ I. CRYSTALLINE COHOMOLOGY 4 0 0 0 We claim that K = K ∩J . The claim proves that K ∩J is preserved by δn, n > 0 by the criterion of Divided Power Algebra, Lemma 4.3 (2)(c) and a computation of δn of the elements listed which we leave to the reader. To prove the claim note that K0 ⊂ K ∩ J 0. Conversely, if h ∈ K ∩ J 0 then, modulo K0 we can write X h = rt(xt − ft) 0 0 P for some rt ∈ B. As h ∈ K ∩ J ⊂ J we see that r0 = rtft ∈ I. Hence (r0, rt) ∈ R and we see that X h = rtxt − r0 0 is in K as desired. 07KE Lemma 2.5. Let (A, I, γ) be a divided power ring. Let B be an A-algebra and IB ⊂ J ⊂ B an ideal. Let xi be a set of variables. Then DB[xi],γ (JB[xi] + (xi)) = DB,γ (J)hxii Proof. One possible proof is to deduce this from Lemma 2.4 as any relation be- tween xi in B[xi] is trivial. On the other hand, the lemma follows from the universal property of the divided power polynomial algebra and the universal property of di- vided power envelopes.
Load more

Recommended publications
  • Noncommutative Counterparts of Celebrated Conjectures
    NONCOMMUTATIVE COUNTERPARTS OF CELEBRATED CONJECTURES GONC¸ALO TABUADA Abstract. In this survey, written for the proceedings of the conference K- theory in algebra, analysis and topology, Buenos Aires, Argentina (satellite event of the ICM 2018), we give a rigorous overview of the noncommuta- tive counterparts of some celebrated conjectures of Grothendieck, Voevodsky, Beilinson, Weil, Tate, Parshin, Kimura, Schur, and others. Contents Introduction1 1. Celebrated conjectures1 2. Noncommutative counterparts5 3. Applications to commutative geometry 10 4. Applications to noncommutative geometry 17 References 21 Introduction Some celebrated conjectures of Grothendieck, Voevodsky, Beilinson, Weil, Tate, Parshin, Kimura, Schur, and others, were recently extended from the realm of alge- braic geometry to the broad noncommutative setting of differential graded (=dg) categories. This noncommutative viewpoint led to a proof of these celebrated con- jectures in several new cases. Moreover, it enabled a proof of the noncommutative counterparts of the celebrated conjectures in many interesting cases. The purpose of this survey, written for a broad mathematical audience, is to give a rigorous overview of these recent developments. Notations. Given a perfect base field k of characteristic p > 0, we will write W (k) for its ring of p-typical Witt vectors and K := W (k)1=p for the fraction field of W (k). For example, when k = Fp, we have W (k) = Zp and K = Qp. 1. Celebrated conjectures In this section, we briefly recall some celebrated conjectures of Grothendieck, Voevodsky, Beilinson, Weil, Tate, Parshin, and Kimura (concerning smooth proper schemes), as well as a conjecture of Schur (concerning smooth schemes). Date: January 14, 2019.
    [Show full text]
  • Cycle Classes in Overconvergent Rigid Cohomology and a Semistable
    CYCLE CLASSES IN OVERCONVERGENT RIGID COHOMOLOGY AND A SEMISTABLE LEFSCHETZ (1, 1) THEOREM CHRISTOPHER LAZDA AND AMBRUS PAL´ ABSTRACT. In this article we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, showing that for k a perfect field of characteristic p, a rational (logarithmic) line bundle on the special fibre of a semistable scheme over kJtK lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham–Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter example to show that the variational Tate conjecture for divisors cannot hold with Qp-coefficients. CONTENTS Introduction 1 1. Cycle class maps in overconvergentrigid cohomology 3 2. Preliminaries on the de Rham–Witt complex 4 3. Morrow’s variational Tate conjecture for divisors 9 4. A semistable variational Tate conjecture for divisors 12 5. Global results 15 6. A counter-example 16 References 19 INTRODUCTION Many of the deepest conjectures in arithmetic and algebraic geometry concern the existence of algebraic cycles on varieties with certain properties. For example, the Hodge and Tate conjectures state, roughly speaking, that on smooth and projective varieties over C (Hodge) or finitely generated fields (Tate) every cohomologyclass which ‘looks like’ the class of a cycle is indeed so. One can also pose variationalforms of arXiv:1701.05017v2 [math.AG] 25 Feb 2019 these conjectures, giving conditions for extending algebraic classes from one fibre of a smooth, projective morphism f : X → S to the whole space.
    [Show full text]
  • A Variational Tate Conjecture in Crystalline Cohomology
    A Variational Tate Conjecture in crystalline cohomology Matthew Morrow Abstract Given a smooth, proper family of varieties in characteristic p > 0, and a cycle z on a fibre of the family, we consider a Variational Tate Conjecture characterising, in terms of the crystalline cycle class of z, whether z extends cohomologically to the entire family. This is a characteristic p analogue of Grothendieck’s Variational Hodge Conjecture. We prove the conjecture for divisors, and an infinitesimal variant of the conjecture for cycles of higher codimension. This can be used to reduce the ℓ-adic Tate conjecture for divisors over finite fields to the case of surfaces. Contents 0 Introduction and statement of main results 1 1 Conjecture 0.1 5 2 Crystalline Th´eor`eme de la Partie Fixe 9 3 Local and infinitesimal forms of Conjecture 0.1 14 3.1 Some remarks on crystalline cohomology . ..... 14 3.2 Proofs of Theorems 0.5 and 0.6 via topological cyclic homology . 15 3.3 Cohomologically flat families over k[[t]] and Artin’s theorem . 22 4 An application to the Tate conjecture 24 0 Introduction and statement of main results arXiv:1408.6783v2 [math.AG] 25 Mar 2015 Let f : X → S be a smooth, proper morphism of smooth varieties over a field k, and let s ∈ S be a closed point. Grothendieck’s Variational Hodge or ℓ-adic Tate Conjecture [25, pg. 359] gives conditions on the cohomology class of an algebraic cycle on Xs under which the cycle conjecturally extends cohomologically to the entire family X. According as k has characteristic 0 or p > 0, the cohomology theory used to formulate Grothendieck’s conjecture is de Rham or ℓ-adic ´etale (ℓ 6= p).
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Arithmetic and Differential Galois Groups
    Mathematisches Forschungsinstitut Oberwolfach Report No. 26/2007 Arithmetic and Differential Galois Groups Organised by David Harbater (Philadelphia) B. Heinrich Matzat (Heidelberg) Marius van der Put (Groningen) Leila Schneps (Paris) May 13th – May 19th, 2007 Abstract. Galois theory is the study of symmetries in solution spaces of polynomial and differential equations and more generally of the relation be- tween automorphism groups (or group schemes respectively) and the struc- ture of algebraic and differential extensions. The lectures of this workshop were focused around the following five main topics: the absolute Galois group GQ and the Grothendieck-Teichm¨uller group, ´etale fundamental groups and the anabelian conjecture, arithmetic Galois realizations and constructive Langlands program, local and global differential modules and the p-curvature conjecture, as well as Galois theory for nonlinear and partial differential equa- tions. Mathematics Subject Classification (2000): 12Fxx,12Gxx,12Hxx, (13Nxx,14Lxx,34Gxx). Introduction by the Organisers Galois theory is the study of symmetries in solution spaces of polynomial and differential equations and more generally of the relation between automorphism groups (or group schemes) and the structure of algebraic and differential exten- sions. Many of the important problems in this area are connected to the classification of (algebraic or differential) Galois extensions and the study of the respective fundamental groups, e. g. via inverse problems. Other interesting points are the direct problem, i. e., the computation of (ordinary or differential) Galois groups, and related constructive aspects. This workshop gave an overview on some important new results and develop- ments. A second goal was the discussion of ideas for proving some open questions and conjectures as well as of new directions of research.
    [Show full text]
  • Holonomic D-Modules and Positive Characteristic
    Japan. J. Math. 4, 1–25 (2009) DOI: 10.1007/s11537-009-0852-x Holonomic -modules and positive characteristic Maxim Kontsevich Received: 8 December 2008 / Revised: 6 January 2009 / Accepted: 11 February 2009 Published online: ????? c The Mathematical Society of Japan and Springer 2009 Communicated by: Toshiyuki Kobayashi Abstract. We discuss a hypothetical correspondence between holonomic -modules on an al- gebraic variety X defined over a field of zero characteristic, and certain families of Lagrangian subvarieties in the cotangent bundle to X. The correspondence is based on the reduction to posi- tive characteristic. Keywords and phrases: rings of differential operators and their modules, characteristic p methods Mathematics Subject Classification (2000): 13N10, 13A35 1. Quantization and -modules 1.1. Quantization Let us recall the heuristic dictionary of quantization. symplectic C∞-manifold (Y,ω) ←→ complex Hilbert space H complex-valued functions on Y ←→ operators in H Lagrangian submanifolds L ⊂ Y ←→ vectors v ∈ H In the basic example of the cotangent bundle Y = T ∗X the space H is L2(X), functions on Y which are polynomial along fibers of the projection Y = T ∗X → X correspond to differential operators, and with the Lagrangian manifold L ⊂ Y This article is based on the 5th Takagi Lectures that the author delivered at the University of Tokyo on October 4 and 5, 2008. M. KONTSEVICH Institut des Hautes Etudes´ Scientifiques, 35 route de Chartres, Bures-sur-Yvette 91440, France (e-mail: [email protected]) 2 M. Kontsevich of the form L = graphdF for some function F ∈ C∞(Y) we associate (approxi- mately) vector exp(iF/) where → 0 is a small parameter (“Planck constant”).
    [Show full text]
  • Introduction
    Chapter 1 Introduction In the late 1940s Weil posed the challenge to create a cohomology theory for algebraic varieties X over an arbitrary field k with coefficients in Z. Weil had a deep arithmetic application in mind, namely to prove a conjecture of his concerning the -function attached to a variety over a finite field: Its first part asserts the rationality of this - function, the second claims the existence of a functional equation, and the third predicts the locations of its zeros and poles. Assuming a well-developed cohomology theory with coefficients in Z, possessing in particular a Lefschetz trace formula, he indicated a proof of these conjectures, cf. [48]. As we now know, for fields k of positive characteristic one cannot hope to con- struct meaningful cohomology groups H i .X; Z/. Nevertheless Weil’s quest provided an important stimulus. Starting with Serre and then with the main driving force of Grothendieck, several good substitutes for such a theory have been constructed. Many later developments in arithmetic algebraic geometry hinge crucially on these theories. Among the most important cohomology theories for algebraic varieties over a field k are the following: If k is of characteristic zero, one may pass from k to the field of complex numbers C and consider the singular cohomology of the associated complex analytic space X.C/. Other invariants in the same situation are given by the algebraic de Rham cohomology of X. For general k and a prime ` different from the characteristic of k, there is the cohomology theory of étale `-adic sheaves due to Grothendieck et al.
    [Show full text]
  • Introduction to Motives
    Introduction to motives Sujatha Ramdorai and Jorge Plazas With an appendix by Matilde Marcolli Abstract. This article is based on the lectures of the same tittle given by the first author during the instructional workshop of the program \number theory and physics" at ESI Vienna during March 2009. An account of the topics treated during the lectures can be found in [24] where the categorical aspects of the theory are stressed. Although naturally overlapping, these two independent articles serve as complements to each other. In the present article we focus on the construction of the category of pure motives starting from the category of smooth projective varieties. The necessary preliminary material is discussed. Early accounts of the theory were given in Manin [21] and Kleiman [19], the material presented here reflects to some extent their treatment of the main aspects of the theory. We also survey the theory of endomotives developed in [5], this provides a link between the theory of motives and tools from quantum statistical mechanics which play an important role in results connecting number theory and noncommutative geometry. An extended appendix (by Matilde Marcolli) further elaborates these ideas and reviews the role of motives in noncommutative geometry. Introduction Various cohomology theories play a central role in algebraic geometry, these co- homology theories share common properties and can in some cases be related by specific comparison morphisms. A cohomology theory with coefficients in a ring R is given by a contra-variant functor H from the category of algebraic varieties over a field k to the category of graded R-algebras (or more generally to a R-linear tensor category).
    [Show full text]
  • A Course on the Weil Conjectures
    A course on the Weil conjectures Tam´asSzamuely Notes by Davide Lombardo Contents 1 Introduction to the Weil conjectures 2 1.1 Statement of the Weil conjectures . .4 1.2 A bit of history . .5 1.3 Grothendieck's proof of Conjectures 1.3 and 1.5 . .6 1.3.1 Proof of Conjecture 1.3 . .6 1.3.2 Proof of Conjecture 1.5 . .8 2 Diagonal hypersurfaces 9 2.1 Gauss & Jacobi sums . 10 2.2 Proof of the Riemann hypothesis for the Fermat curve . 12 3 A primer on ´etalecohomology 14 3.1 How to define a good cohomology theory for schemes . 14 3.1.1 Examples . 15 3.2 Comparison of ´etalecohomology with other cohomology theories . 15 3.3 Stalks of an ´etalesheaf . 16 3.4 Cohomology groups with coefficients in rings of characteristic 0 . 16 3.5 Operations on sheaves . 17 3.6 Cohomology with compact support . 18 3.7 Three basic theorems . 18 3.8 Conjecture 1.11: connection with topology . 19 4 Deligne's integrality theorem 20 4.1 Generalised zeta functions . 20 4.2 The integrality theorem . 21 4.3 An application: Esnault's theorem . 22 4.3.1 Cohomology with support in a closed subscheme . 23 4.3.2 Cycle map . 24 4.3.3 Correspondences . 25 4.3.4 Bloch's lemma . 25 4.3.5 Proof of Theorem 4.11 . 26 5 Katz's proof of the Riemann Hypothesis for hypersurfaces 27 5.1 Reminder on the arithmetic fundamental group . 29 5.2 Katz's proof . 30 1 6 Deligne's original proof of the Riemann Hypothesis 31 6.1 Reductions in the proof of the Weil Conjectures .
    [Show full text]
  • P-ADIC ÉTALE COHOMOLOGY AND
    p-ADIC ETALE´ COHOMOLOGY AND CRYSTALLINE COHOMOLOGY FOR OPEN VARIETIES GO YAMASHITA Contents 1. Introduction 2 2. The main theorems of p-adic Hodge theory 3 3. The main results 7 References 10 This text is a report of a talk “p-adic ´etalecohomology and crystalline cohomology for open varieties” in the symposium “Hodge Theory and Algebraic Geometry” (7-11/Oct/2002 at Hokkaido University). It seemed that more algebraic geometers rather than arithmeti- cians participated in this symposium. Thus, in that talk, the auther began with an in- troduction to the p-adic Hodge theory in the view of the theory of p-adic representations. However, in this report, we do not treat the theory of p-adic representations, and we treat only a review of the main theorems and the main results. The aim of the talk was, roughly speaking, “to extend the main theorems of p-adic Hodge theory for open varieties” by the method of Fontaine-Messing-Kato-Tsuji (see [FM],[Ka2], and [Tsu1]). Here, the main theorems of p-adic Hodge theory are: the Hodge-Tate con- jecture (CHT for short), the de Rham conjecture (CdR), the crystalline conjecture (Ccrys), the semi-stabele conjecture (Cst), and the potentially semi-stable conjecture (Cpst). The theorems CdR, Ccrys, and Cst are called the “comparison theorems”. The section 1 is an introduction to the p-adic Hodge theory. In the section 2, we review the main theorems of the p-adic Hodge theory. In the section 3, we state the main results. In this report, we only announce the main results.
    [Show full text]
  • Twisted Differential Operators and Q-Crystals Michel Gros, Bernard Le Stum, Adolfo Quirós
    Twisted differential operators and q-crystals Michel Gros, Bernard Le Stum, Adolfo Quirós To cite this version: Michel Gros, Bernard Le Stum, Adolfo Quirós. Twisted differential operators and q-crystals. 2020. hal-02559467 HAL Id: hal-02559467 https://hal.archives-ouvertes.fr/hal-02559467 Preprint submitted on 30 Apr 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Twisted differential operators and q-crystals Michel Gros, Bernard Le Stum & Adolfo Quirós∗ Version of April 29, 2020 Abstract We describe explicitly the q-PD-envelopes considered by Bhatt and Scholze in their recent theory of q-crystalline cohomology and explain the relation with our notion of a divided polynomial twisted algebra. Together with an interpretation of crystals on the q-crystalline site, that we call q-crystals, as modules endowed with some kind of stratification, it allows us to associate a module on the ring of twisted differential operators to any q-crystal. Contents Introduction 2 1 δ-structures 3 2 δ-rings and twisted divided powers 5 3 q-divided powers and twisted divided powers 7 4 Complete q-PD-envelopes 14 5 Complete q-PD-envelope of a diagonal embedding 17 6 Hyper q-stratifications 19 7 q-crystals 22 8 Appendix: 1-crystals vs usual crystals 24 References 25 ∗Supported by grant PGC2018-095392-B-I00 (MCIU/AEI/FEDER, UE).
    [Show full text]
  • [Math.AG] 5 Oct 2001 Eddc Htteeaehdewt Seterm10.1)
    EXOTIC TORSION, FROBENIUS SPLITTING AND THE SLOPE SPECTRAL SEQUENCE KIRTI JOSHI Contents 1. Introduction 1 2. Frobenius split varieties 2 3. The slope spectral sequence 3 4. Dominoes and Ekedahl’s duality 3 5. A finiteness result 4 6. On the slope spectral sequence of threefolds 5 7. Exotic Torsion 6 8. Ordinarity of the Albanese Scheme 8 9. Liftings to W2; Hodge-de Rham spectral sequence 9 10. Unirational and Fano Threefolds 9 11. Coda: Chow groups and Abel-Jacobi mappings 10 References 12 1. Introduction In this paper we continue our investigations on some questions of V. B. Mehta on the crystalline aspects of Frobenius splitting which were begun in [12]. In response to a question of Mehta, we show that any F -split smooth projec- tive threefold is Hodge-Witt (see Theorem 6.2). After the first draft of this paper and [12] were written, Rajan and the author showed that there exist arXiv:math/0110069v1 [math.AG] 5 Oct 2001 examples of Frobenius split varieties which are not ordinary in the sense of Bloch-Kato-Illusie-Raynaud (see [3], [11]) and if the dimension is bigger than four then these examples are not even Hodge-Witt (these examples are now included in [12]). Thus the result of Theorem 6.2 on the Hodge-Witt prop- erty of Frobenius split threefolds is best possible. Frobenius split surfaces are ordinary (see [12]) and hence Hodge-Witt. The method of proof uses [11] and [6] to extract an explicit criterion for the degeneration of the slope spec- tral sequence of any smooth projective threefold (see Theorem 6.1).
    [Show full text]
  • A Stacky Approach to Crystals
    A STACKY APPROACH TO CRYSTALS VLADIMIR DRINFELD Abstract. Inspired by a theorem of Bhatt-Morrow-Scholze, we develop a stacky approach to crystals and isocrystals on “Frobenius-smooth” schemes over Fp . This class of schemes goes back to Berthelot-Messing and contains all smooth schemes over perfect fields of char- acteristic p. To treat isocrystals, we prove some descent theorems for sheaves of Banachian modules, which could be interesting in their own right. 1. Introduction Fix a prime p. 1.1. A theorem of Bhatt-Morrow-Scholze. Let X be a smooth scheme over a perfect field k of characteristic p. Let Xperf be the perfection of X, i.e., X := lim(... −→Fr X −→Fr X −→Fr X). perf ←− Let W (Xperf ) be the p-adic formal scheme whose underlying topological space is that of Xperf (or of X) and whose structure sheaf is obtained by applying to OXperf the functor of p-typical Witt vectors. Theorem 1.10 of [BMS] can be reformulated as a canonical isomorphism RΓcris(X, O)= RΓ(W (Xperf)/G , O), where G is a certain flat affine groupoid (in the category of p-adic formal schemes) acting 1 on W (Xperf), and W (Xperf)/G is the quotient stack . If X is affine this gives an explicit complex computing RΓcris(X, O) (the complex is constructed using the nerve of G ). arXiv:1810.11853v1 [math.AG] 28 Oct 2018 One can define G to be the unique groupoid acting on W (Xperf ) such that the morphism G → W (Xperf ) × W (Xperf ) is obtained by taking the divided power envelope of the ideal of the closed subscheme Xperf ×X Xperf ⊂ W (Xperf ) × W (Xperf ).
    [Show full text]