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Crystalline Cohomology FREIE UNIVERSITÄT BERLIN VORLESUNG WS 2017-2018 Crystalline Cohomology Lei Zhang February 2, 2018 INTRODUCTION The purpose of this course is to provide an introduction to the basic theory of crystals and crystalline cohomology. Crystalline cohomology was invented by A.Grothendieck in 1966 to construct a Weil cohomology theory for a smooth proper variety X over a field k of character- istic p 0. Crystals are certain sheaves on the crystalline site. The first main theorem which È we are going to prove is that if there is a lift XW of X to the Witt ring W (k), then the cat- egory of integrable quasi-coherent crystals is equivalent to the category of quasi-nilpotent connection of XW /W . Then we will prove that assuming the existence of the lift the crys- talline cohomology of X /k is "the same" as the de Rham cohomology of XW /W . Following from this we will finally prove a base change theorem of the crystalline cohomology using the very powerful tool of cohomological descent. Along the way we will also see a crystalline version of a "Gauss-Manin" connection. REFERENCES [SP] Authors, Stack Project, https://stacks.math.columbia.edu/download/ crystalline.pdf. [BO] P.Berthelot and A. Ogus, Notes on crystalline cohomology, Mathematical Notes 21, Princeton University Press and University of Tokyo Press, 1978. [B74] P.Berthelot, Cohomologie Cristalline des Schémas de Charactéristique p 0, LNM È 407, Springer Verlag, 1974. 1 1 INTRODUCTION (17/10/2017) In this lecture we will give an introduction to crystals and crystalline cohomology. There will be no proofs, and the purpose is just to get a picture of what is going on. 2 DIVIDED POWERS (24/10/2017) • The Definition of Divided Powers ([BO, §3, 3.1]). • Examples: (a) If A is an algebra over Q; (b) If A W (k), the Witt ring of a perfect field k. Æ • Interlude: The Witt ring of a perfect field k is characterized by the property that it is a complete DVR with uniformizer p and residue field k. n • PD-ideals are nil ideals if A is kill by m NÅ. Easy proof: For any x I, we have x 2 2 Æ n!γ (x) 0 for n m. n Æ ¸ • Definition of sub P.D. ideals ([BO, §3, 3.4]). • Lemma: If (A, I,γ) is a P.D. ring and J A is an ideal, then there is a PD-structure γ¯ on ⊆ I¯: I(A/J) such that (A, I,γ) (A/J, I¯,γ¯) is a PD-map iff J T I I is a sub PD-ideal Æ ! ⊆ ([BO, §3, 3.5]). • Theorem: If (A,M) is a pair, where A is a ring and M is an A-module, then there is triple (¡ (M),¡Å(M),γ˜) with an A-linear map ' : M ¡Å(M) which satisfy the univer- A A ! A sal property that if (B, J,±) is any PD-A-algebra and Ã: M J is A-linear, then there is ! a unique PD-morphism ï :(¡ (M),¡Å(M),γ˜) (B, J,±) A A ! such that ï Á Ã. Moreover, we know that ¡ (M) is graded with ¡ A and ¡ M. ± Æ A 0 Æ 1 Æ • Sketch of the proof: We take G A(M) to be the A-polynomial ring generated by indeter- minates {(x,n) x M,n N} whose grading is given by deg(x,n) n. Let I (M) be the j 2 2 Æ A ideal of G A(M) generated by elements 1.( x,0) 1 ¡ 2.( ¸x,n) ¸n(x,n) for x M and ¸ A ¡ 2 2 (n m)! 3.( x,n)(x,m) Å (x,n m) ¡ n!m! Å P 4.( x y,n) i j n(x,i)(y, j) Å ¡ Å Æ One sees that I (M) is a homogeneous ideal. Define ¡ (M): G (M)/I (M). Now let A A Æ A A x[n] be the image of (x,n). Then we have the following [1] [n] • Lemma: The ideal ¡Å(M) ¡ (M) has a unique PD-structure γ such that γ (x ) x A ½ A i Æ for all i 1 and all x M. ¸ 2 2 • Lemma: If A0 is an A-algebra, A0 A ¡A(M) ¡A (A0 A M). ­ Æ» 0 ­ • Lemma: If {M i I} is a direct system of A-modules, then we have i j 2 lim¡ (M ) ¡ (limM ) A i Æ A i i I i I ¡¡!2 ¡¡!2 • Lemma: ¡A(M) A ¡A(N) ¡A(M N). ­ Æ» © • Lemma: Suppose M is free with basis S : {xi i I}. Then ¡n(M) is free with basis [q ] [q ] Æ j 2 {x 1 x k Pq n}. 1 ¢¢¢ k j i Æ 3 THE PD-ENVELOP (07/11/2017) Theorem 3.1. Let (A, I,γ) be a PD-algebra and let J be an ideal in an A-algebra B such that IB J. Then there exists a B-algebra D (J) with a PD-ideal (J¯,γ¯) such that JD (J) J,¯ ⊆ B,γ B,γ ⊆ such that γ¯ is compatible with γ, and with the following universal property: For any B-algebra containing an ideal K which contains JC and with a PD-structure ± compatible with γ, there is a unique PD-morphism (D (J), J¯,γ¯) (C,K ,±) making the obvious diagrams commute. B,γ ! Proof. First assume that f (I) J. Viewing J as a B-module we get a triple (¡ (J),¡Å(J),γ˜). ⊆ B B Let ': J ¡ (J) be the canonical identification. We define a new ideal J generated by ideals ! 1 of the two forms: 1. '(x) x for x J ¡ 2 2. '(f (y))[n] f (γ (y)) for y I. ¡ n 2 One first has to show the following T Lemma 3.2. The ideal J ¡BÅ(J) is a sub PD-ideal of ¡BÅ(J). T So now we define D (J) to be ¡ (J)/J , J¯ : ¡Å(J)/J ¡Å(J), and γ˜ is the PD-structure B,γ B Æ B B induced by the sub PD-ideal. Now one checks the two things: JD J¯ (come from (1) of the ⊆ definition of J ), and γ is compatible with γ (follows from (2) of the definition of J ). Now it is easy to check that the triple (DB,γ(J), J¯,J ) is universal among all such triples. Here is a list of important properties of PD-envelops. • J¯ is generated, as a PD-ideal, by J. That is J¯ is generated by elements {γ¯ (j) j J,n 1}. n j 2 ¸ Moreover a set of generators of J provides a set of PD-generators of J¯. • If the map (A, I,γ) (B, J) factors as a diagram ! (A, I,γ) / (B, J) 9 & (A0, IA0,γ0) then we have D (J) D (J). B,γ Æ B,γ0 3 • The canonical map B/J D (J) is an isomorphism. Indeed, one just has to consider ! B,γ the PD-triple (B/J,0,0) and play with the universal property of (DB,γ(J), J¯,γ¯). • If M is an A-module, if B Sym (M), and if J¯ is the ideal SymÅ(M), then D (J) Æ A A B,γ Æ ¡A(M). This is clear when man plays with the universal property of the PD-envelop of (B, J). • Lamma: Suppose that J B is an ideal, and (A, I,γ) (B, J) is a morphism. If B 0 is flat ⊆ ! » over B, then there is a canonical isomorphism (D B 0) Æ D (JB 0). B,γ ­B ¡! B 0,γ • Theorem: Let (A, I,γ) be a PD-triple. Then there exists a unique PD-structure ± on the ideal J IA xt t T (A xt t T ) such that Æ h i 2 Å h i 2 Å 1. ± (x ) x[n]; n i Æ i 2. The map (A, I,γ) (A xt t T , J,±) is a PD-morphism. ! h i 2 Moreover, there is a universal property: Whenever (A, I,γ) (C,K ,²) is a PD-map and ! {kt }t T is a family in K , then there exists a unique PD-map (A xt t T , J,±) (C,K ,²) 2 h i 2 ! sending x k . t 7! t • Let (B, I,γ) be a PD-triple, and let J B be an ideal containing I. Choose {ft }t T a ⊆ 2 family in J such that J I ft t T . Then there exists a surjection à :(B xt , J 0,±) Æ Å h i 2 h i ! (D (J), J¯,γ¯) which maps x f¯ , where (B x , J 0,±) is the triple defined in the above B,γ t 7! t h t i theorem, and f¯t is the image of ft . The kernel of à is generated by all elements: 1. x f for f J; t ¡ t t 2 2. ± (P r x r ) whenever P r f r with r I, r B and n 1. n t t t ¡ 0 t t t Æ 0 0 2 t 2 ¸ • Lemma: Let (A, I,γ) be a PD-ring. Let B be an A-algebra, and let IB J B be an ideal. ⊆ ⊆ Then we have (D (JB[x ] x ), JB[x ] x ,γ¯) (D (J) x , J 0,±) B[xt ],γ t Å h t i t Å h t i Æ B,γ h t i 4 THE AFFINE CRYSTALLINE SITE (14/11/2017) Settings: Let p be a prime number, and let (A, I,γ) be a PD-triple in which A is a Z(p)-algebra (i.e.
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