Crystalline Cohomology

Transcription

1 FREIE UNIVERSITÄT BERLIN VORLESUNG WS Crystallne Cohomology Le Zhang February 2, 2018 INTRODUCTION The purpose of ths course s to provde an ntroducton to the basc theory of crystals and crystallne cohomology. Crystallne cohomology was nvented by A.Grothendeck n 1966 to construct a Wel cohomology theory for a smooth proper varety X over a feld k of characterstc p > 0. Crystals are certan sheaves on the crystallne ste. The frst man theorem whch we are gong to prove s that f there s a lft X W of X to the Wtt rng W (k), then the category of ntegrable quas-coherent crystals s equvalent to the category of quas-nlpotent connecton of X W /W. Then we wll prove that assumng the exstence of the lft the crystallne cohomology of X /k s "the same" as the de Rham cohomology of X W /W. Followng from ths we wll fnally prove a base change theorem of the crystallne cohomology usng the very powerful tool of cohomologcal descent. Along the way we wll also see a crystallne verson of a "Gauss-Mann" connecton. REFERENCES [SP] Authors, Stack Project, crystallne.pdf. [BO] P. Berthelot and A. Ogus, Notes on crystallne cohomology, Mathematcal Notes 21, Prnceton Unversty Press and Unversty of Tokyo Press, [B74] P. Berthelot, Cohomologe Crstallne des Schémas de Charactérstque p > 0, LNM 407, Sprnger Verlag,

2 1 INTRODUCTION (17/10/2017) In ths lecture we wll gve an ntroducton to crystals and crystallne cohomology. There wll be no proofs, and the purpose s just to get a pcture of what s gong on. 2 DIVIDED POWERS (24/10/2017) The Defnton of Dvded Powers ([BO, 3, 3.1]). Examples: (a) If A s an algebra over Q; (b) If A = W (k), the Wtt rng of a perfect feld k. Interlude: The Wtt rng of a perfect feld k s characterzed by the property that t s a complete DVR wth unformzer p and resdue feld k. PD-deals are nl deals f A s kll by m N +. Easy proof: For any x I, we have x n = n!γ n (x) = 0 for n m. Defnton of sub P.D. deals ([BO, 3, 3.4]). Lemma: If (A, I,γ) s a P.D. rng and J A s an deal, then there s a PD-structure γ on Ī : = I (A/J) such that (A, I,γ) (A/J, Ī, γ) s a PD-map ff J I I s a sub PD-deal ([BO, 3, 3.5]). Theorem: If (A, M) s a par, where A s a rng and M s an A-module, then there s trple (Γ A (M),Γ + A (M), γ) wth an A-lnear map ϕ : M Γ+ A (M) whch satsfy the unversal property that f (B, J,δ) s any PD-A-algebra and ψ: M J s A-lnear, then there s a unque PD-morphsm ψ: (Γ A (M),Γ + A (M), γ) (B, J,δ) such that ψ φ = ψ. Moreover, we know that Γ A (M) s graded wth Γ 0 = A and Γ 1 = M. Sketch of the proof: We take G A (M) to be the A-polynomal rng generated by ndetermnates {(x,n) x M,n N} whose gradng s gven by deg(x,n) = n. Let I A (M) be the deal of G A (M) generated by elements 1. (x,0) 1 2. (λx,n) λ n (x,n) for x M and λ A 3. (x,n)(x,m) (n+m)! n!m! (x,n + m) 4. (x + y,n) +j =n(x, )(y, j ) One sees that I A (M) s a homogeneous deal. Defne Γ A (M) := G A (M)/I A (M). Now let x [ n] be the mage of (x,n). Then we have the followng Lemma: The deal Γ + A (M) Γ A(M) has a unque PD-structure γ such that γ (x [1] ) = x [n] for all 1 and all x M. 2

3 Lemma: If A s an A-algebra, A A Γ A (M) = Γ A (A A M). Lemma: If {M I } s a drect system of A-modules, then we have lmγ A (M ) = Γ A (lm M ) I Lemma: Γ A (M) A Γ A (N ) = Γ A (M N ). Lemma: Suppose M s free wth bass S := {x I }. Then Γ n (M) s free wth bass {x [q 1] 1 x [q k ] q = n}. k I 3 THE PD-ENVELOP (07/11/2017) Theorem 3.1. Let (A, I,γ) be a PD-algebra and let J be an deal n an A-algebra B such that I B J. Then there exsts a B-algebra D B,γ (J) wth a PD-deal ( J, γ) such that JD B,γ (J) J, such that γ s compatble wth γ, and wth the followng unversal property: For any B-algebra contanng an deal K whch contans JC and wth a PD-structure δ compatble wth γ, there s a unque PD-morphsm (D B,γ (J), J, γ) (C,K,δ) makng the obvous dagrams commute. Proof. Frst assume that f (I ) J. Vewng J as a B-module we get a trple (Γ B (J),Γ + B (J), γ). Let ϕ: J Γ 1 (J) be the canoncal dentfcaton. We defne a new deal J generated by deals of the two forms: 1. ϕ(x) x for x J 2. ϕ(f (y)) [n] f (γ n (y)) for y I. One frst has to show the followng Lemma 3.2. The deal J Γ + B (J) s a sub PD-deal of Γ+ B (J). So now we defne D B,γ (J) to be Γ B (J)/J, J := Γ + B (J)/J Γ + B (J), and γ s the PD-structure nduced by the sub PD-deal. Now one checks the two thngs: JD J (come from (1) of the defnton of J ), and γ s compatble wth γ (follows from (2) of the defnton of J ). Now t s easy to check that the trple (D B,γ (J), J,J ) s unversal among all such trples. Here s a lst of mportant propertes of PD-envelops. J s generated, as a PD-deal, by J. That s J s generated by elements { γ n (j ) j J,n 1}. Moreover a set of generators of J provdes a set of PD-generators of J. If the map (A, I,γ) (B, J) factors as a dagram (A, I,γ) (B, J) (A, I A,γ ) then we have D B,γ (J) = D B,γ (J). 3

4 The canoncal map B/J D B,γ (J) s an somorphsm. Indeed, one just has to consder the PD-trple (B/J,0,0) and play wth the unversal property of (D B,γ (J), J, γ). If M s an A-module, f B = Sym A (M), and f J s the deal Sym + A (M), then D B,γ(J) = Γ A (M). Ths s clear when man plays wth the unversal property of the PD-envelop of (B, J). Lamma: Suppose that J B s an deal, and (A, I,γ) (B, J) s a morphsm. If B s flat over B, then there s a canoncal somorphsm (D B,γ B B ) = DB,γ(JB ). Theorem: Let (A, I,γ) be a PD-trple. Then there exsts a unque PD-structure δ on the deal J = I A x t t T + (A x t t T ) + such that 1. δ n (x ) = x [n] ; 2. The map (A, I,γ) (A x t t T, J,δ) s a PD-morphsm. Moreover, there s a unversal property: Whenever (A, I,γ) (C,K,ɛ) s a PD-map and {k t } t T s a famly n K, then there exsts a unque PD-map (A x t t T, J,δ) (C,K,ɛ) sendng x t k t. Let (B, I,γ) be a PD-trple, and let J B be an deal contanng I. Choose {f t } t T a famly n J such that J = I + f t t T. Then there exsts a surjecton ψ : (B x t, J,δ) (D B,γ (J), J, γ) whch maps x t f t, where (B x t, J,δ) s the trple defned n the above theorem, and f t s the mage of f t. The kernel of ψ s generated by all elements: 1. x t f t for f t J; 2. δ n ( t r t x t r 0 ) whenever t r t f t = r 0 wth r 0 I, r t B and n 1. Lemma: Let (A, I,γ) be a PD-rng. Let B be an A-algebra, and let I B J B be an deal. Then we have (D B[xt ],γ(jb[x t ] + x t ), JB[x t ] + x t, γ) = (D B,γ (J) x t, J,δ) 4 THE AFFINE CRYSTALLINE SITE (14/11/2017) Settngs: Let p be a prme number, and let (A, I,γ) be a PD-trple n whch A s a Z (p) -algebra (.e. any nteger whch s prme to p s nvertble n n A). Let A C be a rng map such that IC = 0 and p s nlpotent n C. (Note that n ths case C s automatcally an A/I -algebra.) Typcal Examples: Keep n mnd the stuaton when and when (A, I,γ) = (W (k),(p),γ) (A, I,γ) = (W n (k),(p),γ) where k s a perfect feld of characterstc p > 0. Defnton A thckenng of C over (A, I,γ) s a PD-map (A, I,γ) (B, J,δ) such that p s nlpotent n B, and an A/I -algebra map C B/J. 4

5 2. A map of PD-thckenngs s a map (B, J,δ) (B, J,δ ) over the thckenng (A, I,γ) whose nduced map B/J B /J s a C -algebra map. 3. We denote CRIS(C /A) the category of PD-thckenngs of C over (A, I,γ). 4. We denote Crs(C /A) the full subcategory of CRIS(C /A) whose objects are PD-thckenngs ((B, J,δ),C B/J) n whch C B/J s an somorphsm. Lemma The category CRIS(C /A) has non-empty products, and the category Crs(C /A) has empty product,.e. the termnal object. 2. The category CRIS(C /A) has all fnte non-empty colmts and the functor commutes wth those. CRIS(C /A) C algebras (B, J,δ) B/J 3. The category Crs(C /A) has all fnte non-empty colmts and the functor commutes wth those. Crs(C /A) CRIS(C /A) Proof. () The empty product of Crs(C /A) s ndeed (C,0,). The product of a famly of thckenngs (B t, J t,δ t ) n CRIS(C /A) s just ( t B t, t J t, t δ t ) wth the A/I -algebra map C t B t comng from each C B t. () Frst note that by to show colmts (resp. lmt) exst we only have to prove that coproducts and pushouts (resp. products and pullbacks) exst. We dvde the proof nto steps. The category of PD-trples admts lmts. The category of PD-trples admts colmts. Coproducts of pars exst n CRIS(C /A). There are also two remarks: (a) If the par s n Crs(C /A), then the coproduct s also n Crs(C /A). (b) The functor commutes wth coproducts. CRIS(C /A) C algebras Coequalzers of pars exst n CRIS(C /A). There are also two remarks: (a) If the par s n Crs(C /A), then the coequalzer s also n Crs(C /A). (b) The functor commutes wth coproducts. Conclude the proof. CRIS(C /A) C algebras 5

6 Defnton 2. Let Ĉrs(C /A) be the category whose objects are PD-trples (B, J,δ), where B s only p-adcally comlete nstead of nlpotent n B, plus an A/I -algebra map C B/J as usual. Clearly that Crs(C /A) s a full subcategory of Ĉrs(C /A), as p n -torson rngs are p-adcally complete. Lemma 4.2. Let (A, I,γ) be a PD-rng. Let p be a prme number. If p s nlpotent n A/I, and f A s a Z (p) -algebra then 1. The p-adc completon  goes surjectvely to A/I. 2. The kernel of  A/I s Î. 3. Each γ n s contnuous for the p-adc topology on I. 4. For e large, the dea p e A I s preserved by γ n and we have (Â, Î, ˆδ) = lm e (A/p e A, I /p e I,γ e ) Lemma 4.3. Let P C be a surjecton of A-algebras wth kernel J. We wrte (D, J, γ) for the PD-envelop of (P, J) wth respect to (A, I,γ). Let ( ˆD, ˆ J, ˆ γ) be the completon of (D, J, γ). For every e 1, set (P e, J e ) := (P/p e P, J/(J p e P)) and (D e, J e, γ e ) the PD-envelop of ths par. Then for large e we have 1. p e D J and p e ˆD ˆ J are preserved by the PD-structures. 2. ˆD/p e ˆD = D/p e D = D e as PD-rngs. 3. (D e, J e, γ e ) Crs(C /A). 4. ( ˆD, ˆ J, ˆ γ ) = lm (D e, J e, γ e ). 5. ( ˆD, ˆ J, ˆ γ) Crs(C ˆ /A). Lemma 4.4. Let P be a polynomal algebra over A, and let P C be a surjecton of A-algebras wth kernel J. Then every object (B, J,δ) of CRIS(C /A) there exsts an e and a morphsm n CRIS(C /A). (D e, J e, γ e ) (B, J,δ) Lemma 4.5. Let P be a polynomal algebra over A, and let P C be a surjecton of A-algebras wth kernel J. Let (D, J, γ) be the p-adc completon of D P,γ (J). For every object (B, J,δ) of Ĉrs(C /A) there exsts a morphsm n Ĉrs(C /A). (D, J, γ) (B, J,δ) 6

7 5 THE DIFFERENTIALS OF PD-STRUCTURES (21/11/2017) Let A be a rng. Let (B, J,δ) be a PD-trple. Let A B be a rng map. Let M be a B- module. A PD-dervaton s a usual A-dervaton θ : B M wth the extra condton that ( ) θ(γ n (x)) = γ n 1 (x)θ(x) for all n 1 and x J. Let Ω B/A,δ : Ω B/A / d(γ n (x)) γ n 1 (x)d x Then Ω B/A,δ has the unversal property that Hom B (Ω B/A,δ, M) PD-Der A (B, M) where M s a B-module. Conceptually, condton ( ) can be thought of as the followng: A basc Lemma: = d( xn n! ) = xn 1 (n 1)! d x Lemma 5.1. Let A be a rng. Let (B, J,δ) be a PD-trple, and A B be a rng map. 1. If we equp B[X ] wth the PD-structure (B[X ], JB[X ],δ ), where γ n (ax m ) = γ n (a)x mn then we have Ω B[X ]/A,δ = Ω B/A,δ B B[X ] B[X ]d X Here B[X ]d X just means a free B[X ]-module. 2. If B X s equpped wth the PD-structure (JB X + B X +,δ ), where δ takes j J to δ n (j ) and j X [m] to (m+n)! m!n! j n X [m+n], then Ω B X /A,δ = Ω B/A,δ B B X B X d X 3. Let K J be an deal preserved by δ n for all n 1. Set B := B/K and denote δ the nduced PD-structure on J/K. Then we an exact sequence: K /K 2 Ω B/A,δ B B Ω B /A,δ 0 Proof. (1) Set B[X ] d Ω B/A,δ B B[X ] B[X ]d X sendng b 0 + b 1 X + b n X n db db 1 X + + db n X n + b 1 d X + + nb n X n 1 d X 7

8 Ths s an A-dervaton. For example we have the dervaton: d(δ n (bx m )) = d(x mn δ n (b)) = δ n (b)d X mn d X + δ n 1 (b) db X mn = mx mn 1 (nδ n (b))d X + δ n 1 (b) db X mn = mx mn 1 δ n 1 (b) b d X + δ n 1 (b) db X mn = (δ n 1 (b) X m(n 1) ) (X m db + mbx m 1 d X ) = (δ n 1 (bx m ) d(bx m )) The unversal property: Usng the unversal property of drect sum the unversal property of d bols down to the unversal property of Ω B/A,δ and the unversal property of the free module B[X ]d X. (2) Almost the same as (1). (3) Look at the dagram: 0 M Ω B/A B B Ω B /A 0 f φ 0 M Ω B/A,δ B B Ω B /A,δ 0 Snce Ker(φ) Ker(ϕ), we see that f s surjectve. Snce K /K 2 M, t follows that K /K 2 M. Defnton: Let (A, I,γ) be a PD-rng. We denote I [n] the deal generated by γ e1 (x 1 ) γ et (x t ) wth e t n and x I. So we have I [0] = A, I [1] = I and I I [ ]. Here s an mportant Proposton: Proposton 5.2. Let a : (A, I,γ) (B, J,δ) be a map of PD-trples. Let (B(1), J(1),δ(1)) be the coproduct of a wth tself. Denote K the kernel of the dagonal map : B(1) B. Then we have Ω B/A,δ = K /(K 2 + (K J(1)) [2] ) Proof. Let s denote the two projectons B(1) B s 0 by s 0, s 1 respectvely. Snce the composton B s 0 s 1 B(1) B s 1 s the dentty, we see that the map B B(1) sendng b s 0 (b) s 1 (b) factors through K. Thus we obtan a map d : B K /(K 2 + (K J(1)) [2] ) ϕ 8

9 Clearly d s addtve and vanshes on A, and d(b 1 b 2 ) = b 1 d(b 2 ) + b 2 d(b 1 ) = s 1 (b 1 )(s 1 (b 1 ) s 0 (b 2 )) + s 0 (b 2 )(s 1 (b 1 ) s 0 (b 1 )) = s 1 (b 1 )s 1 (b 2 ) s 0 (b 2 )s 0 (b 1 ) = s 1 (b 1 b 2 ) s 0 (b 1 b 2 ) Thus d s a dervaton. We have to check that d s a PD-dervaton. Let x J. Set y = s 1 (x), z = s 0 (z) and λ := δ(1). Snce d(λ n (x)) = s 1 (λ n (x)) s 0 (λ n (x)) = λ n (y) λ n (z), and λ n 1 (x) d x = λ n 1 (y)(y z), we need to show that λ n (y) λ n (z) = λ n 1 (y)(y z) for all n 1. If n = 1 ths s clearly true. Let n > 1. We have that λ n (z y) = n ( 1) n λ (z)λ n (y) K 2 + (K J(1)) [2] =0 as z y K J(1) and n 2. Then we have n 1 λ n (y) λ n (z) = λ n (y) + ( 1) n λ (z)λ n (y) Snce we have and =0 = λ n (y) + ( 1) n n 1 λ n (y) + ( 1) n (λ (y) λ 1 (y)(y z))λ n (y) =1 ( ) n λ (y)λ n (y) = λ n (y) ( ) n 1 λ 1 (y)λ n (y) = λ n 1 (y) 1 we can contnue ( ) ( ) λ n (y) λ n (z) = λ n (y) + ( 1) n n 1 λ n (y) + ( 1) n n n 1 λ n (y) ( 1) n n 1 λ 1 (y)(y z)) =1 =1 1 ( ) ( ) n = ( 1) n n n 2 λ n (y) ( 1) n 1 n 1 λ n 1 (y)(y z) =0 =0 = (1 1) n λ n (y) (1 1)λ n 1 (y z) + λ n 1 (y)(y z) = λ n 1 (y)(y z) Let M be any B-module, and let θ : B M be a PD A-dervaton. Set D := B M, where M s an deal of square 0. Defne a PD-structure on J M D by settng δ n (x + m) = δ n (x) + δ n 1 (x)m for all n 1. There are two PD-morphsms: (B, J,δ) t 0 t 1 (D = B M, J M,δ ) 9

10 where t 1 s just the canoncal ncluson b b and t 2 s the map sendng b b + θ(b). Thus by the unversal property we have a commutatve dagram (B(1), J(1),δ(1)) (D, J M,δ ) (B = B(1)/K, J,δ) (B, J,δ) Ths nduces a map K M. Snce M 2 = 0 and M [2] = 0. Thus we get a factorzaton φ := K /(K + (K J(1)) [2] ) M Ths φ s compatble wth d and θ by constructon, and t s unque because K s generated by {s 1 (b) s 0 (b) b B}. Lemma: Let (B, J,δ) CRIS(C /A) and let (B(1), J(1),δ(1)) be the coproduct n CRIS(C /A). Let K be the kernel of the dagonal. Then K J(1) J(1) s preserved by the PD-structure and, Ω B/A,δ = K /(K 2 + (K J(1)) [2] ) 6 THE DE RHAM COMPLEX IN THE AFFINE CASE (28/11/2017) Lemma: Let (A, I,γ) be a PD-trple, and let A B be a rng map. Let I B J B be an deal. Let (D, J, γ) := (D B,γ (J), J, γ). Then we have Ω D/A,δ = Ω B/A B D Proof. Let s frst suppose that A B s flat. Then there s a unque PD-structure (B, I B,γ ) whch s compatble wth (A, I,γ). By a lemma n 3, we see that there s a surjectve morphsm (B x t, J,γ ) (D, J, γ) where J := JB x t + B x t +, whose kernel s generated by elements of the forms: (x t f t ), and γ n ( t r t f t r 0 ) where r t B and r 0 I B. Snce we have that Ω B xt /A = Ω B/A B B x t B x t d x t Thus we have Ω B xt /A B xt D = Ω B/A B D Dd x t By 5.1 there s a canoncal surjecton Ω B xt /A B xt D Ω D/A whose kernel s generated by all {dk 1 k Ker(B x t D)}. 10

11 Clearly the canoncal composton: Ω B/A B D Ω B xt /A B xt D Ω B xt /A B xt D/(d(x t f t )) t T s surjectve. But snce t has a retracton, t s an somorphsm. Now to prove the lemma we only need to show that λ := Ω B xt /A B xt D/(d(x t f t ) t T ) Ω D/A,δ s an somorphsm. Gven an element γ n ( t T r t x t r 0 ) satsfyng the relaton t T r t f t r 0 wth r t B and r 0 I B, we have dγ n ( r t x t r 0 ) = γ n 1 ( r t x t r 0 )d( r t x t r 0 ) t T t T t T = γ n 1 ( r t x t r 0 )( r t d(x t f t ) (x t f t )dr t ) t T t T t T s 0 n Ω B xt /A B xt D/(d(x t f t ) t T ). But snce those elements generate the kernel of λ, we conclude that λ s an somorphsm. In the general case we wrte B as a quotent P B of a polynomal P over A. Let J P be the nverse mage of J, and let (D, J,δ) be the PD-envelop of (P, J ). Then there s a surjecton (D, J,δ) (D, J, γ) whose kernel s generated by {δ n (k) k K := Ker(P B)}. But snce P s flat over A we have Ω D /A,δ = Ω P/A P D The kernel M of Ω P/A P D = Ω D /A,δ D D Ω D/A, γ s generated by {dδ n (k) 1 k K }. Snce dδ n (k) = δ n 1 (k)dk, the kernel M s actually generated by {dk 1 k K }. As Ω B/A s the quotent of Ω P/A P B by the submodule generated by {dk 1 k K }, we have that Ω B/A B D Ω D/A, γ s an somorphsm. Let B be a rng, and let Ω B := Ω B/Z. Let d : B Ω B be the canoncal dervaton. Set Ω B := B Ω B. The we get a complex where the dfferentals d p : Ω p B 0 Ω 0 B Ωp+1 B d 0 Ω 1 B d 1 Ω 2 B s defned by d 2 d(b 0 db 1 db2 dbp ) db 0 db1 db2 dbp ) Clearly we have that d d = 0, so ths s a complex f we can show that d s well-defned. Indeed, the B-module Ω B/Z s the free module on the bass {db b B} modulo the sub B-module M generated by elements of the form d(a + b) d a db and d(ab) adb bd a. If we regard M as a sub abelan group of the free B-module, then M s generated 11

12 by sd(a + b) sd a sdb and sd(ab) sadb sbd a wth s B. These are mapped to 0 by the map we defned. So d 1 s well-defned. The map d 1 defnes for us a map ψ: Ω B Z Ω B Z Z Ω }{{ B Ω p+1 } p tmes sendng w 1 w p ( 1) (+1) w 1 d w w p To show that d p s well-defned we only have to show that ψ sends w 1 f w w p w 1 f w j w p to 0 for all f B. The followng equatons d(f a 1 ) db 1 a 2 db 2 f a 1 db 1 d a 2 db 2 d a 1 db 1 f a 2 db 2 + a 1 db 1 d f a 2 db 2 =(a 2 d f a 1 + f a 1 d a 2 f a 2 d a 1 a 1 d f a 2 ) db 1 db 2 =0 shows wthout the loss of generalty that w 1 f w w p w 1 f w j w p s mapped to 0. So we wn. Lemma: Let B be a rng. Let π : Ω B Ω be a surjecton of B-modules. Denote d : B Ω be the composton of the dervaton d B := B Ω B wth the surjecton. Set Ω = B (Ω). Assume that Ker(π) s generated as a B-module by some elements ω Ω B such that d 1 B (ω) s n the kernel of Ω2 B Ω2. Then there s a (de Rham) complex whose dfferentals are defned by Ω 0 Ω 1 d p : Ω p Ω p+1, d p (f w 1 w p ) d p (f ) w 1 w p Proof. We only have to prove that there exst commutatve dagrams: B d B Ω B d 1 B Ω 2 B d 2 B B π 2 π d Ω d 1 Ω 2 d 2 The left square s gven by defnton. For the second square we have to show that Ker(π) goes to Ker( 2 π) under d 1 B. But Ker(π) s generated by bw, where b B and d 1 B w Ker( 2 π), and d 1 B (bw) = d B b w + bd 1 B w Ker( 2 π) as desred. If > 1, then we have that Ker( π) s equal to the mage of Ker(π) Ω ( 1) Ω 12

13 Now let w 1 Ker(π) and w 2 Ω ( 1). We have B d B (w 1 w 2 ) = d 1 B w 1 w 2 w 1 d ( 1) B w 2 whch s seen by the nducton hypothess to be contaned n Ker( (+1) π). Now we consder a specal case when Ω := Ω B/A,δ, where B s an A-algebra equpped wth a PD-structure (B, J,δ). In ths case the kernel of Ω B/Z Ω B/A,δ s generated by elements of the form d B a for a A and d B δ n (x) δ n 1 (x)d B x for x J. It s enough to show that the mage of these elements under d 1 B s contaned n Ker( 2 π). But we have and, Ths proves everythng. d 1 B (d B a) = 0, a A d 1 B (d B δ n (x) δ n 1 (x)d B x) = d 1 B (δ n 1(x)d B x)) = d B (δ n 1 (x)) d B (x) = δ n 2 (x)d B x d B x = 0 Integrable connectons and the nduced de Rham Complex. 1 The Grothendeck topology 7 THE CRYSTALLINE TOPOS (05/12/2017) The general defnton of Grothendeck topology Examples: (1) The global classcal topology; (2) The global Zarsk topology; (3) The crystallne topology whch we explan now. Defnton: Let X be a topologcal space, and let A be a sheaf of rngs on X. Let I A be an deal of A. A sequence of maps of sets γ n : I I for n 0 s called a PDstructure on I f for each open U X the maps γ n (U ): I (U ) I (U ) s a PD-structure on I (U ). Fact: Let X = Spec(A), and let I A be an deal. Denote Ĩ the quas-coherent deal sheaf assocated wth I. Then to gve a PD-structure on I s equvalent to gvng a PDstructure on the sheaf Ĩ. (Key pont: PD-structure extends along flat maps, so n partcular localzatons.) Stuaton: Let p be a prme number, and let (S, I,γ), or (S 0,S,γ) where S 0 S s a closed subscheme wth kernel I, be a PD-scheme over Z (p). Let X S 0 be a map of schemes and suppose that p s nlpotent on X. The defnton of the bg and the small crystallne ste 13

14 2 The Grothendeck topos The defnton of a topos Examples: (1) The category of sheaves on a topologcal space, n partcular, the category of sets s a topos; (2) The étale topos, the fppf-topos, the fpqc-topos; (3) The crystallne topos whch we explan now: Proposton: A sheaf on Crs(X /S) (resp. CRIS(X /S)) s equvalent to the followng data: For every morphsm u : (U 1,T 1,δ 1 ) (U,T,δ) we are gven a Zarsk sheaf F T on T and a map ρ u : u 1 (F T ) F T subject to the followng condtons: 1. If v : (U 2,T 2,δ 2 ) (U 1,T 1,δ 1 ) s another map, then v 1 (ρ u ) ρ v = ρ u v. 2. If u : T 1 T s an open embeddng, then ρ 1 u s an somorphsm. For a proof see Examples: (1) The structure sheaf O sendng (U,T,δ) O T. (2) The strange sheaf sendng (U,T,δ) O U. 3 Morphsms between topo A morphsm of topo f : X Ỹ conssts of a par of adjont functors (f : X Ỹ, f : Ỹ X ) n whch f commutes wth fnte nverse lmts. Defnton: A functor f 1 : Y X between two stes s called contnuous f for any sheaf F on X the composton F f 1 s a sheaf on Y. Theorem: Suppose that f 1 : Y X s a contnuous functor between two stes, then the functor f 1 : X Ỹ has a left adjont f. Defnton: A functor f 1 : Y X between two stes s called cocontnuous f for any object U Y and every coverng {V f 1 (U )} n X, there exsts a coverng {U j U } n Y such that {f 1 (U j ) f 1 (U )} refnes {V f 1 (U )}, that s for every V f 1 (U ) there exsts a f 1 (U j ) f 1 (U ) whch has a factorzaton f 1 (U j ) V. Theorem: Suppose that f 1 : Y X s cocontnuous, then the nduced map f 1 : Ỹ X has a rght adjont f : Ỹ X and f = ( f, f 1 ) defnes a maps of topo. Theorem: Let X,Y be stes, and let f 1 : Y X be a functor such that 1. f 1 s contnuous and cocontnuous. 2. fbred products and equalzers exst n Y and f 1 commutes wth those. then the nduced functor f : Ỹ X commutes wth fbred products and equalzers. 14

15 Lemma: The category CRIS(X /S) has all fnte non-empty lmts, and the functor commutes wth those. CRIS(X /S) Sch /X (U,T,δ) U Lemma: The category Crs(X /S) has non-empty lmts, and the ncluson commutes wth those. Corollary: There are morphsms of topo: where ĩ = π = ĩ 1. 1 : Crs(X /S) CRIS(X /S) (X /S) Crs (X /S) CRIS π (X /S)Crs Functoralty: Suppose that we have a PD-morphsm (S, I,γ) (S, I,γ ) and a dagram: X X S 0 S 0 where S 0 = Spec(O S /I ). Then we have an obvous functor f : CRIS(X /S) CRIS(X /S ) Ths f s both contnuous and cocontnuous. Ths nduces a map between topo f CRIS (X /S) CRIS (X /S ) CRIS Thus we have a map of topo f Crs obtaned by composton: f CRIS (X /S) Crs (X /S) CRIS (X /S π ) CRIS (X /S ) Crs 1 The global secton functor 8 THE CRYSTALLINE TOPOS (12/12/2017) Let s fx a ste X. We denote ˆX the category of presheaves on X and X the category of sheaves on X. 15

16 Let T be an object n ˆX. Then we defne the functor of "takng T sectons" to the functor: Γ(T, ): X (Sets) F Hom X (T,F ) If T s taken to be the termnal object e of ˆX, then we denote Γ( X, ) or Γ( ) for Γ(e, ), and ths s called the gobal secton functor. The termnal object n ˆX s the sheaf on X whch assocate wth each object n X the sngleton,.e. the set wth only one pont. Examples: 1. If X s a topologcal space equpped wth the usual topology, then the dentty X = X s the termnal object n the category of open embeddngs of X, so the global secton functor assocate wth a sheaf F on Y the global secton Hom ˆX (Y,F ) whch s nothng but the F (Y ) by the by the Yoneda lemma. Moreover, ths termnal object certanly does not depend on the choce of X. 2. If X s our ste Crs(X /S), then there s no termnal object n general. Indeed f we take X to be an affne smooth non-empty scheme over k, and we take (S,I,γ) to be the trple (Spec(W 2 ),(p),γ), then there s always a deformaton X Spec(W 2 ) of X Spec(k). Snce the deal (p) s prncpal, there s a unque PD-structure δ on (X,X ). Snce the PD-structure s unque, any automorphsm of the par (X,X ) (as a deformaton) produces an automorphsm of the trple (X,X,δ). Also any endomorphsm of (X,X ) as a deformaton of nduces an somorphsm of X, because the endomorphsm s radcel (unveral homoemorphsm), fberwse étale (ndeed fberwse somorphsm), and flat (because X Spec(W 2 ) s flat and all the fbres are flat). Now f (U,T,α) was the termnal object then we have morphsm: (X,X,δ) (U,T,α) (X,X,δ) n Crs(X /W 2 ), where the last arrow s obtaned by the smoothness of X Spec(W 2 ). Thus we see that n the unque map (X,X,δ) (U,T,α) the map X T s an mmerson. Hence the par (X,X ) admts only one automorphsm, whch s certanly not the case. Remark: Let X be a topos nduced by a ste X, and let e be the termnal object. Then the global secton functor F Hom X (e,f ) can also be descrbed as follows: It the set of compatble systems {ξ U } U X, where ξ U F (U ). Defnton of rnged topos: A rnged topos s a topos plus a rng object n a topos. Let ˆX be a topos, let O be a rng object. Then we wrte ( ˆX,O) for the rnged topos. Let ( ˆX,O) be a rnged topos. Then we denote O Mod the category of O module objects n X. 16

17 Examples: (1) When we take O to be the sheaf whch assocates to the constant sheaf wth value Z, then O Mod s just the category of abelan sheaves on X. (2) For the crystallne topos (X /S) Crs we take O to be the sheaf assocated wth the constant presheaf of value O S (S). Theorem: For any rnged topos ( X,O), the category O Mod s an abelan category wth enough njectve objects. The global secton functor s left exact, so we defne the rght derved functor to be the crystal cohomology. Suppose that we have a commutatve dagram: X g X Then there s a map of topo (S,I,γ ) (S,I,γ) g Crs : (X /S ) Crs (X /S) Crs Moreover the push-forward nduces the Grothendeck spectral sequence: for any E (X /S ) Crs. E pq 2 = H p ((X /S) Crs,R q g E ) H p+q ((X /S ) Crs,E ) Proposton: There s a natural morphsm of topo u X /S : (X /S) Crs X Zar gven by 1. for any F (X /S) Crs and j : U X open embeddng we defne u (F )(U ) := Γ((U /S) Crs,F U ) 2. for any E X Zar and (U,T,δ) Crs(X /S) we set (u (E)(U,T,δ) := E(U )) Remark: We can actually see u X /S as a map of rnged topo f we equp both (X /S) Crs and X Zar the constant O S (S) rnged topos structure. But we can not equp X Zar wth the O X -structure, otherwse u X /S would not be a map of rnged topo. 17

18 9 THE CRYSTALS AND CALCULUS (19/12/2017) 1 Crystals Defnton: Let C be the ste Crs(X /S). Let F be a sheaf of O X /S -modules on C, where O X /S s the sheaf of rngs (U,T,δ) O T. 1. We say F s a crystal f for all map (U,T,δ ) φ (U,T,δ) n Crs(X /S) the nduced map φ F T F T s an somorphsm. 2. We say that F s a quas-coherent crystal f each F T s a quas-coherent O T - module. 3. We say that F s locally free f for each (U,T,δ) there exsts a coverng {(U,T,δ ) (U,T,δ)} I such that F (U,T,δ ) s a drect sum of O X /S (U,T,δ ). 2 Sheaves of Dfferentals Defnton-Lemma: If (X 0, X,δ) s a PD-scheme over a scheme S wth the structure morphsm f : X S, then there exsts an O X -module Ω X /S,δ and a PD-dervaton d : O X Ω X /S,δ wth the property that for any PD-dervaton ϕ: O X M there exsts a unque O X -lnear map Ω X /S,δ M whch s compatble wth d and φ. Defnton: On Crs(X /S) we have an O X /S -module Ω X /S whose Zarsk sheaf on each object (U,T,δ), namely the sheaf (Ω X /S ) T, s equal to Ω T /S,δ. Moreover, there s a dervaton d : O X /S Ω X /S whch s a PD-dervaton on each object. Ths dervaton s also unversal among all such maps. Lemma: Let (U,T,δ) be an object n Crs(X /S). Let (U (1),T (1),δ(1)) be the product of (U,T,δ) wth tself n Crs(X /S). Let K O T (1) be the deal correspondng to the closed mmerson T T (1). Then K J(1) where J(1) s the deal of U (1) T (1), and we have (Ω X /S ) T = K /K [2] Lemma: The sheaf of dfferentals Ω X /S has the followng propertes: 1. (Ω X /S ) T s quas-coherent, 2. for any morphsm f : (U,T,δ) (U,T,δ ) where T T s a closed embeddng f (Ω X /S ) T (Ω X /S ) T 3 Unversal Thckenng 18

19 Recall: Let (A, I,γ) be a PD-trple, let M be an A-module, and let B : A M be an A- algebra where M s defned to be an deal of square 0. Let J := I M. Set δ n (x + z) := γ n (x) + γ n 1 (x)z for all x I and z M. Then δ s a PD-structure on J and s a PD-map. Now let (U,T,δ) Crs(X /S). Set (A, I,γ) (B, J,δ) T := Spec OT (O T Ω T /S,δ ) wth O T Ω T /S,δ the quas-coherent O T -algebra n whch Ω T /S,δ s a square 0 deal. Let J O T be the deal sheaf of U T. Set J = J Ω T /S,δ. Then as n the affne case one gets a PD-structure on J by settng δ n (f, w) = (δ n(f ),δ n 1 (f )w) Then we get two PD-morphsms: p 0, p 1 := O T O T where p 0 (f ) = (f,0) p 1 (f ) = (f,d f ) or equvalently: p 0, p 1 : (U,T,δ ) (U,T,δ). There s also a map of PD-schemes : (U,T,δ) (U,T,δ ) whch provdes a secton to both p 0 and p 1. 4 Connectons Defnton 3. A Connecton on (X /S) Crs s an O X /S -module F equpped wth an f 1 O S - modules : F F OX /S Ω X /S such that (f s) = f (s)+s d f for all sectons s F and f O X /S. We can contnue defnng : F OX /S Ω n X /S F O X /S Ω n+1 O X /S by sendng f m (f ) m+ f dm. If we wrte (f ) as f a wth f F and a Ω n X /S, then the mage of f m can be wrtten as f (a m) + f dm. We call the connecton ntegrable f we have = 0. In ths case we have the de Rham complex F F Ω 1 X /S F Ω 2 X /S 19

20 Proposton 9.1. Let F be a crystal n O X /S -modules on Crs(X /S). Then F comes wth a canoncal Integrable connecton. Proof. We start wth (U,T,δ) Crs(X /S), then we get a thckenng (U,T,δ ) wth maps (U,T,δ) (U,T,δ ) p 0 = p1 (U,T,δ) Ths provdes us somorphsms: p 0 F T c 0 FT c 1 p 1 F T and the map c := c1 1 c 0 s the dentty of F T va pullng back by. Thus f s F T (T ), then (s) := p1 s c(p 0 s) s 0 when pullback va to T. Ths mples that (s) Ker(p 1 F T F T ) Thus (s) F T OT Ω T /S. The map s f 1 O S -lnear, where f denotes T S, because all the maps F T p1 F, F T p0 F and c are all f 1 O S -lnear. For any f O T we have (f s) = p1 (f s) cp 0 (f s) = (f,d f )p1 s (f,0)c(p 0 (s)) Now let s show that s ntegrable. = f (s) + (0,d f )(s 1) = f (s) + s d f Step 1. Take (U,T,δ) Crs(X /S). We defne where the rng structure s defned as T := Spec OT (O T Ω T /S,δ Ω T /S,δ Ω 2 T /S,δ ) (f, w 1, w 2,η)(f, w 1, w 2,η ) = (f f, f w 1 + f w 1, f w 2 + f w 2, f η + f η + w 1 w 2 + w 1 w2 ) Let We can defne a PD-structure on J" by settng J" := J Ω T /S,δ Ω T /S,δ Ω 2 T /S,δ δ"(f, w 1, w 2,η) = (δ n (f ),δ n 1 (f )w 1,δ n 1 (f )w 2,δ n 1 (f )η + δ n 2 (f )w 1 w2 ) There are 3 maps q 0, q 1, q 2 of PD-trples (U,T,δ") (U,T,δ) defned by q 0 (f ) := (f,0,0,0) q 1 (f ) := (f,d f,0,0) q 2 (f ) := (f,d f,d f,0) 20

21 There are also three projectons O T O T defned by q 01 (f, w) = (f, w,0,0) q 12 (f, w) = (f,d f, w,d w) q 02 (f, w) = (f, w, w,0) These are also PD-maps. Moreover we have the followng relatons. q 0 = q 01 p 0 q 1 = q 01 p 1 q 1 = q 12 p 0 q 2 = q 12 p 1 q 0 = q 02 p 0 q 2 = q 02 p 1 Step 2. Take F a crystal on Crs(X /S). Then there s a commutatve dagram: q 0 F T q 01 c q 1 F T q 02 c q 2 F T q 12 c whose commutatvty comes from the commutatvty of the followng small dagrams: q 0 F T q 01 F T q 1 F q 02 F T F T q 12 F T q 2 F T Step 3. For s Γ(T,F T ) we have c(p 0 s) = p 1 s (s). Wrte (s) = p 1 s w where s F T and w O T. Then we have (q 12 c) (q 01 c)(q 0 s) = (q 12 c) (q 01 c)(q 01 (p 0 s)) = (q 12 c)(q 01 (p 1 s p 1 s w )) = (q 12 c)(q 12 (p 0 s) q 12 (p 0(s ))q 01 (w )) = q 12 (p 1 s p 1 s w ) q 12 (p 1 s (s ))q 01 (w ) = (q 2 s q 2 s q 12 (w )) q 2 s q 01 (w ) + q 12 ( (s )) q 01 (w ) On the other hand one has (9.1) q 02 c(q 0 s) = q 2 s q 2 s q 02 (w ) (9.2) 21

22 Clearly we have q 01 (w ) + q 12 (w ) q 02 (w ) = d w. Thus takng (9.2)-(9.1) we get q2 s d w q12 ( (s )) q 01 (w ) If one looks nto the formula, t s precsely (s). 10 THE EQUIVALENCE BETWEEN CRYSTALS AND CONNECTIONS (16/01/2018) Stuaton: Let p be a prme number, and let (A, I,γ) be a PD-trple n whch A s a Z (p) - algebra. Let A C be a rng map such that IC = 0 and such that p s nlpotent n C. We wrte X = Spec(C ) and S = Spec(A). Choose a polynomal rng P = A[x ] over A and a surjecton P C of A-algebras wth kernel J := Ker(P C ). Set D := lmd P,γ (J)/p e D P,γ (J) e for the p-adcally completed dvded envelop. Ths rng D comes wth a trple (D, J, γ). We have seen n the exercse that (D/p e D, J/ J p e D, γ) s the PD-envelop of (P/p e P, J/p e J) for e large. On the other hand, we have On the other hand we have Ω D = lmω De /A, γ = lmω D/A, γ /p e Ω D/A, γ e Ω D/A, γ = Ω P/A P D as we have seen before. So Ω D/A, γ s a free D-module on the bass {d x } I, and any element n Ω D can be wrtten unquely as a sum (possbly nfnte) of the form I a d x. Defnton: Let D(n) := lmd P A A P,γ(J(n))/p e D P A A P,γ(J(n)) e where J(n) s the kernel of P A P A A P C. We set J(n) := the dvde power deal of D(n) D(n) e := D(n)/p e D(n) Ω D(n) := lmω D(n)e /A, γ = lmω D(n)/A, γ /p e Ω D(n)/A, γ e Quas-nlpotent connectons: Defnton: We call a par (M, ) a quas-nlpotent connecton of D/A f M s a p- adcally complete D-module, s an ntegrable connecton : M M D Ω D and topologcally quas-nlpotent, that s, f we wrte (M) = θ (m)d x for some operators θ : M M, then we have that for any m M there are only fntely many pars (,k) such that θ k (m) pm. e e 22

23 Theorem: There s an equvalence: quas-coherent crystals on Crs(X /S) quas-nlpotent connectons of D/A Proof: We wll construct two functors n two opposte drectons and then clam wthout proof that that they are nverse to each other. The functor from the left to the rght: Gven a quas-coherent crystal F on Crs(X /S), we consder the sequence of objects (X,T e,δ e ) where T e := Spec(D e ). If we take value of F on each T e, then we get a sequence of D-modules M e satsfyng that M e = M e+1 Z/p e+2 Z Z/p e+1 Z Let M := lm e M e then M s a p-adcally complete module. By 9.1 there s a canoncal connecton on : F F OX /S Ω 1 X /S By takng values on each T e and then takng lmt, we get an ntegrable connecton : M M D Ω 1 D We have to show that ths connecton s topologcally quas-nlpotent. We do the same procedure for D(n) and get a p -adcally complete D(n)-module M(n). Snce F s a crystal, we have somorphsms: M D,p0 D(1) M(1) M D,p1 D(1) Let c denote the arrow whch goes drectly from the left to the rght. Pck m M. Wrte ξ := x 1 1 x. Then we have a unque expresson of c(m 1) n terms of ξ : c(m 1) = K θ K (m) ξ [k ] where K runs over all mult-ndces K = (k ) wth k 0 and k <. Ths s due to the followng Lemma: The projecton P P A A P nduces an somorphsm: f f 1 1 D(n) = lm e D ξ (j ) /p e D ξ (j ) where ξ (j ) := x x 1. Proof of the Lemma: Indeed we have P A A P = P[ξ (j )] 23

24 where ξ (j ) are consdered as ndetermnates, and J(n) s generated by J and those ξ (j ). Then we apply the last tem of 3. End of the Proof Set θ = θ K where K has 1 n the -th spot and 0 elsewhere. Recall the constructon of the canoncal connecton on the crystal F. For each thckenng lke (X,T e,δ e ) we construct a thckenng (X,T e,δ e ) wth two projectons: p, q : T e T e. As F s a crystal, there are somorphsms: p c 0 c 1 F Te FT e q F Te We wrote c for the map whch goes drectly from the left to the rght. For any secton s F Te we defned (s) := q s c(p s) We have a unque map φ: T e Spec(D(1) e) whose compostons wth the two canoncal projectons of D(n) e are the two projectons p and q. Indeed ths follows from the followng Lemma: We have D(n) = D D(n) e = j =0,,n D e j =0,,n n Ĉrs(C /A), where e s supposed to be suffcently large. Proof of the Lemma: If (B C, δ) Ĉrs(X /S), then we have HomĈrs(X /S) (D(n) e,b) ={f Hom A ((P e A A P e, J(n)),(B,Ker(B C ))) f nduces dentty on C } = {f Hom A ((P e, J),(B,Ker(B C ))) f nduces dentty on C } n = n HomĈrs(X /S) (D e,b) and we have the same equaton for D(n). Thus (m) =φ (p1 (m) c(p 0 (m))) =φ (m 1 c(m 1)) =φ (m 1 m 1 θ (m)ξ ) End of the Proof = θ (m)d x As n 9.1 we have the equalty: q 02 c = q 12 c q 01 c Applyng t to m 1 we get θ K " (m) ζ " [k "] = K " K,K θ K (θ K (m)) ζ [k ] ζ [k ] 24

25 n M D,q2 D(2), where ζ = x x 1 ζ = 1 x x ζ " = x x We have ζ " = ζ + ζ and that Comparng the coeffcents we get 1. θ θ j = θ j θ D(2) = q 2 (D) ζ,ζ 2. θ K (m) = ( θ k )(m) If we mod p, then there could only be fntely many θ K (m) survve. Thus there are only fntely many θ k (m) whch do not lne n pm. 11 CRYSTALS AND HPD-STRATIFICATIONS (23/01/2018) Fnsh the proof the equvalence between quas-coherent crystals and quas-nlpotent connectons n quas-coherent modules. Defnton: The conventons and notatons are as n the last lecture. Suppose that we have a commutatve dagram X Y S 0 f S Set D, D Y,γ (J) as before. A quas-coherent HPD-stratfcaton assocated wth ths dagram s a p-adcally complete quas-coherent O D -module M equpped wth an somorphsm φ: p 0 M = p 1 M satsfyng the cocycle condton: p 01 φ p 12 φ = p 02 φ where p 0, p 1 are the two projectons D(1) to D and p 01, p 12, p 02 are the three projectons from D(2) to D(1), where the pullbacks are takng by the completed tensor product. Theorem: Assumptons and conventons beng as above, assume further that f s smooth, then there s an equvalence of categores between the category of quas-coherent crystals on Crs(X /S) and the category of HPD-Stratfcatons wth respect to a dagram as as above. 25

26 12 THE COMPARISON THEOREM (I) (30/01/2018) A bref ntroducton to spectral sequences. Frstly, make the defnton of a spectral sequence. Then construct the spectral sequence assocated wth a complex wth a descendng fltraton. Fnally, ntroduce the two spectral sequences comng from a double complex. A bref Introducton to smplcal objects and cosmplcal objects. Introduce the Dold-Kan theorem. Explan how one gets a cochan complex out of a cosmplcal object. Assumng two key lemma: 1. The p-adc poncaré lemma; 2. For any quas-coherent crystal F, the Čech complex assocated wth the Čech coverng D S s quas somorphc to RΓ(Crs(X /S),F ). Note that snce D(n) s actually the n-th product of D n Ĉrs(X /S), the complex s of the form F (D) F (D(1)) F (D(2)) One can prove the comparson theorem usng the spectral sequence assocated wth the followng double complex: M ˆ D Ω p D(q) 13 THE COMPARISON THEOREM (II) (02/02/2018) 1. Fnsh the proof of the two man lemmas. 2. Introduce the comparson theorem n the non-affne case. Note that n ths case, one can only do t assumng that S s klled by a power of p. Ths sucks!!! 26