THE MAIN CONSTRUCTION I

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1 THE MAIN CONSTRUCTION I Contents 1. Introduction/ Notation 1 2. Motivation: the Cartier isomorphism 2 3. Definition of the map 3 4. Small algebras and almost purity 3 5. Cohomology of Z d p 5 6. Décalage 6 7. Computing the map 7 References 8 1. Introduction/ Notation Let O be the ring of integers of a perfectoid field K Q p. Let m denote the maximal ideal of O, and suppose K contains µ p. Let O{1} denote the Breuil-Kisin-Fargues twist O{1} := L O Zp [ 1]. Let X be a smooth p-adic formal scheme, let X denote its generic fibre. Let Ô+ X denote the p-adic completion of the integral structure sheaf ν O + Xét. Let ν : X proét X Zar denote the nearby cycles functor defined previously. For I a sheaf of ideals on X Zar, let η I denote the decalage functor defined last week. We define Ω X := Lη ζp 1(Rν Ô + X ) This will be the mod ξ reduction of the A inf cohomology that will be introduced in the next talk. In this talk we want to explain section 8 of [BMS16], relating the cohomology of Ω X to the completed de Rham complex Ω X := R lim Ω (X/p n ) (O/p n ). The idea of the relation is to consider the transitivity triangle introduced in Kevin s talk (1) LO Zp Ô+ X LÔ+ LÔ+ X Zp O. X (viewed as a distinguished triangle in D(X Zar )), and compare it to the distinguished triangle in D(X proét ) (2) ν LO Zp Ô+ X ν LÔ+ X Zp 1 ν LÔ+ X O.

2 2 THE MAIN CONSTRUCTION I 2. Motivation: the Cartier isomorphism Before we start proving the mixed characteristic Cartier isomorphism, we mention what the actual Cartier isomorphism (even though this doesn t really seem to enter into the proof of Theorem 2). The Cartier isomorphism was originally constructed by Cartier in [Car57]. We follow the exposition of Katz in [Kat70]. Let S be a scheme of characteristic p, and π : Y S a smooth morphism. As usual, let F S and F Y denote their respective absolute Frobenius morphisms, let Y (p) denote the pull-back of Y by absolute Frobenius on S, and relative Frobenius F Y S : Y Y (p) be the factorisation of F S through Y (p). F Y S Y Y (p) Y π S π (p) F S S π Theorem 1 (Cartier Isomorphism). There are unique isomorphism of O Y (p)-modules such that C 1 : Ω i Y (p) S C 1 (1) = 1, C 1 (ω η) = C 1 (ω) C 1 (η), C 1 (d(w 1 (f))) = [f p 1 df].ut H i (F Ω Y S ) Here is a rough sketch of the idea of the proof. We define C 1 on Ω 1 Y (p) S by sending d(f s) to the cohomology class of sf p 1 df. We first need to check that this is well-defined. This is perhaps slightly easier to check by viewing C 1 as a (π (p) ) 1 O S -linear derivation O Y (p) H 1 (F Ω Y S ), or more explicitly an O S -linear derivation O Y (p) O Y OY,F O S H 1 (F Ω Y S ). The computation boils down to the identity (f + g) p 1 d(f + g) f p 1 df g p 1 dg = d( 0<i<p (p 1)! i!(p i)! f i g p i ). Having defined the morphism, we may now check that it s an isomorphism locally. This involves choosing local co-ordinates and writing everything down very explicitly (details may be found in Katz s article). In our setting there will be a similar strategy: once we ve managed to define the morphism, we focus on an explicit local computation.

3 THE MAIN CONSTRUCTION I 3 Let Ω i,cont X O X O denote lim 3. Definition of the map Ωi (X/p n R) (O/p n O). In this section we define a map from to R1 ν Ô + X {1}. The goal of this talk is to show that this factors through a map to H 1 ( Ω X {1}), and that the latter map is an isomorphism. To define the map, we consider the triangles (1) and (2) above. In Kevin s talk we also saw that for any morphism R S of perfectoid rings, L R S = 0, and hence ν LÔ+ X O = 0. We obtain a well-defined morphism in D(X Zar ): L X Zp Rν (ν LÔ+ X Zp) Rν (ν L O Zp Ô+ X ). Recall we defined O{1} to be the O-module L O Zp [ 1]. So we can re-write this as L X Zp Rν (Ô+ X ){1}[1]. Now we take cohomology, to get a morphism H 0 ( L X Zp ) H 1 (Rν (Ô+ X ){1}). Via the transitivity triangle (1) we have an isomorphism and that latter is isomorphic to X O morphism H 0 ( L X Zp ) H 0 ( L X O ), by our assumptions on X. Hence we have a X O H1 (Rν (Ô+ X ){1}). In this talk we will focus on the problem of proving the following result. Theorem 2. The map X O H1 (Rν (Ô+ X ){1}). factors through H 1 ( Ω X ), and induces an isomorphism X O H1 ( Ω X ){1}. More generally, for all i there are isomorphisms Ω i,cont X O Hi ( Ω X ){i}. As explained in [BMS16], the cohomology of Ω X has a natural algebra structure induced from multiplication on Ô+ X, and (as we ll see) Zariski locally the maps for i > 1 are induced from the map when i = Small algebras and almost purity Definition 1. A framing of a formally smooth R-algebra is an étale map Spf(R) Ĝd m = SpfO T ±1 1,... T ±1 d. R is said to be small if it admits a framing. Given a framing define R to be the completion of R O T ±1 : X Ĝd m, ±1/p 1,...T ±1 d O T 1,... T ±1/p d.

4 4 THE MAIN CONSTRUCTION I To get our outlined strategy to work, we need to know that sufficiently small opens are small. Theorem 3. X is (Zariski) covered by small opens. Proof. See Kedlaya [Ked05], or Olsson [Ols09] Reducing to group cohomology. The computation of Rν Ô + X follows Faltings method of proving p-adic comparison theorems. We reduce to a local computation, and we show that locally, étale cohomology can be identified with group cohomology of the fundamental group. locally, the group cohomology of the fundamental group can be replaced with group cohomology of certain Z d p-quotients of the first The first property (étale cohomology being the same as group cohomology of the fundamental group) is equivalent to a space being an étale K(π, 1). Theorem 4. For any connected affinoid X we have an almost isomorphism RΓ(X profét, Ô+ X /p) RΓ(X proét, F p ) O + X. The pro-finite étale cohomology groups are naturally identified with group cohomology, via the identification of the pro-finite étale site with the category of pro-finite π 1 -sets Almost purity and how to use it. In this section we recall Faltings method for proving p-adic comparison theorems using his almost purity theorem [Fal02]. Theorem 5 (Almost purity). Let R,K S,K be a finite étale morphism, and let S denote the normalisation of O[T 1/p 1,..., T 1/p d ] in S,K. Then R S is amost étale. One consequence of almost purity is the following Galois cohomological calculation. Lemma 1. The map is an almost quasi-isomorphism. RΓ(Z p (1) d, R ) RΓ(G R, R) We ll briefly summmarise how to prove this. Via the Hochschild-Serre spectral sequence, it s enough to prove that RΓ(Σ, R) is almost zero. By the following lemma, it s enough to prove that, for all n, is almost zero. RΓ(Σ, R/p n ) Lemma 2. Let (C n ) be an inverse system of complexes of O-modules. Suppose that, for all n, C n is almost zero. Then lim C is almost zero.

5 THE MAIN CONSTRUCTION I 5 This lemma allows us to reduce to proving that RΓ(Σ, R) is almost zero. Since R is a discrete Σ-module, this translates to proving that, for all finite étale Galois covers R[1/p] S with normalisation S 0 S and Galois group G, the cohomology groups H i (G, S 0 ) are almost zero. Lemma 3. Let G be a finite group, A a ring, f : A B a ring homomorphism, ρ : G Aut A (B) a group homomorphism. Let M be a B-module with a semi-linear action of G. Then for all i > 0, H i (G, M) is annihilated by tr(b) := g G ρ(g)(b). Hence, to get H (G, S ) to be almost zero, we need to know that enough elements arise as the trace of elements of S. This is achieved by the almost purity theorem. Lemma 4. Let R S be an almost étale Galois cover. Then the trace map is almost surjective. 5. Cohomology of Z d p Let γ denote a generator of Z p. We recall the following facts: (resolution of Z p -modules): If M is a module with a discrete action of Z p, then RΓ(Z p, M) [M γ 1 M]. (Kunneth formula for group cohomology) If G 1, G 2 are profinite groups, and M 1, M 2 are modules with discrete actions of G 1 and G 2 respectively, then RΓ(G 1 G 2, M 1 M 2 ) RΓ(G 1, M 1 ) L Z RΓ(G 2, M 2 ). (Mittag-Leffler property) If (M i, f ij ) is an inverse system such that, for all i, f ij (M j ) stabilises for j 0, then R lim(m i ) lim M i. Given a profinite group G and a module M = lim M i with a continuous action of G, for all i we have short exact sequences 0 R 1 lim n H i 1 (G, M n ) H i (G, lim n M n ) lim n H i (G, M n ) Koszul complexes and group cohomology of Z p. Given a ring R, and f 1,..., f n in R, the Koszul complex, denoted K R (f 1,..., f n ), is the CDGA whose underlying graded algebra is given by the exterior algebra with differential given by d : i (R n ) i 1 (R n ) (R n ), e 1... e i j ( 1) j+1 f j e 1... e j 1 e j+1... e i More generally, given f 1,..., f d in R, and an R[x 1,..., x d ]-module M, we define K M (f 1,..., f d ) := Hom R[x1...,x d ](K R(f 1,..., f d ), M)

6 6 THE MAIN CONSTRUCTION I where the Koszul complex of f 1,..., f d is viewed as a complex of R[x 1,..., x d ] modules via the map x i f i. Note that, by the lemma above, the cohomology of Z p is computed by Koszul complex of K (γ 1). For us, Koszul complexes will show up because they compute tensor products of complexes. Lemma 5. Given f 1,..., f r, there is a natural quasi-isomorphism of complexes K R(f 1,..., f r ) K (f 1,..., f r 1 ) K R(f r ). Hence we deduce the following Lemma. Lemma 6. Let M be a continuous Z n p there is a quasi-isomorphism module of the form M 1... M n. Then RΓ(Z n p, M) K M (γ 1,..., γ 1) We now describe RΓ(Z p (1) d, R ) in the case R = O T 1 ±1,..., R ±1. We first compute d RΓ(Z p (1), R /p n ) sufficiently explicitly to show that its cohomology satisfies the Mittag-Leffler property, and hence that the inverse limit is quasi-isomorphic to RΓ(Z p (1), R ). Since R /p n i1,...i d Z[1/p]OT i T i d d, it s enough to compute RΓ(Z p(1) d, OT i T i d d ) for each tuple. Finally, since this is a discrete Z p (1) d -module, we may apply Kunneth to reduce to the case d = 1.We find that H (Z p (1), R ) R for r = 0 and R (lots of ζ p 1 torsion) for r = Décalage Finally, we have to add Lη ζp 1. As was explained in Matthew Morrow s talk last week, we have { L ηζp 1 [RΓ(Z p (1), O.T i )] [O.T i 0 OT i ] Define Ω R := Lη ζ 1RΓ(Z p (1) d, R ). Proposition 1. Let R be a formally smooth O-algebra. Then the map is a quasi-isomorphism. Ω R Ω R Recall from last time the following lemma regarding Lη f. Lemma 7. Let A be a ring, I an ideal and f a non-zero-divisor. Let M N be a morphism in D(A), such that the kernel and cokernel of H i (M) H i (N) are killed by f for all i, and such that H i (C) and H i (C)/f have trivial I-torsion. Then Lη g is an isomorphism. We apply this Lemma when I = m and f = ζ p 1. By flat base change and Kunneth, we may reduce to the case R = Ĝm. Recall that since R is perfectoid we have L R Z p L Z/p n Z Hence the Proposition follows from the fact that this map is a quasi-isomorphism, together with our explicit computation of RΓ(Z p (1) d, R ).

7 THE MAIN CONSTRUCTION I 7 The other point is to show that the map X O { 1} H1 (Rν Ô + X ) factors through H 1 ( Ω X ). We return to the setting of a small affine R = Spf(R). By flatness of O T 1 ±1,..., T ±1 d R, we may reduce to the case R = O T 1,..., T d. By the Kunneth formula, we may further reduce to the case R = O T ±1. As above we recall the result about the decalage functor that we need. Lemma 8. A, f as usual, C D a morphism in D(A) such that H i (C) = 0 for i > 1, H i (D) = 0 for i < 0, H 0 (D)[f] = 0. A morphism β : C Lη f D exists iff the map H 1 (C L A A/f) H 1 (D L A A/f) is zero, and this occurs iff the map factors through fh 1 (D). We apply this result to H 1 (C) H 1 (D) L X Zp [ 1]{ 1} Rν Ô + X (when X = Ĝm), and we see that it is enough to show that X Z p H 1 (Z p (1), R ) factors through (ζ p 1)H 1 (Z p (1), R ). Equivalently, it is enough to prove that the corresponding map X Z p Hom(Z p (1), T p (Ω 1 R O )) factors through (ζ p 1) Hom(Z p (1), T p (Ω 1 R O )). 7. Computing the map In this section we will show that dt/t gets sent to the homomorphism sending ζ p n to dζ p n. Note that this is in (ζ p 1) Hom(µ p, Ω O Zp ). To show this it is enough to show that p n ζ p 1 dζ p n = 0, which can be see from the fact that f n(ζ p n)dζ p n = 0. where f n is the p n -th cyclotomic polynomial. By the previous lemma, this gives us the existence of a homomorphism { 1} H 1 ( Ω X ). Moreover, since dlog generates (ζ p 1) Hom(µ p, Ω O Zp ), this shows that the map is an isomorphism. We need to compute the map We have an isomorphism Ω R Zp Z/p n Z H (RΓ(Z p (1), L R Z p L Z/p n Z)). L R Z p L Z/p n Z L (Z/p n Z),

8 8 THE MAIN CONSTRUCTION I and the right hand side is given by the complex concentrated in degrees 1 and 0: p n Proposition 2. Let X = Ĝm. Let dlog Hom(Z p (1), T p (Ω 1 O Z p(1) )) be the homomorphism defined in Darya s talk. Then the map L X Zp H 1 (Z p (1), sends dt/t to dlog 1. First consider the morphism L X Zp L Z/p n Z RΓ(Z p (1), L R Z p L Z Z/p n Z). By definition, the map sends the class of dt/t to the class of dt/t in RΓ(Z p (1), L R Z p L Z Z/p n Z), so we just have to work out what that is. First, we have a resolution of L R Z p L Z p Z/p n Z given by the total complex of γ 1 p n γ 1 p n One can check that dt/t dζ p n/ζ p n, where ζ p n µ p n. is the image of the generator γ in References [BMS16] Bhargav Bhatt, Matthew Morrow, and Peter Scholze. Integral p-adic Hodge theory. arxiv preprint arxiv: , [Car57] Pierre Cartier. Une nouvelle opération sur les formes différentielles. Comptes Rendus., 244(4): , [Fal02] Gerd Faltings. Almost étale extensions. Cohomologies p-adiques et applications arithmétiques, pages , [Kat70] Nicholas M Katz. Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin. Publications Mathématiques de l Institut des Hautes Études Scientifiques, 39(1): , [Ked05] Kiran Kedlaya. More étale covers of affine spaces in positive characteristic. Journal of Algebraic Geometry, 14(1): , [Ols09] Martin C Olsson. On Faltings method of almost étale extensions