INTEGRAL p-adi HODGE THEORY, TALK 3 (RATIONAL p-adi HODGE THEORY I, THE PRIMITIVE OMPARISON THEOREM) RAFFAEL SINGER (NOTES BY JAMES NEWTON) 1. Recap We x /Q p complete and algebraically closed, with tilt and x a pseudouniformiser ϖ with ϖ = p. X will be a proper smooth adic space over. Last week: proétale topology on X, with a map of sites ν : X proét X ét. Sheaves on X proét : O + X = ν O + X ét, O X = ν O Xét Ô+ X = lim O+ X /pn, Ô X = Ô+ X [ 1 p ] Ô+ = lim O + φ X /p, Ô+ = Ô+ [ 1 ϖ ] A inf,x = W (Ô+ ) Proposition 1.1. U = lim U i X proét anoid perfectoid, with U i = Spa(R i, R + i ). Set R + = (lim R + i ) (p-adic completion) and R = R+ [ 1 p ]. We have Ô+ X (U) = R+, Ô X (U) = R, Ô + (U) = R +, Ô (U) = R and A inf,x (U) = A inf (R, R + ) = W (R + ). H i (U, F) = 0 for i > 0 and F {Ô+a X, ÔX, Ô+a, Ô, Aa inf,x, O +a X /pn, Ô+a /p n } 2. Remarks on almost mathematics We have Abelian -categories of almost O -modules, or O a -modules (respectively, almost O - or A inf-modules). Roughly these are O -modules up to m-torsion (respectively, O -modules up to m -torsion or A inf -modules up to [m ]-torsion). Key things to know: The functor M M O from O -modules to -modules factors through the map M M a which takes an O -module to the associated O a -module. The functor ( ) a is exact, so it commutes with taking cohomology. Almost isomorphisms become isomorphisms after inverting p (respectively, ϖ or [ϖ]). Denition 2.1. An O -module is almost nitely generated (afg for short) if for all ɛ > 0 there exists r = r(ɛ) and a map with cokernel killed by p ɛ. O r M 1
2 We make a similar denition for O or A inf modules, replacing p by ϖ or [ϖ]. lassication: Fact. For every afg torsion O -module, there exists a unique sequence ɛ 1 ɛ 2 with ɛ i 0 such that M a = O /p ɛ1 O /p ɛ2 We let γ M = (ɛ 1, ɛ 2,...), which we call the sequence of elementary divisors of M. The function M γ M is subadditive in short exact sequences. More precisely, if we have a short exact sequence 0 M 1 M 2 M 3 0 then γ M2 γ M1 +γ M3, where addition is dened component-wise and the inequality means k k ɛ (2) i ɛ (1) i + ɛ (3) i for all k. i=1 i=1 3. The comparison theorem Theorem 3.1 ([1], Theorem 5.6). The map A inf A inf,x induces, for all i, isomorphisms (of almost A inf -modules) H i (X proét, Z p ) Zp A a inf = H i (X proét, A a inf,x). Here, A inf = A inf () is one of Fontaine's period rings. We'll suppress the proét subscript from now on. This is deduced from a mod p (and ϖ) version (Theorem 5.1 in [2]): H i (X, F p ) Fp O +a /p = H i (X, O +a X /p). ompare with the situation over the complex numbers: H i (X, O X ) (almost) never equals H i (X, ), e.g. by Hodge symmetry. [JN: On the other hand, if k is algebraically closed of characteristic p and X/k is proper then H i (X ét, k) = H i (X, O X ), by an argument (using the ArtinSchreier exact sequence) which essentially appears later in the talk] Raael's interpretation: O + X /p is trying to be a locally constant sheaf. Heuristically, if f is a bounded function on X, then for x y < δ we have f(x) f(y) < ɛ so f mod p is constant on some neighbourhood of x. Idea of proof (in more recent language): tilt X to get a diamond X /. Then we should have a RiemannHilbert correspondence between F p -local systems on X (for example, the constant sheaf F p ) and étale φ-modules over ÔX. Unwinding this should give H i (X, F p ) Fp = H i (X, ÔX ) 3.2. The actual proof. We're going to begin by assuming the H i (X, O + X /p) is an afg O /p-module. We will sketch the proof of this at the end of the talk. Now consider M k = H i (X, Ô+ /ϖ k ). We have exact sequences M 1 M k+1 M k and subadditivity of the elementary divisors implies that γ k+1 (k + 1)γ 1. On the other hand, we have φ k : M k O /ϖ k,φ O /ϖ pk = Mpk
3 induced by Frobenius on O + /ϖ k, so γ pk = pγ k. We deduce that we have γ k = kγ 1 for all k. Now it follows from the classication of torsion almost modules by elementary divisors that 0 M a 1 M a k+1 M a k 0 is short exact for all k. By induction, 0 Mj a Mj+k a Mk a 0 is short exact for all j, k. Dene M = lim M k. Taking the inverse limit of the above exact sequences we k get an exact sequence 0 M a ϖk M a Mk a 0. We deduce that M a is at, and also that M is afg (this follows from M 1 being afg by a successive approximation argument). The maps φ k induce an isomorphism φ : M O,φ O = M, and inverting ϖ we obtain an étale φ-module (M, φ) over. Since is algebraically closed the φ-module is trivial: we have M = ( ) r with φ on the right hand side given by the component-wise Frobenius. Since M is afg, the image M of M in M (which is almost isomorphic to M by atness) satises ϖ m (O )r M ϖ m (O )r for some m. Since M is φ- stable, if we apply φ k and take the limit we deduce that M is almost isomorphic to (O )r, and hence the same is true for M. To sum up, we have: Proposition 3.3. (M a, φ) = ((O a )r, φ), where the Frobenius on the right hand side acts component-wise. Recall that M a = lim H i (X, Ô+a /ϖ k ). Next we are going to show that the k natural map H i (X, Ô+a ) M a is an isomorphism. We use the following general X result: Lemma 3.4. Let F k be an inverse system of sheaves on a site T, and suppose we have a basis U for T such that (1) R i lim F k (U) = 0 for all i > 0 and U U. (2) H i (U, F k ) = 0 for all i > 0 and U U. Then R i lim F k = 0 for all i > 0 and H i (U, lim F k ) = 0 for all i > 0 and U U. We are going to apply this lemma with F k = Ô+a /ϖ k and the basis of anoid perfectoids for X proét. Recall that we have F k (U) = R + a /ϖ k (for R + the perfectoid ring associated to the anoid perfectoid U). The transition maps in the inverse system F k (U) are surjective, so this implies condition (1) of the lemma. The second condition was recalled at the start of today. We have Γ lim = lim Γ so we can compare the two Grothendieck spectral sequences computing the derived functors of this functor. The lemma tells us that R i lim F k = 0 for all i > 0, so on the one hand (using the spectral sequence for Γ lim) we get the cohomology groups H i (X, lim F k ) = H i (X, Ô+a ). On the other hand, we get lim H i (X, Ô+a /ϖ k ) note that the higher derived functors k R i lim H i (X, Ô+a /ϖ k ) vanish for i > 0 because the transition maps M k k+1 M k are surjective. Inverting ϖ, we deduce:
4 Proposition 3.5. We have a φ-equivariant isomorphism H i (X, Ô) = M = ( ) r where the φ action on the right hand side is component-wise. Now we use the ArtinSchreier exact sequence: 0 F p ÔX φ 1 ÔX 0. The associated long exact sequence in cohomology, together with the fact the φ 1 is surjective on H i 1 (X, Ô) = ( ) r (since is algebraically closed), implies that we have H i (X, F p ) = H i (X, ÔX )φ=1 and (since M a = (O a )r ) we have H i (X, F p ) Fp O a = H i (X, Ô+a ). 3.6. Proof of the comparison theorem over A inf. We set F k = A a inf,x /pk. onsider the pair of short exact sequences: 0 Z/p Z/p k+1 Z/p k 0 0 F 1 F k+1 F k 0 omparing the long exact sequences in cohomology for both, by induction (using the ve lemma), we lift the isomorphism H i (X, F p ) Fp O a = H i (X, Ô+a ) to an isomorphism H i (X, Z/p k ) Zp A a inf = H i (X, F k ) for all k. We take the limit over k (using the lemma from earlier as before) to get Note that the sheaf Z p on X proét H i (X, Z p ) Zp A a inf = H i (X, A a inf,x). is (by denition) lim Z/p k. 4. Almost finite generation of H i (X, O + X /p) The idea is to rst show that if V V is a strict rational embedding of anoids (rational subsets of X) then the image of H i (V, O + X /p) Hi (V, O + X /p) is afg. Note that these cohomology groups are for the pro-étale topology. Here a strict embedding means "shrink in all directions". More precisely, there exist topological generators f i of O X (V ) such that f i V are topologically nilpotent. For i = 0 this statement is not so hard: if O + X (V ) is topologically generated (as an O -algebra) by nitely many functions f i which become topologically nilpotent in O + X (V ) then the f i are nilpotent in O + X /p(v ). So the image of O + X /p(v ) in O + X /p(v ) is generated (as an O /p-algebra) by nitely many nilpotent functions, and is hence a nitely generated O /p-module. Now properness of X implies that there are two nite covers by anoids, {V i } and } such that for all i V i V i is a strict rational embedding. Now we compare the two spectral sequences computing the cohomology of O + X /p on X using the covers: {V i
5 and We have E p,q 1 = H q (V (p+1), O + X /p) Hp+q (X, O + X /p) (E p,q 1 ) = H q (V (p+1), O + X /p) Hp+q (X, O + X /p) where V (p+1) denotes the disjoint union of the intersections of p+1 distinct members of the cover {V i }. Our strict embeddings induce maps E p,q 1 (E p,q 1 ) which are compatible with the identity map on the abutment of the spectral sequences. Since the image of E p,q 1 in (E p,q 1 ) is afg for all p and q, we can deduce that H i (X, O + X /p) is afg (in fact the argument is a bit elaborate, see Lemma 5.8 in [2]). References [1] Bhatt, B., Morrow, M. and Scholze, P. Integral p-adic Hodge theory, arxiv:1602.03148. [2] Scholze, P. p-adic Hodge theory for rigid-analytic varieties, arxiv:1205.3463