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INTEGRAL P-ADIC HODGE THEORY INTEGRAL p-ADIC HODGE THEORY BHARGAV BHATT, MATTHEW MORROW, AND PETER SCHOLZE Abstract. We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of Cp. It takes values in a mixed-characteristic analogue of Dieudonn´emod- ules, which was previously defined by Fargues as a version of Breuil{Kisin modules. Notably, this cohomology theory specializes to all other known p-adic cohomology theories, such as crys- talline, de Rham and ´etalecohomology, which allows us to prove strong integral comparison theorems. The construction of the cohomology theory relies on Faltings's almost purity theorem, along with a certain functor Lη on the derived category, defined previously by Berthelot{Ogus. On affine pieces, our cohomology theory admits a relation to the theory of de Rham{Witt complexes of Langer{Zink, and can be computed as a q-deformation of de Rham cohomology. Contents 1. Introduction 1 2. Some examples 13 3. Algebraic preliminaries on perfectoid rings 19 4. Breuil{Kisin{Fargues modules 31 5. Rational p-adic Hodge theory 45 6. The Lη-operator 48 7. Koszul complexes 55 8. The complex ΩeX 60 9. The complex AΩX 68 10. The relative de Rham{Witt complex 79 11. The comparison with de Rham{Witt complexes 85 12. The comparison with crystalline cohomology over Acrys 95 13. Rational p-adic Hodge theory, revisited 103 14. Proof of main theorems 112 References 117 1. Introduction This paper deals with the following question: as an algebraic variety degenerates from char- acteristic 0 to characteristic p, how does its cohomology degenerate? 1.1. Background. To explain the meaning and the history of the above question, let us fix some notation. Let K be a finite extension of Qp, and let OK ⊂ K be its ring of integers. Let X be 1 a proper smooth scheme over OK ; in other words, we consider only the case of good reduction in this paper, although we expect our methods to generalize substantially. Let k be the residue field of O, and let k¯ and K¯ be algebraic closures. There are many different cohomology theories one can associate to this situation. The best understood theory is `-adic cohomology for ` 6= p. In that case, we have ´etalecohomology groups i i H´et(XK¯ ; Z`) and H´et(Xk¯; Z`), and proper smooth base change theorems in ´etalecohomology Date: February 9, 2016. 1We use the fractal letter for consistency with the main body of the paper, where X will be allowed to be a formal scheme. 1 2 BHARGAV BHATT, MATTHEW MORROW, AND PETER SCHOLZE imply that these cohomology groups are canonically isomorphic (once one fixes a specialization of geometric points), i ∼ i H´et(XK¯ ; Z`) = H´et(Xk¯; Z`) : In particular, the action of the absolute Galois group GK of K on the left side factors through the action of the absolute Galois group Gk of the residue field k on the right side; i.e., the action of GK is unramified. Grothendieck raised the question of understanding what happens in the case ` = p. In that i case, one still has well-behaved ´etalecohomology groups H´et(XK¯ ; Zp) of the generic fibre, but the ´etalecohomology groups of the special fibre are usually too small; for example, if i = 1, they capture at best half of the ´etalecohomology of the generic fibre. A related phenomenon i is that the action of GK on H´et(XK¯ ; Zp) is much more interesting than in the `-adic case; in particular, it is usually not unramified. As a replacement for the ´etalecohomology groups of the i special fibre, Grothendieck defined the crystalline cohomology groups Hcrys(Xk=W (k)). These are Dieudonn´emodules, i.e. finitely generated W (k)-modules equipped with a Frobenius operator ' i i which is invertible up to a power of p. However, H´et(XK¯ ; Zp) and Hcrys(Xk=W (k)) are cohomology theories of very different sorts: the first is a variant of singular cohomology, whereas the second is a variant of de Rham cohomology. Over the complex numbers C, integration of differential forms along cycles and the Poincar´elemma give a comparison between the two, but algebraically the two objects are quite unrelated. Grothendieck's question of the mysterious functor was to i i understand the relationship between H´et(XK¯ ; Zp) and Hcrys(Xk=W (k)), and ideally describe each in terms of the other. Fontaine obtained the conjectural answer to this question, using his period rings, after inverting 1 p, in [27]. Notably, he defined a W (k)[ p ]-algebra Bcrys whose definition will be recalled below, which comes equipped with actions of a Frobenius ' and of GK , and he conjectured the existence of a natural '; GK -equivariant isomorphism i ∼ i 1 H (X ¯ ; p) ⊗ Bcrys = H (Xk=W (k))[ ] ⊗ 1 Bcrys : ´et K Q Qp crys p W (k)[ p ] The existence of such an isomorphism was proved by Tsuji, [52], after previous work by Fontaine{ i 1 Messing, [29], Bloch{Kato, [10], and Faltings, [22]. This allows one to recover Hcrys(Xk=W (k))[ p ] i from H´et(XK¯ ; Qp) by the formula i 1 i 1 GK ∼ i GK H (Xk=W (k))[ ] = (H (Xk=W (k))[ ] ⊗ 1 Bcrys) = (H (X ¯ ; p) ⊗ Bcrys) : crys p crys p W (k)[ p ] ´et K Q Qp i i 1 Conversely, Fontaine showed that one can recover H´et(XK¯ ; Qp) from Hcrys(Xk=W (k))[ p ] together i i with the Hodge filtration coming from the identificaton Hcrys(Xk=W (k)) ⊗W (k) K = HdR(XK ). Unfortunately, when p is small or K=Qp is ramified, the integral structure is not preserved by these isomorphisms; only when ie < p−1, where e is the ramification index of K=Qp, most of the story works integrally, roughly using the integral version Acrys of Bcrys instead, as for example in work of Caruso, [15]; cf. also work of Faltings, [23], in the case i < p − 1 with e arbitrary. 1.2. Results. In this paper, we make no restriction of the sort mentioned above: very ramified extensions and large cohomological degrees are allowed throughout. Our first main theorem is the following; it is formulated in terms of formal schemes for wider applicability, and it implies that the torsion in the crystalline cohomology is an upper bound for the torsion in the ´etale cohomology. Theorem 1.1. Let X be a proper smooth formal scheme over OK , where OK is the ring of integers in a complete discretely valued nonarchimedean extension K of Qp with perfect residue field k. Let C be a completed algebraic closure of K, and write XC for the (geometric) rigid- analytic generic fibre of X. Fix some i ≥ 0. (i) There is a comparison isomorphism i ∼ i H´et(XC ; Zp) ⊗Zp Bcrys = Hcrys(Xk=W (k)) ⊗W (k) Bcrys ; compatible with the Galois and Frobenius actions, and the filtration. In particular, i H´et(XC ; Qp) is a crystalline Galois representation. INTEGRAL p-ADIC HODGE THEORY 3 (ii) For all n ≥ 0, we have the inequality length (Hi (X =W (k)) =pn) ≥ length (Hi (X ; ) =pn) : W (k) crys k tor Zp ´et C Zp tor i 2 i In particular, if Hcrys(Xk=W (k)) is p-torsion-free , then so is H´et(XC ; Zp). i i+1 (iii) Assume that Hcrys(Xk=W (k)) and Hcrys(Xk=W (k)) are p-torsion-free. Then one can i i recover Hcrys(Xk=W (k)) with its '-action from H´et(XC ; Zp) with its GK -action. Part (i) is the analogue of Fontaine's conjecture for proper smooth formal schemes over OK . In fact, our methods work more generally: we directly prove the comparison isomorphism in (i) and the inequalities in (ii) (as well as a variant of (iii), formulated below) for any proper smooth formal scheme that is merely defined over OC . For formal schemes over discretely valued base fields, part (i) has also been proved recently by Colmez{Niziol, [16] (in the more general case of semistable reduction), and Tan{Tong, [51] (in the absolutely unramified case, building on previous work of Andreatta{Iovita, [3]). Intuitively, part (ii) says the following. If one starts with a proper smooth variety over the complex numbers C, then the comparison between de Rham and singular (co)homology says that any class in singular homology gives an obstruction to integrating differential forms: the integral over the corresponding cycle has to be zero. However, for torsion classes, this is not an actual obstruction: a multiple of the integral, and thus the integral itself, is always zero. Nevertheless, part (ii) implies the following inequality: i i (1) dimk HdR(Xk) ≥ dimFp H´et(XC ; Fp): In other words, p-torsion classes in singular homology still produce non-zero obstructions to integrating differential forms on any (good) reduction modulo p of the variety. The relation is however much more indirect, as there is no analogue of \integrating a differential form against a cycle" in the p-adic world. Remark 1.2. Theorem 1.1 (ii) \explains" certain pathologies in algebraic geometry in character- istic p. For example, it was observed (by classification and direct calculation, see [35, Corollaire 7.3.4 (a)]) that for any Enriques surface Sk over a perfect field k of characteristic 2, the group 1 HdR(Sk) is never 0, contrary to what happens in any other characteristic. Granting the fact that any such Sk lifts to characteristic 0 (which is known, see [20, 43]), this phenomenon is explained 1 ∼ by Theorem 1.1 (ii): an Enriques surface SC over C has H´et(SC ; F2) = F2 6= 0 as the fundamental 1 group is Z=2, so the inequality (1) above forces HdR(Sk) 6= 0.
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