Building on ideas of Berthelot, we develop a crystalline cohomology formalism over divided power rings $(A, I_0, η)$ for any ring $A$, allowing $\mathbf{Z}$-flat $A$. For a smooth $A$-scheme $Y$ and a closed subscheme $X$ of $Y$ for which $η$ extends to $I_0 \mathscr{O}_X$, a (quasi-coherent) crystal $\mathscr{F}$ on $(X/A)_{\rm{cris}}$ is equivalent to a specific type of module with integrable $A$-linear connection over a certain completion $D_{Y,η}(X)^{\wedge}$ (called "pd-adic") of the divided power envelope $D_{Y,η}(X)$ of $Y$ along $X$ (with divided power structure $δ$)
Our main result, building on ideas of Bhatt and de Jong for $\mathbf{Z}/(p^e)$-schemes (where pd-adic completion has no effect), is a natural isomorphism between ${\rm{R}}Γ((X/A)_{\rm{cris}}, \mathscr{F})$ and the Zariski hypercohomology of the pd-adically completed de Rham complex $\mathscr{F} \widehat{\otimes} \widehatΩ^*_{D_{Y,η}(X)^{\wedge}/A,δ}$ arising from the module with integrable connection over $D_{Y,η}(X)^{\wedge}$ associated to $\mathscr{F}$. By a variant of the same methods, we obtain a representative of the complex $\mathscr{F} \widehat{\otimes} \widehatΩ^*_{D_{Y,η}(X)^{\wedge}/A,δ}$ in the derived category of sheaves of $A$-modules on $X$ in terms of a Čech-Alexander construction.
When $\mathscr{F}=\mathscr{O}_{X/A}$, our comparison theorem implies that in the derived category of sheaves of $A$-modules on $X$, the pd-adic completion of $Ω^*_{D_{Y,η}(X)/A,δ}$ functorially depends only on $X$. Over $\mathbf{Q}$-algebras $A$, so pd-adic completion becomes ideal-adic completion, this recovers a result of Hartshorne.
We prove the finiteness of crystalline cohomology of higher level. An important ingredient is a "higher de Rham complex" and a kind of Poincaré lemma for it.
For a smooth and proper scheme over an artinian local ring with ordinary reduction over the perfect residue field we prove - under some general assumptions - that the relative de Rham-Witt spectral sequence degenerates and the relative crystalline cohomology, equipped with its display structure arising from the Nygaard complexes, has a Hodge-Witt decomposition into a direct sum of (suitably Tate-Twisted) multiplicative displays. As examples our main results include the cases of abelian schemes, complete intersections, varieties of K3 type and some Calabi-Yau $n$-folds.
The goal of this short paper is to give a slightly different perspective on the comparison between crystalline cohomology and de Rham cohomology. Most notably, we reprove Berthelot's comparison result without using pd-stratifications, linearisations, and pd-differential operators.
We provide a simple approach for the crystalline comparison of Ainf-cohomology, and reprove the comparison between crystalline and p-adic etale cohomology for formal schemes in the case of good reduction.
Le Stum and Quirós proved the formal Poincaré lemma in crystalline cohomology of higher level using the jet complex, and applied it to give a de Rham interpretation of this cohomology. In this article, we prove the logarithmic version of the formal Poincaré lemma modulo p. Provided that each term of the log. jet complex is locally free, it gives the logarithmic version of the de Rham interpretation of the crystalline cohomology of higher level.
This note discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties tends to be quite large. In particular, any singular projective variety with at worst ordinary double points has infinitely generated crystalline cohomology in at least two cohomological degrees. These calculations rely critically on comparisons between crystalline and derived de Rham cohomology.
In this article, we prove the comparison theorem between the relative log de Rham-Witt cohomology and the relative log crystalline cohomology for a log smooth saturated morphism of fs log schemes satisfying certain condition. Our result covers the case where the base fs log scheme is etale locally log smooth over a scheme with trivial log structure or the case where the base fs log scheme is hollow, and so it generalizes the previously known results of Matsuue. In Appendix, we prove that our relative log de Rham-Witt complex and our comparison map are compatible with those of Hyodo-Kato.
We prove that rigid cohomology can be computed as the cohomology of a site analogous to the crystalline site. Berthelot designed rigid cohomology as a common generalization of crystalline and Monsky-Washnitzer cohomology. Unfortunately, unlike the former, the functoriality of the theory is not built-in. We defined somewhere else the "overconvergent site" which is functorially attached to an algebraic variety and proved that the category of modules of finite presentation on this ringed site is equivalent to the category of over- convergent isocrystals on the variety. We show here that their coh…
Motivated by applications in point counting algorithms using p-adic cohomology, we give an explicit description of integral lattices in rigid cohomology spaces that p-adically approximate logarithmic crystalline cohomology modules. These lattices are expressed in terms of the global sections of the twisted logarithmic de Rham complex. We prove the main theorem for smooth proper hypersurfaces with a smooth hyperplane section, then deduce the result for the quotient of such a pair in weighted projective space. We show how these results may be used to reduce the necessary p-adic precision with wh…
For a perfect field $k$ of characteristic $p>0$ and a smooth and proper formal scheme $\mathscr{X}$ over the ring of integers of a finite and totally ramified extension $K$ of $W(k)[1/p]$, we propose a cohomological construction of the Breuil-Kisin modules attached to the $p$-adic étale cohomology $H^i_{\mathrm{ét}}(\mathscr{X}_{\overline{K}},\mathbf{Z}_p)$. We then prove that our proposal works when $p>2$, $i < p-1$, and the crystalline cohomology of the special fiber of $\mathscr{X}$ is torsion-free in degrees $i$ and $i+1$.
Let $K$ be a field of characteristic zero complete for a discrete valuation, with perfect residue field of characteristic $p>0$, and let $K^+$ be the valuation ring of $K$. We relate the log-crystalline cohomology of the special fibre of certain affine $K^+$-schemes $X=\text{Spec}(R)$ with semi-stable reduction to the Galois cohomology of the fundamental group of the geometric generic fibre $π_1(X_{\bar{K}})$ with coefficients in a Fontaine ring constructed from $R$. This is based on Faltings' theory of almost étale extensions.
For a strictly semistable log scheme $Y$ over a perfect field $k$ of characteristic $p$ we investigate the canonical Čech spectral sequence $(C)_T$ abutting to the Hyodo-Kato (log crystalline) cohomology $H_{crys}^*(Y/T)_{\mathbb{Q}}$ of $Y$ and beginning with the log convergent cohomology of its various component intersections $Y^i$. We compare the filtration on $H_{crys}^*(Y/T)_{\mathbb{Q}}$ arising from $(C)_T$ with the monodromy operator $N$ on $H_{crys}^*(Y/T)_{\mathbb{Q}}$. We also express $N$ through residue maps and study relations with singular cohomology. If $Y$ lifts to a proper str…
Let $X\to Y^0$ be an abelian prime-to-$p$ Galois covering of smooth schemes over a perfect field $k$ of characteristic $p>0$. Let $Y$ be a smooth compactification of $Y^0$ such that $Y-Y^0$ is a normal crossings divisor on $Y$. We describe a logarithmic $F$-crystal on $Y$ whose rational crystalline cohomology is the rigid cohomology of $X$, in particular provides a natural $W[F]$-lattice inside the latter; here $W$ is the Witt vector ring of $k$. If a finite group $G$ acts compatibly on $X$, $Y^0$ and $Y$ then our construction is $G$-equivariant. As an example we apply it to Deligne-Lusztig va…
Given a smooth, proper family of varieties in characteristic $p>0$, and a cycle $z$ on a fibre of the family, we formulate a Variational Tate Conjecture characterising, in terms of the crystalline cycle class of $z$, whether $z$ extends cohomologically to the entire family. This is a characteristic $p$ analogue of Grothendieck's Variational Hodge Conjecture. We prove the conjecture for divisors, and an infinitesimal variant of the conjecture for cycles of higher codimension. This can be used to reduce the $\ell$-adic Tate conjecture for divisors over finite fields to the case of surfaces.
This paper studies the derived de Rham cohomology of F_p and p-adic schemes, and is inspired by Beilinson's recent work. Generalising work of Illusie, we construct a natural isomorphism between derived de Rham cohomology and crystalline cohomology for lci maps of such schemes, as well logarithmic variants. These comparisons give derived de Rham descriptions of the usual period rings and related maps in p-adic Hodge theory. Placing these ideas in the skeleton of Beilinson's construction leads to a new proof of Fontaine's crystalline conjecture and Fontaine-Jannsen's semistable conjecture.
Given a proper, smooth (formal) scheme over the ring of integers of $\mathbb C_p$, we prove that if the crystalline cohomology of its special fibre is torsion-free then the $p$-adic étale cohomology of its generic fibre is also torsion-free. In this announcement we sketch the proof, which relies on the construction of a new cohomology theory interpolating between crystalline and étale cohomology. Further details and results, including a comparison isomorphism, will be presented in the full forthcoming article.
For a smooth proper scheme or formal scheme over an unramified, complete DVR of mixed characteristics we prove a comparison isomorphism relating etale cohomology of the generic fiber with values in a crystalline etale sheaf to the crystalline cohomology of its special fiber with values in the associated F-isocrystal.
We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for étale cohomology with nontrivial coefficients, as well as in the relative setting, i.e. for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the pro-étale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the pro-étale site. Moreover, we need to prove the Poincaré lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another in…
We prove that algebraic de Rham cohomology as a functor defined on smooth $\mathbb{F}_p$-algebras is formally étale in a precise sense. This result shows that given de Rham cohomology, one automatically obtains the theory of crystalline cohomology as its unique functorial deformation. To prove this, we define and study the notion of a pointed $\mathbb{G}_a^{\mathrm{perf}}$-module and its refinement which we call a quasi-ideal in $\mathbb{G}_a^{\mathrm{perf}}$ -- following Drinfeld's terminology. Our main constructions show that there is a way to "unwind" any pointed $\mathbb{G}_a^{\text{perf}}…
In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of $\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus' prediction for some families of abelian varieties. These families include abelian varieties that have both prime dimension and nontrivial endomorphism ring. The proof is based on …
In 2003, Kedlaya gave an algorithm to compute the zeta function associated to a hyperelliptic curve over a finite field, by computing the rigid cohomology of the curve. Edixhoven remarked that it is actually possible to compute the crystalline cohomology of the curve, which is a lattice in the rigid cohomology. Following a method of Wach, we first explain how to use this lattice to compute the $(φ,Γ)$-module associated to an hyperelliptic curve. We also explain an alternative way to get the $(φ,Γ)$-module mod $p$ that relies on the Deligne-Illusie morphism.
We survey crystalline cohomology, crystals, and formal group laws with an emphasis on geometry. We apply these concepts to K3 surfaces, and especially to supersingular K3 surfaces. In particular, we discuss stratifications of the moduli space of polarized K3 surfaces in positive characteristic, Ogus' crystalline Torelli theorem for supersingular K3 surfaces, the Tate conjecture, and the unirationality of K3 surfaces.
We discuss the notion of a q-PD-envelope considered by Bhatt and Scholze in their recent theory of q-crystalline cohomology and explain the relation with our notion of a divided polynomial twisted algebra. Together with an interpretation of crystals on the q-crystalline site, that we call q-crystals, as modules endowed with some kind of stratification, it allows us to associate a module on the ring of twisted differential operators to any q-crystal. For simplicity, we explain here only the one dimensional case.
Let k be an algebraically closed field of characteristic p>0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates, in the case of abelian schemes, the étale cohomology with $\dbZ_p$ coefficients to the crystalline cohomology with integral coefficients, in the more general context of p-divisible groups endowed with {\it arbitrary} families of crystalline tensors over a finite, discrete valuation ring extension of W(k). This extends a result of Faltings in [Fa2]. As a main new tool we construct global deformations of p-divisible groups…
We construct Λ-adic de Rham and crystalline analogues of Hida's ordinary Λ-adic etale cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of \Q_p, we prove appropriate finiteness and control theorems in each case. We then employ integral p-adic Hodge theory to prove Λ-adic comparison isomorphisms between our cohomologies and Hida's etale cohomology. As applications of our work, we provide a "cohomological" construction of the family of (ϕ,Γ)-modules attached to Hida's ordinary Λ-adic etale cohomology by Dee, and we give a new and purely…
We compute a basis for the p-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give p-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology.
We show that classical Chern classes from higher ($p$-adic) $K$-theory to syntomic cohomology extend to logarithmic syntomic cohomology. These Chern classes are compatible -- in a suitable sense -- with addition, products, and $λ$-operations. They are also compatible with the canonical Gysin sequences and, via period maps, with logarithmic étale Chern classes. Moreover, they induce logarithmic crystalline Chern classes. This uses as a critical new ingredient the recent comparison of syntomic cohomology with $p$-adic nearby cycles and $p$-adic motivic cohomology.
We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of C_p. It takes values in a mixed-characteristic analogue of Dieudonne modules, 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 crystalline, de Rham and etale cohomology, 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 cate…
For a proper, smooth scheme $X$ over a $p$-adic field $K$, we show that any proper, flat, semistable $\mathcal{O}_K$-model $\mathcal{X}$ of $X$ whose logarithmic de Rham cohomology is torsion free determines the same $\mathcal{O}_K$-lattice inside $H^i_{dR}(X/K)$ and, moreover, that this lattice is functorial in $X$. For this, we extend the results of Bhatt--Morrow--Scholze on the construction and the analysis of an $A_{inf}$-valued cohomology theory of $p$-adic formal, proper, smooth $\mathcal{O}_{\overline{K}}$-schemes $\mathfrak{X}$ to the semistable case. The relation of the $A_{inf}$-coho…
Universiteit van Amsterdam（阿姆斯特丹大学）；Imperial College London（帝国理工学院）
In this article we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, stating that a rational (logarithmic) line bundle on the special fibre of a semistable scheme over $k [\![ t ]\!]$ lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham-Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter example to show that the variational Tate conject…
We prove that, for adic étale motives over $\mathbb{C}_p$, the vector bundles on the Fargues-Fontaine curve arising from their Hyodo-Kato cohomology coincide with their de Rham-Fargues-Fontaine cohomologies, where the latter provides an overconvergent refinement of crystalline vector bundles, albeit constructed on the generic fiber. This equivalence is established in the setting of symmetric monoidal $\infty$-categories and respects the full motivic structure. Furthermore, we enrich both realizations with Galois actions, yielding $G_{\breve{\mathbb{Q}}_{p}}$-equivariant solid quasi-coherent sh…
The notion of a $p$-adic superspace is introduced and used to give a transparent construction of the Frobenius map on $p$-adic cohomology of a smooth projective variety over $\zp$ (the ring of $p$-adic integers), as well as an alternative construction of the crystalline cohomology of a smooth projective variety over $\fp$ (finite field with $p$ elements).
In this paper, we show that the infinitesimal Torelli theorem implies the existence of deformations of automorphisms. In the first part, we use Hodge theory and deformation theory to study the deformations of automorphisms of complex projective manifolds. In the second part, we use crystalline cohomology to explore the p-adic analogues of the first part, which generalizes a result of Berthelot and Ogus. The study of the deformations of automorphisms also provides criterions characterizing when the action of the automorphism group of a variety on its cohomology is faithful.
Replacing symmetric powers by divided powers and working over Witt vectors instead of ground fields, I generalize Kawamatas T^1-lifting theorem to characteristic p>0. Combined with the work of Deligne-Illusie on degeneration of the Hodge-de Rham spectral sequences, this gives unobstructedness for certain Calabi-Yau varieties with free crystalline cohomology modules.
Let $K$ be a finite extension of ${\mathbb Q}_p$ and let $X$ be Drinfel'd's symmetric space of dimension $d$ over $K$. Let $Γ\subset {\rm SL}_{d+1}(K)$ be a cocompact discrete (torsionfree) subgroup and let ${X}_Γ=Γ\backslash {X}$, a smooth projective ${K}$-variety. In this paper we investigate the de Rham and log crystalline (log convergent) cohomology of local systems on $X_Γ$ arising from $K[Γ]$-modules. (I) We prove the monodromy weight conjecture in this context. To do so we work out, for a general strictly semistable proper scheme of pure relative dimension $d$ over a cdvr of mixed chara…
