For a smooth and proper curve X over the fraction field K of a discrete valuation ring R, we explain (under very mild hypotheses) how to equip the de Rham cohomology H^1_{dR}(X/K) with a canonical integral structure: i.e. an R-lattice which is functorial in finite (generically etale) K-morphisms of X and which is preserved by the cup-product auto-duality on H^1_{dR}(X/K). Our construction of this lattice uses a certain class of normal proper models of X and relative dualizing sheaves. We show that our lattice naturally contains the lattice furnished by the (truncated) de Rham complex of a regular proper R-model of X and that the index for this inclusion of lattices is a numerical invariant of X (we call it the de Rham conductor). Using work of Bloch and Liu-Saito, we prove that the de Rham conductor of X is bounded above by the Artin conductor, and bounded below by the Efficient conductor. We then study how the position of our canonical lattice inside the de Rham cohomology of X is affected by finite extension of scalars.
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.
For a big class of smooth dagger spaces --- dagger spaces are 'rigid spaces with overconvergent structure sheaf' --- we prove finite dimensionality of de Rham cohomology. This is enough to obtain finiteness of Berthelot's rigid cohomology also in the non-smooth case. We need a careful study of de Rham cohomology in situations of semi-stable reduction.
We prove a Hodge-type decomposition for the de-Rham cohomology of $ p$-adically uniformized varieties by the product of Drinfeld's symmetric spaces. It is based on work of Schneider, Stuhler, Iovita and Spiess on the cohomology of Drinfeld's symmetric space.
We establish a comparison isomorphism between prismatic cohomology and derived de Rham cohomology respecting various structures, such as their Frobenius actions and filtrations. As an application, when $X$ is a proper smooth formal scheme over $\mathcal O_K$ with $K$ being a $p$-adic field, we improve Breuil--Caruso's theory on comparison between torsion crystalline cohomology and torsion étale cohomology.
We define de Rham cohomology groups for rigid spaces over non-archimedean fields of characteristic zero, based on the notion of dagger space. We establish some functorial properties and a finiteness result, and discuss the relation to the rigid cohomology as defined by P. Berthelot.
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.
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 show that Aomoto's $q$-deformation of de Rham cohomology arises as a natural cohomology theory for $Λ$-rings. Moreover, Scholze's $(q-1)$-adic completion of $q$-de Rham cohomology depends only on the Adams operations at each residue characteristic. This gives a fully functorial cohomology theory, including a lift of the Cartier isomorphism, for smooth formal schemes in mixed characteristic equipped with a suitable lift of Frobenius. If we attach $p$-power roots of $q$, the resulting theory is independent even of these lifts of Frobenius, refining a comparison by Bhatt, Morrow and Scholze.
We define an overconvergent version of the Hyodo-Kato complex for semistable varieties $Y$ over perfect fields of positive characteristic, and prove that its hypercohomology tensored with $\mathbb{Q}$ recovers the log-rigid cohomology when $Y$ is quasi-projective. We then describe the monodromy operator using the overconvergent Hyodo-Kato complex. Finally, we show that overconvergent Hyodo-Kato cohomology agrees with log-crystalline cohomology in the projective semistable case.
In their paper which introduced Monsky-Washnitzer cohomology, Monsky and Washnitzer described conditions under which the definition can be adapted to give integral cohomology groups. It seems to be well-known among experts that their construction always gives well-defined integral cohomology groups, but this fact also does not appear to be explicitly written down anywhere. In this paper, we prove that the integral Monsky-Washnitzer cohomology groups are well-defined, for any nonsingular affine variety over a perfect field of characteristic p. We then compare these cohomology groups with overco…
For a natural number $m$, let $\mathcal{S}_m/\mathbb{F}_2$ be the $m$th Suzuki curve. We study the mod $2$ Dieudonné module of $\mathcal{S}_m$, which gives the equivalent information as the Ekedahl-Oort type or the structure of the $2$-torsion group scheme of its Jacobian. We accomplish this by studying the de Rham cohomology of $\mathcal{S}_m$. For all $m$, we determine the structure of the de Rham cohomology as a $2$-modular representation of the $m$th Suzuki group and the structure of a submodule of the mod $2$ Dieudonné module. For $m=1$ and $2$, we determine the complete structure of the …
Milne's correcting factor, which appears in the Zeta-value at $s=n$ of a smooth projective variety $X$ over a finite field $\mathbb{F}_q$, is the Euler characteristic of the derived de Rham cohomology of $X/\mathbb{Z}$ modulo the Hodge filtration $F^n$. In this note, we extend this result to arbitrary separated schemes of finite type over $\mathbb{F}_q$ of dimension at most $d$, provided resolution of singularities for schemes of dimension at most $d$ holds. More precisely, we show that Geisser's generalization of Milne's factor, whenever it is well defined, is the Euler characteristic of the …
We explain how to construct a cohomology theory on the category of separated quasi-compact smooth rigid spaces over $\mathbf{C}_p$ (or more general base fields), taking values in the category of vector bundles on the Fargues-Fontaine curve, which extends (in a suitable sense) Hyodo-Kato cohomology when the rigid space has a semi-stable proper formal model over the ring of integers of a finite extension of $\mathbf{Q}_p$. This cohomology theory factors through the category of rigid analytic motives of Ayoub.
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…
In the late '60s, B. Dwork studied a Frobenius structure compatible with the classical hypergeometric differential equation with parameters $\left(\frac{1}{2},\frac{1}{2} ; 1 \right)$ by analyzing behavior of solutions of the differential equation under Frobenius transformation. Recently, P. Scholze conjectured the existence of $q$-de Rham cohomology groups for any $\mathbb{Z}$-scheme. In this paper, we give a Frobenius structure compatible with the $q$-hypergeometric differential equation with parameters $(q^{\frac12},q^{\frac12};q)$ by showing a $q$-analogue of some results of Dwork. This co…
The author constructs a theory of dagger formal schemes over $R$ and then defines the de Rham cohomology for flat dagger formal schemes $X$ with integral and regular reductions $\bar{X}$ which generalizes the Monsky-Washnitzer cohomology. Finally the author gets Lefschetz' fixed pointed formula for $X$ with certain conditions.
Over any smooth algebraic variety over a $p$-adic local field $k$, we construct the de Rham comparison isomorphisms for the étale cohomology with partial compact support of de Rham $\mathbb Z_p$-local systems, and show that they are compatible with Poincaré duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over $k$. In particular, we prove finiteness of étale cohomology with partial compact support of any $\mathbb Z_p$-local systems, and e…
We introduce differential characters of Drinfeld modules. These are function-field analogues of Buium's p-adic differential characters of elliptic curves and of Manin's differential characters of elliptic curves in differential algebra, both of which have had notable Diophantine applications. We determine the structure of the group of differential characters. This shows the existence of a family of interesting differential modular functions on the moduli of Drinfeld modules. It also leads to a canonical $F$-crystal equipped with a map to the de Rham cohomology of the Drinfeld module. This $F$-…
Let $X$ be a smooth projective $R$-scheme, where $R$ is a smooth $\Z$-algebra. As constructed by Hesselholt, we have the absolute big de Rham-Witt complex $\WΩ^*_X$ of $X$ at our disposal. There is also a relative version $\WΩ^*_{X/R}$ with $\W(R)$-linear differential. In this paper we study the hypercohomology of the relative (big) de Rham-Witt complex after truncation with finite truncation sets $S$. We show that it is a projective $\W_S(R)$-module, provided that the de Rham cohomology is a flat $R$-module. In addition, we establish a Poincaré duality theorem.
In the 1960s, Dwork developed a p-adic cohomology theory of de Rham type for varieties over finite fields, based on a trace formula for the action of a Frobenius operator on certain spaces of p-adic analytic functions. One can consider a purely algebraic analogue of Dwork's theory for varieties over a field of characteristic zero and ask what is the connection between this theory and ordinary de Rham cohomology. N. Katz showed that Dwork cohomology coincides with the primitive part of de Rham cohomology for smooth projective hypersurfaces, but the exact relationship for varieties of higher cod…
Let X be Drinfeld's half space over a p-adic field K. The de Rham cohomology of X was first computed by Schneider and Stuhler. Afterwards there were given different proofs by Alon, de Shalit, Iovita and Spiess. This paper presents yet another approach for the determination of these invariants by analysing the de Rham complex of X from the viewpoint of recent results by the author.
We show that hypergeometric differential equations, unitary and Gauss-Manin connections give rise to de Rham cohomology sheaves which do not admit a Bloch-Ogus resolution. The latter is in contrast to Panin's theorem asserting that corresponding étale cohomology sheaves do fulfill Bloch-Ogus theory.
Building up on work of Epstein, May and Drury, we define and investigate the mod $p$ Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$. We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which $p$ is not a torsion prime, and (special) orthogonal groups when $p=2$.
For any hyperelliptic curve X, we give an explicit basis of the first de-Rham cohomology of X in terms of Čech cohomology. We use this to produce a family of curves in characteristic p>2 for which the Hodge-de-Rham short exact sequence does not split equivariantly; this generalises a result of Hortsch. Further, we use our basis to show that the hyperelliptic involution acts on the first de-Rham cohomology by multiplication by -1, i.e., acts as the identity when p=2.
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 …
We prove that the relative log de Rham cohomology groups of a projective semistable log smooth degeneration admit a natural \textit{limiting} mixed Hodge structure. More precisely, we construct a family of increasing filtrations and a family of nilpotent endomorphisms on the relative log de Rham cohomology groups and show that they satisfy a part of good properties of a nilpotnet orbit in several variables.
We give a definition and study the basic properties of the irregular Hodge filtration on the exponentially twisted de Rham cohomology of a smooth quasi-projective complex variety.
In this short paper, we use Tannakian reconstruction techniques to prove a result that explains how to reconstruct the stacky approach to de Rham cohomology from the classical theory algebraic de Rham cohomology via an application of the adjoint functor theorem.
In order to study $p$-adic étale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect field of characteristic $p>0$, we introduce new $p$-primary torsion sheaves. It is a modification of the logarithmic de Rham-Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$. Then we establish a perfect duality between cohomology groups of the logarithmic de Rham-Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite field, the duality can be used to study wild ramification class field theory for the open su…
In this paper we show that Atkin and Swinnerton-Dyer type of congruences hold for weakly modular forms (modular forms that are permitted to have poles at cusps). Unlike the case of original congruences for cusp forms, these congruences are nontrivial even for congruence subgroups. On the way we provide an explicit interpretation of the de Rham cohomology groups associated to modular forms in terms of "differentials of the second kind". As an example, we consider the space of cusp forms of weight 3 on a certain genus zero quotient of Fermat curve X^N+Y^N=Z^N. We show that the Galois representat…
In this follow-up paper we show that smooth Hodge-proper stacks over $\mathcal O_K$ are $\mathbb Q_p$-locally acyclic: namely the natural map between étale $\mathbb Q_p$-cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the $\mathbb Q_p$-case of general conjectures made in our previous work. As a corollary, we get that if a smooth Artin stack over $K$ has a smooth Hodge-proper model over $\mathcal O_K$, its $\mathbb Q_p$-étale cohomology is a crystalline Galois representation. We then also establish a truncated version of the above results in more gener…
The Cartier isomorphism allows a nice description of the Bockstein spectral sequence of the de Rham complex over the integers. It is used to compute the integral de Rham cohomology of affine spaces. ----- On decrit la suite spectrale de Bockstein issue du complexe de de Rham sur les entiers. L'isomorphisme de Cartier intervient comme un endomorphisme du complexe de de Rham modulo p qui identifie les pages successives de la suite spectrale de Bockstein. On en deduit la cohomologie de de Rham entiere des espaces affines.
This is a review article discussing the de Rham cohomology of period domains of Hodge structures. We explain it as the de Rham cohomology of differentiable stacks as of a moduli space. We also discuss the cohomology of the partial toroidal compactification of these domains using known formulas on cohomology or Chow rings of toric structures. The text is expository and we have tried to connect some existing ideas that probably their relations not processed in the literature of Hodge theory. We state the significance of ideas as they naturally could be related, probably with not serious mathemat…
We construct a functorial decomposition of de Rham cohomology sheaves, called weight decomposition, for smooth analytic spaces over non-Archimedean fields embeddable into $\mathbf{C}_p$, which generalizes a construction of Berkovich and solves a question raised by him. We then investigate complexes of real tropical differential forms and currents introduced by Chambert-Loir and Ducros, by establishing a relation with the weight decomposition and defining tropical cycle maps with values in the corresponding Dolbeault cohomology. As an application, we show that algebraic cycles that are cohomolo…
We show that Illusie's derived de Rham cohomology (Hodge-completed) coincides with Hartshorne's algebraic de Rham cohomology for a finite type map of noetherian schemes in characteristic 0; the case of lci morphisms was a result of Illusie. In particular, the E_1-differentials in the derived Hodge-to-de Rham spectral sequence for singular varieties are often non-zero. Another consequence is a completely elementary description of Hartshorne's algebraic de Rham cohomology: it is computed by the completed Amitsur complex for any variety in characteristic 0.
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}}…
We study equivariant de Rham discriminants associated to arithmetic varieties which support a tame action by a finite group; we form these discriminants by endowing the de Rham cohomology with pairings arising from duality theory. Such equivariant discriminants are shown to break up naturally into a metric part and a signature part. In a previous paper we described the equivariant Arakelov discriminants, obtained by endowing the equivariant determinant of de Rham cohomology with various metrics. In this paper we study the associated equivariant signature information; in particular, we show tha…
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…
Let $k$ be a perfect field of characteristic $p > 0$, $W_n = W_n(k)$. For separated $k$-schemes of finite type, we explain how rigid cohomology with compact supports can be computed as the cohomology of certain de Rham-Witt complexes with coefficients. This result generalizes the classical comparison theorem of Bloch-Illusie for proper and smooth schemes. In the proof, the key step is an extension of the Bloch-Illusie theorem to the case of cohomologies relative to $W_n$ with coefficients in a crystal that is only supposed to be flat over $W_n$.
Let X be a smooth complex algebraic variety with the Zariski topology, and let Y be the underlying complex manifold with the complex topology. Grothendieck's algebraic de Rham theorem asserts that the singular cohomology of Y with complex coefficients can be computed from the complex of sheaves of algebraic differential forms on X. This article gives an elementary proof of Grothendieck's algebraic de Rham theorem, elementary in the sense that we use only tools from standard textbooks as well as Serre's FAC and GAGA papers.
We prove an effective bound for the degrees of generators of the algebraic de Rham cohomology of smooth affine hypersurfaces. In particular, we show that the de Rham cohomology H_dR^p(X) of a smooth hypersurface X of degree d in C^n can be generated by differential forms of degree d^O(pn). This result is relevant for the algorithmic computation of the cohomology, but is also motivated by questions in the theory of ordinary differential equations related to the infinitesimal Hilbert 16th problem.
We define the twisted de Rham cohomology and show how to use it to define the notion of an integral of the form $\int g(x) e^{f(x)}dx$ over an arbitrary ring. We discuss also a definition of a family of integrals and some properties of the homological definition of integral. We show how to use the twisted de Rham cohomology in order to define the Frobenius map on the p-adic cohomology. Finally, we consider two-dimensional topological quantum field theories with general coefficients.
