Related papers: Higher displays arising from filtered de Rham-Witt…

We prove that an $F$-crystal $(M,\vph)$ over an algebraically closed field $k$ of characteristic $p>0$ is determined by $(M,\vph)$ mod $p^n$, where $n\ge 1$ depends only on the rank of $M$ and on the greatest Hodge slope of $(M,\vph)$. We…

We define a de Rham cohomology theory for analytic varieties over a valued field $K^\flat$ of equal characteristic $p$ with coefficients in a chosen untilt of the perfection of $K^\flat$ by means of the motivic version of Scholze's tilting…

Let $\mathcal{X}$ be a smooth $p$-adic formal scheme over a mixed characteristic complete discrete valuation ring $\mathcal{O}_{K}$ with perfect residue field. We introduce a general category $\mathcal{M}\mathcal{F}_{[0,…

Lurie's representability theorem gives necessary and sufficient conditions for a functor to be an almost finitely presented derived geometric stack. We establish several variants of Lurie's theorem, making the hypotheses easier to verify…

We state a conjecture relating de Rham cohomology of a smooth rigid analytic variety to its compactly supported pro-\'etale cohomology. We prove the conjecture in the cases where the variety is a Stein curve of dimension one or a Stein…

Given a smooth projective variety $X$ with a smooth nef divisor $D$ and a positive integer $r$, we construct an $I$-function, an explicit slice of Givental's Lagrangian cone, for Gromov--Witten theory of the root stack $X_{D,r}$. As an…

The classical HKR-theorem gives an isomorphism of the n-th Hochschild cohomology of a smooth algebra and the n-th exterior power of its module of K\"ahler differentials. Here we generalize it for simplicial, graded and anticommutative…

Let $k$ be a perfect field of characteristic $p \geq 3$, and let $K$ be a finite totally ramified extension of $K_0 = W(k)[p^{-1}]$. Let $L_0$ be a complete discrete valuation field over $K_0$ whose residue field has a finite $p$-basis, and…

For a flat proper morphism of finite presentation between schemes with almost coherent structural sheaves (in the sense of Faltings), we prove that the higher direct images of quasi-coherent and almost coherent modules are quasi-coherent…

The saturated de Rham-Witt complex, introduced by Bhatt-Lurie-Mathew in arXiv:1805.05501, is a variant of the classical de Rham-Witt complex which is expected to behave better for singular schemes. We provide partial justification for this…

This article is the first one of a series of three articles devoted to direct images of isocrystals: here we consider isocrystals without Frobenius structure; in the second one (resp. the third one), we will introduce a Frobenius structure…

In this article, we develop a systematic cohomological framework for the study of the rigidity of nilpotent Lie foliations with respect to solvable deformations. We introduce the deformation complex associated to a pair of Lie algebras…

We describe the obstruction to decomposing in degrees $\leq p$ the de Rham complex of a smooth variety over a perfect field $k$ of characteristic $p$ that lifts over $W_2(k)$, and show that there exist liftable smooth projective varieties…

Let $\varphi:X\to S$ be a morphism between smooth complex analytic spaces, and let $f=0$ define a free divisor on $S$. We prove that if the deformation space $T^1_{X/S}$ of $\varphi$ is a Cohen-Macaulay $\mathcal{O}_X$-module of codimension…

In an earlier paper we conjectured a relation between the quantum $\mathcal D$-modules of a smooth variety $X$ and the projectivisation of a direct sum of line bundles over it. In this paper we prove the conjecture when $X$ is a complete…

We characterize the relative prismatic cohomology of Bhatt and Scholze by a universal property by endowing it with the additional structure of a ``derived $\delta$-ring". This involves introducing an analogue of prismatic envelopes in the…

A procedure for constructing bivariant theories by means of Grothendieck duality is developed. This produces, in particular, a bivariant theory of Hochschild (co)homology on the category of schemes that are flat, separated and essentially…

We construct, using geometric invariant theory, a quasi-projective Deligne-Mumford stack of stable graded algebras. We also construct a derived enhancement, which classifies twisted bundles of stable graded A-infinity-algebras. The tangent…

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 $\Gamma\subset {\rm SL}_{d+1}(K)$ be a cocompact discrete (torsionfree) subgroup and let…

We show how the Z_p(r)-cohomologies of a smooth projective algebraic variety can be obtained via its de Rham-Witt complex.