Related papers: De Rham prismatic crystals over $\mathcal{O}_K$

Let $\mathcal{O}_K$ be a mixed characteristic complete discrete valuation ring with perfect residue field. We study $\mathbb{B}_\mathrm{dR}^+$-crystals on the (log-) prismatic site of $\mathcal{O}_K$, which are crystals defined over the de…

Let $\mathcal{O}_K$ be a mixed characteristic complete DVR with perfect residue field $k$ and fraction field $K$. It is a celebrated result of Berthelot and Ogus that for a smooth proper formal scheme $X/\mathcal{O}_K$ there exists a…

Let $Y/S$ be a $p$-completely smooth morphism of $p$-torsion free $p$-adic formal schemes endowed with a Frobenius lift, and let $\overline Y/\overline S$ denote its reduction modulo $p$. We show that the category of crystals on the…

Let $X=\mathrm{Spf}(\mathcal{O}_K)$. We classify perfect complexes of $n$-truncated prismatic crystals on the prismatic site of $X$ when $n\leq 1+\frac{p-1}{e}$ by studying perfect complexes on the $n$-truncated prismatization of $X$, which…

We study Hodge-Tate crystals on the absolute (log-) prismatic site of $\mathcal{O}_K$, where $\mathcal{O}_K$ is a mixed characteristic complete discrete valuation ring with perfect residue field. We first classify Hodge-Tate crystals by…

We prove that a Hodge--Tate prismatic crystal on (O_K)_{\Prism} is uniquely determined by a topologically "nilpotent" operator. Using this operator, we construct a C_p-representation of G_K from a Hodge--Tate crystal in an explicit way. We…

We systematically study relative and absolute ${\Delta}_{\mathrm{dR}}^+$-crystals on the (log-) prismatic site of a smooth (resp.~ semi-stable) formal scheme. Using explicit computation of stratifications, we classify (local) relative…

We construct the crystalline fundamental group of a semi-stable variety over a field of positive characteristic using the log De Rham-Witt complex and Navarro-Aznar's derived Thom-Whitney functor. This approach gives a relatively direct…

Given a perfect field $k$ of characteristic $p>0$, a smooth proper $k$-scheme $Y$, a crystal $E$ on $Y$ relative to $W(k)$ and a finite group $G$ acting on $Y$ and $E$, we show that, viewed as virtual $k[G]$-module, the reduction modulo $p$…

We prove that algebraic de Rham cohomology as a functor defined on smooth $\mathbb{F}_p$-algebras is formally \'etale in a precise sense. This result shows that given de Rham cohomology, one automatically obtains the theory of crystalline…

We consider the crystalline realization of Deligne's 1-motives in positive characteristics and prove a comparison theorem with the De Rham realization of liftings to zero characteristic. We then show that one dimensional crystalline…

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 $K$ be a mixed characteristic complete discrete valuation field with perfect residue field, and let $K_\infty/K$ be a Kummer tower extension by adjoining a compatible system of $p$-power roots of a chosen uniformizer. We use this Kummer…

We develop the formalism of derived divided power algebras, and revisit the theory of derived De Rham and derived crystalline cohomology in this framework. We characterize derived De Rham cohomology of a derived commutative algebra $A$ over…

We define, for each quasi-syntomic ring $R$ (in the sense of Bhatt-Morrow-Scholze), a category $\mathrm{DM}^{\rm adm}(R)$ of \textit{admissible prismatic Dieudonn\'e crystals over $R$} and a natural functor from $p$-divisible groups over…

Let $Y$ be a locally complete intersection over $\mathcal{O}_K$ containing a $p$-power root of unity $\zeta_p$. We classify the derived category of prismatic crystals on the absolute prismatic site of $Y$ by studying quasi-coherent…

Building on ideas of Berthelot, we develop a crystalline cohomology formalism over divided power rings $(A, I_0, \eta)$ for any ring $A$, allowing $\mathbf{Z}$-flat $A$. For a smooth $A$-scheme $Y$ and a closed subscheme $X$ of $Y$ for…

We show that the Poincar\'e lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincar\'e lemma for transversal crystals of level…

Constructing a quantum description of crystals from scattering experiments is of paramount importance to explain their macroscopic properties and to evaluate the pertinence of theoretical ab-initio models. While reconstruction methods of…

We give a new construction of $(\varphi, \hat G)$-modules using the theory of prisms developed by Bhatt and Scholze. As an application, we give a new proof about the equivalence between the category of prismatic $F$-crystals in finite…