Related papers: Revisiting derived crystalline cohomology

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…

We develop prismatic and syntomic cohomology relative to a $\delta$-ring. This simultaneously generalizes Bhatt and Scholze's absolute and relative prismatic cohomology and shows that the latter, which was defined relative to a prism, is in…

Consider a bounded prism $(A,I)$ and a bounded quasi-l.c.i algebra $R$ over $\overline{A}$. In this paper, for any prism $S/A$ with a surjection $S\to R$ such that $\widehat{\mathbb L}_{\overline{S}/\overline{A}}$ is a $p$-completely flat…

We introduce the notion of a prism, which may be regarded as a "deperfection" of the notion of a perfectoid ring. Using prisms, we attach a ringed site -- the prismatic site -- to a $p$-adic formal scheme. The resulting cohomology theory…

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…

We investigate the properties of pure derived categories of module categories, and show that pure derived categories share many nice properties of classical derived categories. In particular, we show that bounded pure derived categories can…

In this note, we study an integral analogue of animated $\delta$-rings: Animated $\lambda$-rings. We define animated $\lambda$-rings in terms of animated rings equipped with a structure of coherently compatible Frobenius lifts and show that…

The geometric models for the module category and derived category of any gentle algebra were introduced to realize the objects in module category and derived category by permissible curves and admissible curves respectively. The present…

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…

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…

The goal of this paper is to study the absolute prismatic cohomology of $p$-adic formal schemes. We do so by recasting the notion of a prismatic crystal on $\mathrm{Spf}(\mathbf{Z}_p)$ in terms of quasicoherent sheaves on a geometric object…

This work focuses on approximation and generation for the derived category of complexes with quasi-coherent cohomology on algebraic stacks. Our methods establish that approximation by compact objects descends along covers that are…

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…

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…

We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for \'etale cohomology with nontrivial coefficients, as well as in the relative setting, i.e.…

In this article, a new construction of derived equivalences is given. It relates different endomorphism rings and more generally cohomological endomorphism rings - including higher extensions - of objects in triangulated categories. These…

For an algebraically closed non-archimedean extension $C/\mathbb{Q}_p$, we define a Tannakian category of $p$-adic Hodge structures over $C$ that is a local, $p$-adic analog of the global, archimedean category of $\mathbb{Q}$-Hodge…

These are notes on derived algebraic geometry in the context of animated rings. More precisely, we recall the proof of To\"en-Vaqui\'e that the derived stack of perfect complexes is locally geometric in the language of $\infty$-categories.…

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…

We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold…