Related papers: A note on effective descent for overconvergent iso…

Let $\Cc$ and $\Dd$ be two corings over a ring $A$ and $\Cc\stackrel{\lambda}{\longrightarrow}\Dd$ be a morphism of corings. We investigate the situation when the associated induced ("corestriction of scalars") functor…

Given a complete, cocomplete category $\mathcal C$, we investigate the problem of describing those small categories $I$ such that the diagonal functor $\Delta:\mathcal C\to {\rm Functors}(I,\mathcal C)$ is a Frobenius functor. This…

We present a characterization of effective descent morphisms in the lax comma category $\mathsf{Ord}//X$ when $X$ is a locally complete ordered set, as well as in the antisymmetric setting.

We study the descent behaviour of homotopy-theoretic properties of smooth complex affine surfaces under finite surjective morphisms. We first examine the Eilenberg-MacLane property and show, by means of an explicit counterexample, that it…

We prove that, under suitable assumptions on a category C, the existence of supercompact cardinals implies that every absolute epireflective class of objects of C is a small-orthogonality class. More precisely, if L is a localization…

We show how the tangent functor extends from ordinary smooth maps to "microformal morphisms" (also called "thick morphisms") of supermanifolds. Microformal morphisms generalize ordinary maps and correspond to formal canonical relations…

The analogous statement to Oppenheim conjecture over a local field of positive characteristic is proved. The dynamical argument is most involved in the case of characteristic 3.

Given a liftable smooth proper variety over $\mathbb{F}_p$, we construct the moduli stacks of crystals and isocrystals on it. We show that the former is a formal algebraic stack over $\mathbb{Z}_p$ and the latter is an adic stack -- Artin…

Up to a translation in the language of arithmetic $\D$-modules, we prove a conjecture of Berthelot on the preservation of the overconvergence under the direct image by a smooth proper morphism of varieties over a perfect field of…

For an arbitrary non-archimedean local field we classify reductive group schemes over the corresponding Fargues-Fontaine curve by group schemes over the category of isocrystals. We then classify torsors under such reductive group schemes by…

We characterise proper morphisms of $\infty$-topoi in terms of a relativised notion of compactness: we show that a geometric morphism of $\infty$-topoi is proper if and only if it commutes with colimits indexed by filtered internal…

We give sufficient conditions for effective descent in categories of (generalized) internal multicategories. Two approaches to study effective descent morphisms are pursued. The first one relies on establishing the category of internal…

We show that the weakly \'etale morphisms, used to define the pro-\'etale site of a scheme, are characterized by a lifting property similar to the one which characterizes formally \'etale morphisms. In order to prove this, we prove a…

This article gives an expository account of quasisyntomic descent and the Nygaard filtration in positive characteristic, complemented by several new applications to $p$-adic cohomology theories. The guiding result is a new approach to…

We develop the deformation-obstruction calculus for morphisms of complexes with a fixed lift of the codomain, to derived categories of flat nilpotent deformations of abelian categories. As an application, we give an alternative proof that…

Let $K$ be a mixed characteristic complete discrete valuation field with perfect residue field $k$. Let $X$ be a variety over $k$, $Y$ be an open of $X$, $Y'$ be an open of $Y$ dense in $X$. We extend Kedlaya's full faithfulness as follows…

We prove that many of the results of the LMMP hold for $3$-folds over fields of characteristic $p>5$ which are not necessarily perfect. In particular, the existence of flips, the cone theorem, the contraction theorem for birational extremal…

Any modality in homotopy type theory gives rise to an orthogonal factorization system of which the left class is stable under pullbacks. We show that there is a second orthogonal factorization system associated to any modality, of which the…

The categories of almost modules and almost algebras are introduced as a convenient setting for the development of Faltings' method of almost etale extensions. After some preliminaries of general "almost homological algebra" we construct…

Let $n$ be a non-negative integer. {Motivated by the universal property of the stable category of Frobenius categories, the authors in \cite{bfss} extended the stabilization of Frobenius categories to $n$-Frobenius categories, and called it…