Related papers: Purity Results on $F$-crystals

We present a comprehensive review of the emerging crystal structure determination method Parameter Space Concept (PSC), which solves and refines either partial or complete crystal structures by mapping each experimental or theoretical…

Spatially resolved neutron diffraction as a function of crystal depth in Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_{3}$ reveals the presence of a distinct near-surface region. A dramatic change in both the lattice constant and the Bragg peak intensity as…

We show that $F$-purity deforms in local $\mathbb{Q}$-Gorenstein rings of prime characteristic $p>0$. Furthermore, we show that $F$-purity is $\mathfrak{m}$-adically stable in local Cohen-Macaulay $\mathbb{Q}$-Gorenstein rings.

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…

Following Johnsen and Verdure (2013), we can associate to any linear code $C$ an abstract simplicial complex and in turn, a Stanley-Reisner ring $R_C$. The ring $R_C$ is a standard graded algebra over a field and its projective dimension is…

We introduce an analogue to Quasi-$F$-splittings, Quasi-$F$-purity, which is definable over rings that are not necessarily $F$-finite. We show that this property is equivalent to being Quasi-$F$-split in the complete local and $F$-finite…

For a $p$-divisible group $G$ over a smooth projective variety $X$ over $k$, where $k$ is a field finitely generated over a perfect field of characteristic $p$, we show that the formal group $R^i f_{\fppf*} G$ is isogenous to a…

Let A be a symmetric monoidal closed exact category. This category is a natural framework to define the notions of purity and flatness. We show that an object F in A is flat if and only if any conflation ending in F is pure. Furthermore, we…

Grothendieck's cohomological purity predicts that the cohomology of a scheme is insensitive to removing a closed subscheme of sufficiently high codimension. In this article, we establish a form of flat cohomological purity over arbitrary…

This article is the second one of a series of three articles devoted to direct images of isocrystals: here we consider convergent isocrystals with Frobenius structure. Let V be a complete discrete valuation ring, with residue field k = V/m…

In this note, we study F-purity of pairs, and show (as is the case with log canonicity) that F-purity is preserved at the F-pure threshold. We also characterize when F-purity is equivalent to sharp F-purity, an alternate notion of purity…

We show that quasi-$F$-pure but not $F$-pure isolated quasi-homogeneous hypersurface singularities necessarily have $F$-pure threshold $1 - \frac{1}{p}$. This extends work of Bhatt and Singh beyond the Calabi-Yau case. We also classify the…

We compute the $F$-pure threshold of some non-principal ideals which satisfy a geometric generic condition about their Newton polyhedron. We also contribute some evidence in favor of the conjectured equality between the $F$-pure threshold…

We introduce the notion of geometric purity in rigidly-compactly generated tt-categories by considering exact triangles that are pure at each tt-stalk. We develop a systematic study of this concept, including examples and applications. In…

The F-pure threshold is a positive characteristic numerical invariant, analogous to the log canonical threshold in characteristic zero. We compute the F-pure threshold of the affine cone over a Calabi-Yau hypersurface, and relate it to the…

A collection $\mathcal{C}$ of $k$-element subsets of $\{1,2,\ldots,m\}$ is weakly separated if for each $I, J \in \mathcal{C}$, when the integers $1,2,\ldots,m$ are arranged around in a circle, there is a chord separating $I \backslash J$…

We introduce the notion of completed $F$-crystals on the absolute prismatic site of a smooth $p$-adic formal scheme. We define a functor from the category of completed prismatic $F$-crystals to that of crystalline \'etale…

In this paper we generalize minimal $p$-divisible groups defined by Oort to $F$-crystal over an algebraically closed field of positive characteristic. We prove a structural theorem and give an explicit formula of the Frobenius endomorphism…

Let X be a smooth variety over an algebraically closed field of characteristic p > 0, Z a smooth divisor, and j : U = X\Z --> X the natural inclusion. An axiomatizing of the properties of a V -filtration on a unit F-crystal is proposed and…

Let $(A, I)$ be a bounded prism, and $X$ be a smooth $p$-adic formal scheme over $\Spf(A/I)$. We consider the notion of crystals on Bhatt--Scholze's prismatic site $(X/A)_{\prism}$ of $X$ relative to $A$. We prove that if $X$ is proper over…