Related papers: Prismatic $G$-displays and descent theory

In this paper, we treat $\mathscr{D}$-modules on the basic affine space $G/U$ and their global sections for a semisimple complex algebraic group $G$. Our aim is to prepare basic results about large non-irreducible modules for the branching…

Let p be an odd prime. We show that the classification of p-divisible groups by Breuil windows and the classification of finite flat group schemes of p-power order by Breuil modules hold over any complete regular local ring with perfect…

We introduce and study the moduli stack $\mathcal{Y}$ of Breuil-Kisin modules with $\hat{G}$-structure and descent data, or Breuil-Kisin $(\Gamma,\hat{G})$-torsors for short. Specifically, for a dominant cocharacter $\mu$, we define the…

We construct derived fundamental group schemes for Tate motives over connected smooth schemes over fields. We show that there exists a pro affine derived group scheme over the rationals such that its category of perfect representations…

We study the affine schemes of modules over gentle algebras. We describe the smooth points of these schemes, and we also analyze their irreducible components in detail. Several of our results generalize formerly known results, e.g. by…

We study prismatics sets analogously to simplical sets except that realization involves prisms, i.e., products of simplices rather than just simplices. Particular examples are the prismatic subdivision of a simplicial set S and the…

Let $K$ be a complete discretely valued field of mixed characteristic $(0,p)$ with perfect residue field, and let $E$ be a finite extension of $\mathbf{Q}_p$ contained in $K$. We show that the category of prismatic $F$-crystals on…

For a smooth $p$-adic formal scheme over the ring of integers of a perfectoid field of mixed characteristic $(0,p)$ containing all $p$-power roots of unity, we prove that the prismatic cohomology of a locally finite free prismatic crystal…

Let $R$ be a commutative unital ring. Given a finitely presented affine $R$-group scheme $G$ acting on a separated scheme $X$ of finite type over $R$, we show that there is a prime $p_0$ such that for any $R$-algebra $k$ which is an…

We prove that for any proper smooth formal scheme $\frak X$ over $\mathcal O_K$, where $\mathcal O_K$ is the ring of integers in a complete discretely valued nonarchimedean extension $K$ of $\mathbb Q_p$ with perfect residue field $k$ and…

It is well-known that the coset spaces G(k((z)))/G(k[[z]]), for a reductive group G over a field k, carry the geometric structure of an inductive limit of projective k-schemes. This k-ind-scheme is known as the affine Grassmannian for G.…

We show an equivalence of categories, over general $p$-adic bases, between finite locally $p^n$-torsion commutative group schemes and $\Int/p^n\Int$-modules in perfect $F$-gauges of Tor amplitude $[-1,0]$ with Hodge-Tate weights $0,1$. By…

In this paper, we apply stack theoretic ideas to the classification problem in Dieudonn\'e theory. First, we use crystalline cohomology of classifying stacks to directly reconstruct the classical Dieudonn\'e module of a finite, $p$-power…

Let $k$ be a field of characteristic zero. Fix a smooth algebraic curve $X$ and a split reductive group $G$ over $k$. We show that the Beilinson--Drinfeld affine Grassmannian $\mathrm{Gr}_{G, \mathrm{Ran}(X)}$ is the presheaf colimit of the…

We develop the Tannakian theory of (analytic) prismatic $F$-crystals on a smooth formal scheme $\mathfrak{X}$ over the ring of integers of a discretely valued field with perfect residue field. Our main result gives an equivalence between…

Among all affine, flat, finitely presented group schemes, we focus on those that are pure, this includes all groups which are extensions of a finite locally free group by a group with connected fibres. We prove that over an arbitrary base…

We study affine group schemes over a discrete valuation ring $R$ using two techniques: Neron blowups and Tannakian categories. We employ the theory developed to define and study differential Galois groups of $\mathcal D$-modules on a scheme…

We show that to every p-divisible group over a p-adic ring one can associate a display by crystalline Dieudonne theory. For an appropriate notion of truncated displays, this induces a functor from truncated Barsotti-Tate groups to truncated…

Let $G$ be the group of rational points of a split connected reductive group over a nonarchimedean local field of residue characteristic $p$. Let $I$ be a pro-$p$ Iwahori subgroup of $G$ and let $R$ be a commutative quasi-Frobenius ring. If…

We study G-valued semi-stable Galois deformation rings where G is a reductive group. We develop a theory of Kisin modules with G-structure and use this to identify the connected components of crystalline deformation rings of minuscule…