Related papers: On retract rationality for finite connected group …

Let $G$ be the linear algebraic group $SL_3$ over a field $k$ of characteristic two. Let $A$ be a finitely generated commutative $k$-algebra on which $G$ acts rationally by $k$-algebra automorphisms. We show that the full cohomology ring…

A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C*-algebras.…

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…

Working over an algebraically closed field of characteristic p > 3, we calculate the orbit closures in the Witt algebra W under the action of its automorphism group G. We also outline how the same techniques can be used to determine…

Let G be a connected reductive linear algebraic group. The aim of this note is to settle a question of J-P. Serre concerning the behaviour of his notion of G-complete reducibility under separable field extensions. Part of our proof relies…

Let $E$ be a number field and $X$ a smooth geometrically connected variety defined over a characteristic $p$ finite field. Given an $n$-dimensional pure $E$-compatible system of semisimple $\lambda$-adic representations of the \'etale…

We establish a McKay correspondence for finite and linearly reductive subgroup schemes of $\mathrm{SL}_2$ in positive characteristic. As an application, we obtain a McKay correspondence for all rational double point singularities in…

We give a geometric characterization of finite rational groups. In particular, we prove that a finite group is rational if and only if there exists a finite geometry $\Gamma$ of type $I$ and action of $G$ on $\Gamma$ as a group of…

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $W_{2}(\mathbb{F}_{q})$ be the ring of Witt vectors of length two over $\mathbb{F}_{q}$. We prove that for any reductive group scheme $\mathbb{G}$ over $\mathbb{Z}$ such…

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group $G$, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a…

In this paper, we establish general categorical frameworks that extend Loewy's classification scheme for finite-dimensional real irreducible representations of groups and Borel--Tits' criterion for the existence of rational forms of…

Let $G$ be a connected reductive algebraic group and $B$ be a Borel subgroup defined over an algebraically closed field of characteristic $p>0$. In this paper, the authors study the existence of generic $G$-cohomology and its stability with…

We apply the Fixed Point Theorem for the actions of finite groups on Bruhat-Tits buildings and their products to establish two results concerning the groups of points of reductive algebraic groups over polynomial rings in one variable,…

A smooth, proper, retract rational variety over a field $k$ is known to be $\mathbb{A}^1$-connected. We improve on this result, in the case when $k$ is infinite, showing that such varieties are naively $\mathbb{A}^1$-connected.

In the present paper we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding…

Given a recollement of three proper dg algebras over a noetherian commutative ring, e.g. three algebras which are finitely generated over the base ring, which extends one step downwards, it is shown that there is a short exact sequence of…

We analyze the structure of a large class of connected algebraic rings over an algebraically closed field of positive characteristic using Greenberg's perfectization functor. We then give applications to rigidity problems for…

A retract variety is defined as a class of algebras closed under isomorphisms, retracts and products. Let a principal retract variety be generated by one algebra and a set-principal retract variety be generated by some set of algebras. It…

We develop a sequential-topological study of rational points of schemes of finite type over local rings typical in higher dimensional number theory and algebraic geometry. These rings are certain types of multidimensional complete fields…

Let $G= \exp(\g)$ be a connected, simply connected nilpotent Lie group. We show that for every $G$-invariant smooth sub-manifold $M$ of $g^*$, there exists an open relatively compact subset $\mathcal{M}$ of $M$ such that for any smooth…