Related papers: Loop group schemes and Abhyankar's lemma

We define the notion of loop torsors under certain group schemes defined over the localization of a regular henselian ring A at a strict normal crossing divisor D. We provide a Galois cohomological criterion for classifying those torsors.…

In the article Categorical Construction of Schemes, arXiv:2511.03433 we gave a natural definition of ordinary schemes based on the fact that the localization of a ring in a maximal ideal is a local representation of the corresponding…

Parahoric group schemes are certain possibly non-reductive, smooth, affine integral models of reductive group schemes defined over a henselian discretely valued field $K$ whose residue field is perfect. We show that any such group scheme…

Let G be a reductive affine group scheme defined over a semilocal ring k. Assume that either G is semisimple or k is normal and noetherian. We show that G has a finite k-subgroup S such that the natural map H^1(R, S) --> H^1(R, G) is…

Let $A$ be the coordinate ring of a projective smooth curve over a finite field minus a closed point. For a nontrivial ideal $I \subset A$, Drinfeld defined the notion of structure of level $I$ on a Drinfeld module. We extend this to that…

In arXiv:2511.04191 we constructed schemes of objects in small categories which contained a set of basepoints with local representing (localizing) objects. Here we prove that the category $\cat{Rings}$ of associative rings with unit has a…

Given a definably amenable approximate subgroup $A$ of a (local) group in some first-order structure, there is a type-definable subgroup $H$ normalised by $A$ and contained in $A^4$ such that every definable superset of $H$ has positive…

In this paper, we prove that, when an abelian scheme has semi-abelian degeneration along normal crossings divisor in a regular base scheme, a finite flat group scheme of torsion points of the abelian scheme degenerates to a log finite group…

The paper was motivated by a question of Vilonen, and the main results have been used by Mirkovic and Vilonen to give a geometric interpretation of the dual group (as a Chevalley group over Z) of a reductive group. We define a…

In the current paper we study the groups, whose subnormal abelian subgroups are normal. We obtained a quite detailed description of such hyperabelian groups with a periodic Baer radical. The description of hyperabelian Lie algebras, whose…

We generalize a well known periodicity lemma from the case of free groups to the case of acylindrically hyperbolic groups. This generalization will be used later to describe solutions of certain equations in acylindrically hyperbolic groups…

We explicitly describe the divisor class groups and semidualizing modules for ladder determinantal rings with coefficients in an arbitrary normal domain for arbitrary ladders, not necessarily connected, and all sizes of minors.

This arXived paper has two independant parts, that are improved and corrected versions of different parts of a single paper once named "On equations in relatively hyperbolic groups". The first part is entitled "Existential questions in…

On the transversals of a subgroup of a group, using the binary operation of the group, structural mappings are defined. Based on these mappings, the notion of the hypergroup over the group is introduced, which generalizes the notion of the…

Oshima's Lemma describes the orbits of parabolic subgroups of irreducible finite Weyl groups on crystallographic root systems. This note generalises that result to all root systems of finite Coxeter groups, and provides a self contained…

We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; i.e., over one variable function fields over complete discretely valued fields. We provide conditions on the group and the…

It is well known that the classical diagram lemmas of homological algebra for abelian groups can be generalized to non-abelian group-like structures, such as groups, rings, algebras, loops, etc. In this paper we establish such a…

We introduce the abelian class group C_{ab}(G) of a reductive group scheme G over a ring A of arithmetical interest and study some of its properties. In particular, we show that if the fraction field of A is a global field without real…

In the present paper, we study a purely inseparable counterpart of Abhyankar's conjecture for the affine line in positive characteristic, and prove its validity for all the finite local non-abelian simple group schemes in characteristic…

We give a detailed description of the torsors that correspond to multiloop algebras. These algebras are twisted forms of simple Lie algebras extended over Laurent polynomial rings. They play a crucial role in the construction of Extended…