arXiv subjects

Egor Voronetsky

Publications and source records attributed to Egor Voronetsky.

At least 19 recordsLinked to original sources

arXiv2026

Nilpotency of locally isotropic $ \mathrm{K}_1 $-functor

We show that the $ \mathrm{K}_1 $-functor modeled on locally isotropic reductive groups is hypoabelian if the base ring has finite Bass--Serre dimension and, for the Tits index $ \mathsf{E}_{8, 2}^{78} $, contains a field. For classical or globally isotropic reductive groups schemes $ \mathrm{K}_1 $ is actually solvable. This implies that the elementary subgroup (or its derived subgroup) is the maximal perfect subgroup of the reductive group.

math.RT↗

arXiv2026

Weyl elements in isotropic reductive groups

We study Weyl elements in isotropic reductive groups over commutative rings. Our main result in an explicit formula for squares of such elements. We also classify these elements in rank one groups and prove basic properties of their loci.

math.RT↗

arXiv2025

Locally isotropic Steinberg groups II. Schur multipliers

We compute Schur multipliers of locally isotropic Steinberg groups and of all root graded Steinberg groups with root systems of rank at least $ 3 $ (excluding the types $ \mathsf H_3 $ and $ \mathsf H_4 $). As an application, we show that locally isotropic Steinberg groups are well defined as abstract groups.

math.GR↗

arXiv2025

The Diophantine problem in isotropic reductive groups

We begin to study model-theoretic properties of non-split isotropic reductive group schemes. In this paper we show that the base ring $K$ is e-interpretable in the point group $G(K)$ of every sufficiently isotropic reductive group scheme $G$. In particular, the Diophantine problems in $K$ and $G(K)$ are equivalent. We also compute the centralizer of the elementary subgroup of $G(K)$ and the common normalizer of all its root subgroups.

math.NT↗

arXiv2024

Locally isotropic Steinberg groups I. Centrality of the $\mathrm K_2$-functor

We begin to study Steinberg groups associated with a locally isotropic reductive group $G$ over a arbitrary ring. We propose a construction of such a Steinberg group functor as a group object in a certain completion of the category of presheaves. We also show that it is a crossed module over $G$ in a unique way, in particular, that the $\mathrm K_2$-functor is central. If $G$ is globally isotropic in a suitable sense, then the Steinberg group functor exists as an ordinary group-valued functor and all such abstract Steinberg groups are crossed modules over the groups of points of $G$.

math.RT↗

arXiv2024

Root graded groups revisited

A group $G$ is called root graded if it has a family of subgroups $G_\alpha$ indexed by roots from a root system $\Phi$ satisfying natural conditions similar to Chevalley groups over commutative unital rings. For any such group there is a corresponding algebraic structure (commutative unital ring, associative unital ring, etc.) encoding the commutator relations between $G_\alpha$. We give a complete description of varieties of such structures for irreducible root systems of rank $\geq 3$ excluding $\mathsf H_3$ and $\mathsf H_4$. Moreover, we provide a construction of root graded groups for all algebraic structures from these varieties.

math.GR↗

arXiv2023

Locally isotropic elementary groups

We construct elementary subgroups of all reductive groups of the local isotropic rank $\geq 2$ over rings and prove their basic properties. In particular, our results may be applied to the automorphism groups of any finitely generated projective modules over commutative unital rings of rank $\geq 3$ at every prime ideal.

math.GR↗

arXiv2023

Groups with $\mathsf{BC}_\ell$-commutator relations

Isotropic odd unitary groups generalize Chevalley groups of classical types over commutative rings and their twisted forms. Such groups have root subgroups parameterized by a root system $\mathsf{BC}_\ell$ and may be constructed by so-called odd form rings with Peirce decompositions. We show the converse: if a group $G$ has root subgroups indexed by roots of $\mathsf{BC}_\ell$ and satisfying natural conditions, then there is a homomorphism $\mathrm{StU}(R, \Delta) \to G$ inducing isomorphisms on the root subgroups, where $\mathrm{StU}(R, \Delta)$ is the odd unitary Steinberg group constructed by an odd form ring $(R, \Delta)$ with a Peirce decomposition. For groups with root subgroups indexed by $\mathsf A_\ell$ (the already known case) the resulting odd form ring is essentially a generalized matrix ring.

math.GR↗

arXiv2023

Cosheaves of Steinberg pro-groups

Steinberg pro-groups are certain pro-groups used to analyze ordinary Steinberg groups locally in Zariski topology. In this paper we show that Steinberg pro-groups associated with general linear groups, odd unitary groups, and Chevalley groups satisfy a Zariski cosheaf property as crossed pro-modules over the base groups. Also, we prove an analogue of the standard commutator formulae for relative Steinberg groups. As an application, we show that the base groups over localized rings naturally act on the corresponding Steinberg pro-groups.

math.GR↗

arXiv2022

A presentation of relative unitary Steinberg groups

We find an explicit presentation of relative odd unitary Steinberg groups constructed by odd form rings and of relative doubly laced Steinberg groups over commutative rings, i.e. the Steinberg groups associated with the Chevalley group schemes of the types $\mathsf B_\ell$, $\mathsf C_\ell$, $\mathsf F_4$ for $\ell \geq 3$. For simply laced root systems such result is already known.

math.GR↗

arXiv2022

Actions of pro-groups and pro-rings

We give an explicit description of internal actions in the semi-abelian categories of pro-groups and non-unital pro-rings in terms of actions of group objects and ring objects in $\mathrm{Pro}(\mathbf{Set})$, as well as in some related categories. Also, we show that a similar result fails for Lie algebras.

math.GR↗

arXiv2022

Groups with $\mathsf A_\ell$-commutator relations

If $A$ is a unital associative ring and $\ell \geq 2$, then the general linear group $\mathrm{GL}(\ell, A)$ has root subgroups $U_\alpha$ and Weyl elements $n_\alpha$ for $\alpha$ from the root system of type $\mathsf A_{\ell - 1}$. Conversely, if an arbitrary group has such root subgroups and Weyl elements for $\ell \geq 4$ satisfying natural conditions, then there is a way to recover the ring $A$. We prove a generalization of this result not using the Weyl elements, so instead of the matrix ring $\mathrm M(\ell, A)$ we construct a non-unital associative ring with a well-behaved Peirce decomposition.

math.GR↗

arXiv2021

On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ modeled on linear and even orthogonal groups

Let $k$ be an arbitrary field. In this paper we show that in the linear case ($\Phi=\mathsf{A}_\ell$, $\ell \geq 4$) and even orthogonal case ($\Phi = \mathsf{D}_\ell$, $\ell\geq 7$, $\mathrm{char}(k)\neq 2$) the unstable functor $\mathrm{K}_2(\Phi, -)$ possesses the $\mathbb{A}^1$-invariance property in the geometric case, i. e. $\mathrm{K}_2(\Phi, R[t]) = \mathrm{K}_2(\Phi, R)$ for a regular ring $R$ containing $k$. As a consequence, the unstable $\mathrm{K}_2$ groups can be represented in the unstable $\mathbb{A}^1$-homotopy category $\mathscr{H}_\bullet(k)$ as fundamental groups of the simply-connected Chevalley--Demazure group schemes $\mathrm{G}(\Phi,-)$. Our invariance result can be considered as the $\mathrm{K}_2$-analogue of the geometric case of Bass--Quillen conjecture. We also show for a semilocal regular $k$-algebra $A$ that $\mathrm{K}_2(\Phi, A)$ embeds as a subgroup into $\mathrm{K}^\mathrm{M}_2(\mathrm{Frac}\,A)$.

math.GR↗

arXiv2021

Explicit presentation of relative Steinberg groups

We find an explicit presentation of relative linear Steinberg groups $\mathrm{St}(n, R, I)$ for any ring $R$ and $n \geq 4$ by generators and relations as abstract groups. We also prove a similar result for relative simply laced Steinberg groups $\mathrm{St}(Φ; R, I)$ for commutative $R$ and $Φ\in \{\mathsf A_\ell, \mathsf D_\ell, \mathsf E_\ell \mid \ell \geq 3\}$.

math.GR↗

arXiv2020

Centrality of odd unitary $K_2$-functor

Let $(R, Δ)$ be an odd form algebra. We show that the unitary Steinberg group $\mathrm{StU}(R, Δ)$ is a crossed module over the odd unitary group $\mathrm U(R, Δ)$ in two major cases: if the odd form algebra has a free orthogonal hyperbolic family satisfying a local stable rank condition and if the odd form algebra is sufficiently isotropic and quasi-finite. The proof uses only elementary localization techniques in terms of pro-groups.

math.GR↗

arXiv2020

Another presentation of orthogonal Steinberg groups

We use the pro-group approach to show that $\mathrm{StO}(M, q)$ admits van der Kallen's "another presentation", where $M$ is a module over a commutative ring with sufficiently isotropic quadratic form $q$. Moreover, we construct an analog of ESD-transvections in orthogonal Steinberg pro-groups under some assumptions on their parameters.

math.GR↗

arXiv2020

Centrality of $\mathrm K_2$ for Chevalley groups: a pro-group approach

We prove the centrality of $\mathrm{K}_2 (\mathsf{F}_4, \,R)$ for an arbitrary commutative ring $R$. This completes the proof of the centrality of $\mathrm K_2(\Phi,\, R)$ for any root system $\Phi$ of rank $\geq 3$. Our proof uses only elementary localization techniques reformulated in terms of pro-groups. Another new result of the paper is the construction of a crossed module on the canonical homomorphism $\mathrm{St}(\Phi, R) \to \mathrm{G}_\mathrm{sc}(\Phi, R)$, which has not been known previouly for exceptional $\Phi$.

math.GR↗

arXiv2020

Centrality of $\mathrm K_2$-functor revisited

We prove that $\mathrm{St}(n, A)$ is a crossed module over $\mathrm{GL}(n, A)$ under a local stable rank condition on an algebra $A$ over a commutative ring. Our proof uses only elementary localization techniques in terms of pro-groups and stability results for $\mathrm K_1$ and $\mathrm K_2$. We also prove similar result for the Steinberg group associated with any sufficiently isotropic general linear group constructed by a quasi-finite algebra.

math.GR↗