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Andrei Lavrenov

Publications and source records attributed to Andrei Lavrenov.

13 recordsLinked to original sources
arXiv2025

Invertible Morava motives in quadrics

We associate to any element in the Milnor K-theory of a field $k$ modulo 2 an invertible Morava K-theory motive over $k$. Specifically, for $α$ in $\mathrm{K}^{\mathrm{M}}_{n+1}(k)/2$ we construct an invertible $\mathrm{K}(n)$-motive $L_α$ in a way that is natural in the base field and additive in $α$. This can be seen as categorification of $\mathrm{K}^{\mathrm{M}}_{n+1}(k)/2$ in motives. The motives $L_α$ are constructed as direct summands of the $\mathrm{K}(n)$-motives of quadrics, and we develop the necessary framework for the study of the latter. We show that passing to the field of functions of quadrics of dimension greater than or equal to $2^{n+1}-1$ does not lose any information about the structure of $\mathrm{K}(n)$-motives. This is based on the study of "decomposition of the diagonal" in Morava K-theory of quadrics. For quadrics of dimension less than $2^{n+1}-1$, we show that their Chow motives can be "reconstructed" from their $\mathrm{K}(n)$-motives, although the latter appear structurally simpler. Our proof of this result relies on the use of the unstable symmetric operations of Vishik on algebraic cobordism. The occurrence of the motive $L_α$ as a direct summand of the $\mathrm{K}(n)$-motive of $X$ can be seen as evidence that $α$ is a cohomological invariant of $X$. We study this occurrence for quadrics and relate it to Kahn's Descent conjecture.

math.AG↗
arXiv2024

Morava $J$-invariant

We compute the co-multiplication of the algebraic Morava K-theory for split orthogonal groups. This allows us to compute the decomposition of the Morava motives of generic maximal orthogonal Grassmannians and to compute a Morava K-theory analogue of the $J$-invariant in terms of the ordinary (Chow) $J$-invariant.

math.KT↗
arXiv2024

Morava K-theory of orthogonal groups and motives of projective quadrics

We compute the algebraic Morava K-theory ring of split special orthogonal and spin groups. In particular, we establish certain stabilization results for the Morava K-theory of special orthogonal and spin groups. Besides, we apply these results to study Morava motivic decompositions of orthogonal Grassmannians. For instance, we determine all indecomposable summands of the Morava motives of a generic quadric.

math.KT↗
arXiv2023

Bounded generation of Steinberg groups over Dedekind rings of arithmetic type

The main result of the present paper is bounded elementary generation of the Steinberg groups $\mathrm{St}(Φ,R)$ for simply laced root systems $Φ$ of rank $\ge 2$ and arbitrary Dedekind rings of arithmetic type. Also, we prove bounded generation of $\mathrm{St}(Φ,\mathbb F_{q}[t,\,t^{-1}])$ for all root systems $Φ$, and bounded generation of $\mathrm{St}(Φ,\mathbb F_{q}[t])$ for all root systems $Φ\neq\mathsf A_1$. The proofs are based on a theorem on bounded elementary generation for the corresponding Chevalley groups, where we provide uniform bounds.

math.KT↗
arXiv2023

Morava K-theory and Rost invariant

We prove that inner forms of a variety of Borel subgroups have isomorphic motives with respect to the second Morava K-theory if and only if the corresponding Tits algebras and Rost invariants coincide. This extends Panin's results on interrelationship of K-theory with Tits algebras to the case of cohomological invariants of degree 3.

math.KT↗
arXiv2023

A Horrocks-type theorem for even orthogonal $K_2$

We prove the Horrocks theorem for unstable even-dimensional orthogonal Steinberg groups. The Horrocks theorem for Steinberg groups is one of the principal ingredients needed for the proof of the $\mathrm{K}_2$-analogue of Serre's problem, whose positive solution is currently known only in the linear case.

math.GR↗
arXiv2022

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 ($Φ=\mathsf{A}_\ell$, $\ell \geq 4$) and even orthogonal case ($Φ= \mathsf{D}_\ell$, $\ell\geq 7$, $\mathrm{char}(k)\neq 2$) the unstable functor $\mathrm{K}_2(Φ, -)$ possesses the $\mathbb{A}^1$-invariance property in the geometric case, i. e. $\mathrm{K}_2(Φ, R[t]) = \mathrm{K}_2(Φ, R)$ 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 $\mathrm{G}(Φ,-)$. 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 $\mathrm{K}_2(Φ, A)$ embeds as a subgroup into $\mathrm{K}^\mathrm{M}_2(\mathrm{Frac}\,A)$.

math.GR↗
arXiv2021

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 $\mathrm K_2(Φ,\, R)$ for any root system $Φ$ 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 $\mathrm{St}(Φ, R) \to \mathrm{G}_\mathrm{sc}(Φ, R)$, which has not been known previouly for exceptional $Φ$.

math.GR↗
arXiv2014

Another presentation for symplectic Steinberg groups

We solve a classical problem of centrality of symplectic $\mathrm K_2$, namely we show that for an arbitrary commutative ring $R$, $l\geq3$ the symplectic Steinberg group $\mathrm{StSp}(2l,\,R)$ as an extension of the elementary symplectic group $\mathrm{Ep}(2l,\,R)$ is a central extension. This allows to conclude that the explicit definition of symplectic $\mathrm{K_2Sp}(2l,\,R)$ as a kernel of this extension, i.e. as a group of non-elementary relations among symplectic transvections, coincides with the usual implicit definition via plus-construction. We proceed from van der Kallen's classical paper, where he shows an analogous result for linear K-theory. We find a new set of generators for the symplectic Steinberg group and a defining system of relations among them. In this new presentation it is obvious that the symplectic Steinberg group is a central extension.

math.KT↗
arXiv2013

On odd unitary Steinberg group

Let $R$ be a ring with pseudo-involution, $\mathfrak L$ be an odd form parameter, $\mathrm U(2n,\,R,\,\mathfrak L)$ be an odd hyperbolic unitary group, $\mathrm{EU}(2n,\,R,\,\mathfrak L)$ be it elementary subgroup and $\mathrm{StU}(2n,\,R,\,\mathfrak L)$ be an odd unitary Steinberg group. We compute the Schur multipliers of $\mathrm{StU}(2n,\,R,\,\mathfrak L)$ and $\mathrm{EU}(R,\,\mathfrak L)$.

math.KT↗