arXiv subjects

Anastasia Stavrova

Publications and source records attributed to Anastasia Stavrova.

At least 19 recordsLinked to original sources

arXiv2025

On the generalized Bass--Quillen conjecture in dimension 2

Let $A$ be a regular ring of dimension $\le 2$. Let $G$ be a reductive group over $A$ such that its derived group is a split, i.e. a Chevalley--Demazure, semisimple group. We prove that every Zariski-locally trivial principal $G$-bundle over $A[x_1,\ldots,x_n]$ is extended from $A$, for any $n\ge 1$. This result generalizes to split reductive groups the dimension $2$ case of the Bass--Quillen conjecture on finitely generated projective modules, settled in positive by M. P. Murthy.

math.AG↗

arXiv2025

Non-Stable $K_1$-Functors of Discrete Valuation Rings Containing a Field

Let $k$ be a field, and let $G$ be a simply connected semisimple k-group which is isotropic and contains a strictly proper parabolic $k$-subgroup $P$. Let $D$ be a discrete valuation ring which is a local ring of a smooth algebraic curve over $k$. Let $K$ be the fraction field of $D$. We show that the corresponding non-stable $K_1$-functor (for $G$ and $P$, also called the Whitehead group of $G$) coincide over $D$ and $K$. As a consequence, $K^G_1 (D)$ coincides with the (generalized) Manin's $R$-equivalence class group of $G(D)$.

math.AG↗

arXiv2025

On a theorem of Harder

We prove that for any simply connected isotropic reductive group G over a Dedekind domain D, any Zariski-locally trivial principal G-bundle over D is trivial. The corresponding result for quasi-split groups was proved in 1967 by G. Harder.

math.AG↗

arXiv2025

On the Gille theorem for the relative projective line

Let $X$ be a Noetherian separated scheme. Let $G$ be a reductive $X$-group scheme, and let $E$ be a principal $G$-bundle over $\mathbb{P}^1_X$. We prove that if the restriction of $E$ to $\infty\times X$ is Zariski locally trivial, then $E$ is itself Zariski locally trivial.

math.AG↗

arXiv2024

Chevalley groups over Laurent polynomial rings

Let $G$ be a simply connected Chevalley--Demazure group scheme without $SL_2$-factors. For any unital commutative ring $R$, we denote by $E(R)$ the standard elementary subgroup of $G(R)$, that is, the subgroup generated by the elementary root unipotent elements. We prove that the map $$ G(R[x_1^{\pm 1},\ldots,x_n^{\pm 1}])/E(R[x_1^{\pm 1},\ldots,x_n^{\pm 1}])\to G\bigl(R((x_1))\ldots((x_n))\bigr)/E\bigl(R((x_1))\ldots((x_n))\bigr) $$ is injective for any $n\ge 1$, if $R$ is either a Dedekind domain or a Noetherian ring that is geometrically regular over a Dedekind domain with perfect residue fields. For $n=1$ this map is also an isomorphism. As a consequence, we show that if $D$ is a PID such that $SL_2(D)=E_2(D)$ (e.g. $D=\mathbb{Z}$), then $G(D[x_1^{\pm 1},\ldots,x_n^{\pm 1}])=E(D[x_1^{\pm 1},\ldots,x_n^{\pm 1}])$. This extends earlier results for special linear and symplectic groups due to A. A. Suslin and V. I. Kopeiko.

math.GR↗

arXiv2024

On the Gille theorem for the relative projective line: I

We prove a relative version of a theorem on torsors on the projective line due to Philippe Gille. As a consequence we obtain a ``weak homotopy invariance'' result for torsors under reductive group schemes defined over arbitrary semi-local regular domains. Specifically, only regular semi-local domains with infinite residue fields are regarded in this preprint. However, all results of the present preprint are true (after minor modifications) for arbitrary semi-local regular domains. This will be the topic of our next preprint.

math.AG↗

arXiv2022

Normal structure of isotropic reductive groups over rings

The paper studies the lattice of subgroups of an isotropic reductive group G(R) over a commutative ring R, normalized by the elementary subgroup E(R). We prove the sandwich classification theorem for this lattice under the assumptions that the reductive group scheme G is defined over an arbitrary commutative ring, its isotropic rank is at least 2, and the structure constants are invertible in R. The theorem asserts that the lattice splits into a disjoint union of sublattices (sandwiches) E(R,q)<=...<=C(R,q) parametrized by the ideals q of R, where E(R,q) denotes the relative elementary subgroup and C(R,q) is the inverse image of the center under the natural homomorphism G(R) to G(R/I). The main ingredients of the proof are the "level computation" by the first author and the universal localization method developed by the second author.

math.GR↗

arXiv2021

A^1-invariance of non-stable K_1-functors in the equicharacteristic case

We apply the techniques developed by I. Panin for the proof of the equicharacteristic case of the Serre-Grothendieck conjecture for isotropic reductive groups (I. Panin, A. Stavrova, N. Vavilov, 2015; I. Panin, 2019) to obtain similar injectivity and A^1-invariance theorems for non-stable K_1-functors associated to isotropic reductive groups. Namely, let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We show that if G has isotropic rank >=2 and R is a regular domain containing a field, then K_1^G(R[x])=K_1^G(R) for any n>=1, where K_1^G(R)=G(R)/E(R) is the corresponding non-stable K_1-functor, also called the Whitehead group of G. If R is, moreover, local, then we show that K_1^G(R)->K_1^G(K) is injective, where K is the field of fractions of R.

math.KT↗

arXiv2020

Torsors of isotropic reductive groups over Laurent polynomials

Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x_1^{\pm 1},...,x_n^{\pm 1}]. We prove that G has isotropic rank >=1 over R iff it has isotropic rank >=1 over the field of fractions k(x_1,...,x_n) of R, and if this is the case, then the natural map H^1_{et}(R,G)\to H^1_{\et}(k(x_1,...,x_n),G) has trivial kernel, and G is loop reductive, i.e. contains a maximal R-torus. In particular, we settle in positive the conjecture of V. Chernousov, P. Gille, and A. Pianzola that H^1_{Zar}(R,G)=* for such groups G. We also deduce that if G is a reductive group over R of isotropic rank >=2, then the natural map of non-stable K_1-functors K_1^G(R)\to K_1^G( k((x_1))...((x_n)) ) is injective, and an isomorphism if G is moreover semisimple.

math.AG↗

arXiv2020

5-Graded simple Lie algebras, structurable algebras, and Kantor pairs

Relying on the classification of simple Lie algebras over algebraically closed fields of characteristic $>3$, we show that any finite-dimensional central simple 5-graded Lie algebra over a field $k$ of characteristic $\neq 2,3$ is a simple Lie algebra of Chevalley type, i.e. a central quotient of the Lie algebra of a simple algebraic $k$-group. As a consequence, we prove that all central simple structurable algebras and Kantor pairs over $k$ arise from 5-gradings on simple Lie algebras of Chevalley type.

math.GR↗

arXiv2019

Chevalley groups of polynomial rings over Dedekind domains

Let R be a Dedekind domain, and let G be a simply connected Chevalley-Demazure group scheme of rank =>2. We prove that G(R[x_1,...,x_n])=G(R)E(R[x_1,...,x_n]) for any n=>1. This extends the corresponding results of A. Suslin and F. Grunewald, J. Mennicke, and L. Vaserstein for G=SL_n, Sp_2n. We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.

math.KT↗

arXiv2018

Isotropic reductive groups over discrete Hodge algebras

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We prove that if G has isotropic rank >=1 and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra A=R[x_1,...,x_n]/I over R, the map H^1_Nis(A,G) -> H^1_Nis(R,G) induced by evaluation at x_1=...=x_n=0, is a bijection. If k has characteristic 0, then, moreover, the map H^1_et(A,G) -> H^1_et(R,G) has trivial kernel. We also prove that if k is perfect, G is defined over k, the isotropic rank of G is >=2, and A is square-free, then K_1^G(A)=K_1^G(R), where K_1^G(R)=G(R)/E(R) is the corresponding non-stable K_1-functor, also called the Whitehead group of G. The corresponging statements for G=GL_n were previously proved by Ton Vorst.

math.KT↗

arXiv2016

Moufang sets and structurable division algebras

A Moufang set is essentially a doubly transitive permutation group such that each point stabilizer contains a normal subgroup which is regular on the remaining vertices; these regular normal subgroups are called the root groups, and they are assumed to be conjugate and to generate the whole group. It has been known for some time that every Jordan division algebra gives rise to a Moufang set with abelian root groups. We extend this result by showing that every structurable division algebra gives rise to a Moufang set, and conversely, we show that every Moufang set arising from a simple linear algebraic group of relative rank one over an arbitrary field k of characteristic different from 2 and 3 arises from a structurable division algebra. We also obtain explicit formulas for the root groups, the τ-map and the Hua maps of these Moufang sets. This is particularly useful for the Moufang sets arising from exceptional linear algebraic groups.

math.GR↗

arXiv2016

On Kazhdan's property (T) for isotropic reductive groups

We show that the standard set of elementary generators of an elementary isotropic reductive group over a connected finitely generated ring is a Kazhdan subset. This generalizes the corresponding result of M. Ershov, A. Jaikin-Zapirain, and M. Kassabov for Chevalley and twisted Chevalley groups.

math.GR↗

arXiv2015

On the Grothendieck--Serre conjecture concerning principal G-bundles over semi-local Dedekind domains

Let R be a semi-local Dedekind domain and let K be the field of fractions of R. Let G be a reductive semisimple simply connected R-group scheme such that every semisimple normal R-subgroup scheme of G contains a split R-torus G_m. We prove that the kernel of the map H^1_et(R,G)-> H^1_et(K,G) induced by the inclusion of R into K, is trivial. This result partially extends a theorem of Nisnevich.

math.AG↗

arXiv2015

Transfers for non-stable K_1-functors of classical type

Let k be a field. Let G be an absolutely almost simple simply connected k-group of type A_l, l>=2, or D_l, l>=4, containing a 2-dimensional split torus. If G is of type D_l, assume moreover that char k is different from 2. We show that the Nisnevich sheafification of the non-stable K_1-functor K_1^G, also called the Whitehead group of G, on the category of smooth k-schemes is A^1-invariant, and has oriented weak transfers for affine varieties in the sense of Panin-Yagunov-Ross. If k has characteristic 0, this implies that the Nisnevich sheafification of K_1^G is birationally invariant. We also prove a rigidity theorem for \A1-invariant torsion presheaves with oriented weak transfers over infinite fields. As a corollary, we conclude that K_1^G(R)=K_1^G(k) whenever R is a Henselian regular local ring with a coefficient field k.

math.KT↗

arXiv2013

Principal bundles of reductive groups over affine schemes

Let R be a semi-local regular domain containing an infinite perfect field k, and let K be the field of fractions of R. Let G be a reductive semi-simple simply connected R-group scheme such that each of its R-indecomposable factors is isotropic. We prove that for any Noetherian affine scheme A over k, the kernel of the map of etale cohomology sets H^1(A\times_k R,G)-> H^1(A\times_ k K,G), induced by the inclusion of R into K, is trivial. If R is the semi-local ring of several points on a k-smooth scheme, then it suffices to require that k is infinite and keep the same assumption concerning G.

math.AG↗

arXiv2013

Homotopy invariance of non-stable K_1-functors

Let G be reductive algebraic group over a field k, such that every semisimple normal subgroup of G has isotropic rank >=2. Let K_1^G be the non-stable K_1-functor associated to G (also called the Whitehead group of G in the field case). We show that K_1^G(k)=K_1^G(k[X_1,...,X_n]) for any n>= 1. This implies that K_1^G is A^1-homotopy invariant on the category of regular k-algebras, if k is perfect. If k is infinite perfect, one also deduces that K_1^G(R)-> K_1^G(K) is injective for any regular local k-algebra R with the fraction field K.

math.AG↗