Vladimir Chernousov - Profile on Academia.edu (original) (raw)
Papers by Vladimir Chernousov
Lower bounds for essential dimensions in characteristic 2 via orthogonal representations
Pacific Journal of Mathematics, 2015
The Michigan Mathematical Journal, Oct 1, 2008
If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greate... more If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct product SL(2,R)^m x SL(2,C)^n$, with m + n > 1. (In geometric terms, this can be interpreted as a statement about the existence of totally geodesic subspaces of finite-volume, noncompact, locally symmetric spaces of higher rank.) Another formulation of the result states that if G is any isotropic, almost simple algebraic group over Q (the rational numbers), such that the real rank of G is greater than 1, then G contains an isotropic, almost simple Q-subgroup H, such that H is quasisplit, and the real rank of H is greater than 1.
Purity of G2-torsors
Comptes Rendus Mathematique, Sep 15, 2007
Let k be a field of characteristic zero, and let G be a split simple algebraic group of type G2 o... more Let k be a field of characteristic zero, and let G be a split simple algebraic group of type G2 over k. We prove that the functor R↦He´t1(R,G) of G-torsors satisfies purity for regular local rings containing k. To cite this article: V. Chernousov, I. Panin, C. R. Acad. Sci. Paris, Ser. I 345 (2007).
In the early 19th century a young French mathematician E. Galois laid the foundations of abstract... more In the early 19th century a young French mathematician E. Galois laid the foundations of abstract algebra by using the symmetries of a polynomial equation to describe the properties of its roots. One of his discoveries was a new type of structure, formed by these symmetries. This structure, now called a "group", is central to much of modern mathematics. The groups that arise in the context of classical Galois theory are finite groups.
We conclude the computation of the essential dimension of split spinor groups and give an applica... more We conclude the computation of the essential dimension of split spinor groups and give an application in algebraic theory of quadratic forms. We also compute essential dimension of quadratic forms with trivial discriminant and Clifford invariant.
Bulletin of Mathematical Sciences, 2014
The conjugacy of split Cartan subalgebras in the finitedimensional simple case (Chevalley) and in... more The conjugacy of split Cartan subalgebras in the finitedimensional simple case (Chevalley) and in the symmetrizable Kac-Moody case (Peterson-Kac) are fundamental results of the theory of Lie algebras. Among the Kac-Moody Lie algebras the affine algebras stand out. This paper deals with the problem of conjugacy for a class of algebras -extended affine Lie algebras-that are in a precise sense higher nullity analogues of the affine algebras. Unlike the methods used by Peterson-Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of Bruhat-Tits on buildings. The main ingredient of our conjugacy proof is the classification of loop torsors over Laurent polynomial rings, a result of its own interest.
Algebra & Number Theory, 2014
We conclude the computation of the essential dimension of split spinor groups and an application ... more We conclude the computation of the essential dimension of split spinor groups and an application in algebraic theory of quadratic forms is given. We also compute essential dimension of split even Clifford group, or equivalently, of the class of quadratic forms with trivial discriminant and Clifford invariant.
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Алгебры с делением, имеющие одинаковые максимальные подполя
Успехи математических наук, 2015
Bulletin of Mathematical Sciences, 2013
Let D be a finite-dimensional central division algebra over a field K . We define the genus gen(D... more Let D be a finite-dimensional central division algebra over a field K . We define the genus gen(D) of D to be the collection of classes [D ] ∈ Br(K ), where D is a central division K -algebra having the same maximal subfields as D. In this paper, we describe a general approach to proving the finiteness of gen(D) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of K . This approach is then implemented in some concrete situations, yielding in particular an extension of the Stability Theorem of A. Rapinchuk and I. Rapinchuk (Manuscr. Math. 132:273-293, 2010) from quaternion algebras to arbitrary algebras of exponent two. We also consider an example where the size of the genus can be estimated explicitly. Finally, we offer two generalizations of the genus problem for division algebras: one deals with absolutely almost simple algebraic Kgroups having the same isomorphism/isogeny classes of maximal K -tori, and the other with the analysis of weakly commensurable Zariski-dense subgroups. • Maximal field • Brauer group • Linear algebraic group • Maximal torus Communicated by Efim Zelmanov.
Владимир Петрович Платонов (к 75-летию со дня рождения)
Успехи математических наук, 2015
Transformation Groups, 2006
Given an algebraic group G defined over a (not necessarily algebraically closed) field F and a co... more Given an algebraic group G defined over a (not necessarily algebraically closed) field F and a commutative ring R we associate the subcategory M(G, R) of the category of Chow motives with coefficients in R that is the Tate pseudo-abelian closure of the category of motives of projective homogeneous G-varieties. We show that M(G, R) is a symmetric tensor category, i.e., the motive of the product of two projective homogeneous G-varieties is a direct sum of twisted motives of projective homogeneous G-varieties. We also study the problem of uniqueness of a direct sum decomposition of objects in M(G, R). We prove that the Krull-Schmidt theorem holds in many cases.
Varieties of representations of fundamental groups of compact nonoriented surfaces
The Michigan Mathematical Journal, 2008
Mathematische Annalen, 2013
We compute the essential p-dimension of split simple groups of type A n−1 in terms of the functor... more We compute the essential p-dimension of split simple groups of type A n−1 in terms of the functor Alg (n, m) of central simple algebras of degree n and exponent dividing m.
Многообразия представлений фундаментальных групп компактных неориентируемых поверхностей
Математический сборник, 1997
manuscripta mathematica, 2008
Let G be a reductive affine group scheme defined over a semilocal ring k. Assume that either G is... more 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 surjective for every semilocal ring R containing k. In other words, G-torsors over Spec(R) admit reduction of structure to S. We also show that the natural map H 1 (X, S) → H 1 (X, G) is surjective in several other contexts, under suitable assumptions on the base ring k, the scheme X/k and the group scheme G/k. These results have already been used to study loop algebras and essential dimension of connected algebraic groups in prime characteristic.
Journal of Algebra, 2006
Let G be a linear algebraic group defined over a field k. We prove that, under mild assumptions o... more Let G be a linear algebraic group defined over a field k. We prove that, under mild assumptions on k and G, there exists a finite k-subgroup S of G such that the natural map H 1 (K, S) -→ H 1 (K, G) is surjective for every field extension K/k. We give two applications of this result in the case where k an algebraically closed field of characteristic zero and K/k is finitely generated. First we show that for every α ∈ H 1 (K, G) there exists an abelian field extension L/K such that αL ∈ H 1 (L, G) is represented by a G-torsor over a projective variety. In particular, we prove that αL has trivial point obstruction. As a second application of our main theorem, we show that a (strong) variant of the algebraic form of Hilbert's 13th problem implies that the maximal abelian extension of K has cohomological dimension ≤ 1. The last assertion, if true, would prove conjectures of Bogomolov and Königsmann, answer a question of Tits and establish an important case of Serre's Conjecture II for the group E8.
Purity for Pfister forms and F4-torsors with trivial g3 invariant
Journal für die reine und angewandte Mathematik (Crelles Journal), 2000
Lower bounds for essential dimensions in characteristic 2 via orthogonal representations
Pacific Journal of Mathematics, 2015
The Michigan Mathematical Journal, Oct 1, 2008
If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greate... more If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct product SL(2,R)^m x SL(2,C)^n$, with m + n > 1. (In geometric terms, this can be interpreted as a statement about the existence of totally geodesic subspaces of finite-volume, noncompact, locally symmetric spaces of higher rank.) Another formulation of the result states that if G is any isotropic, almost simple algebraic group over Q (the rational numbers), such that the real rank of G is greater than 1, then G contains an isotropic, almost simple Q-subgroup H, such that H is quasisplit, and the real rank of H is greater than 1.
Purity of G2-torsors
Comptes Rendus Mathematique, Sep 15, 2007
Let k be a field of characteristic zero, and let G be a split simple algebraic group of type G2 o... more Let k be a field of characteristic zero, and let G be a split simple algebraic group of type G2 over k. We prove that the functor R↦He´t1(R,G) of G-torsors satisfies purity for regular local rings containing k. To cite this article: V. Chernousov, I. Panin, C. R. Acad. Sci. Paris, Ser. I 345 (2007).
In the early 19th century a young French mathematician E. Galois laid the foundations of abstract... more In the early 19th century a young French mathematician E. Galois laid the foundations of abstract algebra by using the symmetries of a polynomial equation to describe the properties of its roots. One of his discoveries was a new type of structure, formed by these symmetries. This structure, now called a "group", is central to much of modern mathematics. The groups that arise in the context of classical Galois theory are finite groups.
We conclude the computation of the essential dimension of split spinor groups and give an applica... more We conclude the computation of the essential dimension of split spinor groups and give an application in algebraic theory of quadratic forms. We also compute essential dimension of quadratic forms with trivial discriminant and Clifford invariant.
Bulletin of Mathematical Sciences, 2014
The conjugacy of split Cartan subalgebras in the finitedimensional simple case (Chevalley) and in... more The conjugacy of split Cartan subalgebras in the finitedimensional simple case (Chevalley) and in the symmetrizable Kac-Moody case (Peterson-Kac) are fundamental results of the theory of Lie algebras. Among the Kac-Moody Lie algebras the affine algebras stand out. This paper deals with the problem of conjugacy for a class of algebras -extended affine Lie algebras-that are in a precise sense higher nullity analogues of the affine algebras. Unlike the methods used by Peterson-Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of Bruhat-Tits on buildings. The main ingredient of our conjugacy proof is the classification of loop torsors over Laurent polynomial rings, a result of its own interest.
Algebra & Number Theory, 2014
We conclude the computation of the essential dimension of split spinor groups and an application ... more We conclude the computation of the essential dimension of split spinor groups and an application in algebraic theory of quadratic forms is given. We also compute essential dimension of split even Clifford group, or equivalently, of the class of quadratic forms with trivial discriminant and Clifford invariant.
[
Алгебры с делением, имеющие одинаковые максимальные подполя
Успехи математических наук, 2015
Bulletin of Mathematical Sciences, 2013
Let D be a finite-dimensional central division algebra over a field K . We define the genus gen(D... more Let D be a finite-dimensional central division algebra over a field K . We define the genus gen(D) of D to be the collection of classes [D ] ∈ Br(K ), where D is a central division K -algebra having the same maximal subfields as D. In this paper, we describe a general approach to proving the finiteness of gen(D) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of K . This approach is then implemented in some concrete situations, yielding in particular an extension of the Stability Theorem of A. Rapinchuk and I. Rapinchuk (Manuscr. Math. 132:273-293, 2010) from quaternion algebras to arbitrary algebras of exponent two. We also consider an example where the size of the genus can be estimated explicitly. Finally, we offer two generalizations of the genus problem for division algebras: one deals with absolutely almost simple algebraic Kgroups having the same isomorphism/isogeny classes of maximal K -tori, and the other with the analysis of weakly commensurable Zariski-dense subgroups. • Maximal field • Brauer group • Linear algebraic group • Maximal torus Communicated by Efim Zelmanov.
Владимир Петрович Платонов (к 75-летию со дня рождения)
Успехи математических наук, 2015
Transformation Groups, 2006
Given an algebraic group G defined over a (not necessarily algebraically closed) field F and a co... more Given an algebraic group G defined over a (not necessarily algebraically closed) field F and a commutative ring R we associate the subcategory M(G, R) of the category of Chow motives with coefficients in R that is the Tate pseudo-abelian closure of the category of motives of projective homogeneous G-varieties. We show that M(G, R) is a symmetric tensor category, i.e., the motive of the product of two projective homogeneous G-varieties is a direct sum of twisted motives of projective homogeneous G-varieties. We also study the problem of uniqueness of a direct sum decomposition of objects in M(G, R). We prove that the Krull-Schmidt theorem holds in many cases.
Varieties of representations of fundamental groups of compact nonoriented surfaces
The Michigan Mathematical Journal, 2008
Mathematische Annalen, 2013
We compute the essential p-dimension of split simple groups of type A n−1 in terms of the functor... more We compute the essential p-dimension of split simple groups of type A n−1 in terms of the functor Alg (n, m) of central simple algebras of degree n and exponent dividing m.
Многообразия представлений фундаментальных групп компактных неориентируемых поверхностей
Математический сборник, 1997
manuscripta mathematica, 2008
Let G be a reductive affine group scheme defined over a semilocal ring k. Assume that either G is... more 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 surjective for every semilocal ring R containing k. In other words, G-torsors over Spec(R) admit reduction of structure to S. We also show that the natural map H 1 (X, S) → H 1 (X, G) is surjective in several other contexts, under suitable assumptions on the base ring k, the scheme X/k and the group scheme G/k. These results have already been used to study loop algebras and essential dimension of connected algebraic groups in prime characteristic.
Journal of Algebra, 2006
Let G be a linear algebraic group defined over a field k. We prove that, under mild assumptions o... more Let G be a linear algebraic group defined over a field k. We prove that, under mild assumptions on k and G, there exists a finite k-subgroup S of G such that the natural map H 1 (K, S) -→ H 1 (K, G) is surjective for every field extension K/k. We give two applications of this result in the case where k an algebraically closed field of characteristic zero and K/k is finitely generated. First we show that for every α ∈ H 1 (K, G) there exists an abelian field extension L/K such that αL ∈ H 1 (L, G) is represented by a G-torsor over a projective variety. In particular, we prove that αL has trivial point obstruction. As a second application of our main theorem, we show that a (strong) variant of the algebraic form of Hilbert's 13th problem implies that the maximal abelian extension of K has cohomological dimension ≤ 1. The last assertion, if true, would prove conjectures of Bogomolov and Königsmann, answer a question of Tits and establish an important case of Serre's Conjecture II for the group E8.
Purity for Pfister forms and F4-torsors with trivial g3 invariant
Journal für die reine und angewandte Mathematik (Crelles Journal), 2000