Kirill Zainoulline

Kirill Zainoulline
University of Ottawa · Department of Mathematics and Statistics

Dr. rer. nat., PhD

About

61
Publications
2,241
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
575
Citations
Additional affiliations
May 2015 - present
University of Ottawa
Position
  • Professor
August 2009 - April 2015
University of Ottawa
Position
  • Professor (Associate)
October 2006 - July 2009
Ludwig-Maximilians-University of Munich
Position
  • Professor (Assistant)
Education
August 2009 - August 2009
April 2000 - April 2000
Saint Petersburg State University
Field of study
  • Mathematics

Publications

Publications (61)
Article
Full-text available
We apply the degree formula for connective $K$-theory to study rational contractions of algebraic varieties. Examples include rationally connected varieties and complete intersections.
Article
Let G be a linear algebraic group over a field F and X be a projective homogeneous G-variety such that G splits over the function field of X. In the present paper we introduce an invariant of G called J-invariant which characterizes the motivic behaviour of X. This generalizes the respective notion invented by A. Vishik in the context of quadratic...
Article
Full-text available
. In the present paper we investigate the question about the injectivity of the map F(R) ! F(K) induced by the canonical inclusion of a local regular ring of geometric type R to its field of fractions K for a homotopy invariant functor F with transfers satisfying some additional properties. As an application we get the proof of Special Unitary Case...
Article
In the present notes we generalize the classical work of Demazure [Invariants sym\'etriques entiers des groupes de Weyl et torsion] to arbitrary oriented cohomology theories and formal group laws. Let G be a split semisemiple linear algebraic group over a field and let T be its split maximal torus. We construct a generalized characteristic map rela...
Article
For an oriented cohomology theory A and a relative cellular space X, we decompose the A-motive of X into a direct sum of twisted motives of the base spaces. We also obtain respective decompositions of the A-cohomology of X. Applying them, one can compute A(X), where X is an isotropic projective homogeneous variety and A means algebraic K-theory, mo...
Article
The present paper is devoted to twisted foldings of root systems that generalize the involutive foldings corresponding to automorphisms of Dynkin diagrams. A motivating example is Lusztig’s projection of the root system of type E 8 E_8 onto the subring of icosians of the quaternion algebra, which gives the root system of type H 4 H_4 . By using mom...
Preprint
Full-text available
We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the $K$-theory and hyperbolic cohomology theory of flag varieties. We first show that, in $K$-theory, the two different choices of Kazhdan-Lusztig bases produce dual bases, one of which can be interpreted as characteristic classes of the intersection homology mixed Hod...
Article
Full-text available
We define the formal affine Demazure algebra and formal affine Hecke algebra associated to a Kac-Moody root system. We prove the structure theorems of these algebras, hence, extending several result and construction (presentation in terms of generators and relations, coproduct and product structures, filtration by codimension of Bott-Samelson class...
Preprint
In the present paper we extend the Riemann-Roch formalism to structure algebras of moment graphs. We introduce and study the Chern character and pushforwards for twisted fibrations of moment graphs. We prove an analogue of the Riemann-Roch theorem for moment graphs. As an application, we obtain the Riemann-Roch type theorem for equivariant $K$-theo...
Preprint
In the present paper we provide a general algorithm to compute multiplicative cohomological operations on algebraic oriented cohomology of projective homogeneous G-varieties, where G is a split reductive algebraic group over a field of characteristic 0. More precisely, we extend such operations to the respective T-equivariant (T is a maximal split...
Article
We introduce and study various categories of (equivariant) motives of (versal) flag varieties. We relate these categories with certain categories of parabolic (Demazure) modules. We show that the motivic decomposition type of a versal flag variety depends on the direct sum decomposition type of the parabolic module. To do this we use localization t...
Article
Full-text available
In the present paper we introduce and study the push pull operators on the formal affine Demazure algebra and its dual. As an application we provide a non-degenerate pairing on the dual of the formal affine Demazure algebra which serves as an algebraic counterpart of the natural pairing on the T-equivariant oriented cohomology of G/B induced by mul...
Preprint
In the present paper we introduce and study twisted foldings of root systems which generalize usual involutive foldings corresponding to automorphisms of Dynkin diagrams. Our motivating example is the celebrated projection of the root system of type E8 onto the subring of icosians of the quaternion algebra which gives the root system of type H4. Us...
Article
Let $G$ be a split semisimple linear algebraic group over a field and let $X$ be a generic twisted flag variety of $G$. Extending the Hilbert basis techniques to Laurent polynomials over integers we give an explicit presentation of the Grothendieck ring $K_0(X)$ in terms of generators and relations in the case $G=G^{sc}/\mu_2$ is of Dynkin type ${\...
Article
In the present paper we extend the theory of sheaves on moment graphs due to Braden-MacPherson and Fiebig to the context of an arbitrary oriented equivariant cohomology $h$ (e.g. to algebraic cobordism). We introduce and investigate structure $h$-sheaves on double moment graphs to describe equivariant oriented cohomology of products of flag varieti...
Article
Studying geodesics in Cayley graphs of groups has been a very active area of research over the last decades. We introduce the notion of a uniquely labelled geodesic, abbreviated with u.l.g. These will be studied first in finite Coxeter groups of type $A_n$. Here we introduce a generating function, and hence are able to precisely describe how many u...
Article
Full-text available
We study the equivariant oriented cohomology ring $h_T(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott-Samelson classes in $h_{T}(G/P)$ can be obtained by applying this action to the fundamental class of the identity point, hence general...
Article
Full-text available
In the present paper we generalize the coproduct structure on nil Hecke rings introduced and studied by Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory and its associated formal group law. We then construct an algebraic model of the T-equivariant oriented cohomology of the variety of complete flags.
Article
Full-text available
We address the problem of defining Schubert classes independently of a reduced word in equivariant elliptic cohomology, based on the Kazhdan-Lusztig basis of a corresponding Hecke algebra. We study some basic properties of these classes, and make two important conjectures about them: a positivity conjecture, and the agreement with the topologically...
Article
Full-text available
Given an equivariant oriented cohomology theory $h$, a split reductive group $G$, a maximal torus $T$ in $G$, and a parabolic subgroup $P$ containing $T$, we explain how the $T$-equivariant oriented cohomology ring $h_T(G/P)$ can be identified with the dual of a coalgebra defined using exclusively the root datum of $(G,T)$, a set of simple roots de...
Article
Full-text available
Motivated by the motivic Galois group and the Kostant-Kumar results on equivariant cohomology of flag varieties, we provide a uniform description of motivic (direct sum) decompositions with integer coefficients of versal flag varieties in terms of integer representations of the associated affine nil-Hecke algebra $H$. More generally, we establish a...
Article
Full-text available
We prove that the group of normalized cohomological invariants of degree 3 modulo the subgroup of semidecomposable invariants of a semisimple split linear algebraic group G is isomorphic to the torsion part of the Chow group of codimension 2 cycles of the respective versal G-flag. In particular, if G is simple, we show that this factor group is iso...
Article
Full-text available
An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Willems for the localization of Schubert classes at torus fixed points. These formulas work uniformly in all Lie types, and are based on the concept of a root polynomial. In this paper we define formal root polyn...
Article
Let W be the Weyl group of a crystallographic root system acting on the associated weight lattice by reflections. In the present notes we extend the notion of an exponent of the W-action introduced in [Baek-Neher-Zainoulline, arXiv:1106.4332] to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel, Panin-Smirnov and the...
Article
Full-text available
Present notes can be viewed as an attempt to extend the notion of Schubert/Grothendieck polynomial to the context of an arbitrary algebraic oriented cohomology theory and, hence, of a commutative one-dimensional formal group law.
Article
Full-text available
In the erratum we correct a mistake (due to a wrong choice of basic polynomial invariants over Z[1/2]) in the original paper (v1). Using the correct basic polynomial invariants we improve our results and bounds on the annihilator. We also simplify some of the proofs.
Article
We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G of rank 2 over afield. We exhibit exceptional collections of the expected length for types A(2) and B-2 = C-2 and prove that no such collection exists for type G(2). This settles the question of the e...
Article
Full-text available
In the present paper we generalize the construction of the nil Hecke ring of Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel and Panin-Smirnov, e.g. to Chow groups, Grothendieck's K_0, connective K-theory, elliptic cohomology, and algebraic cobordism. The resulting object, which we call a formal (af...
Article
Full-text available
In the present paper we introduce and study the twisted γ-filtration on K0(Gs ), where Gs is a split simple linear algebraic group over a field k of characteristic prime to the order of the center of Gs . We apply this filtration to construct nontrivial torsion elements in γ-rings of twisted flag varieties.
Article
In the present paper we set up a connection between the indices of the Tits algebras of a simple linear algebraic group $G$ and the degree one parameters of its motivic $J$-invariant. Our main technical tool are the second Chern class map and Grothendieck's $\gamma$-filtration. As an application we recover some known results on the $J$-invariant of...
Article
Full-text available
Consider a crystallographic root system together with its Weyl group $W$ acting on the weight lattice $M$. Let $Z[M]^W$ and $S^*(M)^W$ be the $W$-invariant subrings of the integral group ring $Z[M]$ and the symmetric algebra $S^*(M)$ respectively. A celebrated theorem of Chevalley says that $Z[M]^W$ is a polynomial ring over $Z$ in classes of funda...
Article
Full-text available
In the present paper we generalise one Quillen's Lemma.
Article
Full-text available
We prove that rational injectivity holds for homotopy invariant pretheories defined over a smooth base scheme. As an application we get injectivity of etale cohomology. The purpose of this paper is to generalize one result of V.Voevodsky about pretheories (Corollary 4.19, [10]) to the relative case. Namely, let F be a homotopy invariant pretheory d...
Article
Full-text available
In the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a Rost cycle module and provide a version of Merkurjev's (equiv...
Article
Full-text available
Let X be the variety of Borel subgroups of a simple and strongly inner linear algebraic group G over a field k. We prove that the torsion part of the second quotient of Grothendieck's gamma-filtration on X is a cyclic group of order the Dynkin index of G. As a byproduct of the proof we obtain an explicit cycle that generates this cyclic group; we p...
Chapter
Full-text available
We provide a general algorithm used to prove purity for functors with transfers. As a basic example we consider the Witt group of an algebraic variety.
Article
Full-text available
We compute the essential dimension of Hermitian forms in the sense of O. Izhboldin. Apart from this we investigate the Chow motives of twisted incidence varieties and prove their incompressibility in dimensions 2 r − 1. Let W be an n-dimensional vector space over a field L which is a quadratic extension of some subfield F with char F ̸ = 2. Given a...
Article
Full-text available
In the present paper, we generalize the Quillen presentation lemma. As an application, for a given functor with transfers, we prove the exactness of its Gersten complex with support.
Article
Full-text available
This an extended version of the previous preprint dated by February 2005. We prove that the Chow motive of an anisotropic projective homogeneous variety of type F4 is isomorphic to the direct sum of twisted copies of a generalized Rost motive. In particular, we provide an explicit construction of a generalized Rost motive for a generically splittin...
Article
Full-text available
We prove that the function field of a variety which possesses a special correspondence in the sense of M. Rost preserves rationality of cycles of small codimensions. This fact was proven by Vishik in the case of quadrics and played the crucial role in his construction of fields with u-invariant 2r + 1. The main technical tools are the algebraic cob...
Article
Full-text available
This is an essentially extended version of the preprint dated by August 2005 (this includes now the varieties of types F_4, E_6 and E_7). Let k be a field of characteristic not 2 and 3. Let G be an exceptional simple algebraic group of type F_4, inner type E_6 or E_7 with trivial Tits algebras. Let X be a projective G-homogeneous variety. If G is o...
Article
Full-text available
Let M be a Chow motive over a field F. Let X be a smooth projective variety over F and N be a direct summand of the motive of X. Assume that over the generic point of X the motives M and N become isomorphic to a direct sum of twisted Tate motives. The main result of the paper says that if a morphism f : M → N splits over the generic point of X then...
Article
Full-text available
In the present notes we provide a new uniform way to compute a canonical p-dimension of a split algebraic group G for a torsion prime p using degrees of basic polynomial invariants described by V.Kac. As an application, we compute the canonical p-dimensions for all split exceptional algebraic groups.
Article
Full-text available
We give a complete classification of anisotropic projective homogeneous varieties of dimension less than 6 up to motivic isomorphism. We give several criteria for anisotropic flag varieties of type A_n to have isomorphic motives.
Article
Full-text available
Let G be an adjoint simple algebraic group of inner type. We express the Chow motive (with integral coefficients) of some anisotropic projective G-homogeneous varieties in terms of motives of simpler G-homogeneous varieties, namely, those that correspond to maximal parabolic subgroups of G. We decompose the motive of a generalized Severi-Brauer var...
Article
Full-text available
Let A and B be two central simple algebras of a prime degree n over a field F generating the same subgroup in the Brauer group. We show that the Chow motive of a Severi-Brauer variety SB(A) is a direct summand of the motive of a generalized Severi-Brauer variety SB_d(B) if and only if [A]=d[B] or [A]=-d[B] in the Brauer group. The proof uses method...
Article
Full-text available
We prove Knebusch’s Norm Principle for finite extensions of semi-local regular rings containing a field of characteristic 0. As an application we prove the Grothendieck-Serre conjecture on principal homogeneous spaces for the case of spinor groups of regular quadratic forms over a field of characteristic 0.
Article
Full-text available
We prove a version of Knebusch's Norm Principle for finite \'etale extensions of semi-local Noetherian domains with infinite residue fields of characteristic different from 2. As an application we prove Grothendieck's conjecture on principal homogeneous spaces for the spinor group of a quadratic space.
Article
Full-text available
For an oriented cohomology theory A and a relative cellular space X, we decompose the A-motive of X into a direct sum of twisted motives of the base spaces. We also obtain respective decompositions of the A-cohomology of X. Applying them, one can compute A(X), where X is an isotropic projective homogeneous variety and A means algebraic K-theory, mo...
Article
Let k be a field. We call W a smooth semi-local k-scheme if there exists a smooth affine k-scheme Y and finitely many closed points y1,..., yn on Y such that W is the inverse limit of all Zariski open neighbourhoods of {y1,..., yn} in Y. The objective of this paper is to show the following Theorem 1 Let W be a connected smooth semi-local scheme ove...
Article
Full-text available
In the present paper we discuss questions concerning the arithmetic resolution for etale cohomology. In particular, consider a smooth quasi-projective variety X over a field k together with the local scheme at a point x X. Let p : Y be a smooth projective morphism and let Y k(u) denote its fiber over the generic point of a subvariety u of . We prov...
Article
Full-text available
The present paper is devoted to the purity problem for functors endowed with a structure of transfer map. Namely let R be a local regular ring of geometric type and let \mathfrakF\mathfrak{F} be a covariant functor from the category of R-algebras to Abelian groups that has a structure of transfer map. We prove that purity holds for the functor...
Article
Full-text available
Let R be a local regular ring of geometric type and K be its field of fractions. Let F be a covariant functor from the category of R-algebras to abelian groups satisfying some additional properties (continuity, existence of well behaving transfer map). We show that the following equality holds for the subgroups of the group F(K): # p#SpecR,htp=1 im...
Article
Full-text available
The goal of this paper is to introduce some applications and consequences of the injectivity of the map F(R) ! F(K) induced by the canonical inclusion of a local regular ring R of geometric type to its field of fractions R ,! K, where F is a homotopy invariant functor with transfers. In particular, we get an original proof of a Grothendieck Conject...
Article
Full-text available
In the present paper we investigate the question about injectivity of the map F(R) ! F(K) induced by the canonical inclusion of a local regular ring of geometric type R to its field of fractions K for a homotopy invariant functor F with transfers. Typeset by A M S-T E X Let R be some local regular ring of geometric type, i.e. R is a local ring at s...
Article
Full-text available
Let G be an anisotropic linear algebraic group over a field F which splits by a field extension of a prime degree. Let X be a projective homogeneous G-variety such that G splits over the function field of X. We prove that under certain conditions the Chow motive of X is isomorphic to a direct sum of twisted copies of an indecomposable mo- tive RX....
Article
Full-text available
Chow motives. We define the category of Chow motives (over k) follow-ing [Ma68]. Fisrt, we define the category of correspondences (over k). Its objects are smooth projective varieties over k. For morphisms, called cor-respondences, we set Mor(X, Y ) := CH,(X 脳 Y ). The pseudo-abelian completion of the category of correspondences is called the categ...

Network

Cited By