# Kirill ZainoullineUniversity of Ottawa · Department of Mathematics and Statistics

Kirill Zainoulline

Dr. rer. nat., PhD

## About

72

Publications

2,691

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691

Citations

Introduction

Additional affiliations

May 2015 - present

August 2009 - April 2015

October 2006 - July 2009

Education

August 2009 - August 2009

November 1997 - April 2000

**Steklov Mathematical Institute at St.Petersburg**

Field of study

- Mathematics

## Publications

Publications (72)

We apply the degree formula for connective $K$-theory to study rational
contractions of algebraic varieties. Examples include rationally connected
varieties and complete intersections.

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...

. 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...

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...

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...

We prove that the structure algebra of a Bruhat moment graph of a finite real root system is a Hopf algebroid with respect to the Hecke and the Weyl actions. We introduce new techniques (reconstruction and push-forward formula of a product, twisted coproduct, double quotients of bimodules) and apply them together with our main result to linear alge...

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...

We produce a short and elementary algorithm to compute an upper bound for the canonical dimension of a spit semisimple linear algebraic group. Using this algorithm we confirm previously known bounds by Karpenko [6–9] and Devyatov [5] as well as we produce new bounds (e.g., for groups of types $F_4$, adjoint $E_6$, for some semisimple groups).

We produce a short and elementary algorithm to compute an upper bound for the canonical dimension of a spit semisimple linear algebraic group. Using this algorithm we confirm previously known bounds by Karpenko and Devyatov as well as we produce new bounds (e.g. for groups of types $F_4$, adjoint $E_6$, for some semisimple groups).

We define a diagrammatic monoidal category, together with a full and essentially surjective monoidal functor from this category to the category of modules over the exceptional Lie algebra of type $F_4$. In this way, we obtain a set of diagrammatic tools for studying type $F_4$ representation theory that are analogous to those of the oriented and un...

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 k of characteristic 0. More precisely, we extend such operations to the respective T-equivariant (T is a maximal spli...

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...

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...

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...

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...

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...

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...

In the present paper we study twisted foldings of root systems which generalize usual involutive foldings corresponding to automorphisms of Dynkin diagrams. Our motivating example is the Lusztig projection of the root system of type $E_8$ onto the subring of icosians of the quaternion algebra which gives the root system of type $H_4$. Using moment...

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 ${\...

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...

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...

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...

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.

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...

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...

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...

International audience
An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Graham-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. We define f...

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...

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...

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...

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.

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.

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...

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...

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.

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...

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...

In the present paper we generalise one Quillen's Lemma.

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...

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...

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...

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.

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...

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.

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...

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...

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...

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...

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.

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.

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...

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...

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.

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.

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...

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...

Let R be a semi-local regular ring of geometric type over a field k . Let \mathcal U=\operatorname{Spec} R be the semi-local scheme. Consider a smooth proper morphism p:Y \to \mathcal U . Let Y_{k(u)} be the fiber over the generic point of a subvariety u of \mathcal U . We prove that the Gersten-type complex for étale cohomology
0\to H_{\mathrm{ét}...

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...

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...

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...

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...

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....

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...