### Directions:

• Equivariant oriented cohomology theories.
• Elliptic cohomology and Schubert calculus
• Representations of Hecke-type algebras and motives of twisted flag varieties.

Slides of talks/Lecture notes:

Papers (theory and techniques):

• Calmes, B.; Zainoulline, K.; Zhong, C. Equivariant oriented cohomology of flag varieties. Documenta Math. Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015), 113--144.

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 defining P and the formal group law of h. In two papers [Push-pull operators on the formal affine Demazure algebra and its dual, arXiv:1312.0019] and [A coproduct structure on the formal affine Demazure algebra, arXiv:1209.1676], we studied the properties of this dual and of some related operators by algebraic and combinatorial methods, without any reference to geometry. The present paper can be viewed as a companion paper, that justifies all the definitions of the algebraic objects and operators by explaining how to match them to equivariant oriented cohomology rings endowed with operators constructed using push-forwards and pull-backs along geometric morphisms. Our main tool is the pull-back to the T-fixed points of G/P which injects the cohomology ring in question into a direct product of a finite number of copies of the T-equivariant oriented cohomology of a point.

• Calmes, B., Zainoulline, K., Zhong, C. Push-pull operators on the formal affine Demazure algebra and its dual. Preprint 2013, 25pp.

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 multiplication and push-forward to a point. This paper can be viewed as the next step towards the `algebraization program' for equivariant oriented cohomology theories started in arXiv:0905.1341 and continued in arXiv:1208.4114 and arXiv:1209.1676; the general idea being to match cohomology rings of algebraic varieties and elements of classical interest in them (such as classes of Schubert varieties) with algebraic and combinatorial objects that can be introduced in the spirit of [Demazure, Invariants sym\'etriques entiers des groupes de Weyl et torsion, Invent. Math. 21:287-301, 1973] and [Kostant, Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G, Advances in Math. 62:187-237, 1986].

• Calmes, B., Zainoulline, K., Zhong, C. A coproduct structure on the formal affine Demazure algebra. To appear in Math. Zeitschrift, 27pp.

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.

• Hoffnung, A., Malagon-Lopez, J., Savage, A., Zainoulline, K. Formal Hecke algebras and algebraic oriented cohomology theories. Selecta Math. 20 (2014), no.4, 1213--1245.

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 (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and the 0-Hecke rings respectively. We also introduce a deformed version of the formal (affine) Demazure algebra, which we call a formal (affine) Hecke algebra. We show that the specialization of the formal (affine) Hecke algebra to the additive and multiplicative periodic formal group laws gives completions of the degenerate (affine) Hecke algebra and the usual (affine) Hecke algebra respectively. We show that all formal affine Demazure algebras (and all formal affine Hecke algebras) become isomorphic over certain coefficient rings, proving an analogue of a result of Lusztig.

• Calmes, B.; Petrov, V.; Zainoulline, K. Invariants, torsion indices and cohomology of complete flags. Ann. Sci. Ecole Norm. Sup. (4) 46 (2013), no.3, 36pp.

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 relating the so called formal group ring of the character group of T with the cohomology of the variety of Borel subgroups of G. The main result of the paper says that the kernel of this map is generated by W-invariant elements, where W is the Weyl group of G. As one of the applications we provide an algorithm (realized as a Macaulau2 package) which can be used to compute the ring structure of an oriented cohomology (algebraic cobordism, Morava $K$-theories, connective K-theory, Chow groups, K_0, etc.) of a complete flag variety.

• Gille, S.; Zainoulline, K. Equivariant pretheories and invariants of torsors. Transf. Groups 17 (2012), no.2, 471-498.

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 (equivariant K-theory) spectral sequence for such a theory. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information concerning the motivic J-invariant of E and in the case of Grothendieck's K_0 -- indexes of the respective Tits algebras.

• Nenashev, A.; Zainoulline, K. Oriented cohomology and motivic decompositions of relative cellular spaces. J. of Pure and Applied Algebra 205 (2006), no.2, 323-340.

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, motivic cohomology or algebraic cobordism MGL.

Applications to elliptic cohomology and Schubert calculus:

• Lenart, C., Zainoulline, K., Zhong, C. Parabolic Kazhdan-Lusztig basis, Schubert classes and equivariant elliptic cohomology. Preprint 2015, 17pp.
• Lenart, C., Zainoulline, K. A Schubert basis in equivariant elliptic cohomology. Preprint 2015, 17pp.
• Lenart, C., Zainoulline, K. Towards generalized cohomology Schubert calculus via formal root polynomials. Preprint 2014, 18pp.

Applications to motives and representations:

• Neshitov, A., Petrov, V., Semenov, N., Zainoulline, K. Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras. Preprint 2015, 21pp.

### Project: Topological and the gamma-filtration on twisted flag varieties. Applications to cohomological invariants and Chow motives.

Directions:

• Topological and gamma-filtration of twisted flag varieties.
• Applications to cohomological invariants (the Rost invariant)
• Applications to the torsion problem in Chow groups.

Slides of talks:

Papers:

• Merkurjev, A., Neshitov, A., Zainoulline, K. Cohomological Invariants in degree 3 and torsion in the Chow group of a versal flag. Compositio Math. 151, Issue 8, (2015), 1416--1432.

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 isomorphic to the group of indecomposable invariants of G. As an application, we construct nontrivial cohomological classes for indecomposable central simple algebras.

•
• Malagon-Lopez, J.; Zainoulline, K.; Zhong, C. Invariants, exponents and formal group laws. J. of Algebra 409, July 1, (2014), 244-264.

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 associated formal group law. From this point of view the classical Dynkin index of the associated Lie algebra will be the second exponent of the deformation map from the multiplicative to the additive formal group law. We apply this generalized exponent to study the torsion part of an arbitrary oriented cohomology theory of a twisted flag variety.

• Baek, S., Zainoulline, K., Zhong, C. On the torsion of Chow groups of twisted Spin-flags. Math. Research Letters 20 (2013), no.4, 601-614.

In the present paper we provide a uniform bound for the annihilators of the torsion of the Chow groups of the variety of Borel subgroups of a strongly inner linear algebraic group of orthogonal type.

• Garibaldi, S.; Zainoulline, K. The gamma-filtration and the Rost invariant. J. Reine und Angew. Math. 696 (2014), 225--244.

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 provide an upper bound for the torsion of the Chow group of codimension-3 cycles on X; we relate the generating cycle with the Rost invariant and the torsion of the respective generalized Rost motives; we use this cycle to obtain a uniform lower bound for the essential dimension of (almost) all simple linear algebraic groups.

• Zainoulline, K. Twisted gamma-filtration of a linear algebraic group. Compositio Math. 148 (2012), no.5, 1645-1654.

In the present notes we introduce and study the twisted gamma-filtration on K_0(G), where G is a split simple linear algebraic group over a field of characteristic prime to the order of the center of G. We apply this filtration to construct torsion elements in the gamma-ring of an inner form of the variety of Borel subgroups of G.

• Queguiner-Mathieu A.; Semenov, N.; Zainoulline, K. The J-invariant, Tits algebras and Triality. J. of Pure and Applied Algebra 216 (2012), 2614-2628.

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 quadratic forms of small dimension; we describe all possible values of the J-invariant of an algebra with orthogonal involution up to degree 8 and give explicit examples; we establish several relations between the J-invariant of an algebra A with orthogonal involution and the J-invariant of the corresponding quadratic form over the function field of the Severi-Brauer variety of A.

• Baek, S.; Neher, E.; Zainoulline, K. Basic polynomial invariants, fundamental representations and the Chern class map. Documenta Math. 17 (2012), 135--150.

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 fundamental representations w1,...,wn and S(M)^W over rational numbers is a polynomial ring in basic polynomial invariants q1,...,qn, where n is the rank. In the present paper we establish and investigate the relationship between wi's and qi's over the integers. As an application we provide an annihilator of the torsion part of the 3rd and the 4th quotients of the Grothendieck gamma-filtration on the variety of Borel subgroups of the associated linear algebraic group.

### Project: Motivic decompositions and algebraic cycles on twisted flag varieties

Directions:

• Motivic decomposition of generically split twisted flag varieties.
• Discrete motivic invariants of torsors. The J-invariant of a linear algebraic group.
• The canonical dimension of a linear algebraic group.

Maple package to work with algebraic cycles

Papers:

• Zainoulline, K. Degree formula for connective K-theory. Invent. Math. 179 (2010), no.3, 507-522.

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

• Zainoulline, K. Special correspondences and Chow traces of Landweber-Novikov operations. J. Reine und Angew. Math. 628 (2009), 195-204.

We prove that the function field of a variety which possesses a special correspondence in the sense of M. Rost preserves the 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 2^r+1. The main technical tools are algebraic cobordism of Levine-Morel, generalized Rost degree formula and divisibility of Chow traces of certain Landweber-Novikov operations. As a direct application of our methods we prove the Vishik's Theorem for all F_4-varieties.

• Semenov, N.; Zainoulline, K. Essential dimension of Hermitian spaces. Math. Annalen 346 (2009), no.2, 499-503.

Given an hermitian space we compute its essential dimension, Chow motive and prove its incompressibility in certain dimensions.

• Nikolenko, S.; Semenov, N.; Zainoulline, K. Motivic decompositions of anisotropic varieties of type F4 into generalized Rost motives. J. of K-theory 3 (2009), no.1, 85-102.

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 splitting variety for a symbol in K_3^M(k)/3. We also establish a motivic isomorphism between two anisotropic non-isomorphic projective homogeneous varieties of type F4. All our results hold for Chow motives with integral coefficients.

• Petrov, V.; Semenov, N.; Zainoulline, K. J-invariant of linear algebraic groups. Ann. Sci. Ecole Norm. Sup. (4) 41 (2008), no.6, 1023-1053.

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 forms. As a main application we obtain a uniform proof of all known motivic decompositions of generically split projective homogeneous varieties (Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians, G2- and F4-varieties) as well as provide new examples (exceptional varieties of types E6, E7 and E8). We also discuss relations with torsion indices, canonical dimensions and cohomological invariants of the group G.

• Vishik, A.; Zainoulline, K. Motivic Splitting Lemma. Documenta Math. 13 (2008), 81-96.

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 \to N splits over the generic point of X then it splits over F, i.e., N is a direct summand of M. We apply this result to various examples of motives of projective homogeneous varieties.

• Petrov, V.; Semenov, N.; Zainoulline, K. Zero-cycles on a twisted Cayley plane. Canadian Math. Bull. 51 (2008), no.1, 114-124.

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 of type E_7 we assume in addition that the respective parabolic subgroup is of type P_7. The main result of the paper says that the degree map on the group of zero cycles of X is injective.

• Zainoulline, K. Canonical p-dimensions of algebraic groups and degrees of basic polynomial invariants. London Math. Bull. 39 (2007), 301-304.

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.

• Calmes, B.; Petrov, V.; Semenov, N.; Zainoulline, K. Chow motives of twisted flag varieties. Compositio Math. 142 (2006), no.4, 1063-1080.

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 variety SB_2(A), where A is a division algebra of degree 5, into a direct sum of two indecomposable motives. As an application we provide another counter-example to the uniqueness of a direct sum decomposition in the category of motives with integral coefficients.

• Semenov, N.; Zainoulline, K. A classification of projective homogeneous varieties up to motivic isomorphism (English. Russian original). J. Math. Sciences 140 (2007), no.5, 692-698; transl. Zap. Nauchn. Semin. POMI 330 (2006), 158-172.

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.

### Project: The Grothendieck-Serre conjecture, the Purity and the Gersten conjecture.

(an old project related to my PhD thesis)

The conjecture has been recently proven by Fedorov and Panin

(see arXiv.org preprints http://arxiv.org/abs/1211.2678, http://arxiv.org/abs/1406.0241, http://arxiv.org/abs/1406.0247 and http://arxiv.org/abs/1406.1129)

Directions:

• Gersten resolutions, Gersten conjecture and Purity problem for presheaves with transfers
• Geometric presentation lemmas via general position arguments
• Grothendieck-Serre's conjecture on G-torsors.  (short overview on what was done before 2007)

Papers:

• Panin, I.; Zainoulline, K. Gersten resolution with support. Manuscripta Math. 128 (2009), no. 4, 443-452.

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.

• Ojanguren, M.; Panin, I.; Zainoulline, K. On the norm principle for quadratic forms. J. Ramanujan Math. Soc. 19 (2004), no.4, 1-12.

We prove a version of the Knebusch 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. (this result generalizes the previous paper)

• Zainoulline, K. On Knebusch's Norm Principle for quadratic forms over semi-local rings. Math. Zeitschrift 251 (2005), no.2, 415-425.

We prove the Knebusch 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.

• Panin, I.; Zainoulline, K. Variations on the Bloch-Ogus Theorem. Documenta Math. 8 (2003), 51-67.

In the present paper we discuss arithmetic resolutions for etale cohomology. Namely, consider a smooth quasi-projective variety X over a field k together with the local scheme U at a point x. Let Y be a smooth proper scheme over U. We prove there is the Gersten-type exact sequence for etale cohomology with coefficients in a locally constant etale sheaf F of Z/nZ-modules on Y which has finite stalks and (n,char(k))=1.

• Schmidt, A.; Zainoulline, K. Generic injectivity for etale cohomology and pretheories. J. of Algebra 263 (2003), no.2, 215-227.

Let W be a connected smooth semi-local scheme over a field k and let \eta be its generic point. Let X->W be a proper smooth morphism, n an integer prime to char(k) and K a complex of sheaves of Z/nZ full-modules on X_{et} whose cohomology sheaves are locally constant constructible and bounded below. In the present paper we show that for every integer q the canonical map H^q_{et}(X,K) -> H^q_{et}(X_\eta,K) is a universal monomorphism.

• Zainoulline, K. The Purity Theorem for functors with transfers. K-Theory J. 22 (2001), no.4, 303-333. (came out of PhD thesis)

Let R be a local regular ring obtained by localizing a smooth k-algebra A and let K be its field of fractions. Let F be a covariant functor from the category of A-algebras to abelian groups that satisfies some additional properties (continuity, existence of a well behaved transfer map). In the paper we show that the subgroup im(F(R) -> F(K)) of the group F(K) coincides with the intersection of the subgroups im(F(R_p) -> F(K)), where all maps are induced by the canonical inclusions and R_p runs through localizations of the local ring R at all prime ideals p of height 1.

• Zainullin, K. On Grothendieck's conjecture about principal homogeneous spaces for some classical algebraic groups. (English. Russian original) St. Petersburg Math. J. 12 (2001), no.1, 117-143; translation from Algebra i Analiz 12 (2000), no.1, 150-184. (came out of PhD thesis)

In the present paper we investigate 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 obtain the proof of Special Unitary Group case of the Grothendieck-Serre conjecture about principal homogeneous spaces.