Showing papers by "Kirill Zainoulline published in 2014"

Equivariant oriented cohomology of flag varieties

Abstract: 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.

Formal Hecke algebras and algebraic oriented cohomology theories

Abstract: In the present paper, we generalize the construction of the nil Hecke ring of Kostant–Kumar to the context of an arbitrary formal group law, in particular, to 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 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.

Towards generalized cohomology Schubert calculus via formal root polynomials

Abstract: 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. In this paper we define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory). We focus on the case of the hyperbolic formal group law (corresponding to elliptic cohomology). We study some of the properties of formal root polynomials. We give applications to the efficient computation of the transition matrix between two natural bases of the formal Demazure algebra in the hyperbolic case. As a corollary, we rederive in a transparent and uniform manner the formulas of Billey and Graham-Willems. We also prove the corresponding formula in connective $K$-theory, which seems new, and a duality result in this case. Other applications, including some related to the computation of Bott-Samelson classes in elliptic cohomology, are also discussed.

The γ-filtration and the Rost invariant

Abstract: 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 γ-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 θ 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 cycle θ with the Rost invariant and the torsion of the respective generalized Rost motives; we use θ to obtain a uniform lower bound for the essential dimension of (almost) all simple linear algebraic groups. Grothendieck’s celebrated γ-filtration of the ring K0(X) gives a way to estimate the Chow group CH(X) of algebraic cycles on X modulo rational equivalence when X is a smooth projective variety over a field k. Namely, by the RiemannRoch theorem without denominators [8, §15.3] the i-th Chern class provides a well-defined group homomorphism ci : γ (X) → CH(X), i ≥ 0 from the i-th quotient of the γ-filtration to the Chow group of codimension-i cycles onX . Observe that for i = 0 it is the identity map and for i = 1 it is an isomorphism identifying CH(X) with the Picard group of X . In the present paper we study these homomorphisms in the cases i = 2, 3 and X is a generically split projective homogeneous variety under a semisimple linear algebraic group G. Our core determines and bounds respectively the torsion subgroup of γ(B) and γ(B) for the variety of Borel subgroups B of strongly inner G (Theorem 3.1). For instance, we show that the torsion subgroup Torsγ(B) is cyclic of order the Dynkin index ofG and exhibit a generator θ for it (Definition 3.3). This fact together with the Riemann-Roch theorem imply (see §4) that the surjection 〈θ〉 = Tors γ(B) c2 // // TorsCH(B) can be viewed as a substitute of the key map Q(V ) → H(k,Q/Z(2)) in the definition of the Rost invariant [10, pp. 126-127]. Indeed, a theorem of PeyreMerkurjev [27] shows that TorsCH(B) can be identified with the kernel of the restriction H(k,Q/Z(2)) → H(k(B),Q/Z(2)). Furthermore, the order of c2(θ) in TorsCH(B) equals to the order of the Rost invariant of G (see Prop. 3.2). Our result gives bounds for the torsion in CH for generically split X (see §5) and provides explicit generators of torsion subgroups of CH of certain generalized Rost-Voevodsky motives. Note that typically, one does not even know a priori if the torsion subgroup of CH(X), i ≥ 3, is finitely generated. However, determining the torsion subgroup determines CH(X) as an abelian group, since the dimension of its free part CH(X)⊗Q can be easily computed.

Erratum to: Formal Hecke algebras and algebraic oriented cohomology theories

Abstract: Proposition 6.8(d) contains a few sign errors. We thank Marc-Antoine Leclerc for bringing this to our attention. The correct statement is as follows: If 〈α∨ i , α j 〉 = −1 and 〈α∨j , αi 〉 = −3 so that mi j = 6, then Δ j i j i j i −Δi j i j i j = Δi j i j (κ j,i + κ2i+3 j,−i−2 j + κ−i−3 j,i+2 j + κi+2 j,− j ) −Δ j i j i (κi, j + κ−2i−3 j,i+2 j + κ−i−2 j,i+3 j + κi+ j, j ) +Δ j i j ( i (κi, j + κ−2i−3 j,i+2 j + κ−i−2 j,i+3 j + κi+ j, j ) )

Chow motives of generically split varieties

Abstract: 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 motive RX. This covers all known examples of motivic decompositions of generically split projective homogeneous varieties (Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians) as well as provides new ones (exceptional varieties of types E6 and E8).

On formal Schubert polynomials

Abstract: 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.