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Motivic structures in noncommutative geometry. Available at arXiv:1003.3210
 the Proceedings of the ICM
, 2010
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Generalized Schubert Calculus
"... Dedicated to C.S. Seshadri on the occasion of his 80th birthday In this paper we study the Tequivariant generalized cohomology of flag varieties using two models, the Borel model and the moment graph model. We study the differences between the Schubert classes and the BottSamelson classes. After s ..."
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Dedicated to C.S. Seshadri on the occasion of his 80th birthday In this paper we study the Tequivariant generalized cohomology of flag varieties using two models, the Borel model and the moment graph model. We study the differences between the Schubert classes and the BottSamelson classes. After setup of the general framework we compute, for classes of Schubert varieties of complex dimension � 3 in rank 2 (including A2,), moment graph representatives, PieriChevalley formulas and products of Schubert classes. These computations generalize the computations in equivariant Ktheory for rank 2 cases which are given in GriffethRam [GR]. B2, G2 and A (1)
NORM VARIETIES AND THE CHAIN LEMMA (AFTER MARKUS ROST)
"... The goal of this paper is to present proofs of two results of Markus Rost, the Chain Lemma 0.1 and the Norm Principle 0.3. These are the steps needed to complete the published verification of the BlochKato conjecture, that the norm residue maps are isomorphisms KM n (k)/p ≃ → Hn et(k, Z/p) for eve ..."
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The goal of this paper is to present proofs of two results of Markus Rost, the Chain Lemma 0.1 and the Norm Principle 0.3. These are the steps needed to complete the published verification of the BlochKato conjecture, that the norm residue maps are isomorphisms KM n (k)/p ≃ → Hn et(k, Z/p) for every prime p, every n and every field k containing 1/p. Throughout this paper, p is a fixed prime, and k is a field of characteristic 0, containing the pth roots of unity. We fix an integer n ≥ 2 and an ntuple (a1,..., an) of units in k, such that the symbol {a} = {a1,..., an} is nontrivial in the Milnor Kgroup KM n (k)/p. Associated to this data are several notions. A field F over k is a splitting field for {a} if {a}F = 0 in KM n (F)/p. A variety X over k is called a splitting variety if its function field is a splitting field; X is pgeneric if any splitting field F has a finite extension E/F of degree prime to p with X(E) ̸ = ∅. A norm variety for {a} is a smooth projective pgeneric splitting variety for {a} of dimension pn−1−1. The following sequence of theorems reduces the BlochKato conjecture to the
Cartier isomorphism and Hodge theory in the noncommutative case
 in Arithmetic geometry, Clay Math. Proc. 8, AMS
, 2009
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SHEAVES, GRADINGS, AND THE EXACT FUNCTOR THEOREM
"... MU → K induces an isomorphism K ∗ ⊗MU ∗ MU∗(X) → K∗(X). K homology is thus algebraically determined from complex cobordism ..."
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MU → K induces an isomorphism K ∗ ⊗MU ∗ MU∗(X) → K∗(X). K homology is thus algebraically determined from complex cobordism