arXiv subjects
Publications and source records attributed to Changlong Zhong.
We introduce Milnor-Witt $K$-groups of local rings and show that the $n$th Milnor-Witt $K$-group of a local ring $R$ which contains an infinite field of characteristic not $2$ is the pull-back of the $n$th power of the fundamental ideal in the Witt ring of $R$ and the $n$th Milnor $K$-group of $R$ over the $n$th Milnor $K$-group of $R$ modulo $2$. This generalizes the work of Morel-Hopkins on Milnor-Witt $K$-groups of fields.
For any formal group law, there is a formal affine Hecke algebra defined by Hoffnung, Malagón-López, Savage, and Zainoulline. Coming from this formal group law, there is also an oriented cohomology theory. We identify the formal affine Hecke algebra with a convolution algebra coming from the oriented cohomology theory applied to the Steinberg variety. As a consequence, this algebra acts on the corresponding cohomology of the Springer fibers. This generalizes the action of classical affine Hecke algebra on the $K$-theory of the Springer fibers constructed by Lusztig. We also give a residue interpretation of the formal affine Hecke algebra, which coincides with the residue construction of Ginzburg, Kapranov, and Vasserot when the formal group law comes from a 1-dimensional algebraic group.
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 symétriques 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].
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.
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.
We study the action of the formal affine Hecke algebra on the formal group algebra, and show that the the formal affine Hecke algebra has a basis indexed by the Weyl group as a module over the formal group algebra. We also define a concept called the normal formal group law, which we use to simplify the relations of the generators of the formal affine Demazure algebra and the formal affine Hecke algebra.
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.
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 study the characteristic map of algebraic oriented cohomology of complete spin-flags and the ideal of invariants of formal group algebra. As an application, we provide an annihilator of the torsion part of the $γ$-filtration. Moreover, if the formal group law determined by the oriented cohomology is congruent to the additive formal group law modulo 2, then at degree 2 and 3, the $γ$-filtration of complete spin-flags is torsion free.
We prove that there is a map from Bloch's cycle complex to Kato's complex of Milnor K-theory, which induces a quasi-isomorphism from étale sheafified cycle complex to the Gersten complex of logarithmic de Rham--Witt sheaves. Next we show that the truncation of Bloch's cycle complex at -3 is quasi-isomorphic to Spiess' dualizing complex.