Cyclic homology for schemes

Using hypercohomology, we can extend cyclic homology from algebras to all schemes over a ring . By `extend' we mean that the usual cyclic homology of any commutative algebra agrees with the cyclic homology of its corresponding affine scheme.

     

Krichever Maps,Faà di Bruno Polynomials,and Cohomology in KP Theory

We study the geometrical meaning of the Faà di Bruno polynomials in the context of KP theory. They provide a basis in a subspace W of the universal Grassmannian associated to the KP hierarchy. When W comes from geometrical data via the Krichever map, the Faà di Bruno recursion relation turns out to be the cocycle condition for (the Welters hypercohomology group describing) the deformations of the dynamical line bundle on the spectral curve together with the meromorphic sections which give rise to the Krichever map. Starting from this, one sees that the whole KP hierarchy has a similar cohomological meaning.     

The Hodge Filtration and Cyclic Homology

We relate the Hodge filtration of the cohomology of a complex algebraic variety X to the Hodge decomposition of its cyclic homology. If X is smooth and projective, is the quotient of the Betti cohomology by the piece of the Hodge filtration.     

等变上同调与ech超上同调的同构定理

首先证明了任意一个等变微分流形都存在等变良好开覆盖,且等变良好开覆盖集所组成的集合在全部开覆盖组成的集合中共尾.在此基础上,证明了等变上同调与ech超上同调的同构.此定理可应用于实代数簇的Deligne上同调研究.     

Hasse's norm theorem forK2

In this paper, we prove Hasse's principle onK ₂ for a certain two-dimensional semi-global field. And, as a bi-product, we have the isomorphism theorem of the class field theory of such a field.