微分K理论中的λ-环结构

Cohomology ring of differential operator rings

We compute the multiplicative structure in the Hocshchild cohomology ring of a differential operators ring and the cap product of Hochschild cohomology on the Hochschild homology.

微分分次环的 Ore 局部化；迈向微分分次代数的 Goldie 定理

Ore Localisation for differential graded rings; Towards Goldie's theorem for differential graded algebras

本文研究微分分次代数的 Ore 局部化，定义 dg-素环、dg-半素环、dg-幂零根、dg-本质子模与 dg-一致维数，并推广 Goldie 定理至素 dg-环。

Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory

We develop differential algebraic K-theory for rings of integers in number fields and we construct a cycle map from geometrized bundles of modules over such a ring to the differential algebraic K-theory. We also treat some of the foundational aspects of differential cohomology, including differential function spectra and the differential Becker-Gottlieb transfer. We then state a transfer index conjecture about the equality of the Becker-Gottlieb transfer and the analytic transfer defined by Lott. In support of this conjecture, we derive some non-trivial consequences which are provable by indep…

Relative Milnor $K$-groups and differential forms of split nilpotent extensions

Let $R$ be a commutative ring and $I\subset R$ be a nilpotent ideal such that the quotient $R/I$ splits out of $R$ as a ring. Let $N$ be a natural number such that ${I^N=0}$. We establish a canonical isomorphism between the relative Milnor $K$-group $K^{M}_{n+1}(R,I)$ and the quotient of the relative module of differential forms $Ω^n_{R,I}/d\,Ω^{n-1}_{R,I}$ assuming that $N!$ is invertible in $R$ and that the ring $R$ is weakly $5$-fold stable. The latter means that any $4$-tuple of elements in $R$ can be shifted by an invertible element to become a $4$-tuple of invertible elements.

Differential algebras with Banach-algebra coefficients I: From C*-algebras to the K-theory of the spectral curve

We present an operator-coefficient version of Sato's infinite-dimensional Grassmann manifold, and tau-function. In this context, the Burchnall-Chaundy ring of commuting differential operators becomes a C*-algebra, to which we apply the Brown-Douglas-Fillmore theory, and topological invariants of the spectral ring become readily available. We construct KK classes of the spectral curve of the ring and, motivated by the fact that all isospectral Burchnall-Chaundy rings make up the Jacobian of the curve, we compare the (degree-1) K-homology of the curve with that of its Jacobian. We show how the B…