Algebras of operations in K-theory

Operations in connective K-theory

In this article we classify additive operations in connective K-theory with various torsion-free coefficients. We discover that the answer for the integral case requires understanding of the $\hat{\mathbb{Z}}$ one. Moreover, although integral additive operations are topologically generated by Adams operations, these are not reduced to infinite linear combinations of the latter ones. We describe a topological basis for stable operations and relate it to a basis of stable operations in graded K-theory. We classify multiplicative operations in both theories and show that homogeneous additive stab…

A Fundamental Theorem for the $K$-theory of connective $\mathbb{S}$-algebras

Invoking the density argument of Dundas-Goodwillie-McCarthy, we extend the Fundamental Theorem of $K$-theory from the category of simplicial rings to the category of $\mathbb{S}$-algebras. As an intermediate step, we prove the Fundamental Theorem for simplicial rings appealing to recent results from the first author's thesis. This recovers as a special case the Fundamental Theorem for the $K$-theory of spaces appearing in Hüttemann-Klein-Vogell-Waldhausen-Williams.

Fundamental theorems for the K-theory of ${\bf S}$-algebras, I: the connective case

We extend the Bass-Quillen Fundamental Theorem of Algebraic K-theory to the Waldhausen K-theory of connective ${\bf S}$-algebras. The same technique used in this extension also yields two localization theorems for the relative K-theory of a 1-connected morphism of connective ${\bf S}$-algebras.

再论连通K-理论谱的上同调

The cohomology of the connective spectra for {K}-theory revisited

针对连通实与复K-理论谱的稳定模2上同调现有基表示复杂的问题，本文基于Adem关系，用Steenrod运算的可容许序列构造了更简单的基，并明确了复情形基的次数限制条件。

Connective C*-algebras

Connectivity is a homotopy invariant property of separable C*-algebras which has three notable consequences: absence of nontrivial projections, quasidiagonality and a more geometric realization of KK-theory for nuclear C*-algebras using asymptotic morphisms. The purpose of this paper is to further explore the class of connective C*-algebras. We give new characterizations of connectivity for exact and for nuclear separable C*-algebras and show that an extension of connective separable nuclear C*-algebras is connective. We establish connectivity or lack of connectivity for C*-algebras associated…

连通K-理论与Adams运算

Connective K-theory and Adams operations

该研究探讨代数簇的凝聚模Grothendieck群与Chow群的挠现象关联，借助Adams运算在相关谱序列及连通K₀-理论上的作用分析，以初等论证完成两类代数对象的连通K₀-理论计算。