加法范畴的Nil项消失与负K-理论

Non-connective K- and Nil-spectra of additive categorie

We present an elementary construction of the non-connective algebraic K-theory spectrum associated to an additive category following the contracted functor approach due to Bass. It comes with a universal property that easily allows us to identify it with other constructions, for instance with the one of Pedersen-Weibel in terms of Z^i-graded objects and bounded homomorphisms.

关于幂零群同调的一些注记

Some remarks on the homology of nilpotent groups

本文研究幂零群的同调与上同调，证明了其特定的消没结果，为相关理论提供注记。

Splitting the relative assembly map, Nil-terms and involutions

We show that the relative Farrell-Jones assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for algebraic K-theory is split injective in the setting where the coefficients are additive categories with group action. This generalizes a result of Bartels for rings as coefficients. We give an explicit description of the relative term. This enables us to show that it vanishes rationally if we take coefficients in a regular ring. Moreover, it is, considered as a Z[Z/2]-module by the involution coming from taking dual modules, an extended module and in particu…

Vanishing of negative $K$-theory in positive characteristic

We show how a theorem of Gabber on alterations can be used to apply work of Cisinski, Suslin, Voevodsky, and Weibel to prove that $K_n(X)[1/p] = 0$ for $n < - \dim X$ where $X$ is a quasi-excellent noetherian scheme, $p$ is a prime that is nilpotent on $X$, and $K_n$ is the $K$-theory of Bass-Thomason-Trobaugh. This gives a partial answer to a question of Weibel.

正则正合范畴与代数K-理论

Regular exact categories and algebraic K-theory

本文通过为环和正合范畴引入新的正则性概念，证明了Nil群的强消失定理，并构造出比Waldhausen类Cl更大的一类群，使其白头群均为平凡。

On nilpotent extensions of $\infty$-categories and the cyclotomic trace

We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $\infty$-categories) for additive $\infty$-categories, (2) define the notion of nilpotent extensions for suitable $\infty$-categories and furnish interesting examples such as categorical square-zero extensions, and (3) use (1) and (2) to extend the Dundas-Goodwillie-McCarthy theorem for stable $\infty$-categories which are not monogenically generated (such as the stable $\infty$-category of Voevodsky's motives or the stable $\infty$-category of perfect complexes on s…