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

Abstract

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 some algebraic stacks). The key input in our paper is Bondarko's notion of weight structures which provides a "ring-with-many-objects" analog of a connective ${\mathbb{E}}_1$-ring spectrum. As applications, we prove cdh descent results for truncating invariants of stacks extending the work of Hoyois-Krishna for homotopy $K$-theory, and establish new cases of Blanc's lattice conjecture.

arXiv comment38 pages; final version; to appear in IMRN

References in corpus (8)

• On the algebraic K-theory of higher categories
• Weights for relative motives; relation with mixed complexes of sheaves
• Algebraic groups and compact generation of their derived categories of representations
• Differential Graded Categories are k-linear Stable Infinity Categories
• The étale local structure of algebraic stacks
• Categorical Milnor squares and K-theory of algebraic stacks
• Weight structures and the algebraic $K$-theory of stable $\infty$-categories
• Nilpotent invariance of semi-topological K-theory of dg-algebras and the lattice conjecture

Cited by in corpus (1)

• Categorical Milnor squares and K-theory of algebraic stacks