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arXiv · 2010.09155

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

Elden Elmanto·Vladimir Sosnilo
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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.

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BibTeXRIS

Elden Elmanto, Vladimir Sosnilo. 2020-10-19. On nilpotent extensions of $\infty$-categories and the cyclotomic trace. https://arxiv.org/abs/2010.09155

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