[Paper Review] On nilpotent extensions of $\infty$-categories and the cyclotomic trace

Elden Elmanto, Vladimir Sosnilo|arXiv (Cornell University)|Oct 19, 2020

Homotopy and Cohomology in Algebraic Topology44 references4 citations

TL;DR

This paper extends the Dundas-Goodwillie-McCarthy (DGM) theorem on the cyclotomic trace to stable ∞-categories that are not monogenically generated, using nilpotent extensions and weight structures. By introducing nilpotent extensions in additive ∞-categories and leveraging Bondarko’s weight structures, the authors establish cdh descent for truncating invariants on stacks and prove new cases of Blanc’s lattice conjecture for derived stacks over ℂ.

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.

Motivation & Objective

• To generalize the Dundas-Goodwillie-McCarthy theorem beyond ring spectra to stable ∞-categories lacking a single compact generator.
• To define and study nilpotent extensions in additive ∞-categories, including categorical square-zero extensions, as a generalization of classical nilpotent ideals.
• To establish cdh descent for truncating invariants on algebraic stacks using the extended DGM theorem.
• To prove new cases of Blanc’s lattice conjecture for derived stacks over ℂ via the developed framework.
• To unify various K-theory invariants (direct sum, Waldhausen, BGT) in the context of weight hearts of stable ∞-categories.

Proposed method

• Introduce the notion of nilpotent extensions in additive ∞-categories, generalizing ideal-based nilpotence in ring spectra.
• Use Bondarko’s weight structures to model 'ring-with-many-objects' analogs of connective E1-ring spectra.
• Construct localization sequences in additive ∞-categories via Verdier quotients and idempotent completion.
• Apply Morita theory and ind-completions to relate additive ∞-categories to stable ∞-categories with compact generators.
• Establish the DGM theorem for nilpotent extensions in additive ∞-categories by reducing to known cases via weight structures.
• Use cdh descent and Nisnevich descent techniques to extend invariants to stacks, particularly derived and ANS stacks.

Experimental results

Research questions

• RQ1 Can the DGM theorem on the cyclotomic trace be extended to stable ∞-categories that are not monogenically generated?
• RQ2 What is the appropriate generalization of nilpotent ideals in ∞-categories, and how can such extensions be defined and studied?
• RQ3 Does cdh descent hold for truncating invariants on algebraic stacks beyond the classical case?
• RQ4 Under what conditions does Blanc’s lattice conjecture hold for derived stacks over ℂ?
• RQ5 How do different K-theory invariants (direct sum, Waldhausen, BGT) relate in the context of weight hearts of stable ∞-categories?

Key findings

• The DGM theorem is extended to nilpotent extensions in additive ∞-categories, providing a framework for non-monogenic stable ∞-categories.
• The authors prove cdh excision for truncating invariants on derived stacks satisfying certain geometric conditions, including quotient stacks and ANS stacks.
• New cases of Blanc’s lattice conjecture are verified for derived stacks of the form [Y/G] with Y affine and G reductive, and for smooth ANS stacks over ℂ.
• For boundedly weighted stable ∞-categories, the direct sum, Waldhausen, and BGT K-theory invariants on the weight heart are canonically equivalent.
• The connective K-theory of the weight heart is shown to be a connective cover of the nonconnective K-theory of the full category.
• The framework enables the application of trace methods to Voevodsky’s motives and perfect complexes on algebraic stacks, previously outside the scope of classical DGM theorems.