K-theoretic obstructions to bounded t-structures

TL;DR

This paper proves vanishing results for negative K-groups of small stable ∞-categories with bounded t-structures, extending Schlichting's conjecture and applying to various algebraic and geometric contexts.

Contribution

It establishes new vanishing theorems for negative K-theory in the setting of stable ∞-categories with bounded t-structures, generalizing previous results.

Findings

K_{-1}(E) vanishes for small stable ∞-categories with bounded t-structures

K_{-n}(E) vanishes for all n≥1 when the heart is noetherian

Barwick's theorem of the heart holds for nonconnective K-theory spectra under these conditions

Abstract

Schlichting conjectured that the negative K-groups of small abelian categories vanish and proved this for noetherian abelian categories and for all abelian categories in degree $-1$. The main results of this paper are that $K_{-1}(E)$ vanishes when $E$ is a small stable $\infty$-category with a bounded t-structure and that $K_{-n}(E)$ vanishes for all $n\geq 1$ when additionally the heart of $E$ is noetherian. It follows that Barwick's theorem of the heart holds for nonconnective K-theory spectra when the heart is noetherian. We give several applications, to non-existence results for bounded t-structures and stability conditions, to possible K-theoretic obstructions to the existence of the motivic t-structure, and to vanishing results for the negative K-groups of a large class of dg algebras and ring spectra.