K-theoretic obstructions to bounded t-structures

Benjamin Antieau; David Gepner; Jeremiah Heller

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arXiv · 1610.07207

K-theoretic obstructions to bounded t-structures

Benjamin Antieau·David Gepner·Jeremiah Heller

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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.

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BibTeX RIS

Benjamin Antieau, David Gepner, Jeremiah Heller. 2016-10-23. K-theoretic obstructions to bounded t-structures. https://arxiv.org/abs/1610.07207

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