K-theoretic obstructions to bounded t-structures

2016/10/23 by Benjamin Antieau, David Gepner, Antieau, Benjamin +3 · 1 citation
Mathematics · #16E45 #18E30 #19D35 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1610.07207

openalex publication_date 2016/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∞-category with a bounded t-structure and that K_-n(E) vanishes for all n≥ 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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