K-theoretic obstructions to bounded t-structures

By Benjamin Antieau, D. Gepner, J. Heller. Inventiones Mathematicae, 2016

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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 $$-1$$-1. The main results of this paper are that $$\mathrm {K}_{-1}(E)$$K-1(E) vanishes when E is a small stable $$\infty $$∞-category with a bounded t-structure and that $$\mathrm {K}_{-n}(E)$$K-n(E) vanishes for all $$n\geqslant 1$$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.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/46a345a8978b543e2491dda07159182a7a06b27b