K-theoretic obstructions to bounded t-structures

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.