Bounded $t$-structures, finitistic dimensions, and singularity categories of triangulated categories

Abstract

Recently, Amnon Neeman settled a bold conjecture by Antieau, Gepner, and Heller regarding the relationship between the regularity of finite-dimensional noetherian schemes and the existence of bounded $t$-structures on their derived categories of perfect complexes. In this paper, using different methods, we prove some very general results about the existence of bounded $t$-structures on (not necessarily algebraic or topological) triangulated categories and their invariance under completion. We show that if the opposite category of an essentially small triangulated category has finite finitistic dimension in our sense, then the existence of a bounded t-structure on it forces it to be equal to its completion. We also prove a parallel result regarding the equivalence of all bounded t-structures on any intermediate triangulated category between the starting category and its completion. Our general treatment, when specialized to the case of schemes, immediately gives us Neeman's theorem as an application and significantly generalizes another remarkable theorem by Neeman about the equivalence of bounded $t$-structures on the bounded derived categories of coherent sheaves. When specialized to other cases like associative rings, nonpositive DG-rings, connective $\mathbb{E}_1$-rings, triangulated categories without models, etc., we get many other applications. Under mild finiteness assumptions, these results not only give a categorical obstruction (the singularity category in our sense) to the existence of bounded $t$-structures on a triangulated category, but also provide plenty of triangulated categories on which all bounded $t$-structures are equivalent. The strategy used in our treatment is introducing a new concept of finitistic dimension for triangulated categories and lifting $t$-structures along completions of triangulated categories.