Bounded t-structures, finitistic dimensions, and singularity categories of triangulated categories
arxiv(2023)
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, 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. 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 (not necessarily commutative) rings, nonpositive DG-rings,
connective 𝔼_1-rings, triangulated categories without models, etc.,
we get many other applications. Under mild finiteness assumptions, these
results give a categorical obstruction, the singularity category in our sense,
to the existence of bounded t-structures on a triangulated category. The two
key tools used in our treatment are the finitistic dimension for a triangulated
category (a new concept introduced in the paper) and lifting t-structures
along completions of triangulated categories.
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