Finitistic dimension via modules over the singularity category
Abstract
We combine results of Rickard and Shaul with methods of intrinsic homological algebra and the theory of purity to prove that the finiteness of the big finitistic dimension of an Artin algebra is an intrinsic property of the singularity category of its opposite algebra, via its category of modules. This `object-free' approach complements work of Dey--Šťovíček and arrives at the same conclusion: the finiteness of the finitistic dimension of Artin algebras is a singular invariant (of the opposite algebras). In fact, our characterisation can be used to prove that finite finitistic dimension descends along certain fully faithful functors between singularity categories. We include an appendix where we explain how our methods can be used to prove non-existence of bounded t-structures on singularity categories.
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BibTeX
@misc{kostas2026finitistic,
title = {{Finitistic dimension via modules over the singularity category}},
author = {Panagiotis Kostas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.15482},
eprint = {2608.15482},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.15482},
}
RIS
TY - UNPB
TI - Finitistic dimension via modules over the singularity category
AU - Panagiotis Kostas
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2608.15482
UR - https://arxiv.org/abs/2608.15482
AB - We combine results of Rickard and Shaul with methods of intrinsic homological algebra and the theory of purity to prove that the finiteness of the big finitistic dimension of an Artin algebra is an intrinsic property of the singularity category of its opposite algebra, via its category of modules. This `object-free' approach complements work of Dey--Šťovíček and arrives at the same conclusion: the finiteness of the finitistic dimension of Artin algebras is a singular invariant (of the opposite algebras). In fact, our characterisation can be used to prove that finite finitistic dimension descends along certain fully faithful functors between singularity categories. We include an appendix where we explain how our methods can be used to prove non-existence of bounded t-structures on singularity categories.
ER -