Homological degrees of representations of categories with shift functors

Abstract

Let $k$ be a commutative Noetherian ring and $\underline{\mathcal{C}}$ be a locally finite $k$-linear category equipped with a self-embedding functor of degree 1. We show under a moderate condition that finitely generated torsion representations of $\underline{\mathcal{C}}$ are super finitely presented (that is, they have projective resolutions each term of which is finitely generated). In the situation that these self-embedding functors are genetic functors, we give upper bounds for homological degrees of finitely generated torsion modules. These results apply to quite a few categories recently appearing in representation stability theory. In particular, when $k$ is a field of characteristic 0, we obtain another upper bound for homological degrees of finitely generated $\mathrm{FI}$-modules.

Major changes include: A stronger upper bound for homological degrees of torsion modules; a new proof of the Koszulity of categories with shift functors; a new upper bound for homological degrees of FI-modules, etc

References in corpus (6)

• Representation stability and finite linear groups
• Homology of FI-modules
• FI-modules and the cohomology of modular representations of symmetric groups
• Noetherian property of infinite EI categories
• Koszulity of directed categories in representation stability theory
• A remark on FI-module homology

Cited by in corpus (11)

• A long exact sequence for homology of FI-modules
• Homological Invariants of FI-modules and FI_G-modules
• Upper bounds of homological invariants of $F{I}_G$-modules
• Depth and the Local Cohomology of FI_G-modules
• On the degree-wise coherence of FI_G-modules
• Filtrations and Homological degrees of FI-modules
• Two homological proofs of the Noetherianity of $F{I}_G$
• FI${}^m$-modules over Noetherian rings
• Local cohomology and the multi-graded regularity of FI${}^m$-modules
• A Survey of Representation Stability Theory
• An inductive machinery for representations of categories with shift functors