Filtrations and Homological degrees of FI-modules

Abstract

Let $k$ be a commutative Noetherian ring. In this paper we consider filtered modules of the category FI firstly introduced by Nagpal. We show that a finitely generated FI-module $V$ is filtered if and only if its higher homologies all vanish, and if and only if a certain homology vanishes. Using this homological characterization, we characterize finitely generated FI-modules $V$ whose projective dimension is finite, and describe an upper bound for it. Furthermore, we give a new proof for the fact that $V$ induces a finite complex of filtered modules, and use it as well as a result of Church and Ellenberg to obtain another upper bound for homological degrees of $V$.

Minor changes following the suggestion of the referee

References in corpus (4)

• FI-modules and the cohomology of modular representations of symmetric groups
• Representation Stability
• Homological Invariants of FI-modules and FI_G-modules
• Homological dimensions of crossed products

Cited by in corpus (8)

• 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
• Bounding the homology of FI-modules
• Two homological proofs of the Noetherianity of $F{I}_G$
• Generalized Representation Stability and FI_d-modules