Homology of complexes over finite tensor categories

TL;DR

This paper extends a recent result to finite tensor categories with finitely generated cohomology, demonstrating the existence of infinitely many non-isomorphic complexes with small homology under certain conditions.

Contribution

It generalizes Carlson's result to broader categories and establishes new conditions for the existence of complexes with specific properties.

Findings

Existence of infinitely many non-isomorphic complexes with small homology

Results apply to finite tensor categories with large cohomology Krull dimension

Includes a version for finite dimensional algebras with finitely generated cohomology

Abstract

We generalize a recent result by J.F. Carlson to finite tensor categories having finitely generated cohomology. Specifically, we show that if the Krull dimension of the cohomology ring is sufficiently large, then there exist infinitely many non-isomorphic and nontrivial bounded complexes of projective objects, and with small homology. We also prove a version for finite dimensional algebras with finitely generated cohomology.