Algebras with non-periodic bounded modules

TL;DR

This paper investigates weakly symmetric special biserial algebras of infinite representation type, showing that most have non-periodic bounded modules unless their Brauer graph is a tree without multiple edges, impacting Hochschild cohomology properties.

Contribution

It demonstrates that most such algebras possess non-periodic bounded modules, except when their Brauer graph is a tree with no multiple edges, linking module properties to graph structure.

Findings

Most weakly symmetric special biserial algebras have non-periodic bounded modules.

Algebras with a tree-like Brauer graph lack non-periodic bounded modules.

Presence of non-periodic bounded modules implies Hochschild cohomology does not satisfy (Fg).

Abstract

We study weakly symmetric special biserial algebras of infinite representation type. We show that usually some socle deformation of such an algebra has non-periodic bounded modules. The exceptions are precisely the algebras whose Brauer graph is a tree with no multiple edges. If the algebra has a non-periodic bounded module then its Hochschild cohomology cannot satisfy the finite generation property (Fg).