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Finitely-generated modules over Special Biserial Algebras
: A combinatorial model using strips and belts

  • Luke A Kershaw

Student thesis: Doctoral Thesis › Doctor of Philosophy (PhD)

Abstract

We present a discrete model of band modules of special biserial (SB) algebras; complementing existing models for string modules. We provide efficient theoretical algorithms for calculating syzygies of band modules in terms of this discrete model, along with other functors relating to the delooping level for both string and band modules.

We first use these tools to prove some small results about the delooping levels of string and band modules for a general SB algebra; then we use these ideas specifically with radical-cube-zero SB algebras, where we show that all such algebras are either syzygy-finite or satisfy a very strict structural condition.

We also use these ideas to characterise, for a given SB algebra A, the string and band modules, M ∈ mod-A, with Ext1A(M, A) = 0, which (along with the syzygy algorithms for these models) can be used when the algebra has small dimension to identify all Gorenstein-projective modules. For example, we classify the Gorenstein-projective modules for a handful of example SB algebras of dimension ≤ 20. We build on this by classifying the Gorenstein-projective band modules for any SB algebra, and then we determine some sufficient and/or necessary conditions for certain Gorenstein-homological properties, including an equivalent condition for an SB algebra to have finitely many indecomposable Gorenstein-projective modules.
Date of Award3 Oct 2023
Original languageEnglish
Awarding Institution
  • University of Bristol
SupervisorJeremy C Rickard (Supervisor)

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