Functorial filtrations for homotopy categories of some generalisations of gentle algebras

TL;DR

This paper classifies complexes in the homotopy category of certain gentle algebras over a noetherian ring, using functorial filtrations to describe indecomposables and verify their uniqueness.

Contribution

It introduces a functorial filtration approach to classify indecomposable complexes in the homotopy category of generalized gentle algebras over a noetherian ring.

Findings

Complexes with finitely-generated components decompose into string and band complexes.

The classification of indecomposables is achieved using functorial filtrations.

The Krull-Remak-Schmidt-Azumaya property is verified for these categories.

Abstract

We consider algebras defined over a complete, local and noetherian ground ring. They are gentle algebras in case the ground ring is a field. The unbounded homotopy category of complexes of projective modules is considered. Complexes with finitely-generated homogeneous components are shown to be isomorphic to direct sums of indecomposable string and band complexes. The corresponding isoclasses are described, and the Krull-Remak-Schmidt-Azumaya property is verified. This classification problem is solved using the idea of functorial filtrations.