$\Sigma$-pure-injectives in homotopy categories for gentle algebras

TL;DR

This paper characterizes indecomposable $ ext{Sigma}$-pure-injective objects in the homotopy category of complexes over gentle algebras, showing they are shifts of string or band complexes, using classification techniques and model theory tools.

Contribution

It provides a classification of $ ext{Sigma}$-pure-injective objects in the homotopy category of gentle algebras, connecting purity, triangulated categories, and model theory.

Findings

Indecomposable $ ext{Sigma}$-pure-injective objects are shifts of string or band complexes.

The paper adapts the functorial filtrations method for this classification.

It describes the compact objects in the homotopy category of gentle algebras.

Abstract

We consider the homotopy category of complexes of projective modules over any gentle algebra. We prove that indecomposable $\Sigma$-pure-injective objects in s must be shifts of string or band complexes. We begin with a survey of purity in compactly generated triangulated categories, recalling some characterisations of $\Sigma$-pure-injective objects that mimic classical results from the model theory of modules. We then specify to the aforementioned homotopy category, and describe the compact objects in this case. Our proof uses a recent adaptation of a classification technique known as the functorial filtrations method. One of the key steps in our employment of this method is an interpretation of the appropriate linear relations in terms of pp formulas in a canonical multi-sorted language.