Recollements and Gorenstein projective modules for gentle algebras

TL;DR

This paper establishes a bijection between certain Gorenstein projective modules over gentle algebras and specific recollements, revealing how tensor functors interact with these modules under particular cycle conditions.

Contribution

It introduces a novel correspondence between Gorenstein projective modules and recollements in gentle algebras, detailing the behavior of tensor functors in this context.

Findings

Bijection between non-projective indecomposable Gorenstein projective modules and special recollements.

Tensor functor preserves Gorenstein projective modules under cycle conditions.

Characterization of Gorenstein projective modules via cycle properties in gentle algebras.

Abstract

Let $A={\rm \mathbb{k}}Q/\mathcal{I}$ be a gentle algebra. We provide a bijection between non-projective indecomposable Gorenstein projective modules over $A$ and special recollements induced by an arrow $a$ on any full-relational oriented cycle $\mathscr{C}$, which satisfies some interesting properties, for example, the tensor functor $-\otimes_A A/A\varepsilon A$ sends Gorenstein projective module $aA$ to an indecomposable projective $A/A\varepsilon A$-module; and $-\otimes_A A/A\varepsilon A$ preserves Gorenstein projective objects if any two full-relational oriented cycles do not have common vertex.