Resolving subcategories for gentle algebras I: Monogeneous resolving subcategories for gentle trees

TL;DR

This paper advances the understanding of resolving subcategories in gentle algebras, especially for gentle quivers with tree structures, by improving algorithms and characterizing monogeneous resolving subcategories.

Contribution

It refines an existing algorithm for resolving closure calculations and characterizes monogeneous resolving subcategories for gentle quivers with tree structures.

Findings

Improved algorithm for resolving closure calculations.

Computed resolving subcategories for gentle quivers with finite global dimension.

Characterized monogeneous resolving subcategories as join-irreducible elements.

Abstract

This paper is the first part of a series that intends to study the resolving subcategories for gentle algebras over an algebraically closed field $\mathbb{K}$. In a general setting, we improve the precision of an algorithm from Takahashi for resolving closure calculations in well-behaved abelian categories. Then, we modify the geometric model of Baur--Coelho-Sim\~oes and Opper--Plamondon--Schroll to compute such subcategories for gentle quivers that have a finite global dimension. Finally, we focus on gentle quivers $(Q,R)$ such that $Q$ is a directed tree, and we study the monogeneous resolving subcategories, which are the ones generated by a single non-projective indecomposable $\mathbb{K}Q/\langle R \rangle$-module. By the way, we prove that these subcategories are the join-irreducible elements of the poset of all the resolving subcategories ordered by inclusion.