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

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.