Resolving subcategories for gentle algebras II: Resolving subcategories for gentle trees

Abstract

This paper is the second part of a series that intends to study the resolving subcategories for gentle algebras over an algebraically closed field $\mathbb{K}$. As in the first part, we continue to focus on gentle quivers $(Q,R)$, where $Q$ is a directed tree, known as gentle trees. In our previous work, via a modified surface model for gentle algebras with finite global dimension, we studied the join-irreducible elements of the lattice of resolving subcategories of $\mathbb{K}Q/\langle R \rangle - \text{mod}$, which happen to be those generated by a non-projective indecomposable object. In this paper, we notice that this lattice is not semidistributive in general and, accordingly, introduce a so-called upper join-decomposition, replacing the canonical one. Together with the techniques we develop in our geometric model, it allows us to describe the resolving subcategories of any gentle tree. These same techniques let us explicitly construct the resolving subcategory generated by any collection of indecomposable $\mathbb{K}Q/\langle R\rangle$-modules.