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

TL;DR

This work develops a combinatorial and geometric framework for classifying resolving subcategories of gentle-tree algebras. It introduces an upper join-decomposition to manage non-semidistributive lattices and employs closure operators to organize resolving subcategories, connecting lattice theory with the geometry of accordions on marked discs. Monogeneous resolving subcategories are identified as the join-irreducibles, and every resolving subcategory is expressible as a union of such monogeneous pieces via an explicit algorithm grounded in the geometric model. The results bridge representation-theoretic structures with a concrete surface-model toolkit, enabling explicit computation of resolving closures for collections of indecomposable modules in gentle-tree settings.

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.

Figures (9)

• Figure 1: Drawings representing the rules that an accordion must satisfy: on the left, $\delta$ satisfies the rule $(a)$ and $(b)$; on the right, $\delta$ does not satisfy the rule $(b)$.
• Figure 2: Illustration of a crossing corresponding to a basis element of $\operatorname{Hom}(\operatorname{M}(\delta), \operatorname{M}(\eta))$. The shaded part is a part where all the segments of $\delta$ and all the ones of $\eta$, given by the cutting of $\pmb{\Sigma}$ with $\pmb{\Gamma}(\Delta^{{\color{dark-green}{\circ}}})$, are homotopic.
• Figure 3: The two types of accordions $\kappa_i$ representing the indecomposable summands of $\operatorname{Ker}(f)$.
• Figure 6: An example of a gentle quiver, which is a gentle tree (see \ref{['def:gentle trees']})
• Figure 7: An example of $\operatorname{N}_{\operatorname{proj}}(\delta)$.

Theorems & Definitions (44)

• Theorem 1.1: DS251
• Theorem 1.2: DS251
• Theorem 1.3
• Definition 2.2
• Lemma 2.3
• Definition 2.5
• Definition 2.6
• Theorem 2.7: BCS21OPS18PPP182
• Definition 2.8: HKK17OPS18
• Remark 2.9