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-Simões 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.

Figures (10)

• Figure 1: Figures showing a substring $\sigma$ of $\rho$ on top of $\rho$ (above), and a substring $\sigma$ of $\rho$ at the bottom of $\rho$ (below).
• Figure 2: Figure illustrating combinatorially the map $\varphi_{(\sigma, \sigma')}$ defined in \ref{['thm:CB']}, for $\sigma = \sigma'$ a substring on the top of $\rho$ and at the bottom of $\rho'$.
• Figure 7: An example of a $p$-cover of a string $\mu$
• Figure 8: Illustration of \ref{['lem:covertripletdecomp']} on the reduced $4$-cover seen in \ref{['ex:cover']}.
• Figure 9: Illustration of the construction of the string $\nu_0$ (left), and the string $\nu_i$ for $i \in \{1,\ldots,p-1\}$ (right), pictured as orange dotted lines, thanks to the strings $\rho_1,\ldots,\rho_p$ (purple dashed lines), and $\mu$ (blue line). On the right, the gray part corresponds to the common substring $\upsilon_{i,i+1}$ of $\mu, \rho_{i}, \rho_{i+1}$ and $\nu_i$.

Theorems & Definitions (62)

• Theorem 1.1: \ref{['thm:Monoareallthejoinirred']}
• Theorem 1.2
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• Example 2.5
• Definition 2.6
• Corollary 2.7
• Lemma 2.8
• Definition 2.9