Resolving subcategories for gentle algebras I: Monogeneous resolving subcategories for gentle trees
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914072031657984 |
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| author | Dequêne, Benjamin Schoonheere, Michaël |
| author_facet | Dequêne, Benjamin Schoonheere, Michaël |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_20994 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Resolving subcategories for gentle algebras I: Monogeneous resolving subcategories for gentle trees Dequêne, Benjamin Schoonheere, Michaël Representation Theory Combinatorics 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. |
| title | Resolving subcategories for gentle algebras I: Monogeneous resolving subcategories for gentle trees |
| topic | Representation Theory Combinatorics |
| url | https://arxiv.org/abs/2502.20994 |