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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.03998 |
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| _version_ | 1866914936808013824 |
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| author | Chang, Wen |
| author_facet | Chang, Wen |
| contents | We give a geometric model for any algebraic heart in the derived category of a gentle algebra, which is equivalent to the module category of some gentle algebra. To do this, we deform the geometric model for the module category of a gentle algebra given in [BC21], and then embed it into the geometric model of the derived category given in [OPS18], in the sense that each so-called zigzag curve on the surface represents an indecomposable module as well as the minimal projective resolution of this module. A key point of this embedding is to give a geometric explanation of the duality between the simple modules and the projective modules.
Such a blend of two geometric models provides us with a handy way to describe the homological properties of a module within the framework of the derived category. In particular, we realize any higher Yoneda-extension as a polygon on the surface, and realize the Yoneda-product as gluing of these polygons. As an application, we realize any algebraic heart in the derived category of a gentle algebra on the marked surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_03998 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Geometric models for the algebraic hearts in the derived category of a gentle algebra Chang, Wen Representation Theory We give a geometric model for any algebraic heart in the derived category of a gentle algebra, which is equivalent to the module category of some gentle algebra. To do this, we deform the geometric model for the module category of a gentle algebra given in [BC21], and then embed it into the geometric model of the derived category given in [OPS18], in the sense that each so-called zigzag curve on the surface represents an indecomposable module as well as the minimal projective resolution of this module. A key point of this embedding is to give a geometric explanation of the duality between the simple modules and the projective modules. Such a blend of two geometric models provides us with a handy way to describe the homological properties of a module within the framework of the derived category. In particular, we realize any higher Yoneda-extension as a polygon on the surface, and realize the Yoneda-product as gluing of these polygons. As an application, we realize any algebraic heart in the derived category of a gentle algebra on the marked surface. |
| title | Geometric models for the algebraic hearts in the derived category of a gentle algebra |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2305.03998 |