Super-decomposable pure-injective modules over some Jacobian algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908866213576704 |
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| author | Sardar, Shantanu |
| author_facet | Sardar, Shantanu |
| contents | Existence of superdecomposable pure-injective modules reflects complexity in the category of finite-dimensional representations over an algebra. Such an existence occurs when an algebra is non-domestic; a conjecture due to M. Prest. G. Puniski confirms the conjecture for non-domestic string algebras. Geiß, Labardini-Fragoso and Schröer show that every Jacobian algebra associated with a triangulation of a closed surface with marked points is finite-dimensional and tame. We show that, excluding only the case of a sphere with four (or fewer) punctures, there exists a special family of pointed modules, called an independent pair of dense chains of pointed modules. In the process, we show the existence of such an independent pair in a non-domestic skew-gentle algebra and (skew) Brauer graph algebras by showing that the Galois semi-covering functor and trivial extension preserve such pairs. Then it follows from a result of M. Ziegler that there exists a superdecomposable pure-injective module if the algebraically closed field is countable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_04387 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Super-decomposable pure-injective modules over some Jacobian algebras Sardar, Shantanu Representation Theory Category Theory 16, 18 Existence of superdecomposable pure-injective modules reflects complexity in the category of finite-dimensional representations over an algebra. Such an existence occurs when an algebra is non-domestic; a conjecture due to M. Prest. G. Puniski confirms the conjecture for non-domestic string algebras. Geiß, Labardini-Fragoso and Schröer show that every Jacobian algebra associated with a triangulation of a closed surface with marked points is finite-dimensional and tame. We show that, excluding only the case of a sphere with four (or fewer) punctures, there exists a special family of pointed modules, called an independent pair of dense chains of pointed modules. In the process, we show the existence of such an independent pair in a non-domestic skew-gentle algebra and (skew) Brauer graph algebras by showing that the Galois semi-covering functor and trivial extension preserve such pairs. Then it follows from a result of M. Ziegler that there exists a superdecomposable pure-injective module if the algebraically closed field is countable. |
| title | Super-decomposable pure-injective modules over some Jacobian algebras |
| topic | Representation Theory Category Theory 16, 18 |
| url | https://arxiv.org/abs/2603.04387 |