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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.13029 |
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| _version_ | 1866914562692874240 |
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| author | Bobiński, Grzegorz Schröer, Jan |
| author_facet | Bobiński, Grzegorz Schröer, Jan |
| contents | We study projective presentations of finite-dimensional modules over finite-dimensional algebras. We discuss if projective presentations of maximal rank behave additively. More precisely, we ask if the direct sum of copies of a projective presentation of maximal rank is again of maximal rank. The modules which have a projective presentation of maximal rank are exactly the $τ$-regular modules. This class of modules can be seen as a generalization of modules of projective dimension at most one, and of $τ$-rigid modules. The $τ$-regular modules form open subsets of module varieties. Their closures are therefore unions of irreducible components, which are called generically $τ$-regular. We discuss when a $τ$-regular module or a generically $τ$-regular component can be reduced to a module or component of projective dimension at most one, and we show that this is closely related to the question on the additivity of maximal rank presentations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13029 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the additivity of projective presentations of maximal rank Bobiński, Grzegorz Schröer, Jan Representation Theory We study projective presentations of finite-dimensional modules over finite-dimensional algebras. We discuss if projective presentations of maximal rank behave additively. More precisely, we ask if the direct sum of copies of a projective presentation of maximal rank is again of maximal rank. The modules which have a projective presentation of maximal rank are exactly the $τ$-regular modules. This class of modules can be seen as a generalization of modules of projective dimension at most one, and of $τ$-rigid modules. The $τ$-regular modules form open subsets of module varieties. Their closures are therefore unions of irreducible components, which are called generically $τ$-regular. We discuss when a $τ$-regular module or a generically $τ$-regular component can be reduced to a module or component of projective dimension at most one, and we show that this is closely related to the question on the additivity of maximal rank presentations. |
| title | On the additivity of projective presentations of maximal rank |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2605.13029 |