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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.15425 |
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| _version_ | 1866911633986551808 |
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| author | Positselski, Leonid |
| author_facet | Positselski, Leonid |
| contents | As a special case of Bass' theory of perfect rings, one obtains the assertion that, over a finite-dimensional associative algebra over a field, all flat modules are projective. In this paper we prove the following relative version of this result. Let $R\rightarrow A$ be a homomorphism of associative rings such that $A$ is a finitely generated projective right $R$-module. Then every flat left $A$-module is a direct summand of an $A$-module filtered by $A$-modules $A\otimes_RF$ induced from flat left $R$-modules $F$. In other words, a left $A$-module is cotorsion if and only if its underlying left $R$-module is cotorsion. The proof is based on the cotorsion periodicity theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15425 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A relative version of Bass' theorem about finite-dimensional algebras Positselski, Leonid Rings and Algebras As a special case of Bass' theory of perfect rings, one obtains the assertion that, over a finite-dimensional associative algebra over a field, all flat modules are projective. In this paper we prove the following relative version of this result. Let $R\rightarrow A$ be a homomorphism of associative rings such that $A$ is a finitely generated projective right $R$-module. Then every flat left $A$-module is a direct summand of an $A$-module filtered by $A$-modules $A\otimes_RF$ induced from flat left $R$-modules $F$. In other words, a left $A$-module is cotorsion if and only if its underlying left $R$-module is cotorsion. The proof is based on the cotorsion periodicity theorem. |
| title | A relative version of Bass' theorem about finite-dimensional algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2507.15425 |