A contramodule generalization of Neeman's flat and projective module theorem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918194047877120 |
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| author | Positselski, Leonid |
| author_facet | Positselski, Leonid |
| contents | This paper builds on top of arXiv:2306.02734. We consider a complete, separated topological ring $\mathfrak R$ with a countable base of neighborhoods of zero consisting of open two-sided ideals. The main result is that the homotopy category of projective left $\mathfrak R$-contramodules is equivalent to the derived category of the exact category of flat left $\mathfrak R$-contramodules, and also to the homotopy category of flat cotorsion left $\mathfrak R$-contramodules. In other words, a complex of flat $\mathfrak R$-contramodules is contraacyclic (in the sense of Becker) if and only if it is an acyclic complex with flat $\mathfrak R$-contramodules of cocycles, and if and only if it is coacyclic as a complex in the exact category of flat $\mathfrak R$-contramodules. These are contramodule generalizations of theorems of Neeman and of Bazzoni, Cortes-Izurdiaga, and Estrada. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10928 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A contramodule generalization of Neeman's flat and projective module theorem Positselski, Leonid Rings and Algebras Category Theory This paper builds on top of arXiv:2306.02734. We consider a complete, separated topological ring $\mathfrak R$ with a countable base of neighborhoods of zero consisting of open two-sided ideals. The main result is that the homotopy category of projective left $\mathfrak R$-contramodules is equivalent to the derived category of the exact category of flat left $\mathfrak R$-contramodules, and also to the homotopy category of flat cotorsion left $\mathfrak R$-contramodules. In other words, a complex of flat $\mathfrak R$-contramodules is contraacyclic (in the sense of Becker) if and only if it is an acyclic complex with flat $\mathfrak R$-contramodules of cocycles, and if and only if it is coacyclic as a complex in the exact category of flat $\mathfrak R$-contramodules. These are contramodule generalizations of theorems of Neeman and of Bazzoni, Cortes-Izurdiaga, and Estrada. |
| title | A contramodule generalization of Neeman's flat and projective module theorem |
| topic | Rings and Algebras Category Theory |
| url | https://arxiv.org/abs/2408.10928 |