Compact generators of the contraderived category of contramodules
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916545390706688 |
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| author | Positselski, Leonid Stovicek, Jan |
| author_facet | Positselski, Leonid Stovicek, Jan |
| contents | We consider the contraderived category of left contramodules over a right linear topological ring $\mathfrak R$ with a countable base of neighborhoods of zero. Equivalently, this is the homotopy category of unbounded complexes of projective left $\mathfrak R$-contramodules. Assuming that the abelian category of discrete right $\mathfrak R$-modules is locally coherent, we show that the contraderived category of left $\mathfrak R$-contramodules is compactly generated, and describe its full subcategory of compact objects as the opposite category to the bounded derived category of finitely presentable discrete right $\mathfrak R$-modules. Under the same assumptions, we also prove the flat and projective periodicity theorem for $\mathfrak R$-contramodules. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_20494 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Compact generators of the contraderived category of contramodules Positselski, Leonid Stovicek, Jan Category Theory Rings and Algebras We consider the contraderived category of left contramodules over a right linear topological ring $\mathfrak R$ with a countable base of neighborhoods of zero. Equivalently, this is the homotopy category of unbounded complexes of projective left $\mathfrak R$-contramodules. Assuming that the abelian category of discrete right $\mathfrak R$-modules is locally coherent, we show that the contraderived category of left $\mathfrak R$-contramodules is compactly generated, and describe its full subcategory of compact objects as the opposite category to the bounded derived category of finitely presentable discrete right $\mathfrak R$-modules. Under the same assumptions, we also prove the flat and projective periodicity theorem for $\mathfrak R$-contramodules. |
| title | Compact generators of the contraderived category of contramodules |
| topic | Category Theory Rings and Algebras |
| url | https://arxiv.org/abs/2412.20494 |