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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2306.02734 |
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| _version_ | 1866909377260158976 |
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| author | Positselski, Leonid |
| author_facet | Positselski, Leonid |
| contents | This paper is a follow-up to arXiv:2212.09639. We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring $\mathcal C$ over a noncommutative ring $A$, we show that all $A$-flat $\mathcal C$-comodules are $\aleph_1$-directed colimits of $A$-countably presentable $A$-flat $\mathcal C$-comodules. In the context of a complete, separated topological ring $\mathfrak R$ with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat $\mathfrak R$-contramodules are $\aleph_1$-directed colimits of countably presentable flat $\mathfrak R$-contramodules. We also describe arbitrary complexes, short exact sequences, and pure acyclic complexes of $A$-flat $\mathcal C$-comodules and flat $\mathfrak R$-contramodules as $\aleph_1$-directed colimits of similar complexes of countably presentable objects. The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in arXiv:2310.16773. Applications to cotorsion periodicity and coderived categories of flat objects in the respective settings are discussed. In particular, in any acyclic complex of cotorsion $\mathfrak R$-contramodules, all the contramodules of cocycles are cotorsion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_02734 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Flat comodules and contramodules as directed colimits, and cotorsion periodicity Positselski, Leonid Rings and Algebras Algebraic Geometry This paper is a follow-up to arXiv:2212.09639. We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring $\mathcal C$ over a noncommutative ring $A$, we show that all $A$-flat $\mathcal C$-comodules are $\aleph_1$-directed colimits of $A$-countably presentable $A$-flat $\mathcal C$-comodules. In the context of a complete, separated topological ring $\mathfrak R$ with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat $\mathfrak R$-contramodules are $\aleph_1$-directed colimits of countably presentable flat $\mathfrak R$-contramodules. We also describe arbitrary complexes, short exact sequences, and pure acyclic complexes of $A$-flat $\mathcal C$-comodules and flat $\mathfrak R$-contramodules as $\aleph_1$-directed colimits of similar complexes of countably presentable objects. The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in arXiv:2310.16773. Applications to cotorsion periodicity and coderived categories of flat objects in the respective settings are discussed. In particular, in any acyclic complex of cotorsion $\mathfrak R$-contramodules, all the contramodules of cocycles are cotorsion. |
| title | Flat comodules and contramodules as directed colimits, and cotorsion periodicity |
| topic | Rings and Algebras Algebraic Geometry |
| url | https://arxiv.org/abs/2306.02734 |