Resolutions as directed colimits
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917276965404672 |
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| author | Positselski, Leonid |
| author_facet | Positselski, Leonid |
| contents | A general principle suggests that "anything flat is a directed colimit of countably presentable flats". In this paper, we consider resolutions and coresolutions of modules over a countably coherent ring $R$ (e.g., any coherent ring or any countably Noetherian ring). We show that any $R$-module of flat dimension $n$ is a directed colimit of countably presentable $R$-modules of flat dimension at most $n$, and any flatly coresolved $R$-module is a directed colimit of countably presentable flatly coresolved $R$-modules. If $R$ is a countably coherent ring with a dualizing complex, then any F-totally acyclic complex of flat $R$-modules is a directed colimit of F-totally acyclic complexes of countably presentable flat $R$-modules. The proofs are applications of an even more general category-theoretic principle going back to an unpublished 1977 preprint of Ulmer. Our proof of the assertion that every Gorenstein-flat module over a countably coherent ring is a directed colimit of countably presentable Gorenstein-flat modules uses a different technique, based on results of Saroch and Stovicek. We also discuss totally acyclic complexes of injectives and Gorenstein-injective modules, obtaining various cardinality estimates for the accessibility rank under various assumptions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_07197 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Resolutions as directed colimits Positselski, Leonid Commutative Algebra Category Theory Rings and Algebras A general principle suggests that "anything flat is a directed colimit of countably presentable flats". In this paper, we consider resolutions and coresolutions of modules over a countably coherent ring $R$ (e.g., any coherent ring or any countably Noetherian ring). We show that any $R$-module of flat dimension $n$ is a directed colimit of countably presentable $R$-modules of flat dimension at most $n$, and any flatly coresolved $R$-module is a directed colimit of countably presentable flatly coresolved $R$-modules. If $R$ is a countably coherent ring with a dualizing complex, then any F-totally acyclic complex of flat $R$-modules is a directed colimit of F-totally acyclic complexes of countably presentable flat $R$-modules. The proofs are applications of an even more general category-theoretic principle going back to an unpublished 1977 preprint of Ulmer. Our proof of the assertion that every Gorenstein-flat module over a countably coherent ring is a directed colimit of countably presentable Gorenstein-flat modules uses a different technique, based on results of Saroch and Stovicek. We also discuss totally acyclic complexes of injectives and Gorenstein-injective modules, obtaining various cardinality estimates for the accessibility rank under various assumptions. |
| title | Resolutions as directed colimits |
| topic | Commutative Algebra Category Theory Rings and Algebras |
| url | https://arxiv.org/abs/2312.07197 |