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| Main Authors: | , |
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| Format: | Preprint |
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2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2011.06989 |
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| _version_ | 1866909625564004352 |
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| author | Pol, Luca Williamson, Jordan |
| author_facet | Pol, Luca Williamson, Jordan |
| contents | Given a commutative ring $R$ and finitely generated ideal $I$, one can consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes. Under a mild assumption on the ideal $I$ called weak pro-regularity, these three notions of completions interact well. We consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes and prove that they present the same homotopy theory. Given a ring homomorphism $R \to S$, we then give necessary and sufficient conditions for the categories of complete $R$-complexes and the categories of complete $S$-complexes to have equivalent homotopy theories. This recovers and generalizes a result of Sather-Wagstaff and Wicklein on extended local (co)homology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_06989 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The homotopy theory of complete modules Pol, Luca Williamson, Jordan Commutative Algebra Algebraic Topology 13B35, 13D09, 18N40 Given a commutative ring $R$ and finitely generated ideal $I$, one can consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes. Under a mild assumption on the ideal $I$ called weak pro-regularity, these three notions of completions interact well. We consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes and prove that they present the same homotopy theory. Given a ring homomorphism $R \to S$, we then give necessary and sufficient conditions for the categories of complete $R$-complexes and the categories of complete $S$-complexes to have equivalent homotopy theories. This recovers and generalizes a result of Sather-Wagstaff and Wicklein on extended local (co)homology. |
| title | The homotopy theory of complete modules |
| topic | Commutative Algebra Algebraic Topology 13B35, 13D09, 18N40 |
| url | https://arxiv.org/abs/2011.06989 |