Finitistic dimensions over commutative DG-rings
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866913534726635520 |
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| author | Bird, Isaac Shaul, Liran Sridhar, Prashanth Williamson, Jordan |
| author_facet | Bird, Isaac Shaul, Liran Sridhar, Prashanth Williamson, Jordan |
| contents | In this paper we study the finitistic dimensions of commutative noetherian non-positive DG-rings with finite amplitude. We prove that any DG-module $M$ of finite flat dimension over such a DG-ring satisfies $\mathrm{projdim}_A(M) \leq \mathrm{dim}(\mathrm{H}^0 (A)) - \inf(M)$. We further provide explicit constructions of DG-modules with prescribed projective dimension and deduce that the big finitistic projective dimension satisfies the bounds $\mathrm{dim}(\mathrm{H}^0 (A)) - \mathrm{amp}(A) \leq \mathsf{FPD}(A) \leq \mathrm{dim}(\mathrm{H}^0(A))$. Moreover, we prove that DG-rings exist which achieve either bound. As a direct application, we prove new vanishing results for the derived Hochschild (co)homology of homologically smooth algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_06865 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Finitistic dimensions over commutative DG-rings Bird, Isaac Shaul, Liran Sridhar, Prashanth Williamson, Jordan Commutative Algebra K-Theory and Homology Rings and Algebras Primary: 13D05, 16E45, Secondary: 13D09, 13H10 In this paper we study the finitistic dimensions of commutative noetherian non-positive DG-rings with finite amplitude. We prove that any DG-module $M$ of finite flat dimension over such a DG-ring satisfies $\mathrm{projdim}_A(M) \leq \mathrm{dim}(\mathrm{H}^0 (A)) - \inf(M)$. We further provide explicit constructions of DG-modules with prescribed projective dimension and deduce that the big finitistic projective dimension satisfies the bounds $\mathrm{dim}(\mathrm{H}^0 (A)) - \mathrm{amp}(A) \leq \mathsf{FPD}(A) \leq \mathrm{dim}(\mathrm{H}^0(A))$. Moreover, we prove that DG-rings exist which achieve either bound. As a direct application, we prove new vanishing results for the derived Hochschild (co)homology of homologically smooth algebras. |
| title | Finitistic dimensions over commutative DG-rings |
| topic | Commutative Algebra K-Theory and Homology Rings and Algebras Primary: 13D05, 16E45, Secondary: 13D09, 13H10 |
| url | https://arxiv.org/abs/2204.06865 |