Derived equivalences via Tate resolutions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909994314629120 |
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| author | Ganapathy, K. Sane, Sarang |
| author_facet | Ganapathy, K. Sane, Sarang |
| contents | For any finite sequence of elements $s_1, \ldots , s_d$ in a commutative noetherian ring $R$, we show that for $n \gg 0$, the natural map from the Koszul complex $K(s_1^n, \ldots , s_d^n)$ to the Koszul complex $K(s_1, \ldots , s_d)$ factors through the Tate resolution on $s_1^n, \ldots , s_d^n$. Using this, for any resolving subcategory $\mathcal A$ of mod($R$) and any ideal $I$ such that it has a filtration $\{ I_n \}$ which is equivalent to the $I$-adic filtration and $\textrm{dim}_{\mathcal A}(R/I_n) < \infty$, we show a derived equivalence between the bounded derived category of finitely generated modules supported on $V(I)$ having finite $\mathcal A$-dimension and the bounded derived category of $\mathcal A$ with homologies supported on $V(I)$. As a special case, when $R$ is of prime characteristic and $I$ is of finite projective dimension, we obtain a derived equivalence between the bounded derived category of finite projective dimension modules supported on $V(I)$ and the bounded derived category of projective modules with homologies supported on $V(I)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_12531 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Derived equivalences via Tate resolutions Ganapathy, K. Sane, Sarang Commutative Algebra 13D02, 13D09 (Primary), 13C13, 13E05 (Secondary) For any finite sequence of elements $s_1, \ldots , s_d$ in a commutative noetherian ring $R$, we show that for $n \gg 0$, the natural map from the Koszul complex $K(s_1^n, \ldots , s_d^n)$ to the Koszul complex $K(s_1, \ldots , s_d)$ factors through the Tate resolution on $s_1^n, \ldots , s_d^n$. Using this, for any resolving subcategory $\mathcal A$ of mod($R$) and any ideal $I$ such that it has a filtration $\{ I_n \}$ which is equivalent to the $I$-adic filtration and $\textrm{dim}_{\mathcal A}(R/I_n) < \infty$, we show a derived equivalence between the bounded derived category of finitely generated modules supported on $V(I)$ having finite $\mathcal A$-dimension and the bounded derived category of $\mathcal A$ with homologies supported on $V(I)$. As a special case, when $R$ is of prime characteristic and $I$ is of finite projective dimension, we obtain a derived equivalence between the bounded derived category of finite projective dimension modules supported on $V(I)$ and the bounded derived category of projective modules with homologies supported on $V(I)$. |
| title | Derived equivalences via Tate resolutions |
| topic | Commutative Algebra 13D02, 13D09 (Primary), 13C13, 13E05 (Secondary) |
| url | https://arxiv.org/abs/2601.12531 |