Product-complete tilting complexes and Cohen-Macaulay hearts
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911891421396992 |
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| author | Hrbek, Michal Martini, Lorenzo |
| author_facet | Hrbek, Michal Martini, Lorenzo |
| contents | We show that the cotilting heart associated to a tilting complex $T$ is a locally coherent and locally coperfect Grothendieck category (i.e. an Ind-completion of a small artinian abelian category) if and only if $T$ is product-complete. We then apply this to the specific setting of the derived category of a commutative noetherian ring $R$. If $\dim(R)<\infty$, we show that there is a derived duality $\mathcal{D}^b_{fg}(R) \cong \mathcal{D}^b(\mathcal{B})^{op}$ between $\mathrm{mod} R$ and a noetherian abelian category $\mathcal{B}$ if and only if $R$ is a homomorphic image of a Cohen--Macaulay ring. Along the way, we obtain new insights about t-structures in $\mathcal{D}^b_{fg}(R)$. In the final part, we apply our results to obtain a new characterization of the class of those finite-dimensional Noetherian rings that admit a Gorenstein complex. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_16722 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Product-complete tilting complexes and Cohen-Macaulay hearts Hrbek, Michal Martini, Lorenzo Representation Theory Commutative Algebra Primary: 13D09, 14F08, Secondary: 16D90, 13H10 We show that the cotilting heart associated to a tilting complex $T$ is a locally coherent and locally coperfect Grothendieck category (i.e. an Ind-completion of a small artinian abelian category) if and only if $T$ is product-complete. We then apply this to the specific setting of the derived category of a commutative noetherian ring $R$. If $\dim(R)<\infty$, we show that there is a derived duality $\mathcal{D}^b_{fg}(R) \cong \mathcal{D}^b(\mathcal{B})^{op}$ between $\mathrm{mod} R$ and a noetherian abelian category $\mathcal{B}$ if and only if $R$ is a homomorphic image of a Cohen--Macaulay ring. Along the way, we obtain new insights about t-structures in $\mathcal{D}^b_{fg}(R)$. In the final part, we apply our results to obtain a new characterization of the class of those finite-dimensional Noetherian rings that admit a Gorenstein complex. |
| title | Product-complete tilting complexes and Cohen-Macaulay hearts |
| topic | Representation Theory Commutative Algebra Primary: 13D09, 14F08, Secondary: 16D90, 13H10 |
| url | https://arxiv.org/abs/2307.16722 |