Generation of singularity categories and infinite injective dimension locus via annihilation of cohomologies
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915547447296000 |
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| author | Dey, Souvik Liu, Jian Mifune, Yuki Otake, Yuya |
| author_facet | Dey, Souvik Liu, Jian Mifune, Yuki Otake, Yuya |
| contents | Let R be a commutative Noetherian ring. We establish a close relationship between the strong generation of the singularity category of R and the nonvanishing of the annihilator of the singularity category of R. As an application, we prove that the singularity category of R has a strong generator if and only if the annihilator of the singularity category of R is nonzero when R is a Noetherian domain with Krull dimension at most one. We introduce the notion of the co-cohomological annihilator of modules. If the category of finitely generated R-modules has a strong generator, we show that the infinite injective dimension locus of a finitely generated R-module M is closed, with the defining ideal given by the co-cohomological annihilator of M. Finally, we provide a connection between the existence of an extension generator of the category of finitely generated R-modules and the finiteness of the Krull dimension of R. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_24186 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generation of singularity categories and infinite injective dimension locus via annihilation of cohomologies Dey, Souvik Liu, Jian Mifune, Yuki Otake, Yuya Commutative Algebra Representation Theory 2020: 13D09 (primary), 13C60, 13D05, 13D07, 18G80 (secondary) Let R be a commutative Noetherian ring. We establish a close relationship between the strong generation of the singularity category of R and the nonvanishing of the annihilator of the singularity category of R. As an application, we prove that the singularity category of R has a strong generator if and only if the annihilator of the singularity category of R is nonzero when R is a Noetherian domain with Krull dimension at most one. We introduce the notion of the co-cohomological annihilator of modules. If the category of finitely generated R-modules has a strong generator, we show that the infinite injective dimension locus of a finitely generated R-module M is closed, with the defining ideal given by the co-cohomological annihilator of M. Finally, we provide a connection between the existence of an extension generator of the category of finitely generated R-modules and the finiteness of the Krull dimension of R. |
| title | Generation of singularity categories and infinite injective dimension locus via annihilation of cohomologies |
| topic | Commutative Algebra Representation Theory 2020: 13D09 (primary), 13C60, 13D05, 13D07, 18G80 (secondary) |
| url | https://arxiv.org/abs/2503.24186 |