Linkage and $F$-Regularity of Determinantal Rings
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913235987333120 |
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| author | Pandey, Vaibhav Tarasova, Yevgeniya |
| author_facet | Pandey, Vaibhav Tarasova, Yevgeniya |
| contents | In this paper, we prove that the generic link of a generic determinantal ring defined by maximal minors is strongly $F$-regular. In the process, we strengthen a result of Chardin and Ulrich in the graded setting. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that if the said complete intersection is defined by homogeneous elements and is $F$-rational, then in fact, its generic residual intersections are strongly $F$-regular in positive prime characteristic.
Hochster and Huneke showed that determinantal rings are strongly $F$-regular; however, their proof is quite involved. Our techniques allow us to give a new and simple proof of the strong $F$-regularity of determinantal rings defined by maximal minors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_07922 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Linkage and $F$-Regularity of Determinantal Rings Pandey, Vaibhav Tarasova, Yevgeniya Commutative Algebra 13C40, 13A35 (Primary) 14M06, 14M10, 14M12 (Secondary) In this paper, we prove that the generic link of a generic determinantal ring defined by maximal minors is strongly $F$-regular. In the process, we strengthen a result of Chardin and Ulrich in the graded setting. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that if the said complete intersection is defined by homogeneous elements and is $F$-rational, then in fact, its generic residual intersections are strongly $F$-regular in positive prime characteristic. Hochster and Huneke showed that determinantal rings are strongly $F$-regular; however, their proof is quite involved. Our techniques allow us to give a new and simple proof of the strong $F$-regularity of determinantal rings defined by maximal minors. |
| title | Linkage and $F$-Regularity of Determinantal Rings |
| topic | Commutative Algebra 13C40, 13A35 (Primary) 14M06, 14M10, 14M12 (Secondary) |
| url | https://arxiv.org/abs/2211.07922 |