Homological properties of pinched Veronese rings
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916282766458880 |
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| author | Maddox, Kyle Pandey, Vaibhav |
| author_facet | Maddox, Kyle Pandey, Vaibhav |
| contents | Pinched Veronese rings are formed by removing an algebra generator from a Veronese subring of a polynomial ring. We study the homological properties of such rings, including the Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco and Martino classified Cohen-Macaulayness of pinched Veronese rings by the maximum entry of the exponent vector of the pinched monomial; we re-prove their results with semigroup methods and correct an omission of a small class of examples of Cohen-Macaulay pinched Veronese rings. When the underlying field is of prime characteristic, we show that pinched Veronese rings exhibit a variety of F-singularities, including F-regular, F-injective, and F-nilpotent. We also compute upper bounds on the Frobenius test exponents of pinched Veronese rings, a computational invariant which controls the Frobenius closure of all parameter ideals simultaneously. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_05810 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Homological properties of pinched Veronese rings Maddox, Kyle Pandey, Vaibhav Commutative Algebra 13F65 (Primary) 13A35, 13D45 (Secondary) Pinched Veronese rings are formed by removing an algebra generator from a Veronese subring of a polynomial ring. We study the homological properties of such rings, including the Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco and Martino classified Cohen-Macaulayness of pinched Veronese rings by the maximum entry of the exponent vector of the pinched monomial; we re-prove their results with semigroup methods and correct an omission of a small class of examples of Cohen-Macaulay pinched Veronese rings. When the underlying field is of prime characteristic, we show that pinched Veronese rings exhibit a variety of F-singularities, including F-regular, F-injective, and F-nilpotent. We also compute upper bounds on the Frobenius test exponents of pinched Veronese rings, a computational invariant which controls the Frobenius closure of all parameter ideals simultaneously. |
| title | Homological properties of pinched Veronese rings |
| topic | Commutative Algebra 13F65 (Primary) 13A35, 13D45 (Secondary) |
| url | https://arxiv.org/abs/2111.05810 |