On the rate of generic Gorenstein $K$-algebras
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866915724253986816 |
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| author | Boij, Mats De Negri, Emanuela De Stefani, Alessandro Rossi, Maria Evelina |
| author_facet | Boij, Mats De Negri, Emanuela De Stefani, Alessandro Rossi, Maria Evelina |
| contents | The rate of a standard graded $K$-algebra $A$ is a measure of the growth of the shifts in a minimal free resolution of $K$ as an $A$-module. In particular $A$ has rate one if and only if it is Koszul. It is known that a generic Artinian Gorenstein algebra of embedding dimension $n \geq 3$ and socle degree $s=3$ is Koszul. We prove that a generic Artinian Gorenstein algebra with $n\geq 4$ and $s \ge 3 $ has rate $ \lfloor \frac{s}{2} \rfloor. $ In the process we show that such an algebra is generated in degree $\lfloor \frac{s}{2} \rfloor +1. $ This gives a partial positive answer to a longstanding conjecture stated by the first author on the minimal free resolution of a generic Artinian Gorenstein ring of odd socle degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_14842 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the rate of generic Gorenstein $K$-algebras Boij, Mats De Negri, Emanuela De Stefani, Alessandro Rossi, Maria Evelina Commutative Algebra The rate of a standard graded $K$-algebra $A$ is a measure of the growth of the shifts in a minimal free resolution of $K$ as an $A$-module. In particular $A$ has rate one if and only if it is Koszul. It is known that a generic Artinian Gorenstein algebra of embedding dimension $n \geq 3$ and socle degree $s=3$ is Koszul. We prove that a generic Artinian Gorenstein algebra with $n\geq 4$ and $s \ge 3 $ has rate $ \lfloor \frac{s}{2} \rfloor. $ In the process we show that such an algebra is generated in degree $\lfloor \frac{s}{2} \rfloor +1. $ This gives a partial positive answer to a longstanding conjecture stated by the first author on the minimal free resolution of a generic Artinian Gorenstein ring of odd socle degree. |
| title | On the rate of generic Gorenstein $K$-algebras |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2304.14842 |