Quadratic Gorenstein Rings and the Koszul Property II
| Autor: | Michael Stillman, Matthew Mastroeni, Hal Schenck |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Property (philosophy) Mathematics::Commutative Algebra General Mathematics Mathematics::Rings and Algebras 010102 general mathematics Mathematics - Commutative Algebra Commutative Algebra (math.AC) 16. Peace & justice 01 natural sciences 010101 applied mathematics Quadratic equation FOS: Mathematics 0101 mathematics Mathematics |
| Zdroj: | International Mathematics Research Notices. 2023:1461-1482 |
| ISSN: | 1687-0247 1073-7928 |
| DOI: | 10.1093/imrn/rnab297 |
| Popis: | A question of Conca, Rossi, and Valla asks whether every quadratic Gorenstein ring $R$ of regularity three is Koszul. In a previous paper, we use idealization to answer their question, proving that in nine or more variables there exist quadratic Gorenstein rings of regularity three which are not Koszul. In this paper, we study the analog of the Conca-Rossi-Valla question when the regularity of $R$ is four or more. Let $R$ be a quadratic Gorenstein ring having $\mathrm{codim}\, R = c$ and $\mathrm{reg}\, R = r \ge 4$. We prove that if $c = r+1$ then $R$ is always Koszul, and for every $c \geq r+2$, we construct quadratic Gorenstein rings that are not Koszul, answering questions of Matsuda and Migliore-Nagel concerning the $h$-vectors of quadratic Gorenstein rings. Comment: v2 - Minor changes based on referee comments |
| Databáze: | OpenAIRE |
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