1. Quadratic Gorenstein Rings and the Koszul Property II
- Author
-
Michael Stillman, Matthew Mastroeni, and Hal Schenck
- Subjects
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 - Abstract
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
- Published
- 2021