Quadratic Gorenstein Rings and the Koszul Property II

AbstractConca–Rossi–Valla [6] ask if every quadratic Gorenstein ring $R$ of regularity three is Koszul. In [15], 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 ${\operatorname {codim}} \ R = c$ and ${\operatorname {reg}} \ R = r \ge 4$. 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 [16] and Migliore–Nagel [19].