# Quadratic Gorenstein rings and the Koszul property II

## TL;DR

This paper investigates the Koszul property of quadratic Gorenstein rings with regularity four or more, establishing conditions under which they are Koszul and providing counterexamples for higher codimensions.

## Contribution

It extends the understanding of the Koszul property for quadratic Gorenstein rings to regularity four and higher, proving specific cases where the rings are Koszul and constructing counterexamples.

## Key findings

- If codimension c equals regularity r plus one, the ring is always Koszul.
- For codimension c at least r plus two, there exist quadratic Gorenstein rings that are not Koszul.
- Answers open questions about the $h$-vectors of quadratic Gorenstein rings.

## 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.