# Quadratic Gorenstein rings and the Koszul property I

## TL;DR

This paper investigates the Koszul property of quadratic Gorenstein rings, providing conditions under which certain non-Koszul rings of regularity 3 exist, thus answering an open question in the field.

## Contribution

It establishes sufficient conditions for non-Koszul quadratic Gorenstein rings and constructs examples with regularity 3 for all codimensions at least 9.

## Key findings

- Constructed non-Koszul quadratic Gorenstein rings of regularity 3 for all codimension ≥ 9
- Provided conditions linking Cohen-Macaulay rings and their Nagata idealizations to the Koszul property
- Negatively answered the open question about Koszulness of quadratic Gorenstein rings with regularity 3

## Abstract

Let $R$ be a standard graded Gorenstein algebra over a field presented by quadrics. Conca, Rossi, and Valla have shown that such a ring is Koszul if $\mathrm{reg}\, R \leq 2$ or if $\mathrm{reg}\, = 3$ and $c= \mathrm{codim}\, R \leq 4$, and they ask whether this is true for $\mathrm{reg}\, R = 3$ in general. We determine sufficient conditions on a non-Koszul quadratic Cohen-Macaulay ring $R$ that guarantee the Nagata idealization $\tilde{R} = R \ltimes \omega_R(-a-1)$ is a non-Koszul quadratic Gorenstein ring. We use this to negatively answer the question of Conca-Rossi-Valla, constructing non-Koszul quadratic Gorenstein rings of regularity 3 for all $c \geq 9$.