Abstract: Let $R$ be a standard graded Gorenstein algebra over a field presented by quadrics. Conca, Rossi, and Valla showed that such a ring is Koszul if $\mathrm{reg}\, R \leq 2$ or if $\mathrm{reg}\, R = 3$ and $\mathrm{codim}\, R \leq 4$, and they ask whether this is true for $\mathrm{reg}\, R = 3$ in general. We give a negative answer to their question in almost all codimensions at least 9 by finding suitable conditions on a non-Koszul quadratic Cohen-Macaulay ring $R$ that guarantee the Nagata idealization $\widetilde{R} = R \ltimes \omega_R(-a - 1)$ is a non-Koszul quadratic Gorenstein ring. This is joint work with Hal Schenck and Mike Stillman.

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