Quadratic Gorenstein rings and the Koszul property I

Mastroeni, Matthew; Schenck, Hal; Stillman, Mike

doi:10.1090/tran/8214

Quadratic Gorenstein rings and the Koszul property I
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Abstract:

Let $R$ be a standard graded Gorenstein algebra over a field presented by quadrics. In [Compositio Math. 129 (2001), no. 1, 95–121], Conca-Rossi-Valla show that such a ring is Koszul if $\mathrm {reg} R \leq 2$ or if $\mathrm {reg} R = 3$ 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 $\widetilde {R} = R \ltimes \omega _R(-a-1)$ is a non-Koszul quadratic Gorenstein ring. We prove there exist rings of regularity $3$ satisfying our conditions for all $c \ge 9$; this yields a negative answer to the question from the above-mentioned paper.

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