Quadratic Gorenstein rings and the Koszul property I

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 \tilde {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.