DMS Algebra/Linear Algebra Seminar

Speaker: Hal Schenck

Title: Quadratic Gorenstein Rings and the Koszul Property I

Abstract: Let $R$ be a standard graded Gorenstein algebra over a field presented by quadrics. Conca-Rossi-Valla showed that such a ring is Koszul if $\reg R \leq 2$ or if $\reg R = 3$ and $c=\codim R \leq 4$, and they ask whether this is true for $\reg R = 3$ 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 \cite{Gröbner:flags:and:Gorenstein:algebras}, constructing non-Koszul quadratic Gorenstein rings of regularity 3 for all $c \geq 9$.