Quadratic Gorenstein rings and the Koszul property I

2019/03/19 by Mastroeni, Matthew, Schenck, Hal, Stillman, Mike
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1903.08265

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 reg R ≤ 2 or if reg = 3 and c= codim R ≤ 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 R = R \ltimes ω_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 ≥ 9.

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