Measuring the non-Gorenstein locus of Hibi rings and normal affine semigroup rings

Abstract

The trace of the canonical module of a Cohen-Macaulay ring describes its non-Gorenstein locus. We study the trace of the canonical module of a Segre product of algebras, and we apply our results to compute the non-Gorenstein locus of toric rings. We provide several sufficient and necessary conditions for Hibi rings and normal semigroup rings to be Gorenstein on the punctured spectrum.

Stable set rings which are Gorenstein on the punctured spectrum

“…The aim of the present paper is to provide in Theorem 1.1 a simple combinatorial description of the perfect graphs G such that the ring R = Stab K (G) is Gorenstein on the punctured spectrum. It turns out that this situation is also equivalent to tr(ω R ) being a power of the graded maximal ideal in R. Similar characterizations have been obtained by Herzog, Mohammadi and Page in [6] for the non-Gorenstein locus of Hibi rings. Also, modeling [6, Corollary 3.12] and using simple techniques on combinatorics on finite posets, we construct, for any integers 4 ≤ a < b, a finite simple connected perfect graph G such that height(tr(ω R )) = a and dim R = b (Proposition 1.3).…”

“…Naturally, the height of the latter indicates how far R is from a Gorenstein ring. For some classes of toric rings the non-Gorenstein locus has been studied in [6], [13].…”

Gorenstein on the punctured spectrum and nearly Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph

“…Next we improve [HMP,Proposition 2.2]. In [HMP,Proposition 2.2], it is assumed that all rings involved are standard graded.…”

Regularity and the Gorenstein property of $L$-convex Polyominoes

“…Here we use that the coordinate ring of a a Ferrer diagram may be viewed as a Hibi ring. Then we can apply a recent result of Herzog et al [14] which characterizes the Hibi rings which are Gorenstein on the punctured spectrum.…”