Non-Gorenstein loci of Ehrhart rings of chain and order polytopes

Abstract

Let P be a finite poset, K a field, and O(P ) (resp. C (P )) the order (resp. chain) polytope of P . We study the non-Gorenstein locus of E K [O(P )] (resp. E K [C (P )]), the Ehrhart ring of O(P ) (resp. C (P )) over K, which are each normal toric rings associated P . In particular, we show that the dimension of non-Gorenstein loci of E K [O(P )] and E K [C (P )] are the same. Further, we show that E K [C (P )] is nearly Gorenstein if and only if P is the disjoint union of pure posets P 1 , . . . , P s with |rankP i − rankP j | ≤ 1 for any i and j.

Toric non-Gorenstein loci

“…They prove that k[P ] is almost Gorenstein if and only if all connected components P i of P are pure, and the difference in their ranks is bounded by one. In [MP20] (Theorem 4.5), Miyazaki and Page give a description of the radical of tr(ω) in combinatorial terms. By proving Theorem 3.8 and Theorem 3.6, we give a new characterization of tr(ω) in a more conceptual, simple, and natural manner that unifies the above results.…”

“…In the paper [MP20], non-Gorenstein loci of Hibi rings are studied. In particular, a family of graded ideals is described in Theorem 4.5, that intersect in the radical ideal tr(ω).…”

Stable set rings which are Gorenstein on the punctured spectrum

“…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].…”