The trace of the canonical module

Abstract

The trace of the canonical module (the canonical trace) determines the non-Gorenstein locus of a local Cohen-Macaulay ring. We call a local Cohen-Macaulay ring nearly Gorenstein, if its canonical trace contains the maximal ideal. Similar definitions can be made for positively graded Cohen-Macaulay K-algebras. We study the canonical trace for tensor products and Segre products of algebras, as well as of (squarefree) Veronese subalgebras. The results are used to classify the nearly Gorenstein Hibi rings. We also consider the canonical trace of onedimensional rings with a focus on numerical semigroup rings.By using a result of Goto and Watanabe [15] one deduces then from Theorem 4.6 that all Veronese subalgebras of a standard graded K-algebra R over an infinite field K are nearly Gorenstein, if R is Gorenstein.The situation for squarefree Veronese subalgebras is more complicated. Given integers n ≥ d > 0, the d-th squarefree Veronese subalgebra R n,d of the polynomial ring S in n variables over a field K is the K-algebra generated by the squarefree monomials in S of degree d. Based on a theorem of Bruns, Vasconcelos and Villarreal [5] we give in Theorem 4.12 an explicit description of the anti-canonical ideal of R n,d , and we use this result to show in Theorem 4.14 that the following conditions are equivalent: (i) R n,d is nearly Gorenstein, (ii) R n,d is Gorenstein, (iii) d = 1 or d = n − 1 or n = 2d.By another result of Goto and Watanabe [15], the canonical module for the Segre product T = R#S of positively graded Cohen-Macaulay K-algebras R and S of Krull dimension at least 2 is just the Segre product of the respective canonical modules, assuming that T is Cohen-Macaulay. We use this result in Theorem 4.15 to compute tr(ω T ) in the case that R and S are standard graded Gorenstein Kalgebras. It is shown in Theorem 4.15 that m |r−s| T ⊆ tr(ω T ), where m T is the graded maximal ideal of T and where r and s are the respective a-invariants of R and S. Equality holds, if T is a domain. If follows that under these conditions T is nearly Gorenstein if and only if |r − s| ≤ 1, see Corollary 4.16. The section ends with Proposition 4.18 in which the anti-canonical ideal of the Segre product is computed in the case that R and S are polynomial rings.These results on Segre products are used in Section 5 to give a complete classification of all nearly Gorenstein Hibi rings. Given a finite distributive lattice L and a field K, the Hibi ring of L defined over K is the toric ring R K [L] whose relations are the meet-join relations of L. By a fundamental theorem of Birkhoff, L is the ideal lattice J (P ) of its poset of join irreducible elements. It is shown in Theorem 5.4 3

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

“…In particular, when each R i is Gorenstein, we get the following result, which is a slight generalization of Theorem 4.15 in [HHS19].…”

“…In this section we develop some of the algebraic tools which will be used in the next section, and introduce trace ideals. We refer to [HHS19] for a more complete introduction of the trace of the canonical module.…”

“…Let R be a local or graded Cohen-Macaulay ring which admits a canonical module ω R . The trace of an R-module M, denoted tr(M), is the sum of all ideals ϕ(M), where the sum is taken over all R-module homomorphisms ϕ : M → R. It is noticed in [HHS19] that the non-Gorenstein locus of R is the closed subset of Spec(R) which is given by the set of prime ideals containing tr(ω R ). It follows that the height of tr(ω R ) is a good measure for the non-Gorenstein locus of R. For example, R is Gorenstein on the punctured spectrum (the open subset Spec(R)\{m} of Spec(R)) if and only if tr(ω R ) is primary to the (graded) maximal ideal m of R. In this note, we study the trace of the canonical module of Segre products of algebras, Hibi rings, and normal affine semigroup rings.…”

“…A classical result of Hibi [Hib87, Corollary 3.d] says that K[P ] is Gorenstein if and only if P is a pure poset, that is, all maximal chains of P have the same length. In [HHS19], Herzog, Hibi, and Stamate call a ring as above nearly Gorenstein if tr(ω R ) = m, and they classify all nearly Gorenstein Hibi rings. Indeed, they show that K[P ] is nearly Gorenstein if and only if all connected components P i of P are pure and | rank P i − rank P j | ≤ 1 for all i and j.…”

Rings with Canonical Reductions

“…The concepts of generalized Gorenstein and nearly Gorenstein rings are defined in [12] and [20], respectively. Here we prove that nearly Gorenstein rings have canonical reductions.…”