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 study connections between the class of nearly Gorenstein rings and that of almost Gorenstein rings. We show that in dimension one, the former class includes the latter.

The trace of the canonical module

Let $A$ be a graded algebra. In this paper we develop a generalized Koszul theory by assuming that $A_0$ is self-injective instead of semisimple and generalize many classical results. The application of this generalized theory to directed categories and finite EI categories is described.

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A generalized Koszul theory and its application

We survey results related to the magnitude of the Betti numbers of numerical semigroup rings and of their tangent cones.

Betti numbers for numerical semigroup rings

Let $A$ be a graded algebra In this paper we develop a generalized Koszul theory by assuming that $A_0$ is self-injective instead of semisimple and generalize many classical results The application of this generalized theory to directed categories and finite EI categories is described

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid S, consider the family of “shifted” monoids Mn obtained by adding n to each generator of S. In this paper, we examine minimal relations among the generators of Mn when n is sufficiently large, culminating in a description that is periodic in the shift parameter n. We explore several applications to computation and factorization theory, and improve a recent result of Thanh Vu from combinatorial commutative algebra.

Minimal presentations of shifted numerical monoids

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 study connections between the class of nearly Gorenstein rings and that of almost Gorenstein rings. We show that in dimension one, the former class includes the latter.

Dumitru I. Stamate

Papers

The trace of the canonical module

On the defining equations of the tangent cone of a numerical semigroup ring

The trace of the canonical module

Betti numbers for numerical semigroup rings

Koszul incidence algebras, affine semigroups, and Stanley–Reisner ideals