Gorenstein Property Research Articles

Almost Gorenstein simplicial semigroup rings

We give a criterion for almost Gorenstein property for semigroup rings associated with simplicial semigroups. We extend Nari's theorem for almost symmetric numerical semigroups to simplicial semigroups with higher rank. By this criterion, we determine 2-dimensional normal semigroup rings which have “Ulrich elements” defined in [8].

Comparing generalized Gorenstein properties in semi-standard graded rings

Semi-standard graded rings are a generalized notion of standard graded rings. In this paper, we compare generalized notions of the Gorenstein property in semi-standard graded rings. We discuss the commonalities between standard graded rings and semi-standard graded rings, as well as elucidate distinctive phenomena present in semi-standard graded rings that are absent in standard graded rings.

Remarks on almost Gorenstein rings

This paper investigates the relation between the almost Gorenstein properties for graded rings and for local rings. Once R is an almost Gorenstein graded ring, the localization RM of R at the graded maximal ideal M is almost Gorenstein as a local ring. The converse does not hold true in general ([7, Theorems 2.7, 2.8], [8, Example 8.8]). However, it does for one-dimensional graded domains with mild conditions, which we clarify in the present paper. We explore the defining ideals of almost Gorenstein numerical semigroup rings as well.

One-sided Gorenstein rings

Abstract Distinctive characteristics of Iwanaga–Gorenstein rings are typically understood through their intrinsic symmetry. We show that several of those that pertain to the Gorenstein global dimensions carry over to the one-sided situation, even without the noetherian hypothesis. Our results yield new relations among homological invariants related to the Gorenstein property, not only Gorenstein global dimensions but also the suprema of projective/injective dimensions of injective/projective modules and finitistic dimensions.

Non-Gorenstein Locus and Almost Gorenstein Property of the Ehrhart Ring of the Stable Set Polytope of a Cycle Graph

Non-Gorenstein Locus and Almost Gorenstein Property of the Ehrhart Ring of the Stable Set Polytope of a Cycle Graph

F. S. Macaulay: From plane curves to Gorenstein rings

Francis Sowerby Macaulay began his career working on Brill and Noether’s theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation. Though he never spoke of the connection, the threads of Macaulay’s work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected.

Hilbert–Poincaré Series and Gorenstein Property for Some Non-simple Polyominoes

Let $$\mathcal {P}$$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of $$K[\mathcal {P}]$$ , showing that it is the rook polynomial of $$\mathcal {P}$$ . It is known by Rinaldo and Romeo (J Algebr Comb 54:607–624, 2021), that if $$\mathcal {P}$$ is a simple thin polyomino, then the h-polynomial is equal to the rook polynomial of $$\mathcal {P}$$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert–Poincaré series of the coordinate ring attached to a closed path $$\mathcal {P}$$ having no zig-zag walks, as a combination of the Hilbert–Poincaré series of convenient simple thin polyominoes. As a consequence, we prove that the Krull dimension is equal to $$\vert V(\mathcal {P})\vert -\textrm{rank}\, \mathcal {P}$$ and the regularity of $$K[\mathcal {P}]$$ is the rook number of $$\mathcal {P}$$ . Finally, we characterize the Gorenstein prime closed paths, proving that $$K[\mathcal {P}]$$ is Gorenstein if and only if $$\mathcal {P}$$ consists of maximal blocks of length three.

Betti sequence of the projective closure of affine monomial curves

We introduce the notion of star gluing of numerical semigroups and show that this preserves the arithmetically Cohen-Macaulay and Gorenstein properties of the projective closure. Next, we give a sufficient condition involving Gröbner basis for the matching of Betti sequences of the affine curve and its projective closure. We also study the effect of simple gluing on Betti sequences of the projective closure. Finally, we construct numerical semigroups by gluing, such that for every positive integer n, the last Betti number of the corresponding affine curve and its projective closure are both n.

Almost Gorenstein determinantal rings of symmetric matrices

We provide a characterization of the almost Gorenstein property of determinantal rings of a symmetric matrix of indeterminates over an infinite field. We give an explicit formula for ranks of the last two modules in the resolution of determinantal rings using Schur functors.

The reduction number of canonical ideals

In this article, we introduce an invariant of Cohen-Macaulay local rings in terms of the reduction number of canonical ideals. The invariant can be defined in arbitrary Cohen-Macaulay rings and it measures how close to being Gorenstein. First, we clarify the relation between almost Gorenstein rings and nearly Gorenstein rings by using the invariant in dimension one. We next characterize the idealization of trace ideals over Gorenstein rings in terms of the invariant. It provides better prospects for a result on the almost Gorenstein property of idealization.

Simplicial Affine Semigroups with Monomial Minimal Reduction Ideals

We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has a monomial minimal reduction. When this is the case, we study the Cohen–Macaulay and Gorenstein properties of the associated graded ring and provide several bounds for the reduction number with respect to the monomial minimal reduction.

Base change along the Frobenius endomorphism and the Gorenstein property

Let R be a local ring of positive characteristic and X a complex with nonzero finitely generated homology and finite injective dimension. We prove that if the derived base change of X via the Frobenius (or more generally, via a contracting) endomorphism has finite injective dimension then R is Gorenstein. In particular, we give an affirmative answer to a question by Falahola and Marley [7, Question 3.9].

ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH

In this paper, we give a criterion of the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ is Gorenstein if and only if (1) sizes of maximal cliques are constant (say $n$) and (2) (a) $n=1$, (b) $n=2$ and there is no odd cycle without chord and length at least 7 or (c) $n\geq 3$ and there is no odd cycle without chord and length at least 5.

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

We study the coordinate ring of an $L$-convex polyomino, determine its regularity in terms of the maximal number of rooks that can be placed in the polyomino. We also characterize the Gorenstein $L$-convex polyominoes and those which are Gorenstein on the punctured spectrum, and compute the Cohen–Macaulay type of any $L$-convex polyomino in terms of the maximal rectangles covering it. Though the main results are of algebraic nature, all proofs are combinatorial.

Gorenstein property for phylogenetic trivalent trees

We study the Gorenstein property for phylogenetic group-based models. We prove that the varieties associated to a trivalent tree and any of the groups Z3 and Z2×Z2 are Gorenstein Fano varieties, extending the results of Buczyńska and Wiśniewski for the group Z2.

The monoid of monotone functions on a poset and quasi-arithmetic multiplicities for uniform matroids

We describe the structure of the monoid of natural-valued monotone functions on an arbitrary poset. For this monoid we provide a presentation, a characterization of prime elements, and a description of its convex hull. We also study the associated monoid ring, proving that it is normal, and thus Cohen-Macaulay. We determine its Cohen-Macaulay type, characterize the Gorenstein property, and provide a Gröbner basis of the defining ideal. Then we apply these results to the monoid of quasi-arithmetic multiplicities on a uniform matroid. Finally we state some conjectures on the number of irreducibles for the monoid of multiplicities on an arbitrary matroid.

Nearly Gorenstein cyclic quotient singularities

We investigate the nearly Gorenstein property among d-dimensional cyclic quotient singularities Bbbk llbracket x_1,dots ,x_drrbracket ^G, where Bbbk is an algebraically closed field and Gsubseteq {text {GL}}(d,Bbbk ) is a finite small cyclic group whose order is invertible in Bbbk . We prove a necessary and sufficient condition to be nearly Gorenstein that also allows us to find several new classes of such rings.

Level Algebras and $\mathbf {s}$-Lecture Hall Polytopes

Given a family of lattice polytopes, a common endeavor in Ehrhart theory is the classification of those polytopes in the family that are Gorenstein, or more generally level. In this article, we consider these questions for ${\boldsymbol s}$-lecture hall polytopes, which are a family of simplices arising from $\mathbf {s}$-lecture hall partitions. In particular, we provide concrete classifications for both of these properties purely in terms of ${\boldsymbol s}$-inversion sequences. Moreover, for a large subfamily of ${\boldsymbol s}$-lecture hall polytopes, we provide a more geometric classification of the Gorenstein property in terms of its tangent cones. We then show how one can use the classification of level ${\boldsymbol s}$-lecture hall polytopes to construct infinite families of level ${\boldsymbol s}$-lecture hall polytopes, and to describe level ${\boldsymbol s}$-lecture hall polytopes in small dimensions.

Extremal growth of Betti numbers and trivial vanishing of (co)homology

A Cohen–Macaulay local ring R R satisfies trivial vanishing if Tor i R ⁡ ( M , N ) = 0 \operatorname {Tor}_i^R(M,N)=0 for all large i i implies that M M or N N has finite projective dimension. If R R satisfies trivial vanishing, then we also have that Ext R i ⁡ ( M , N ) = 0 \operatorname {Ext}^i_R(M,N)=0 for all large i i implies that M M has finite projective dimension or N N has finite injective dimension. In this paper, we establish obstructions for the failure of trivial vanishing in terms of the asymptotic growth of the Betti and Bass numbers of the modules involved. These, together with results of Gasharov and Peeva, provide sufficient conditions for R R to satisfy trivial vanishing; we provide sharpened conditions when R R is generalized Golod. Our methods allow us to settle the Auslander–Reiten conjecture in several new cases. In the last part of the paper, we provide criteria for the Gorenstein property based on consecutive vanishing of Ext. The latter results improve similar statements due to Ulrich, Hanes–Huneke, and Jorgensen–Leuschke.

Almost Gorenstein rings arising from fiber products

AbstractThe purpose of this paper is, as part of the stratification of Cohen–Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product$R \times _T S$of Cohen–Macaulay local ringsR,Sof the same dimension$d>0$over a regular local ringTwith$\dim T=d-1$is an almost Gorenstein ring if and only if so areRandS. In addition, the other generalizations of Gorenstein properties are also explored.