Affine Semigroup Rings Research Articles

COUNTING GEOMETRIC BRANCHES VIA THE FROBENIUS MAP AND F-NILPOTENT SINGULARITIES

AbstractWe give an explicit formula to count the number of geometric branches of a curve in positive characteristic using the theory of tight closure. This formula readily shows that the property of having a single geometric branch characterizes F-nilpotent curves. Further, we show that a reduced, local F-nilpotent ring has a single geometric branch; in particular, it is a domain. Finally, we study inequalities of Frobenius test exponents along purely inseparable ring extensions with applications to F-nilpotent affine semigroup rings.

On the depth of simplicial affine semigroup rings

AbstractWe recall and delve into the different characterizations of the depth of an affine semigroup ring, providing an original characterization of depth two in three and four dimensional cases which are closely related to the existence of a maximal element in certain Apéry sets.

Graded local cohomology of modules over semigroup rings

We give a combinatorial description of local cohomology modules of a graded module over a semigroup ring, with support at the graded maximal ideal. This combinatorial framework yields Hochster-type formulas for the Hilbert series of such local cohomology modules in terms of the homology of finitely many polyhedral cell complexes. A Cohen–Macaulay criterion immediately follows. We also provide an alternative proof of a result of [18] characterizing Cohen–Macaulay affine semigroup rings.

Levelness versus nearly Gorensteinness of homogeneous rings

Levelness and nearly Gorensteinness are well-studied properties of graded rings as a generalized notion of Gorensteinness. In this paper, we compare the strength of these properties. For any Cohen-Macaulay homogeneous affine semigroup ring R, we give a necessary condition for R to be non-Gorenstein and nearly Gorenstein in terms of the h-vector of R and we show that if R is nearly Gorenstein with Cohen-Macaulay type 2, then it is level. We also show that if Cohen-Macaulay type is more than 2, there are 2-dimensional counterexamples. Moreover, we characterize nearly Gorensteinness of Stanley-Reisner rings of low-dimensional simplicial complexes.

Differentially fixed ideals in affine semigroup rings

We give a complete characterization of when monomial ideals are fixed by differential operators of affine semigroup rings over [Formula: see text]. Perhaps surprisingly, every monomial ideal is fixed by an infinite set of homogeneous differential operators and is in fact determined by them. This opens up a new tool for studying monomial ideals. We explore applications of this to (mixed) multiplier ideals and other variants as well as give examples of detecting ideal membership in integrally closed powers and symbolic powers of squarefree monomial ideals.

Publisher Correction: On the algebraic invariants of certain affine semigroup rings

Publisher Correction: On the algebraic invariants of certain affine semigroup rings

On the algebraic invariants of certain affine semigroup rings

On the algebraic invariants of certain affine semigroup rings

Affine semigroups of maximal projective dimension

A submonoid of \( {\mathbb {N}}^d \) is of maximal projective dimension (\({\text {MPD}}\)) if the associated affine semigroup ring has the maximum possible projective dimension. Such submonoids have a nontrivial set of pseudo-Frobenius elements. We generalize the notion of symmetric semigroups, pseudo-symmetric semigroups, and row-factorization matrices for pseudo-Frobenius elements of numerical semigroups to the case of \({\text {MPD}}\)-semigroups in \({\mathbb {N}}^d\). Under suitable conditions, we prove that these semigroups satisfy the generalized Wilf’s conjecture. We prove that the generic nature of the defining ideal of the associated semigroup ring of an \({\text {MPD}}\)-semigroup implies uniqueness of row-factorization matrix for each pseudo-Frobenius element. Further, we give a description of pseudo-Frobenius elements and row-factorization matrices of gluing of \({\text {MPD}}\)-semigroups. We prove that the defining ideal of gluing of \({\text {MPD}}\)-semigroups is never generic.

On the Gorenstein locus of simplicial affine semigroup rings

We study the Gorenstein locus of simplicial affine semigroup rings in terms of some Apéry sets. The results come from an analysis of Cohen-Macaulay type of homogeneous localizations at monomial prime ideals. Characterizing when the homogeneous localization at a monomial prime ideal is Gorenstein, we can read the dimension of the non-Gorenstein locus. In particular, we determine when the semigroup ring is Gorenstein on the punctured spectrum.

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.

Type and conductor of simplicial affine semigroups

We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring K [ S ] . We define the type of S , type ( S ) , in terms of some Apéry sets of S and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when K [ S ] is Cohen-Macaulay. If K [ S ] is a d -dimensional Cohen-Macaulay ring of embedding dimension at most d + 2 , then type ( S ) ≤ 2 . Otherwise, type ( S ) might be arbitrary large and it has no upper bound in terms of the embedding dimension. Finally, we present a generating set for the conductor of S as an ideal of its normalization.

Standard pairs of monomial ideals over nonnormal affine semigroups in SageMath

We present stdPairs.spyx, a SageMath library to compute standard pairs of a monomial ideal over a pointed (non-normal) affine semigroup ring. Moreover, stdPairs.spyx provides the associated prime ideals, the corresponding multiplicities, and an irredundant irreducible primary decomposition of a monomial ideal. The library expands on the standardPairs function on Macaulay2 over polynomial rings, and is based on algorithms from arXiv:2005.10968v2 [math.AC]. We also provide methods that allow the outputs from this library to be compatible with the Normaliz package of Macaulay2 and SageMath.

A constructive approach to one-dimensional Gorenstein ${\mathbf {k}}$-algebras

Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $\Gamma. $ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulay's correspondence characterizing the $R$-submodules of $\Gamma $ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein $k$-algebras of dimension one and any codimension. This has been achieved through a deep analysis of the $G$-admissible submodules of $\Gamma. $ Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed.

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

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.

Acyclic digraphs giving rise to complete intersections

We call a directed acyclic graph a CI-digraph if a certain affine semigroup ring defined by it is a complete intersection. We show that if $D$ is a 2-connected CI-digraph with cycle space of dimension at least 2, then it can be decomposed into two subdigraphs, one of which can be taken to have only one cycle, that are CI-digraphs and are glued together on a directed path. If the arcs of the digraph are the covering relations of a poset, this is the converse of a theorem of Boussicault, Feray, Lascoux and Reiner. The decomposition result shows that CI-digraphs can be easily recognized.

The Cohen-Macaulay property of affine semigroup rings in dimension 2

Consider the subring of , where k is a field and x, y indeterminates over k. We calculate the Hilbert polynomial of . The multiplicity of this ideal provides part of a criterion for the ring to be Cohen-Macaulay. Next, we prove a simple numerical criterion for R to be Cohen-Macaulay in the case when t = 2. We also provide a simple algorithm which identifies the monomial k-basis of . Finally, these methods are specialized to the case of projective monomial curves in .

Uniform Harbourne–Huneke bounds via flat extensions

Over an arbitrary field F, Harbourne [3] conjectured that the symbolic power I(N(r−1)+1)⊆Ir for all r>0 and all homogeneous ideals I in S=F[PN]=F[x0,…,xN]. The conjecture has been disproven for select values of N≥2: first by Dumnicki, Szemberg, and Tutaj-Gasińska in characteristic zero [7], and then by Harbourne and Seceleanu in positive characteristic [13]. However, the ideal containments above do hold when, e.g., I is a monomial ideal in S[3, Ex. 8.4.5].As a sequel to [21], we present criteria for containments of type I(N(r−1)+1)⊆Ir for all r>0 and certain classes of ideals I in a prodigious class of normal rings. Of particular interest is a result for monomial primes in tensor products of affine semigroup rings. Indeed, we explain how to give effective multipliers N in several cases including: the D-th Veronese subring of any polynomial ring F[x1,…,xn](n≥1); and the extension ring F[x1,…,xn,z]/(zD−x1⋯xn) of F[x1,…,xn].

Doset Hibi rings with an application to invariant theory

ABSTRACTWe define the concept of a doset Hibi ring and a generalized doset Hibi ring which are subrings of a Hibi ring and are normal affine semigroup rings. We apply the theory of (generalized) doset Hibi rings to analyze the rings of absolute orthogonal invariants and absolute special orthogonal invariants and show that these rings are normal and Cohen-Macaulay and has rational singularities if the characteristic of the base field is zero and is F-rational otherwise. We also state criteria of Gorenstein property of these rings.

Affine semigroup rings are of finite F-representation type

ABSTRACTIt is known that normal affine semigroup rings are of finite F-representation type. In this paper, we prove that non-normal affine semigroup rings are also of finite F-representation type.

Dualizing complexes of seminormal affine semigroup rings and toric face rings

We characterize the seminormality of an affine semigroup ring in terms of the dualizing complex, and the normality of a Cohen–Macaulay semigroup ring by the “shape” of the canonical module. We also characterize the seminormality of a toric face ring in terms of the dualizing complex. A toric face ring is a simultaneous generalization of Stanley–Reisner rings and affine semigroups.