Narrow Results

A good way of parameterizing zero-dimensional schemes in an affine space ${𝔸}_K^n$ has been developed in the last 20 years using border basis schemes. Given a multiplicity $\mu$, they provide an open covering of the Hilbert scheme ${\mathrm{\text{Hilb}}}^{\mu }({𝔸}_K^n)$ and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose rational points represent zero-dimensional $K$-algebras sharing a given property. The main focus of this paper is on giving effective answers to this general problem. The properties considered here are the locally Gorenstein, strict Gorenstein, strict complete intersection, Cayley–Bacharach, and strict Cayley–Bacharach properties. The key characteristic of our approach is that we describe these loci by exhibiting explicit algorithms to compute their defining ideals. All results are illustrated by nontrivial, concrete examples.

Let (R,𝔪) be a 1-dimensional Cohen–Macaulay local ring of multiplicity e and embedding dimension ν ≥ 2. Let B denote the blowing-up of R along 𝔪 and let I be the conductor of R in B. Let x ∈ 𝔪 be a superficial element in 𝔪 of degree 1 and , . We assume that the length . This class of local rings contains the class of 1-dimensional Gorenstein local rings (see 1.5). In Sec. 1, we prove that (see 1.6) if the associated graded ring G = gr_𝔪(R) is Cohen–Macaulay, then I ⊆ 𝔪^s + xR, where s is the degree of the h-polynomial h_R of R. In Sec. 2, we give necessary and sufficient conditions (see Corollaries 2.4, 2.5, 2.9 and Theorem 2.11) for the Cohen–Macaulayness of G. These conditions are numerical conditions on the h-polynomial h_R, particularly on its coefficients and the degree in comparison with the difference e - ν. In Sec. 3, we give some conditions (see Propositions 3.2, 3.3 and Corollary 3.4) for the Gorensteinness of G. In Sec. 4, we give a characterization (see Proposition 4.3) of numerical semigroup rings which satisfy the condition .

A ring with a test module of finite upper complete intersection dimension is complete intersection.

In this paper, we give combinatorial formulas for the Hilbert coefficients, h-polynomial and the Cohen–Macaulay type of Schubert varieties in Grassmannians in terms of the posets associated with them. As a consequence, necessary conditions for a Schubert variety to be a complete intersection and combinatorial criteria are given for a Schubert variety to be Gorenstein and almost Gorenstein, respectively.

Let (R, 𝔪, k) be a complete Gorenstein local ring of dimension n. Let be the local cohomology module with respect to a pair of ideals I, J and . In this paper we will show that the endomorphism ring is a commutative ring. In particular if for all i ≠ t, then B is isomorphic to R. Also we prove that, B is a finite R-module if and only if is an Artinian R-module, where d = n - t. Moreover we will show that in the case that for all i ≠ t the natural homomorphism is nonzero which gives a positive answer to a conjecture due to Hellus–Schenzel (see [On cohomologically complete intersections, J. Algebra 320 (2008) 3733–3748]).

This work introduces a new kind of semigroup of ${\mathbb{N}}^p$ called proportionally modular affine semigroup. These semigroups are defined by modular Diophantine inequalities and they are a generalization of proportionally modular numerical semigroups. We give an algorithm to compute their minimal generating sets, and we specialize when $p=2$. For this case, we also provide a faster algorithm to compute their minimal system of generators, prove they are Cohen–Macaulay and Buchsbaum, and determinate their (minimal) Frobenius vectors. Besides, Gorenstein proportionally modular affine semigroups are characterized.

A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic complex of injective modules is totally acyclic. We extend this characterization, which is due to Iyengar and Krause, to arbitrary commutative noetherian rings, i.e. we remove the assumption about a dualizing complex. In this context Gorenstein, of course, means locally Gorenstein at every prime.

Let ${𝒜}$ be an abelian category. In this paper, we investigate the global $({𝒳},{𝒴})$-Gorenstein projective dimension ${\mathrm{\text{gl.GPD}}}_{({𝒳},{𝒴})}({𝒜})$, associated to a GP-admissible pair $({𝒳},{𝒴})$. We give homological conditions over $({𝒳},{𝒴})$ that characterize it. Moreover, given a GI-admissible pair $({𝒵},{𝒲})$, we study conditions under which ${\mathrm{\text{gl.GID}}}_{({𝒵},{𝒲})}({𝒜})$ and ${\mathrm{\text{gl.GPD}}}_{({𝒳},{𝒴})}({𝒜})$ are the same.

Let $H$ be a simple undirected graph. The family of all matchings of $H$ forms a simplicial complex called the matching complex of $H$. Here, we give a classification of all graphs with a Gorenstein matching complex. Also we study when the matching complex of $H$ is Cohen–Macaulay and, in certain classes of graphs, we fully characterize those graphs which have a Cohen–Macaulay matching complex. In particular, we characterize when the matching complex of a graph with girth at least five or a complete graph is Cohen–Macaulay.

Recall that a ring $R$ is a Hilbert ring if any maximal ideal of $R[X]$ contracts to a maximal ideal of $R$. The main purpose of this paper is to characterize the prime ideals of a commutative ring $R$ which are traces of the maximal ideals of the polynomial ring $R[X]$. In this context, we prove that if $p$ is a prime ideal of $R$ such that $R/p$ is a semi-local domain of (Krull) dimension $\le 1$, then $p$ is the trace of a maximal ideal of $R[X]$. Whereas, if $R$ is Noetherian and either ($dim(R/p)\ge 2$) or (the quotient field of $R/p$ is algebraically closed, $dim(R/p)=1$ and $R/p$ is not semi-local), then $p$ is never the trace of a maximal ideal of $R[X]$. Putting these results into use in investigating the Ext-index of Noetherian rings, we establish connections between the finiteness of the Ext-index of localizations of the polynomial rings $R[X]$ and the finiteness of the Ext-index of localizations of $R$. This allows us to provide a new class of rings satisfying some known conjectures on Ext-index of Noetherian rings as well as to build bridges between these conjectures.

Let denote an ideal of a d-dimensional Gorenstein local ring R, and M and N two finitely generated R-modules with pd M < ∞. It is shown that if and only if for all .

Let (R, 𝔪) be a commutative Noetherian local ring and M an R-module which is relative Cohen-Macaulay with respect to a proper ideal 𝔞 of R, and set n := ht_M𝔞. We prove that injdim M < ∞ if and only if and that . We also prove that if R has a dualizing complex and Gid_RM < ∞, then . Moreover if R and M are Cohen-Macaulay, then Gid_RM < ∞ whenever . Next, for a finitely generated R-module M of dimension d, it is proved that if is Cohen-Macaulay and , then . The above results have consequences which improve some known results and provide characterizations of Gorenstein rings.