Gorenstein Local Ring Research Articles

Vanishing of relative homology and depth of tensor products

For finitely generated modules M and N over a Gorenstein local ring R, one has depthM+depthN=depth(M⊗RN)+depthR, i.e., the depth formula holds, if M and N are Tor-independent and Tate homology Torˆi(M,N) vanishes for all i∈Z. We establish the same conclusion under weaker hypotheses: if M and N are G-relative Tor-independent, then the vanishing of Torˆi(M,N) for all i⩽0 is enough for the depth formula to hold. We also analyze the depth of tensor products of modules and obtain a necessary condition for the depth formula in terms of G-relative homology.

A measure of non-sequential Cohen–Macaulayness of finitely generated modules

Let M be a finitely generated module over a Noetherian local ring R. In this paper we introduce the notion of sequential polynomial type of M, which is denoted by sp(M), in order to measure how far M is different from the sequential Cohen–Macaulayness. We study the sequential polynomial type under localization and m-adic completion. We investigate an ascent–descent property of sequential polynomial type between M and M/xM for certain parameter x of M. When R is a quotient of a Gorenstein local ring, we describe sp(M) in term of the deficiency modules of M.

Conditions for the Yoneda algebra of a local ring to be generated in low degrees

The powers mn of the maximal ideal m of a local Noetherian ring R are known to satisfy certain homological properties for large values of n. For example, the homomorphism R→R/mn is Golod for n≫0. We study when such properties hold for small values of n, and we make connections with the structure of the Yoneda Ext algebra, and more precisely with the property that the Yoneda algebra of R is generated in degrees 1 and 2. A complete treatment of these properties is pursued in the case of compressed Gorenstein local rings.

On the cofiniteness of Artinian local cohomology modules

Let [Formula: see text] be a commutative Noetherian local ring, which is a homomorphic image of a Gorenstein local ring and [Formula: see text] an ideal of [Formula: see text]. Let [Formula: see text] be a nonzero finitely generated [Formula: see text]-module and [Formula: see text] be an integer. In this paper we show that, the [Formula: see text]-module [Formula: see text] is nonzero and [Formula: see text]-cofinite if and only if [Formula: see text]. Also, several applications of this result will be included.

The almost Gorenstein Rees algebras of parameters

There is given a characterization for the Rees algebras of parameters in a Gorenstein local ring to be almost Gorenstein graded rings. A characterization is also given for the Rees algebras of socle ideals of parameters. The latter one shows almost Gorenstein Rees algebras rather rarely exist for socle ideals, if the dimension of the base local ring is greater than two.

A counterexample to a conjecture of Ding

We give a counterexample to a conjecture posed by S. Ding in [4] regarding the index of a Gorenstein local ring by exhibiting several examples of one dimensional local complete intersections of embedding dimension three with index 5 and generalized Löewy length 6.

Endomorphism ring of local cohomology modules with respect to a pair of ideals

Let (R, 𝔪, k) be a complete Gorenstein local ring of dimension n. Let [Formula: see text] be the local cohomology module with respect to a pair of ideals I, J and [Formula: see text]. In this paper we will show that the endomorphism ring [Formula: see text] is a commutative ring. In particular if [Formula: see text] for all i ≠ t, then B is isomorphic to R. Also we prove that, B is a finite R-module if and only if [Formula: see text] is an Artinian R-module, where d = n - t. Moreover we will show that in the case that [Formula: see text] for all i ≠ t the natural homomorphism [Formula: see text] 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]).

Maximal Cohen–Macaulay Approximations and Serre’s Condition

This paper studies the relationship between Serre's condition $(\R_n)$ and Auslander--Buchweitz's maximal Cohen--Macaulay approximations. It is proved that a Gorenstein local ring satisfies $(\R_n)$ if and only if every maximal Cohen--Macaulay module is a direct summand of a maximal Cohen--Macaulay approximation of a (Cohen--Macaulay) module of codimension $n+1$.

Construction of Auslander–Gorenstein local rings as Frobenius extensions

Construction of Auslander–Gorenstein local rings as Frobenius extensions

LOCAL COHOMOLOGY WITH RESPECT TO A COHOMOLOGICALLY COMPLETE INTERSECTION PAIR OF IDEALS

Let (R;m;k) be a local Gorenstein ring of dimension n. Let H i (R) be the local cohomology with respect to a pair of ideals I;J and c be the inffijH i (R) 6 0g. A pair of ideals I;J is called co- homologically complete intersection if H i (R) = 0 for all i 6 c. It is shown that, when H i (R) = 0 for all i 6 c, (i) a minimal injec- tive resolution of H c (R) presents like that of a Gorenstein ring; (ii) Hom R(H c (R); H c (R))' R, where (R;m) is a complete ring. Also we get an estimate of the dimension of H i (R).

Analytic isomorphisms of compressed local algebras

In this paper we consider Artin local K-algebras with maximal length in the class of Artin algebras with given embedding dimension and socle type. They have been widely studied by several authors, among others by Iarrobino, Froberg and Laksov. If the local K-algebra is Gorenstein of socle degree 3, then the authors proved that it is canonically graded, i.e. analytically isomorphic to its associated graded ring, see (6). This unexpected result has been extended to compressed level K-algebras of socle degree 3 in (4). In this paper we end the investigation proving that extremal Artin Gorenstein local K-algebras of socle degree s � 4 are canonically graded, but the result does not extend to extremal Artin Gorenstein local rings of socle degree 5 or to compressed level local rings of socle degree 4 and type > 1. As a consequence we present results on Artin compressed local K-algebras having a specified socle type.

Almost Gorenstein rings – towards a theory of higher dimension

The notion of almost Gorenstein local ring introduced by V. Barucci and R. Fröberg for one-dimensional Noetherian local rings which are analytically unramified has been generalized by S. Goto, N. Matsuoka and T.T. Phuong to one-dimensional Cohen–Macaulay local rings, possessing canonical ideals. The present purpose is to propose a higher-dimensional notion and develop the basic theory. The graded version is also posed and explored.

Attached primes of local cohomology modules under localization and completion

Let (R,m) be a Noetherian local ring and M a finitely generated R-module. Following I.G. Macdonald [8], the set of all attached primes of the Artinian local cohomology module Hmi(M) is denoted by AttR(Hmi(M)). In [13, Theorem 3.7], R.Y. Sharp proved that if R is a quotient of a Gorenstein local ring then the shifted localization principle holds true for any local cohomology modules Hmi(M), i.e.(1)AttRp(HpRpi−dim⁡(R/p)(Mp))={qRp|q∈AttRHmi(M),q⊆p} for any p∈Spec(R). In this paper, we improve Sharp's result as follows: the shifted localization principle holds true if and only if R is universally catenary and all its formal fibers are Cohen–Macaulay, if and only if the shifted completion principle(2)AttRˆ(Hmi(M))=⋃p∈AttR(Hmi(M))AssRˆ(Rˆ/pRˆ) holds true for any local cohomology module Hmi(M). This also improves the main result of the paper by T.D.M. Chau and the first author [2].

Vanishing of Tate homology and depth formulas over local rings

Auslander's depth formula for pairs of Tor-independent modules over a regular local ring, depth(M⊗RN)=depthM+depthN−depthR, has been generalized in several directions; most significantly it has been shown to hold for pairs of Tor-independent modules over complete intersection rings.In this paper we establish a depth formula that holds for every pair of Tate Tor-independent modules over a Gorenstein local ring. It subsumes previous generalizations of Auslander's formula and yields new results on vanishing of cohomology over certain Gorenstein rings.

Some loci related to Cohen–Macaulayness

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module. The pseudo Cohen–Macaulayness (respectively, generalized Cohen–Macaulayness) was introduced by Cuong–Nhan [Pseudo Cohen–Macaulay and pseudo generalized Cohen–Macaulay modules, J. Algebra267 (2003) 156–177] as an extension of the Cohen–Macaulayness (respectively, generalized Cohen–Macaulayness). In this paper, we describe the pseudo Cohen–Macaulay (pseudo CM) locus and pseudo generalized Cohen–Macaulay (pseudo generalized CM) locus of M. We also study the non-Cohen–Macaulay locus and the non-generalized Cohen–Macaulay locus of the canonical module K(M) of M in case where R is a quotient of a Gorenstein local ring.

Poincaré series of modules over compressed Gorenstein local rings

Given two positive integers e and s we consider Gorenstein Artinian local rings R whose maximal ideal m satisfies ms≠0=ms+1 and rankR/m(m/m2)=e. We say that R is a compressed Gorenstein local ring when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If s≠3, we prove that the Poincaré series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When s is even this formula depends only on the integers e and s. Note that for s=3 examples of compressed Gorenstein local rings with transcendental Poincaré series exist, due to Bøgvad.

The Frobenius functor and injective modules

We investigate commutative Noetherian rings of prime characteristic such that the Frobenius functor applied to any injective module is again injective. We characterize the class of one-dimensional local rings with this property and show that it includes all one-dimensional F F -pure rings. We also give a characterization of Gorenstein local rings in terms of T o r i R ( R f , E ) \mathrm {Tor}_i^R(R^{f},E) , where E E is the injective hull of the residue field and R f R^{f} is the ring R R whose right R R -module action is given by the Frobenius map.

On the Asymptotic Behavior of the Bass Numbers of Modules

Let (R, 𝔪) be a Noetherian Gorenstein local ring and I be a principal ideal of R. In this article we show that the Bass numbers of the R-modules R/I n take constant values for large n.

Vanishing of Tate Cohomology and Gorenstein Injective Dimension

In this note, some sufficient conditions are given for the equality Gid R RHom R (M, N) = G-dim R M + Gid R N to hold via vanishing of Tate cohomology. For instance, let R be a commutative Gorenstein local ring, and let M and N be finite R-modules. If one has for all i ∈ ℤ, then Gid R RHom R (M, N) = G-dim R M + Gid R N. In addition, some Auslander–Buchsbaum type width (depth) formulas are given.

The negative side of cohomology for Calabi-Yau categories

We study Z-graded cohomology rings defined over Calabi–Yau categories. We show that the cohomology in negative degree is a trivial extension of the cohomology ring in non-negative degree, provided the latter admits a regular sequence of central elements of length 2. In particular, the products of elements of negative degrees are zero. As corollaries, we apply this to Tate–Hochschild cohomology rings of symmetric algebras, and to Tate cohomology rings over group algebras. We also prove similar results for Tate cohomology rings over commutative local Gorenstein rings.