Gorenstein Local Ring Research Articles

On Duality in the Homology Algebra of a Koszul Complex

The homology algebra of the Koszul complex K(x 1, ..., x n ; R) of a Gorenstein local ring R has Poincare duality if the ideal I = (x 1, ..., x n ) of R is strongly Cohen-Macaulay (i.e., all homology modules of the Koszul complex are Cohen-Macaulay) and under the assumption that dim R - grade I ⩽ 4 the converse is also true.

On Vanishing of Generalized Local Cohomology Modules

Let [Formula: see text] 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 [Formula: see text] if and only if [Formula: see text] for all [Formula: see text].

Maximal Buchsbaum modules over Gorenstein local rings

We clarify the structure of a maximal Buchsbaum module over a Gorenstein local ring by resolving the dual of its minimal free resolution into the mapping cones of successive chain maps from free complexes to direct sums of finite copies of the minimal free resolution of the residue field.

STRONGLY TORSION-FREE MODULES AND LOCAL COHOMOLOGY OVER COHEN–MACAULAY RINGS#

ABSTRACT In this paper we investigate the relationship between strongly torsion-free, maximal Cohen–Macaulay, and strongly cotorsion modules over Cohen–Macaulay local rings via local cohomology. Results are given analogous to those known for Gorenstein flat, totally reflexive, and Gorenstein injective modules over a Gorenstein local ring.

The Gorenstein and complete intersection properties of associated graded rings

Let I be an m -primary ideal of a Noetherian local ring ( R , m ) . We consider the Gorenstein and complete intersection properties of the associated graded ring G ( I ) and the fiber cone F ( I ) of I as reflected in their defining ideals as homomorphic images of polynomial rings over R / I and R / m respectively. In case all the higher conormal modules of I are free over R / I , we observe that: (i) G ( I ) is Cohen–Macaulay iff F ( I ) is Cohen–Macaulay, (ii) G ( I ) is Gorenstein iff both F ( I ) and R / I are Gorenstein, and (iii) G ( I ) is a relative complete intersection iff F ( I ) is a complete intersection. In case ( R , m ) is Gorenstein, we give a necessary and sufficient condition for G ( I ) to be Gorenstein in terms of residuation of powers of I with respect to a reduction J of I with μ ( J ) = dim R and the reduction number r of I with respect to J . We prove that G ( I ) is Gorenstein if and only if J : I r - i = J + I i + 1 for 0 ⩽ i ⩽ r - 1 . If ( R , m ) is a Gorenstein local ring and I ⊆ m is an ideal having a reduction J with reduction number r such that μ ( J ) = ht ( I ) = g > 0 , we prove that the extended Rees algebra R [ It , t - 1 ] is quasi-Gorenstein with a-invariant a if and only if J i : I r = I i + a - r + g - 1 for every i ∈ Z . If, in addition, dim R = 1 , we show that G ( I ) is Gorenstein if and only if J i : I r = I i for 1 ⩽ i ⩽ r .

Gorenstein Injective Dimension and Dual Auslander–Bridger Formula#

An R-module M satisfies the dual Auslander–Bridger formula if Gid R M + Tor − depth M = depth R. Let R be a complete Cohen–Macaulay local ring with residue field k and M be a non-injective Ext-finite R-module such that for all i ≥ 1 and all indecomposable injective R-modules E ≠ E(k). If M has finite Gorenstein injective dimension, then we will prove that M satisfies the dual Auslander–Bridger formula if either Ext-depth M > 0 or Ext-depth M = 0 and Gid R M ≠ 1. We denote Gorenstein injective envelope of M by G(M). If R is a Gorenstein local ring and M is a non-Gorenstein injective finitely generated R-module and G(M) is reduced, then this formula holds for and its cosyzygy. As the last result, if R is a regular local ring, then dual Auslander–Bridger holds for any non-injective Ext-finite R − module.

Ω-Gorenstein Projective and Flat Covers and Ω-Gorenstein Injective Envelopes#

In this paper, we show that if R is a local Cohen–Macaulay ring admitting a dualizing module Ω, then Ω-Gorenstein projective and flat covers and Ω-Gorenstein injective envelopes exist for certain modules. These results generalize the well known results for local Gorenstein rings.

The Gorenstein Injective Envelope of the Residue Field of a Local Ring#

Let R be a commutative Noetherian local Gorenstein ring with residue field k. We show that G(k), the Gorenstein injective envelope of k, is an artinian R-module, and we compute G(k) in the case where R = k[[S]] is a semigroup ring and S is symmetric. We also show that a certain subring of the endomorphism ring of G(k) is a complete local (but possibly non-commutative) ring.

Nonvanishing cohomology and classes of Gorenstein rings

We give counterexamples to the following conjecture of Auslander: given a finitely generated module M over an Artin algebra Λ, there exists a positive integer n M such that for all finitely generated Λ-modules N, if Ext Λ i ( M, N)=0 for all i≫0, then Ext Λ i ( M, N)=0 for all i⩾ n M . Some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local Gorenstein rings.

Hilbert coefficients and the Gorenstein property of the associated graded ring

In this article we investigate the Gorenstein property of the associated graded ring G A ( I) of an ideal I in a Gorenstein local ring (A, m) of positive dimension. We especially concentrate on the case where I is m -primary. Assuming that the Rees algebra of I is Cohen–Macaulay we then give necessary and sufficient conditions for the Gorensteiness of G A ( I) in terms of the Hilbert coefficients of I.

Some characterizations of Gorenstein local rings in terms of G-dimension

We characterize Gorenstein local rings by the existence of special modules of finite G-dimension.

Symmetry in the vanishing of Ext over Gorenstein rings

We investigate symmetry in the vanishing of Ext for finitely generated modules over local Gorenstein rings. In particular, we define a class of local Gorenstein rings, which we call AB rings, and show that for finitely generated modules $M$ and $N$ over an AB ring $R$, $\mathrm{Ext}^i_R(M,N)=0$ for all $i\gg 0$ if and only if $\mathrm{Ext}^i_R(N,M)=0$ for all $i\gg 0$.

Kustin--Miller unprojection with complexes

A main ingredient for Kustin-Miller unprojection, as developed in (S. Papadakis and M. Reid, Kustin-Miller unprojection without complexes, math.AG/0011094), is the module Hom_R(I, \om_R), where R is a local Gorenstein ring and I a codimension one ideal with R/I Gorenstein. We prove a method of calculating it in a relative setting using resolutions. We give three applications. In the first we generalise a result of (F. Catanese et al., Embeddings of curves and surfaces, Nagoya Math. J. 154 (1999), 185-220). The second and the third are about Tom and Jerry, two families of Gorenstein codimension four rings with 9x16 resolutions.

Good Ideals in Artinian Gorenstein Local Rings Obtained by Idealization

Good Ideals in Artinian Gorenstein Local Rings Obtained by Idealization

Vanishing of cohomology over Gorenstein rings of small codimension

We prove that if M, N are finite modules over a Gorenstein local ring R of codimension at most 4, then the vanishing of Ext n R(M, N) for n >> 0 is equivalent to the vanishing of Ext n R(N, M) for n >> 0. Furthermore, if R has no embedded deformation, then such vanishing occurs if and only if M or N has finite projective dimension.

NOTE ON GOOD IDEALS IN GORENSTEIN LOCAL RINGS

Let I be an ideal in a Gorenstein local ring A with the maximal ideal m and d = dim A. Then we say that I is a good ideal in A, if I contains a reduction <TEX>$Q=(a_1,a_2,...,a_d)$</TEX> generated by d elements in A and <TEX>$G(I)=\bigoplus_{n\geq0}I^n/I^{n+1}$</TEX> of I is a Gorenstein ring with a(G(I)) = 1-d, where a(G(I)) denotes the a-invariant of G(I). Let S = A[Q/a<TEX>$_1$</TEX>] and P = mS. In this paper, we show that the following conditions are equivalent. (1) <TEX>$I^2$</TEX> = QI and I = Q:I. (2) <TEX>$I^2S$</TEX> = <TEX>$a_1$</TEX>IS and IS = <TEX>$a_1$</TEX>S：sIS. (3) <TEX>$I^2$</TEX>Sp = <TEX>$a_1$</TEX>ISp and ISp = <TEX>$a_1$</TEX>Sp ：sp ISp. We denote by <TEX>$X_A(Q)$</TEX> the set of good ideals I in <TEX>$X_A(Q)$</TEX> such that I contains Q as a reduction. As a Corollary of this result, we show that <TEX>$I\inX_A(Q)\Leftrightarrow\IS_P\inX_{SP}(Qp)$</TEX>.

Equimultiple good ideals

Let $I$ be an ideal in a Gorenstein local ring A with the maximal ideal $\mathfrak{m}$. Then $I$ is said to be an equimultiple good ideal if I contains a reduction $Q = (a_{1}, a_{2}, \ldots ,a_{s})$ generated by s elements in $A$ and if the associated graded ring $\mathrm{G}(I) = \bigoplus _{n\geq 0} I^{n}/I^{n+1}$ of $I$ is a Gorenstein ring with $\mathrm{a}(\mathrm{G}(I))=1-s$, where $s = \mathrm{ht}_{A}I$ and $\mathrm{a}(\mathrm{G}(I))$ denotes the $a$-invariant of $\mathrm{G}(I)$. The purpose is to explore the structure of the sets $\chi _{A,s}$ $(s\geq 0)$ of equimultiple good ideals I with $\mathrm{ht}_{A}I=s$. Goto, Iai and Watanabe [GIW] already studied $\mathfrak{m}$-primary good ideals $I$, that is the case where $\mathrm{ht}_{A}I = \dim A$. Some of their results are successfully generalized to those of equimultiple case with improvements.

EQUIMULTIPLE GOOD IDEALS WITH HEIGHT 1

Let I be an ideal in a Gorenstein local ring A with the maximal ideal m. Then we say that I is an equimultiple good ideal in A, if I contains a reduction Q = ( <TEX>$a_1$</TEX>, <TEX>$a_2$</TEX>,ㆍㆍㆍ, <TEX>$a_{s}$</TEX> ) generated by s elements in A and G(I) =(equation omitted)<TEX>$_{n 0}$</TEX> <TEX>$I^{n}$</TEX> / <TEX>$I^{n＋1}$</TEX> of I is a Gorenstein ring with a(G(I)) = 1 - s, where s = h <TEX>$t_{A}$</TEX> I and a(G(I)) denotes the a-invariant of G(I). Let <TEX>$X_{A}$</TEX><TEX>$^{s}$</TEX> denote the set of equimultiple good ideals I in A with h <TEX>$t_{A}$</TEX> I = s, R(I) = A [It] be the Rees algebra of I, and <TEX>$K_{R(I)}$</TEX> denote the canonical module of R(I). Let a I such that <TEX>$I^{n＋l}$</TEX> = a <TEX>$I^{n}$</TEX> for some n<TEX>$\geq$</TEX>0 and <TEX>$\mu$</TEX><TEX>$_{A}$</TEX>(I)<TEX>$\geq$</TEX>2, where <TEX>$\mu$</TEX><TEX>$_{A}$</TEX>(I) denotes the number of elements in a minimal system of generators of I. Assume that A/I is a Cohen-Macaulay ring. We show that the following conditions are equivalent. (1) <TEX>$K_{R(I)}$</TEX>(equation omitted)R(I)＋as graded R(I)-modules. (2) <TEX>$I^2$</TEX> = aI and aA : I<TEX>$\in$</TEX> <TEX>$X^1$</TEX><TEX>$_{A}$</TEX>._{A}$</TEX>./.

On the dimension and multiplicity of local cohomology modules

AbstractThis paper is concerned with a finitely generated module M over a (commutative Noetherian) local ring R. In the case when R is a homomorphic image of a Gorenstein local ring, one can use the well-known associativity formula for multiplicities, together with local duality and Matlis duality, to produce analogous associativity formulae for the local cohomology modules of M with respect to the maximal ideal. The main purpose of this paper is to show that these formulae also hold in the case when R is universally catenary and such that all its formal fibres are Cohen–Macaulay.These formulae involve certain subsets of the spectrum of R called the pseudosupports of M; these pseudo-supports are closed in the Zariski topology when R is universally catenary and has the property that all its formal fibres are Cohen–Macaulay. However, examples are provided to show that, in general, these pseudo-supports need not be closed. We are able to conclude that the above-mentioned associativity formulae for local cohomology modules do not hold over all local rings.

Good ideals in Gorenstein local rings obtained by idealization

The structure of certain equimultiple good ideals in Gorenstein local rings obtained by idealization is explored.