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1.
Abstract

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.  相似文献   

2.
3.
In this note we prove two theorems. In theorem 1 we prove that if M andN are two non-zero reflexive modules of finite projective dimensions over a Gorenstein local ring, such that Hom (M, N) is a third module of syzygies, then the natural homomorphismM* ⊗N → Hom (M, N) is an isomorphism. This extends the result in [7]. In theorem 2, we prove that projective dimension of a moduleM over a regular local ringR is less than or equal ton if and only if ExtR n (M, R) ⊗M → ExtR n (M, M) is surjective; in which case it is actually bijective. This extends the usual criterion for the projectivity of a module.  相似文献   

4.
    
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.  相似文献   

5.
    

Let be an -primary ideal in a Gorenstein local ring (, ) with , and assume that contains a parameter ideal in as a reduction. We say that is a good ideal in if is a Gorenstein ring with . The associated graded ring of is a Gorenstein ring with if and only if . Hence good ideals in our sense are good ones next to the parameter ideals in . A basic theory of good ideals is developed in this paper. We have that is a good ideal in if and only if and . First a criterion for finite-dimensional Gorenstein graded algebras over fields to have nonempty sets of good ideals will be given. Second in the case where we will give a correspondence theorem between the set and the set of certain overrings of . A characterization of good ideals in the case where will be given in terms of the goodness in their powers. Thanks to Kato's Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show that the structure of the set of good ideals in heavily depends on . The set may be empty if , while is necessarily infinite if and contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring in three variables over a field . Examples are given to illustrate the theorems.

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6.
    

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

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7.
    
Let be an integrally closed ideal in a commutative Noetherian ring . Then the local ring is regular (resp. Gorenstein) for every if the projective dimension of is finite (resp. the Gorenstein dimension of is finite and satisfies Serre's condition (S)).

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8.
    
For a one-dimensional local domain constructed by Akizuki, we find residue maps which give rise to a local duality. The completion of is described using these residue maps.

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9.
    
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10.
In this paper we determine the possible Hilbert functions ofa Cohen–Macaulay local ring of dimension d and multiplicitye, in the case where the embedding dimension v satisfies v =e + d – 3 and the Cohen–Macaulay type is less thanor equal to e – 3. 1991 Mathematics Subject Classification:primary 13D40; secondary 13P99.  相似文献   

11.
    
Let C be a class of R-modules closed under isomorphisms and finite direct sums. It is first shown that the finite direct sum of almost C-precovers is an almost C-precover and the direct sum of an almost C-cover and a weak C-cover is a weak C-cover. Then the notion of almost C-preenvelopes is introduced and studied.  相似文献   

12.
    
Let be a regular local ring and let be a filtration of ideals in such that is a Noetherian ring with . Let and let be the -invariant of . Then the theorem says that is a principal ideal and for all if and only if is a Gorenstein ring and . Hence , if is a Gorenstein ring, but the ideal is not principal.

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13.
《代数通讯》2013,41(11):4415-4432
Abstract

Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions.  相似文献   

14.
Let (R, m) be a Cohen–Macaulay local ring, and let ? = {F i } i∈? be an F 1-good filtration of ideals in R. If F 1 is m-primary we obtain sufficient conditions in order that the associated graded ring G(?) be Cohen–Macaulay. In the case where R is Gorenstein, we use the Cohen–Macaulay result to establish necessary and sufficient conditions for G(?) to be Gorenstein. We apply this result to the integral closure filtration ? associated to a monomial parameter ideal of a polynomial ring to give necessary and sufficient conditions for G(?) to be Gorenstein. Let (R, m) be a Gorenstein local ring, and let F 1 be an ideal with ht(F 1) = g > 0. If there exists a reduction J of ? with μ(J) = g and reduction number u: = r J (?), we prove that the extended Rees algebra R′(?) is quasi-Gorenstein with a-invariant b if and only if J n : F u  = F n+b?u+g?1 for every n ∈ ?. Furthermore, if G(?) is Cohen–Macaulay, then the maximal degree of a homogeneous minimal generator of the canonical module ω G(?) is at most g and that of the canonical module ω R′(?) is at most g ? 1; moreover, R′(?) is Gorenstein if and only if J u : F u  = F u . We illustrate with various examples cases where G(?) is or is not Gorenstein.  相似文献   

15.
《代数通讯》2013,41(9):4371-4385
Abstract

We study Gorenstein injective and projective modules over Zariski filtered rings and obtain relations between the Gorenstein dimensions on the category of filtered modules from the associated category of graded modules over the associated graded ring.  相似文献   

16.
Jonathan Shick 《代数通讯》2013,41(4):1371-1388
The local cohomology modules HJ I(M) of a Matlis reflexive module are shown to be I-cofinite when j >= 1 and have finite Bass numbers when j >= 0, where I is an ideal satisfying any one of a list of properties. In addition, we show that the completion of a Matlis reflexive module is finitely generated over the completion of the ring and we classify Matlis reflexive modules over a one dimensional ring.  相似文献   

17.
    
This note proves that if is an unramified regular local ring and proper ideals of height at least two, then is never Gorenstein.

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18.
We extend the notion of virtually Gorenstein rings to the setting of arbitrary rings, and prove that all rings R of finite Gorenstein weak global dimension are virtually Gorenstein such that all Gorenstein projective R-modules are Gorenstein flat. For such a ring R, we introduce the notion of relative homology functors of complexes with respect to Gorenstein projective (resp., flat) modules, and establish a balanced and a vanishing result for the homology functor.  相似文献   

19.
高增辉 《中国科学:数学》2013,43(10):1037-1046
设n 是正整数, 本文引入并研究n- 强Gorenstein FP- 内射模. 对于正整数n > m, 给出例子说明n- 强Gorenstein FP- 内射模未必是m- 强Gorenstein FP- 内射的, 并讨论n- 强Gorenstein FP-内射模的诸多性质. 最后, 利用n- 强Gorenstein FP- 内射模刻画n- 强Gorenstein Von Neumann 正则环.  相似文献   

20.
    
Liu Dajun  Jiaqun Wei 《代数通讯》2020,48(9):3846-3858
Abstract

Let A be an n-Gorenstein ring. Employing the theory developed by Enochs on the existence of Gorenstein preenvelopes and precovers, we introduce the concept of Gorenstein tilting pair. Moreover, we give a simple characterization on Gorenstein tilting pair, which shows that Gorenstein cotilting and tilting modules are special examples of Gorenstein tilting pair.  相似文献   

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