Modules of generalized fractions

Sharp, Rodney Y.; Zakeri, H.

doi:10.1112/s0025579300012134

Cited by 43 publications

(47 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We show that U n is a triangular subset of A" for each neN. Now U x is a multiplicatively closed and hence [13,3.1] triangular subset of A 1 . Assume, inductively, that n>l and U n -t is known to be a triangular subset of A"" 1 .…”

mentioning

confidence: 82%

See 1 more Smart Citation

Cousin complexes and generalized fractions

Riley

Sharp

Zakeri³

1985

Glasgow Math. J.

View full text Add to dashboard Cite

Let A be a commutative Noetherian ring (with non-zero identity). The Cousin complex C(A) for A is described in [6, §2]: it is a complex of A-modules and A-homomorphismswith the property that, for each n≥0,Cohen-Macaulay rings may be characterized in terms of the Cousin complex: A is a Cohen-Macaulay ring if and only if C(A) is exact [6, (4.7)]. Also the Cousin complex provides a natural minimal injective resolution for a Gorenstein ring: see [6, (5.4)].

show abstract

mentioning

confidence: 82%

“…, U,, l)e U i+1 ; (iv) whenever (u u ... ,u i )eU i with K i e M , then (wj,..., Uj_i)6 t/ c _ x . Each L/j leads to a module of generalized fractions UJ l M [13], and we can, in fact, arrange these modules into a complex denoted by C(% M), for which e°(m) = -^-for all meM and i, . .…”

mentioning

confidence: 99%

Cousin complexes and generalized fractions

Riley

Sharp

Zakeri³

1985

Glasgow Math. J.

View full text Add to dashboard Cite

show abstract

“…The main purpose of this note is to establish connections between modules of generalized fractions introduced in [7] and the above mentioned sequences. This paper is divided in two sections.…”

Section: §0 Introductionmentioning

confidence: 99%

Characterizations of filter regular sequences and unconditioned strong d-sequences

Khashyarmanesh

Salarian

Zakeri³

1998

Nagoya Mathematical Journal

Self Cite

View full text Add to dashboard Cite

Abstract. The first part of the paper is concerned, among other things, with a characterization of filter regular sequences in terms of modules of generalized fractions. This characterization leads to a description, in terms of generalized fractions, of the structure of an arbitrary local cohomology module of a finitely generated module over a notherian ring. In the second part of the paper, we establish homomorphisms between the homology modules of a Koszul complex and the homology modules of a certain complex of modules of generalized fractions. Using these homomorphisms, we obtain a characterization of unconditioned strong ^-sequences. §0. IntroductionThroughout this note A is a commutative ring (with non-zero identity), α is an ideal of A and M is an A-module.There is a lot of current interest in the theory of filter regular sequences and unconditioned strong d-sequences (u.s.d-sequences) in commutative algebra; and, in recent years, there have appeared many papers concerned with the role of these sequences in the theory of local cohomology. The main purpose of this note is to establish connections between modules of generalized fractions introduced in [7] and the above mentioned sequences. This paper is divided in two sections. In the first section we provide a characterization of α-filter regular sequences in terms of modules of generalized fractions. This result is a slight generalization of the Exactness Theorem [5, 3.1]. In [9], Sharp and the third author proved, in certain situation, for a finitely generated module N over a (Noetherian) local ring R having maximal ideal m, that the i-th local cohomology module H^(N) is isomorphic to the i-th homology module of a certain complex of modules of generalized fractions. Our characterization of α-filter regular sequences, in this paper, yields improved forms of the above theorem and the results

show abstract

“…In [7], Z. Tang and H. Zakeri introduced the concept of co-Cohen-Macaulay Artinian module over a quasi-local commutative ring R (with identity): a non-zero Artinian R-module A is said to be a co-Cohen-Macaulay module if and only if codepth A = dim ,4, where codepth ^4 is the length of a maximal /4-cosequence and dim/4 is the Krull dimension of A as defined by R. N. Roberts in [2]. Tang and Zakeri obtained several properties of co-Cohen-Macaulay Artinian ^-modules, including a characterization of such modules by means of the modules of generalized fractions introduced by Zakeri and the present second author in [6]; this characterization is explained as follows.Let m denote the maximal ideal of R, and let A be a non-zero Artinian /?-module of Krull dimension d > 0. Roberts [2, Theorem 6] proved that d is equal to the least integer /' for which there exists a proper ideal q of R generated by i elements such that (O^q) Tang and Zakeri did not extend the notion of co-Cohen-Macaulay module to Artinian modules over arbitrary commutative rings, and so we make here the following obvious definition.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Co-Cohen-Macaulay Artinian modules over commutative rings

Denizler

Sharp

1996

Glasgow Math. J.

Self Cite

View full text Add to dashboard Cite

0. Introduction. In [7], Z. Tang and H. Zakeri introduced the concept of co-Cohen-Macaulay Artinian module over a quasi-local commutative ring R (with identity): a non-zero Artinian R-module A is said to be a co-Cohen-Macaulay module if and only if codepth A = dim ,4, where codepth ^4 is the length of a maximal /4-cosequence and dim/4 is the Krull dimension of A as defined by R. N. Roberts in [2]. Tang and Zakeri obtained several properties of co-Cohen-Macaulay Artinian ^-modules, including a characterization of such modules by means of the modules of generalized fractions introduced by Zakeri and the present second author in [6]; this characterization is explained as follows.Let m denote the maximal ideal of R, and let A be a non-zero Artinian /?-module of Krull dimension d > 0. Roberts [2, Theorem 6] proved that d is equal to the least integer /' for which there exists a proper ideal q of R generated by i elements such that (O^q) Tang and Zakeri did not extend the notion of co-Cohen-Macaulay module to Artinian modules over arbitrary commutative rings, and so we make here the following obvious definition. 0.1. DEFINITION. Let A be a non-zero Artinian module over the commutative ring R (with identity). Observe that A m is an Artinian /? m -module for each maximal ideal m E Supp(/l). We say that A is a co-Cohen-Macaulay 7?-module precisely when A m is a co-Cohen-Macaulay fl m -module for each maximal ideal m e Supp(y4).It should be noted that, in the situation of this definition, the support of the Artinian -module A consists of maximal ideals; furthermore, it is an elementary exercise to show that Supp(/4) is equal to the finite set of maximal ideals m of R for which Soc(/4), the socle of A, has a submodule isomorphic to R/m.One of the aims of this paper is to provide a generalization of Tang's and Zakerfs

show abstract

Modules of generalized fractions

Cited by 43 publications

References 9 publications

Cousin complexes and generalized fractions

Cousin complexes and generalized fractions

Characterizations of filter regular sequences and unconditioned strong d-sequences

Co-Cohen-Macaulay Artinian modules over commutative rings

Contact Info

Product

Resources

About