Some homological properties of complete modules

1990 ◽  
Vol 108 (2) ◽  
pp. 231-246 ◽  
Author(s):  
Anne-Marie Simon

In this paper A is a commutative noetherian ring, a an ideal of A and the A- modules are given the a-adic topology.It is a general feeling that completeness is a kind of finiteness condition. We make precise that feeling and, after a result concerning the homology of a complex of complete modules which can be used in place of Nakayama's Lemma, we establish analogies between complete modules and finitely generated ones, with respect to flat dimension, injective dimension, Bass numbers and the Koszul complex. This is particularly clear in the local case, where we have also some partial information on the support of a complete module. With respect to dimension however, the analogy fails, as shown by an example.

2018 ◽  
Vol 61 (4) ◽  
pp. 865-877 ◽  
Author(s):  
Liran Shaul

AbstractLet A be a commutative noetherian ring, let a ⊆ A be an ideal, and let I be an injective A-module. A basic result in the structure theory of injective modules states that the A-module Γa(I) consisting of ɑ-torsion elements is also an injective A-module. Recently, de Jong proved a dual result: If F is a flat A-module, then the ɑ-adic completion of F is also a flat A-module. In this paper we generalize these facts to commutative noetherian DG-rings: let A be a commutative non-positive DG-ring such that H0(A) is a noetherian ring and for each i < 0, the H0(A)-module Hi(A) is finitely generated. Given an ideal ⊆ H0(A), we show that the local cohomology functor R associated with does not increase injective dimension. Dually, the derived -adic completion functor LΛ does not increase flat dimension.


2013 ◽  
Vol 12 (07) ◽  
pp. 1350039 ◽  
Author(s):  
JIANGSHENG HU ◽  
DONGDONG ZHANG

Let S and R be rings and SCR a semidualizing bimodule. We define and study GC-FP-injective R-modules, and these modules are exactly C-Gorenstein injective R-modules defined by Holm and Jørgensen provided that S = R is a commutative Noetherian ring. We mainly prove that the category of GC-FP-injective R-modules is a part of a weak AB-context, which is dual of weak AB-context in the terminology of Hashimoto. In particular, this allows us to deduce the existence of certain Auslander–Buchweitz approximations for R-modules with finite GC-FP-injective dimension. As an application, a new model structure in Mod R is given.


2011 ◽  
Vol 54 (4) ◽  
pp. 619-629 ◽  
Author(s):  
Mohammad T. Dibaei ◽  
Alireza Vahidi

AbstractLet M be a finite module over a commutative noetherian ring R. For ideals a and b of R, the relations between cohomological dimensions of M with respect to a, b, a ⋂ b and a + b are studied. When R is local, it is shown that M is generalized Cohen–Macaulay if there exists an ideal a such that all local cohomology modules of M with respect to a have finite lengths. Also, when r is an integer such that 0 ≤ r < dimR(M), any maximal element q of the non-empty set of ideals ﹛a : (M) is not artinian for some i, i ≥ r} is a prime ideal, and all Bass numbers of (M) are finite for all i ≥ r.


2014 ◽  
Vol 21 (01) ◽  
pp. 167-180
Author(s):  
Lingguang Li

Let R be a Noetherian ring, M an Artinian R-module, and 𝖒 ∈ Cos RM. Then cograde R𝔭 Hom R (R𝔭,M) = inf {i | πi(𝔭,M) > 0} and [Formula: see text] where πi(𝔭,M) is the i-th dual Bass number of M with respect to 𝔭, cograde R𝔭 Hom R (R𝔭,M) is the common length of any maximal Hom R (R𝔭, M)-quasi co-regular sequence contained in 𝔭 R𝔭, and fd R𝔭 Hom R (R𝔭, M) is the flat dimension of the R𝔭-module Hom R (R𝔭, M). We also study the relations among cograde, co-dimension and flat dimension of co-localization modules.


1994 ◽  
Vol 116 (3) ◽  
pp. 401-414
Author(s):  
D. Katz

In [R] D. Rees introduced the notions of reduction and integral closure for modules over a commutative Noetherian ring and proved the following remarkable result. Let R be a locally quasi-unmixed Noetherian ring and I an ideal generated by n elements. Suppose that height (I) = h. Then the ith module of cycles in the Koszul complex on a set of n generators for I is contained in the integral closure of the ith module of boundaries for i > n − h. This result should be considered a dimension-theoretic analogue of the famous depth sensitivity property of the Koszul complex demonstrated by Serre and Auslander-Buschsbaum in the 1950s. At roughly the same time, Hoschster and Huneke introduced the notion of tight closure and thereafter gave a number of theorems in the same (though considerably broader) vein for tight closure. In particular, in [HH] they showed that if R is an equidimensional local ring of characteristic p > 0, which is a homomorphic image of a Gorenstein ring, then for all i > 0, the ith module of cycles is contained in the tight closure of the ith module of boundaries for any complex satisfying the so-called standard rank and height conditions (see the definitions below). Since the tight closure is contained in the integral closure for such rings, the result of Hochster and Huneke extends (in characteristic p) considerably the result of Rees. In fact, their result could be considered a dimension-theoretic analogue of the Buchsbaum-Eisenbud exactness theorem ([BE]), which in a certain sense is the ultimate depth sensitivity theorem. Moreover, using the technique of reduction to characteristic p, Hochster and Huneke have shown that their results hold in equicharacteristic zero as well, whenever the tight closure is defined.


2016 ◽  
Vol 15 (08) ◽  
pp. 1650147 ◽  
Author(s):  
H. Roshan-Shekalgourabi ◽  
D. Hassanzadeh-Lelekaami

Let [Formula: see text] be a commutative Noetherian ring with identity and [Formula: see text] be an ideal of [Formula: see text]. Assume that [Formula: see text] is a finite [Formula: see text]-module and [Formula: see text] and [Formula: see text] are minimax [Formula: see text]-modules such that [Formula: see text]. In this paper, among other things, we show that [Formula: see text] is minimax for all [Formula: see text] and [Formula: see text] when one of the following conditions holds: [Formula: see text](i) [Formula: see text]; [Formula: see text] (ii) [Formula: see text]; or (iii) [Formula: see text]. As a consequence, we obtain that the Bass numbers and Betti numbers of [Formula: see text] are finite for all [Formula: see text] when one of the above conditions holds.


2015 ◽  
Vol 58 (3) ◽  
pp. 664-672 ◽  
Author(s):  
Alireza Vahidi

AbstractAssume that R is a commutative Noetherian ring with non-zero identity, 𝔞 is an ideal of R, and X is an R-module. In this paper, we first study the finiteness of Betti numbers of local cohomology modules . Then we give some inequalities between the Betti numbers of X and those of its local cohomology modules. Finally, we present many upper bounds for the flat dimension of X in terms of the flat dimensions of its local cohomology modules and an upper bound for the flat dimension of in terms of the flat dimensions of the modules , and that of X.


2013 ◽  
Vol 12 (07) ◽  
pp. 1350036 ◽  
Author(s):  
HERO SAREMI ◽  
AMIR MAFI

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M, N two nonzero finitely generated R-modules. Let t be a non-negative integer. It is shown that dim Supp [Formula: see text] for all i < t if and only if there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and [Formula: see text] for all i < t. As a consequence all Bass numbers and all Betti numbers of generalized local cohomology modules [Formula: see text] are finite for all i < t, provided that the projective dimension pd (M) is finite.


2016 ◽  
Vol 15 (03) ◽  
pp. 1650043 ◽  
Author(s):  
Pye Phyo Aung

Given a semidualizing module [Formula: see text] over a commutative Noetherian ring, Holm and Jørgensen [Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205(2) (2006) 423–445] investigate some connections between [Formula: see text]-Gorenstein dimensions of an [Formula: see text]-complex and Gorenstein dimensions of the same complex viewed as a complex over the “trivial extension” [Formula: see text]. We generalize some of their results to a certain type of retract diagram. We also investigate some examples of such retract diagrams, namely D’Anna and Fontana’s amalgamated duplication [An amalgamated duplication of a ring along an ideal: The basic properties, J. Algebra Appl. 6(3) (2007) 443–459] and Enescu’s pseudocanonical cover [A finiteness condition on local cohomology in positive characteristic, J. Pure Appl. Algebra 216(1) (2012) 115–118].


1985 ◽  
Vol 28 (3) ◽  
pp. 289-299 ◽  
Author(s):  
K. R. Goodearl ◽  
D. A. Jordan

The question of whether an injective module E over a noncommutative noetherian ring R remains injective after localization with respect to a denominator set X⊆R is addressed. (For a commutative noetherian ring, the answer is well-known to be positive.) Injectivity of the localization E[X-1] is obtained provided either R is fully bounded (a result of K. A. Brown) or X consists of regular normalizing elements. In general, E [X-1] need not be injective, and examples are constructed. For each positive integer n, there exists a simple noetherian domain R with Krull and global dimension n+1, a left and right denominator set X in R, and an injective right R-module E such that E[X-1 has injective dimension n; moreover, E is the injective hull of a simple module.


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