FP-INJECTIVE COMPLEXES AND FP-INJECTIVE DIMENSION OF COMPLEXES

2011 ◽  
Vol 91 (2) ◽  
pp. 163-187 ◽  
Author(s):  
ZHANPING WANG ◽  
ZHONGKUI LIU
Keyword(s):  
Close Connection ◽  
Injective Dimension ◽  
Arbitrary Ring ◽  
Flat Dimension ◽  
Injective Modules

AbstractIn this paper we extend the notion of FP-injective modules to that of complexes and characterize such complexes. We show that some characterizations similar to those for injective complexes exist for FP-injective complexes. We also introduce and study the notion of an FP-injective dimension associated to every complex of left R-modules over an arbitrary ring. We show that there is a close connection between the FP-injective dimension of complexes and flat dimension.

scholarly journals Homological Dimensions of Local (Co)homology Over Commutative DG-rings

2018 ◽  
Vol 61 (4) ◽  
pp. 865-877 ◽  
Author(s):  
Liran Shaul
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Basic Result ◽  
Structure Theory ◽  
Injective Dimension ◽  
Commutative Noetherian Ring ◽  
Flat Dimension ◽  
Injective Modules ◽  
Homological Dimensions ◽  
Adic Completion

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.

Balance for relative cohomology of complexes

2019 ◽  
Vol 19 (01) ◽  
pp. 2050005
Author(s):  
Zhenxing Di ◽  
Bo Lu ◽  
Junxiu Zhao
Keyword(s):  
Projective Dimension ◽  
Cohomology Groups ◽  
Injective Dimension ◽  
Projective Generator ◽  
Arbitrary Ring ◽  
Relative Cohomology ◽  
Injective Modules ◽  
Injective Cogenerator ◽  
Semidualizing Modules

Let [Formula: see text] be an arbitrary ring. We use a strict [Formula: see text]-resolution [Formula: see text] of a complex [Formula: see text] with finite [Formula: see text]-projective dimension, where [Formula: see text] denotes a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits an injective cogenerator [Formula: see text], to define the [Formula: see text]th relative cohomology functor [Formula: see text] as [Formula: see text]. If a complex [Formula: see text] has finite [Formula: see text]-injective dimension, then one can use a dual argument to define a notion of a relative cohomology functor [Formula: see text], where [Formula: see text] is a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits a projective generator. Under several orthogonal conditions, we show that there exists an isomorphism [Formula: see text] of relative cohomology groups for each [Formula: see text]. This result simultaneously encompasses a balance result of Holm on Gorenstein projective and injective modules, a balance result of Sather-Wagstaff, Sharif and White on Gorenstein projective and injective modules with respect to semidualizing modules, and a balance result of Liu on Gorenstein projective and injective complexes. In particular, as an application of this result, we extend the above balance result of Sather-Wagstaff, Sharif and White to the setting of complexes.

WEAK AB-CONTEXT FOR FP-INJECTIVE MODULES WITH RESPECT TO SEMIDUALIZING MODULES

2013 ◽  
Vol 12 (07) ◽  
pp. 1350039 ◽  
Author(s):  
JIANGSHENG HU ◽  
DONGDONG ZHANG
Keyword(s):  
Noetherian Ring ◽  
Model Structure ◽  
Injective Dimension ◽  
Commutative Noetherian Ring ◽  
New Model ◽  
Injective Modules ◽  
Semidualizing Modules ◽  
Semidualizing Bimodule ◽  
Gorenstein Injective

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.

Complexes of FP-injective Modules

2010 ◽  
Vol 17 (04) ◽  
pp. 667-684
Author(s):  
Yuxian Geng
Keyword(s):  
Injective Dimension ◽  
Cotorsion Pair ◽  
Injective Modules ◽  
Coherent Ring

In this paper, we study the class [Formula: see text] of complexes of FP-injective left R-modules. It is shown that the pair [Formula: see text] is a complete cotorsion pair. If R is a left coherent ring, it is proved that every complex has an [Formula: see text]-cover. We also introduce the FP-injective dimension of complexes. A special attention is paid to the dimension of homologically bounded above R-complexes over a left coherent ring which has a nice functorial description.

Stability of Gorenstein Classes of Modules

2013 ◽  
Vol 20 (04) ◽  
pp. 623-636 ◽  
Author(s):  
Samir Bouchiba
Keyword(s):  
Exact Sequence ◽  
Projective Modules ◽  
Natural Question ◽  
Alternate Proof ◽  
Similar Treatment ◽  
Arbitrary Ring ◽  
Injective Modules ◽  
Gorenstein Projective Modules ◽  
Category Of Modules ◽  
Gorenstein Injective

The purpose of this paper is to give, via totally different techniques, an alternate proof to the main theorem of [18] in the category of modules over an arbitrary ring R. In effect, we prove that this theorem follows from establishing a sequence of equalities between specific classes of R-modules. Actually, we tackle the following natural question: What notion emerges when iterating the very process applied to build the Gorenstein projective and Gorenstein injective modules from complete resolutions? In other words, given an exact sequence of Gorenstein injective R-modules G= ⋯ → G1→ G0→ G-1→ ⋯ such that the complex Hom R(H,G) is exact for each Gorenstein injective R-module H, is the module Im (G0→ G-1) Gorenstein injective? We settle such a question in the affirmative and the dual result for the Gorenstein projective modules follows easily via a similar treatment to that used in this paper. As an application, we provide the Gorenstein versions of the change of rings theorems for injective modules over an arbitrary ring.

Triangulated Equivalences Involving Gorenstein Projective Modules

2017 ◽  
Vol 60 (4) ◽  
pp. 879-890 ◽  
Author(s):  
Yuefei Zheng ◽  
Zhaoyong Huang
Keyword(s):  
Noetherian Ring ◽  
Derived Category ◽  
Projective Modules ◽  
Injective Dimension ◽  
Stable Category ◽  
Dualizing Complex ◽  
Injective Modules ◽  
Gorenstein Projective Modules ◽  
Left And Right

AbstractFor any ring R, we show that, in the bounded derived category Db(Mod R) of left R-modules, the subcategory of complexes with finite Gorenstein projective (resp. injective) dimension modulo the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category (resp. ) of Gorenstein projective (resp. injective) modules. As a consequence, we get that if R is a left and right noetherian ring admitting a dualizing complex, then and are equivalent.

scholarly journals On Injective Sheaves

1964 ◽  
Vol 7 (3) ◽  
pp. 415-423
Author(s):  
H. Kleisli ◽  
Y.C. Wu
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Crucial Point ◽  
Arbitrary Ring ◽  
Injective Modules ◽  
Group A ◽  
Group B ◽  
Divisible Groups ◽  
Group D ◽  
Divisible Abelian Group

A divisible abelian group D can be characterized by the following property: Every homomorphism from an abelian group A to D can be extended to every abelian group B containing A. This together with the result that every abelian group can be embedded in a divisible group is a crucial point in many investigations on abelian groups. It was Baer, [1], who extended this result to modules over an arbitrary ring, replacing divisible groups by injective modules, that is, modules with the property mentioned above. Another proof was found later by Eckmann and Schopf, [3]. This proof assumes the proposition to hold for abelian groups and transfers it in a very simple and elegant manner to modules. In the sequel, we shall refer to this proof as to the Eckmann-Schopf proof.

Some homological properties of complete modules

1990 ◽  
Vol 108 (2) ◽  
pp. 231-246 ◽  
Author(s):  
Anne-Marie Simon
Keyword(s):  
Noetherian Ring ◽  
Partial Information ◽  
Finiteness Condition ◽  
Koszul Complex ◽  
Injective Dimension ◽  
Commutative Noetherian Ring ◽  
Flat Dimension ◽  
Local Case ◽  
General Feeling ◽  
Bass Numbers

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.

scholarly journals Rings characterized by their weakly-injective modules

1992 ◽  
Vol 34 (3) ◽  
pp. 349-353 ◽  
Author(s):  
Sergio R. López-Permouth
Keyword(s):  
Injective Hull ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Arbitrary Ring ◽  
Injective Modules

The notation in this paper will be standard and it may be found in [2] or [8]. Throughout the paper, the notation A ⊂' B will mean that A is an essential submodule of the module B. Given an arbitrary ring R and R-modules M and N, we say that M is weakly N-injective if and only if every map φ:N → E(M) from N into the injective hull E(M) of M may be written as a composition σ〫 , where :N→M and σ:M→E(M) is a monomorphism. This is equivalent to saying that for every map φ:N→E(M), there exists a submodule X of E(M), isomorphic to M, such that φ(N) is contained in X. In particular, M is weakly R-injective if and only if, for every x ∈ E(M), there exists X ⊂ E(M) such that x ∈ X ≌ M. We say that M is weakly-injective if and only if it is weakly N-innjective for every finitely generated module N. Clearly, M is weakly-injective if and only if, for every finitely generated submodule N of E(M), there exists X ⊂ E(M) such that N ⊂ X ≌ M.

scholarly journals Continuous homomorphisms and rings of injective dimension one

2012 ◽  
Vol 110 (2) ◽  
pp. 181
Author(s):  
Shou-Te Chang ◽  
I-Chiau Huang
Keyword(s):  
Power Series ◽  
Injective Dimension ◽  
One Dimensional ◽  
Injective Modules ◽  
Valuation Rings ◽  
Dimension One ◽  
Injective Hulls ◽  
Essential Extensions ◽  
Formal Power ◽  
The Ideal

Let $S$ be an $R$-algebra and $\mathfrak a$ be an ideal of $S$. We define the continuous hom functor from $R$-mod to $S$-mod with respect to the $\mathfrak a$-adic topology on $S$. We show that the continuous hom functor preserves injective modules iff the ideal-adic property and ideal-continuity property are satisfied for $S$ and $\mathfrak a$. Furthermore, if $S$ is $\mathfrak a$-finite over $R$, we show that the continuous hom functor also preserves essential extensions. Hence, the continuous hom functor can be used to construct injective modules and injective hulls over $S$ using what we know about $R$. Using the continuous hom functor we can characterize rings of injective dimension one using symmetry for a special class of formal power series subrings. In the Noetherian case, this enables us to construct one-dimensional local Gorenstein domains. In the non-Noetherian case, we can apply the continuous hom functor to a generalized form of the $D+M$ construction. We may construct a class of domains of injective dimension one and a series of almost maximal valuation rings of any complete DVR.

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