Semidualizing bimodules and related Gorenstein homological dimensions

2016 ◽  
Vol 15 (10) ◽  
pp. 1650193 ◽  
Author(s):  
Aimin Xu ◽  
Nanqing Ding
Keyword(s):  
Projective Dimension ◽  
Negative Integer ◽  
Homological Dimensions ◽  
Semidualizing Bimodule ◽  
Finitely Presented

Let [Formula: see text] be a semidualizing bimodule with [Formula: see text] left coherent and [Formula: see text] right coherent. For a non-negative integer [Formula: see text], it is shown that [Formula: see text]-[Formula: see text]-[Formula: see text] if and only if every finitely presented left [Formula: see text]-module has [Formula: see text]-projective dimension at most [Formula: see text] if and only if every finitely presented right [Formula: see text]-module has [Formula: see text]-projective dimension at most [Formula: see text]. As applications, some well-known results are extended.

Homological Dimensions Relative to Special Subcategories

2021 ◽  
Vol 28 (01) ◽  
pp. 131-142
Author(s):  
Weiling Song ◽  
Tiwei Zhao ◽  
Zhaoyong Huang
Keyword(s):  
Abelian Category ◽  
Projective Dimension ◽  
Homological Dimensions ◽  
Category Of Modules ◽  
The Right

Let [Formula: see text] be an abelian category, [Formula: see text] an additive, full and self-orthogonal subcategory of [Formula: see text] closed under direct summands, [Formula: see text] the right Gorenstein subcategory of [Formula: see text] relative to [Formula: see text], and [Formula: see text] the left orthogonal class of [Formula: see text]. For an object [Formula: see text] in [Formula: see text], we prove that if [Formula: see text] is in the right 1-orthogonal class of [Formula: see text], then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical; if the [Formula: see text]-projective dimension of [Formula: see text] is finite, then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical. We also prove that the supremum of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension and that of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension coincide. Then we apply these results to the category of modules.

REGULAR LOCAL ALGEBRAS OVER VALUATION DOMAINS: WEAK DIMENSION AND REGULAR SEQUENCES

2008 ◽  
Vol 07 (05) ◽  
pp. 575-591
Author(s):  
HAGEN KNAF
Keyword(s):  
Projective Dimension ◽  
Regular Sequence ◽  
Homological Dimension ◽  
Finitely Generated ◽  
Prüfer Domain ◽  
Regular Sequences ◽  
Local Algebras ◽  
Weak Dimension ◽  
Valuation Domains ◽  
Finitely Presented

A local ring O is called regular if every finitely generated ideal I ◃ O possesses finite projective dimension. In the article localizations O = Aq, q ∈ Spec A, of a finitely presented, flat algebra A over a Prüfer domain R are investigated with respect to regularity: this property of O is shown to be equivalent to the finiteness of the weak homological dimension wdim O. A formula to compute wdim O is provided. Furthermore regular sequences within the maximal ideal M ◃ O are studied: it is shown that regularity of O implies the existence of a maximal regular sequence of length wdim O. If q ∩ R has finite height, then this sequence can be chosen such that the radical of the ideal generated by its members equals M. As a consequence it is proved that if O is regular, then the factor ring O/(q ∩ R)O, which is noetherian, is Cohen–Macaulay. If in addition (q ∩ R)Rq ∩ R is not finitely generated, then O/(q ∩ R)O itself is regular.

Gorenstein homological dimensions and some duality results

2017 ◽  
Vol 10 (03) ◽  
pp. 1750048
Author(s):  
Fatemeh Mohammadi Aghjeh Mashhad
Keyword(s):  
Local Ring ◽  
Duality Theorem ◽  
Projective Dimension ◽  
Injective Dimension ◽  
Duality Results ◽  
Dualizing Complex ◽  
Matlis Duality ◽  
Homological Dimensions ◽  
Gorenstein Injective Dimension ◽  
Gorenstein Injective

Let [Formula: see text] be a local ring and [Formula: see text] denote the Matlis duality functor. Assume that [Formula: see text] possesses a normalized dualizing complex [Formula: see text] and [Formula: see text] and [Formula: see text] are two homologically bounded complexes of [Formula: see text]-modules with finitely generated homology modules. We will show that if G-dimension of [Formula: see text] and injective dimension of [Formula: see text] are finite, then [Formula: see text] Also, we prove that if Gorenstein injective dimension of [Formula: see text] and projective dimension of [Formula: see text] are finite, then [Formula: see text] These results provide some generalizations of Suzuki’s Duality Theorem and the Herzog–Zamani Duality Theorem.

scholarly journals Ascent properties for test modules

2021 ◽  
pp. 153-167
Author(s):  
Keri Sather-Wagstaff
Keyword(s):  
Local Ring ◽  
Projective Dimension ◽  
Graded Algebras ◽  
Ring Homomorphisms ◽  
Flat Dimension ◽  
Finite Dimensional ◽  
Differential Graded Algebras ◽  
Homological Dimensions ◽  
Tor Modules

We investigate modules for which vanishing of Tor-modules implies finiteness of homological dimensions (e.g., projective dimension and G-dimension). In particular, we answer a question of O. Celikbas and Sather-Wagstaff about ascent properties of such modules over residually algebraic flat local ring homomorphisms. To accomplish this, we consider ascent and descent properties over local ring homomorphisms of finite flat dimension, and for flat extensions of finite dimensional differential graded algebras.

Homological Dimensions with Respect to a Semidualizing Module and Tensor Products of Algebras

2015 ◽  
Vol 22 (02) ◽  
pp. 215-222
Author(s):  
Maryam Salimi ◽  
Elham Tavasoli ◽  
Siamak Yassemi
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Tensor Products ◽  
Projective Dimension ◽  
Injective Dimension ◽  
Semidualizing Module ◽  
Homological Dimensions

Let C be a semidualizing module for a commutative ring R. It is shown that the [Formula: see text]-injective dimension has the ability to detect the regularity of R as well as the [Formula: see text]-projective dimension. It is proved that if D is dualizing for a Noetherian ring R such that id R(D) = n < ∞, then [Formula: see text] for every flat R-module F. This extends the result due to Enochs and Jenda. Finally, over a Noetherian ring R, it is shown that if M is a pure submodule of an R-module N, then [Formula: see text]. This generalizes the result of Enochs and Holm.

scholarly journals CLASSIFYING PROJECTIVE MODULES OVER SOME SEMILOCAL RINGS

2007 ◽  
Vol 06 (05) ◽  
pp. 839-865 ◽  
Author(s):  
NIKOLAY DUBROVIN ◽  
GENA PUNINSKI
Keyword(s):  
Projective Dimension ◽  
Projective Modules ◽  
Module Theory ◽  
Finitely Presented

We investigate the module theory of a certain class of semilocal rings connected with nearly simple uniserial domains. For instance, we classify finitely presented and pure-projective modules over these rings and calculate their projective dimension.

n-Matlis cotorsion modules and n-matlis domains

2019 ◽  
Vol 19 (07) ◽  
pp. 2050139
Author(s):  
Yongyan Pu ◽  
Gaohua Tang ◽  
Fanggui Wang
Keyword(s):  
Projective Dimension ◽  
Negative Integer ◽  
Dimension Formula

Let [Formula: see text] be a domain with its field [Formula: see text] of quotients, [Formula: see text] an [Formula: see text]-module and [Formula: see text] a fixed non-negative integer. Then [Formula: see text] is called [Formula: see text]-Matlis cotorsion if [Formula: see text] for any integer [Formula: see text]. Also [Formula: see text] is said to be [Formula: see text]-Matlis flat if [Formula: see text] for any [Formula: see text]-Matlis cotorsion [Formula: see text]-module [Formula: see text]. We proved that [Formula: see text] is a complete hereditary cotorsion theory, where [Formula: see text] (respectively, [Formula: see text]) denotes the class of all [Formula: see text]-Matlis flat (respectively, [Formula: see text]-Matlis cotorsion) [Formula: see text]-modules. In this paper, it is proved that [Formula: see text] is an [Formula: see text]-Matlis domain if and only if epic images of [Formula: see text]-Matlis cotorsion [Formula: see text]-modules are again [Formula: see text]-Matlis cotorsion if and only if [Formula: see text]-Matlis flat [Formula: see text]-modules are of projective dimension [Formula: see text] if and only if [Formula: see text] if and only if [Formula: see text].

scholarly journals ON n-PERFECT RINGS AND COTORSION DIMENSION

2009 ◽  
Vol 08 (02) ◽  
pp. 181-190 ◽  
Author(s):  
DRISS BENNIS ◽  
NAJIB MAHDOU
Keyword(s):  
Projective Dimension ◽  
Flat Module ◽  
Homological Dimensions

A ring is called n-perfect (n ≥ 0), if every flat module has projective dimension less or equal than n. In this paper, we show that the n-perfectness relates, via homological approach, some homological dimensions of rings. We study n-perfectness in some known ring construction. Finally, several examples of n-perfect rings satisfying special conditions are given.

A QUILLEN MODEL STRUCTURE APPROACH TO HOMOLOGICAL DIMENSIONS OF COMPLEXES

2013 ◽  
Vol 13 (03) ◽  
pp. 1350106
Author(s):  
REN WEI ◽  
ZHONGKUI LIU
Keyword(s):  
Projective Dimension ◽  
Model Structure ◽  
Cotorsion Pair ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Homological Dimensions ◽  
Von Neumann Regular ◽  
Category Of Modules ◽  
Bounded Below ◽  
Quillen Model

In this paper, we first give an alternative characterization of the derived functor Ext via the Quillen model structure on the category of complexes induced by a given cotorsion pair [Formula: see text] in the category of modules, then based on this, we consider homological dimensions of complexes related to [Formula: see text]. As applications, we extend Gorenstein projective dimension of homologically bounded below complexes (in the sense of Christensen and coauthors) to unbounded complexes whenever R is Gorenstein. Moreover, we extend Stenström's FP-injective dimension from modules to complexes, define FP-projective dimension for complexes, and characterize Noetherian and von Neumann regular rings by these dimensions.

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