Gorenstein Projective Dimension Relative to a Semidualizing Bimodule

2013 ◽  
Vol 41 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Zengfeng Liu ◽  
Zhaoyong Huang ◽  
Aimin Xu
Keyword(s):  
Projective Dimension ◽  
Semidualizing Bimodule

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.

scholarly journals Duality pairs induced by Auslander and Bass classes

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Zhaoyong Huang
Keyword(s):  
Projective Dimension ◽  
Duality Pairs ◽  
Semidualizing Bimodule

Abstract Let R and S be arbitrary rings and let C S R {{}_{R}C_{S}} be a semidualizing bimodule, and let 𝒜 C ⁢ ( R op ) {\mathcal{A}_{C}(R^{\mathrm{op}})} and ℬ C ⁢ ( R ) {\mathcal{B}_{C}(R)} be the Auslander and Bass classes, respectively. Then both pairs ( 𝒜 C ⁢ ( R op ) , ℬ C ⁢ ( R ) )   and   ( ℬ C ⁢ ( R ) , 𝒜 C ⁢ ( R op ) ) (\mathcal{A}_{C}(R^{\mathrm{op}}),\mathcal{B}_{C}(R))\quad\text{and}\quad(% \mathcal{B}_{C}(R),\mathcal{A}_{C}(R^{\mathrm{op}})) are coproduct-closed and product-closed duality pairs and both 𝒜 C ⁢ ( R op ) {\mathcal{A}_{C}(R^{\mathrm{op}})} and ℬ C ⁢ ( R ) {\mathcal{B}_{C}(R)} are covering and preenveloping; in particular, the former duality pair is perfect. Moreover, if ℬ C ⁢ ( R ) {\mathcal{B}_{C}(R)} is enveloping in Mod ⁡ R {\operatorname{Mod}R} , then 𝒜 C ⁢ ( S ) {\mathcal{A}_{C}(S)} is enveloping in Mod ⁡ S {\operatorname{Mod}S} . Also, some applications to the Auslander projective dimension of modules are given.

Modules of finite projective dimension and cocovers

1996 ◽  
Vol 306 (1) ◽  
pp. 445-457 ◽  
Author(s):  
Dieter Happel ◽  
Luise Unger
Keyword(s):  
Projective Dimension ◽  
Finite Projective Dimension

scholarly journals Projective dimension and Brown-Peterson homology

Topology ◽  
1973 ◽  
Vol 12 (4) ◽  
pp. 327-353 ◽  
Author(s):  
David Copeland Johnson ◽  
W.Stephen Wilson
Keyword(s):  
Projective Dimension

Cellular actions and groups of finite quasi-projective dimension

1980 ◽  
Vol 170 (1) ◽  
pp. 85-90 ◽  
Author(s):  
James Howie ◽  
Hans Rudolf Schneebeli
Keyword(s):  
Projective Dimension

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.

scholarly journals PROPERLY STRATIFIED ALGEBRAS AND TILTING

2005 ◽  
Vol 92 (1) ◽  
pp. 29-61 ◽  
Author(s):  
ANDERS FRISK ◽  
VOLODYMYR MAZORCHUK
Keyword(s):  
Projective Dimension ◽  
Finitistic Dimension ◽  
Tilting Module ◽  
Tilting Modules ◽  
Category Of Modules ◽  
Cotilting Modules ◽  
Contravariantly Finite ◽  
Properly Stratified Algebras ◽  
Stratified Algebra ◽  
Generalized Tilting Module

We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

scholarly journals Projective dimension and regularity of the path ideal of the line graph

2018 ◽  
Vol 17 (04) ◽  
pp. 1850068 ◽  
Author(s):  
Guangjun Zhu
Keyword(s):  
Line Graph ◽  
Quotient Ring ◽  
Projective Dimension ◽  
Algebraic Properties

By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen–Macaulay and also provide some exact formulas for the projective dimension and the regularity of these ideals. As some consequences, we give some exact formulas for the depth of these ideals.

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