Gorenstein-Projective Modules over Morita Rings

2021 ◽  
Vol 28 (03) ◽  
pp. 521-532
Author(s):  
Dadi Asefa
Keyword(s):  
Projective Dimension ◽  
Projective Modules ◽  
Artin Algebra ◽  
Gorenstein Algebra ◽  
Finite Projective Dimension ◽  
Gorenstein Projective Modules

Let [Formula: see text] be a Morita ring which is an Artin algebra. In this paper we investigate the relations between the Gorenstein-projective modules over a Morita ring [Formula: see text] and the algebras [Formula: see text] and [Formula: see text]. We prove that if [Formula: see text] is a Gorenstein algebra and both [Formula: see text] and [Formula: see text] (resp., both [Formula: see text] and [Formula: see text]) have finite projective dimension, then [Formula: see text] (resp., [Formula: see text]) is a Gorenstein algebra. We also discuss when the CM-freeness and the CM-finiteness of a Morita ring [Formula: see text] is inherited by the algebras [Formula: see text] and [Formula: see text].

scholarly journals Relative Gorenstein Dimensions over Triangular Matrix Rings

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2676
Author(s):  
Driss Bennis ◽  
Rachid El Maaouy ◽  
Juan Ramón García Rozas ◽  
Luis Oyonarte
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Projective Dimension ◽  
Global Dimension ◽  
Projective Modules ◽  
Matrix Rings ◽  
Weakly Compatible ◽  
Gorenstein Algebra ◽  
Finite Projective Dimension ◽  
Triangular Matrix Ring

Let A and B be rings, U a (B,A)-bimodule, and T=A0UB the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over T using the corresponding ones over A and B. We show that when U is relative (weakly) compatible, we are able to describe the structure of GC-projective modules over T. As an application, we study when a morphism in T-Mod is a special GCP(T)-precover and when the class GCP(T) is a special precovering class. In addition, we study the relative global dimension of T. In some cases, we show that it can be computed from the relative global dimensions of A and B. We end the paper with a counterexample to a result that characterizes when a T-module has a finite projective dimension.

Endomorphism algebras of Gorenstein projective modules

2018 ◽  
Vol 17 (09) ◽  
pp. 1850177 ◽  
Author(s):  
Aiping Zhang
Keyword(s):  
Projective Modules ◽  
Text Representation ◽  
Artin Algebra ◽  
Gorenstein Algebra ◽  
Endomorphism Algebras ◽  
Gorenstein Projective Modules

Let [Formula: see text] be an Artin algebra, [Formula: see text] be a Gorenstein projective [Formula: see text]-module and [Formula: see text]. We give a characterization of modules on [Formula: see text] and show that if [Formula: see text] is [Formula: see text]-representation-finite, then [Formula: see text] is also [Formula: see text]-representation-finite. As an application, we prove if [Formula: see text] is a CM-finite [Formula: see text]-Gorenstein algebra, then [Formula: see text] is a [Formula: see text]-Igusa-Todorov algebra.

THE FINITISTIC DIMENSION CONJECTURE AND RELATIVELY PROJECTIVE MODULES

2013 ◽  
Vol 15 (02) ◽  
pp. 1350004 ◽  
Author(s):  
CHANGCHANG XI ◽  
DENGMING XU
Keyword(s):  
Left Ideal ◽  
General Setting ◽  
Projective Dimension ◽  
Homological Dimension ◽  
Projective Modules ◽  
Finitistic Dimension ◽  
Artin Algebra ◽  
Finite Dimensional ◽  
New Formulation ◽  
Dimension Conjecture

The famous finitistic dimension conjecture says that every finite-dimensional 𝕂-algebra over a field 𝕂 should have finite finitistic dimension. This conjecture is equivalent to the following statement: If B is a subalgebra of a finite-dimensional 𝕂-algebra A such that the radical of B is a left ideal in A, and if A has finite finitistic dimension, then B has finite finitistic dimension. In the paper, we shall work with a more general setting of Artin algebras. Let B be a subalgebra of an Artin algebra A such that the radical of B is a left ideal in A. (1) If the category of all finitely generated (A, B)-projective A-modules is closed under taking A-syzygies, then fin.dim (B) ≤ fin.dim (A) + fin.dim (BA) + 3, where fin.dim (A) denotes the finitistic dimension of A, and where fin.dim (BA) stands for the supremum of the projective dimensions of those direct summands of BA that have finite projective dimension. (2) If the extension B ⊆ A is n-hereditary for a non-negative integer n, then gl.dim (A) ≤ gl.dim (B) + n. Moreover, we show that the finitistic dimension of the trivially twisted extension of two algebras of finite finitistic dimension is again finite. Also, a new formulation of the finitistic dimension conjecture in terms of relative homological dimension is given. Our approach in this paper is completely different from the one in our earlier papers.

GORENSTEIN MODULES AND GORENSTEIN MODEL STRUCTURES

2017 ◽  
Vol 59 (3) ◽  
pp. 685-703 ◽  
Author(s):  
AIMIN XU
Keyword(s):  
Projective Dimension ◽  
Model Structure ◽  
Projective Modules ◽  
Cotorsion Pair ◽  
Chain Complexes ◽  
Gorenstein Projective Modules ◽  
Coherent Ring ◽  
Gorenstein Injective Dimension ◽  
Gorenstein Modules ◽  
Model Structures

AbstractGiven a complete hereditary cotorsion pair$(\mathcal{X}, \mathcal{Y})$, we introduce the concept of$(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integern, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤nover a left Noetherian ringR. Similarly, ifRis a left coherent ring in which all flat leftR-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤n. These structures have their analogous in the category of chain complexes.

GORENSTEIN-PROJECTIVE MODULES OVER TRIANGULAR MATRIX ARTIN ALGEBRAS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250066 ◽  
Author(s):  
BAO-LIN XIONG ◽  
PU ZHANG
Keyword(s):  
Triangular Matrix ◽  
Projective Modules ◽  
Artin Algebra ◽  
Gorenstein Projective Modules ◽  
Artin Algebras

Let [Formula: see text] be an Artin algebra. Under suitable conditions, we describe all the modules in ⊥Λ, and obtain criteria for the Gorensteinness of Λ. As applications, we determine explicitly all the Gorenstein-projective Λ-modules if Λ is Gorenstein, and all the Gorenstein-projective Tn(A)-modules if A is Gorenstein.

scholarly journals From subcategories to the entire module categories

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Rasool Hafezi
Keyword(s):  
Representation Theory ◽  
Triangular Matrix ◽  
Projective Modules ◽  
Artin Algebra ◽  
Finite Representation ◽  
Triangular Matrix Algebra ◽  
Gorenstein Projective Modules ◽  
Category Of Modules ◽  
Almost Split Sequences ◽  
Auslander Algebra

AbstractIn this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the {2\times 2} lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ.

scholarly journals Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Dadi Asefa
Keyword(s):  
Fundamental Problem ◽  
Homological Algebra ◽  
Triangular Matrix ◽  
Singularity Theory ◽  
Projective Modules ◽  
Artin Algebra ◽  
Gorenstein Projective Modules ◽  
Projective Resolutions ◽  
Representation Theory Of Algebras ◽  
Necessary And Sufficient

Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra A , how to construct all the Gorenstein-projective A -modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra Λ = A M B A 0 B . We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over Λ = A M B A 0 B .

Pure-injectivity in the category of Gorenstein projective modules

2016 ◽  
Vol 16 (08) ◽  
pp. 1750146
Author(s):  
Peng Yu ◽  
Zhaoyong Huang
Keyword(s):  
Projective Modules ◽  
Artin Algebra ◽  
Gorenstein Algebras ◽  
Gorenstein Projective Modules ◽  
Equivalent Characterization

In this paper, we introduce and study (weak) pure-injective Gorenstein projective modules. Let [Formula: see text] be an Artin algebra. We prove that the category of weak pure-injective Gorenstein projective left [Formula: see text]-modules coincides with the intersection of the category of pure-injective left [Formula: see text]-modules and that of Gorenstein projective left [Formula: see text]-modules. Then, we get an equivalent characterization of virtually Gorenstein algebras (being CM-finite). Furthermore, we prove that the category of weak pure-injective Gorenstein projective left [Formula: see text]-modules is enveloping in the category of left [Formula: see text]-modules; and if [Formula: see text] is virtually Gorenstein, then it is precovering in the category of pure-injective left [Formula: see text]-modules.

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

Noncommutative G-semihereditary rings

2018 ◽  
Vol 17 (01) ◽  
pp. 1850014 ◽  
Author(s):  
Jian Wang ◽  
Yunxia Li ◽  
Jiangsheng Hu
Keyword(s):  
Associative Ring ◽  
Sufficient Condition ◽  
Projective Modules ◽  
The Third ◽  
Flat Modules ◽  
Gorenstein Projective Modules ◽  
Direct Limits ◽  
Gorenstein Flat ◽  
Finitely Presented ◽  
Semihereditary Rings

In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.

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