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.

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.

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].

ENDOMORPHISM ALGEBRAS OF PROJECTIVE MODULES OVER LAURA ALGEBRAS

2004 ◽  
Vol 03 (01) ◽  
pp. 49-60 ◽  
Author(s):  
IBRAHIM ASSEM ◽  
FLÁVIO U. COELHO
Keyword(s):  
Projective Modules ◽  
Artin Algebra ◽  
Endomorphism Algebras ◽  
Tilted Algebras

Let A be a connected artin algebra, and e be an idempotent in A such that B=eAe is connected. We show here that if A is laura, left (or right) glued or weakly shod, so is B, respectively. Our proof yields also similar (and known) results for shod and quasi-tilted algebras.

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 .

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.

scholarly journals Gorenstein Projective Modules over Triangular Matrix Rings

2016 ◽  
Vol 23 (01) ◽  
pp. 97-104 ◽  
Author(s):  
H. Eshraghi ◽  
R. Hafezi ◽  
Sh. Salarian ◽  
Z. W. Li
Keyword(s):  
Triangular Matrix ◽  
Projective Modules ◽  
Matrix Rings ◽  
Matrix Algebras ◽  
Gorenstein Projective Modules ◽  
Artin Algebras ◽  
Acyclic Complexes ◽  
Matrix Extension

Let R and S be Artin algebras and Γ be their triangular matrix extension via a bimodule SMR. We study totally acyclic complexes of projective Γ-modules and obtain a complete description of Gorenstein projective Γ-modules. We then construct some examples of Cohen-Macaulay finite and virtually Gorenstein triangular matrix algebras.

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