The MCM-approximation of the trivial module over a category algebra

For a finite free EI category, we construct an explicit module over its category algebra. If in addition the category is projective over the ground field, the constructed module is a maximal Cohen–Macaulay approximation of the trivial module and is the tensor identity of the stable category of Gorenstein-projective modules over the category algebra. We give conditions on when the trivial module is Gorenstein-projective.

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The Gorenstein-projective modules over a monomial algebra

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210518000185 ◽

2018 ◽

Vol 148 (6) ◽

pp. 1115-1134 ◽

Cited By ~ 5

Author(s):

Xiao-Wu Chen ◽

Dawei Shen ◽

Guodong Zhou

Keyword(s):

Projective Modules ◽

Stable Category ◽

Monomial Algebra ◽

Gorenstein Projective Modules ◽

The Given

We introduce the notion of a perfect path for a monomial algebra. We classify indecomposable non-projective Gorenstein-projective modules over the given monomial algebra via perfect paths. We apply the classification to a quadratic monomial algebra and describe explicitly the stable category of its Gorenstein-projective modules.

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Triangulated Equivalences Involving Gorenstein Projective Modules

Canadian Mathematical Bulletin ◽

10.4153/cmb-2017-045-2 ◽

2017 ◽

Vol 60 (4) ◽

pp. 879-890 ◽

Cited By ~ 2

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.

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Symmetric Auslander and Bass categories

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004110000629 ◽

2011 ◽

Vol 150 (2) ◽

pp. 227-240 ◽

Cited By ~ 2

Author(s):

PETER JØRGENSEN ◽

KIRIKO KATO

Keyword(s):

Abelian Category ◽

Triangulated Categories ◽

Projective Modules ◽

Stable Category ◽

Gorenstein Projective Modules ◽

Left And Right ◽

Acyclic Complexes ◽

Image Position

AbstractWe define the symmetric Auslander category As(R) to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right-tails of totally acyclic complexes of projective modules.The symmetric Auslander category contains A(R), the ordinary Auslander category. It is well known that A(R) is intimately related to Gorenstein projective modules, and our main result is that As(R) is similarly related to what can reasonably be called Gorenstein projective homomorphisms. Namely, there is an equivalence of triangulated categories where GMor(R) is the stable category of Gorenstein projective objects in the abelian category Mor(R) of homomorphisms of R-modules.This result is set in the wider context of a theory for As(R) and Bs(R), the symmetric Bass category which is defined dually.

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Noncommutative G-semihereditary rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498818500147 ◽

2018 ◽

Vol 17 (01) ◽

pp. 1850014 ◽

Cited By ~ 2

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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Gorenstein Projective Modules over Triangular Matrix Rings

Algebra Colloquium ◽

10.1142/s1005386716000122 ◽

2016 ◽

Vol 23 (01) ◽

pp. 97-104 ◽

Cited By ~ 1

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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