The Global Dimension of the Endomorphism Algebra of a Generator-cogenerator for a Subcategory

2013 ◽  
Vol 20 (03) ◽  
pp. 443-456
Author(s):  
Jingjing Guo
Keyword(s):  
Global Dimension ◽  
Endomorphism Algebra ◽  
Artin Algebra ◽  
Hereditary Artin Algebra

Let A be a hereditary Artin algebra and T a tilting A-module. The possibilities for the global dimension of the endomorphism algebra of a generator-cogenerator for the subcategory T⊥ in A-mod are determined in terms of relative Auslander-Reiten orbits of indecomposable A-modules in T⊥.

scholarly journals On the Auslander–Reiten quiver of a P1-hereditary artin algebra

2000 ◽  
Vol 151 (1) ◽  
pp. 11-29 ◽  
Author(s):  
Lidia Angeleri Hügel ◽  
Flávio U. Coelho
Keyword(s):  
Artin Algebra ◽  
Hereditary Artin Algebra

On Algebras Stably Equivalent to an Hereditary Artin Algebra

1978 ◽  
Vol 30 (4) ◽  
pp. 817-829 ◽  
Author(s):  
María Inés Platzeck
Keyword(s):  
Finitely Generated ◽  
Artin Algebra ◽  
Finitely Generated Module ◽  
Artin Ring ◽  
Finitely Generated Modules ◽  
Hereditary Artin Algebra

Let Λ be an artin algebra, that is, an artin ring that is a finitely generated module over its center C which is also an artin ring. We denote by mod Λ the category of finitely generated left Λ-modules. We recall that the category of finitely generated modules modulo projectives is the category given by the following data: the objects are the finitely generated Λ-modules.

Gorenstein global dimension and Hopf algebroid actions

2016 ◽  
Vol 15 (05) ◽  
pp. 1650090
Author(s):  
Yong Wang ◽  
Fang Li
Keyword(s):  
Triangular Matrix ◽  
Global Dimension ◽  
Smash Product ◽  
Artin Algebra ◽  
Representation Dimension ◽  
Hopf Algebroid ◽  
Hopf Algebroids ◽  
The Stability ◽  
Gorenstein Injective ◽  
The Relationship

Let [Formula: see text] be a Hopf algebroid and [Formula: see text] a left [Formula: see text]-module algebra. In this paper, we mainly present the duality theorem for the smash product [Formula: see text], and making use of integral theory for Hopf algebroids, we investigate the stability of Gorenstein injective pre-envelopes and Gorenstein projective precovers between the category of [Formula: see text]-modules and the category of [Formula: see text]-modules. Moreover, we establish the relationship between Gorenstein global dimension of [Formula: see text] and that of [Formula: see text], and prove that [Formula: see text] has finite representation type, resp. is selfinjective, resp. is CM-finite [Formula: see text]-Gorenstein, if and only if [Formula: see text] has the same property under suitable conditions. As an application, we investigate the representation dimension of the lower triangular matrix Artin algebra [Formula: see text].

scholarly journals The finiteness of the Gorenstein dimension for Artin algebras

2019 ◽  
Vol 18 (06) ◽  
pp. 1950112
Author(s):  
René Marczinzik
Keyword(s):  
Projective Dimension ◽  
Indecomposable Module ◽  
Global Dimension ◽  
Injective Dimension ◽  
Artin Algebra ◽  
Artinian Rings ◽  
Gorenstein Dimension ◽  
Finite Global Dimension ◽  
Artin Algebras

In [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478], the authors proved that an Artin algebra [Formula: see text] with infinite global dimension has an indecomposable module with infinite projective and infinite injective dimension, giving a new characterization of algebras with finite global dimension. We prove in this paper that an Artin algebra [Formula: see text] that is not Gorenstein has an indecomposable [Formula: see text]-module with infinite Gorenstein projective dimension and infinite Gorenstein injective dimension, which gives a new characterization of algebras with finite Gorenstein dimension. We show that this gives a proper generalization of the result in [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478] for Artin algebras.

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