The representation dimension of Artin algebras

2017 ◽  
Vol 11 (2) ◽  
pp. 348-360
Author(s):  
Flávio U. Coelho
Keyword(s):  
Representation Dimension ◽  
Artin Algebras

scholarly journals Gabriel–Roiter Measure, Representation Dimension and Rejective Chains

2020 ◽  
Vol 71 (2) ◽  
pp. 619-635
Author(s):  
Teresa Conde
Keyword(s):  
Global Dimension ◽  
Alternative Proof ◽  
Representation Dimension ◽  
Finite Global Dimension ◽  
Measure Representation ◽  
Artin Algebras

Abstract The Gabriel–Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel–Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel–Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel–Roiter measure $\mu$ in an abelian length category $\mathcal{A}$, there exists an object $X^{\prime}$ that depends on $X$ and $\mu$, such that $\Gamma =\operatorname{End}_{\mathcal{A}}(X\oplus X^{\prime})$ has finite global dimension. Analogously to Iyama’s original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.

scholarly journals AUSLANDER GENERATORS AND HOMOLOGICAL CONJECTURES

2013 ◽  
Vol 56 (3) ◽  
pp. 503-506 ◽  
Author(s):  
JIAQUN WEI
Keyword(s):  
Artin Algebra ◽  
Representation Dimension ◽  
Homological Conjectures

AbstractLet A be an artin algebra with representation dimension not more than 3. Assuming that AV is an Auslander generator and M ∈ addAV, we show that both findim(EndAM) and findim(EndAM)op are finite, and consequently the Gorenstein symmetry conjecture, the Wakamatsu-tilting conjecture and the generalized Nakayama conjecture hold for EndAM.

scholarly journals Maximal representation dimension of finitep-groups

2011 ◽  
pp. ---
Author(s):  
Shane Cernele ◽  
Masoud Kamgarpour ◽  
Zinovy Reichstein
Keyword(s):  
Representation Dimension ◽  
Maximal Representation

scholarly journals Projective resolutions over Artin algebras with zero relations

1985 ◽  
Vol 29 (1) ◽  
pp. 180-190 ◽  
Author(s):  
E. L. Green ◽  
D. Happel ◽  
D. Zacharia
Keyword(s):  
Artin Algebras ◽  
Projective Resolutions

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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