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 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.

scholarly journals The graded Cartan matrix and global dimension of 0-relations algebras

1987 ◽  
Vol 30 (3) ◽  
pp. 351-362 ◽  
Author(s):  
W. D. Burgess
Keyword(s):  
Artinian Ring ◽  
Global Dimension ◽  
Cartan Matrix ◽  
Integral Matrix ◽  
Finite Global Dimension ◽  
Composition Factors

The Cartan matrix C of a left artinian ring A, with indecomposable projectives P1,…,Pn and corresponding simples Si=Pi/JPi, is an n×n integral matrix with entries Cij, the number of copies of the simple sj which appear as composition factors of Pi. A relationship between the invertibility of this matrix (as an integral matrix) and the finiteness of the global dimension has long been known: gl dim A < ∞⇒det C = ± 1 (Eilenberg [3]). More recently Zacharia [9] has shown that gl dim A ≦ 2⇒det C = 1, and in fact no rings of finite global dimension are known with det C = −1. The converse, det C = l⇒gl dim A < ∞, is false, as easy examples show ([[1) or [3]). However if A is left serial, gl dim A < ∞iff det C = l [1]. If A = ⊕n ≧ 0 An is ℤ-graded and the radical J = ⊕n ≧ 0 An, Wilson [8] calls such rings positively graded. Here there is a graded Cartan matrix with entries from ℤ[X] and gl dim A < ∞⇒det = 1 and, hence, det C = l [8, Prop. 2.2].

scholarly journals Algebras stably equivalent to selfinjective algebras whose Auslander-Reiten quivers consist only of generalized standard components

1998 ◽  
Vol 40 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Zygmunt Pogorzały
Keyword(s):  
Unit Element ◽  
Global Dimension ◽  
Closed Field ◽  
Modular Representation Theory ◽  
Stable Category ◽  
Finite Global Dimension ◽  
Finite Dimensional ◽  
Stable Equivalences ◽  
Stable Categories

Throughout the paper K denotes a fixed algebraically closed field. All algebras considered are finite-dimensional associative K-algebras with a unit element. Moreover, they are assumed to be basic and connected. For an algebra A we denote by mod(A) the category of all finitely generated right A-modules, and mod(A) denotes the stable category of mod(A), i.e. mod(A)/℘ where ℘ is the two-sided ideal in mod(A) of all morphisms that factorize through projective A-modules. Two algebras A and B are said to be stably equivalent if the stable categories mod(A) and mod(B) are equivalent. The study of stable equivalences of algebras has its sources in modular representation theory of finite groups. It is of importance in this theory whether two stably equivalent algebras have the same number of pairwise non-isomorphic nonprojective simple modules. Another motivation for studying stable equivalences appears in the following context. If E is a K-algebra of finite global dimension then its derived category Db(E) is equivalent to the stable category mod(Ê) of the repetitive category Ê of E [15]. Thus the problem of a classification of derived equivalent algebras leads in many cases to a classification of stably equivalent selfinjective algebras.

Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor

2015 ◽  
Vol 67 (1) ◽  
pp. 28-54 ◽  
Author(s):  
Javad Asadollahi ◽  
Rasool Hafezi ◽  
Razieh Vahed
Keyword(s):  
Noetherian Ring ◽  
Global Dimension ◽  
Derived Categories ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
Finite Global Dimension

AbstractWe study bounded derived categories of the category of representations of infinite quivers over a ring R. In case R is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left (resp. right) rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.

scholarly journals Anick resolution and Koszul algebras of finite global dimension

2017 ◽  
Vol 45 (12) ◽  
pp. 5380-5383 ◽  
Author(s):  
Vladimir Dotsenko ◽  
Soutrik Roy Chowdhury
Keyword(s):  
Global Dimension ◽  
Koszul Algebras ◽  
Finite Global Dimension

scholarly journals Matrix subrings having finite global dimension

1987 ◽  
Vol 109 (1) ◽  
pp. 74-92 ◽  
Author(s):  
Ellen Kirkman ◽  
James Kuzmanovich
Keyword(s):  
Global Dimension ◽  
Finite Global Dimension

scholarly journals Noncommutative motives of Azumaya algebras

2014 ◽  
Vol 14 (2) ◽  
pp. 379-403 ◽  
Author(s):  
Gonçalo Tabuada ◽  
Michel Van den Bergh
Keyword(s):  
Commutative Ring ◽  
Global Dimension ◽  
Azumaya Algebras ◽  
Finite Global Dimension ◽  
Finite Dimensional ◽  
Noncommutative Motives ◽  
Finite Dimensional Algebras ◽  
The Way

AbstractLet $k$ be a base commutative ring, $R$ a commutative ring of coefficients, $X$ a quasi-compact quasi-separated $k$-scheme, and $A$ a sheaf of Azumaya algebras over $X$ of rank $r$. Under the assumption that $1/r\in R$, we prove that the noncommutative motives with $R$-coefficients of $X$ and $A$ are isomorphic. As an application, we conclude that a similar isomorphism holds for every $R$-linear additive invariant. This leads to several computations. Along the way we show that, in the case of finite-dimensional algebras of finite global dimension, all additive invariants are nilinvariant.

scholarly journals A class of maximal orders integral over their centres

1983 ◽  
Vol 24 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Andy J. Gray
Keyword(s):  
Prime Ring ◽  
Global Dimension ◽  
Local Rings ◽  
Maximal Orders ◽  
Finite Global Dimension ◽  
Krull Domain ◽  
Rank One

In a recent paper [1], Brown, Hajarnavis and MacEacharn have considered non-commutative Noetherian local rings of finite global dimension which are integral over their centres. For such a ring Rthey have shown:(i) R is a prime ring whose Krull and global dimensions coincide;(ii) R = ∩ RP where p runs through the set of rank one primes of the centre of R, and each Rp is hereditary;(iii) the centre of R is a Krull domain.

Global dimension of Harada rings and serial rings

2015 ◽  
Vol 14 (07) ◽  
pp. 1550115
Author(s):  
Kazutoshi Koike
Keyword(s):  
Global Dimension ◽  
Serial Ring ◽  
Finite Global Dimension

Baba [On Harada rings and quasi-Harada rings with left global dimension at most 2. Comm. Algebra28(6) (2000) 2671–2684] proved that every left Harada rings with global dimension at most 2 is a serial ring. In this paper, improving the result, we show that every left Harada ring with global dimension at most 3 is a serial ring. We also prove that if a left Harada ring A of finite global dimension is of type (*) or has homogeneous right socle, then A is serial. Finally, we give an example of a non-serial left Harada ring of finite global dimension.

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