scholarly journals On the finitistic dimension conjecture II: Related to finite global dimension

2006 ◽  
Vol 201 (1) ◽  
pp. 116-142 ◽  
Author(s):  
Changchang Xi
1987 ◽  
Vol 30 (3) ◽  
pp. 351-362 ◽  
Author(s):  
W. D. Burgess

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].


1998 ◽  
Vol 40 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Zygmunt Pogorzały

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.


2015 ◽  
Vol 67 (1) ◽  
pp. 28-54 ◽  
Author(s):  
Javad Asadollahi ◽  
Rasool Hafezi ◽  
Razieh Vahed

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.


2017 ◽  
Vol 45 (12) ◽  
pp. 5380-5383 ◽  
Author(s):  
Vladimir Dotsenko ◽  
Soutrik Roy Chowdhury

2008 ◽  
Vol 320 (1) ◽  
pp. 253-258 ◽  
Author(s):  
Aiping Zhang ◽  
Shunhua Zhang

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

1996 ◽  
Vol 67 (6) ◽  
pp. 448-456 ◽  
Author(s):  
F. H. Membrillo-Hern�ndez ◽  
L. Salmer�n

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