On the Cartan matrix of an artin algebra of global dimension two

Abstract

The purpose of this paper is to show that if n is an artin algebra of g!obal dimension (81 dim) 2, then the determinant of its Cartan matrix equals + I. This generalizes previous results of Donovan and Freislich [2], Igusa and Todorov ]4 ] and Wilson [5]. We recall that if n is an artin algebra (for instance, a finite-dimensional algebra over a field), there are finitely many nonisomorphic indecomposable projective rl modules P,, P ,,..., P,,. We denote by /End,(P.) Hom,,(P,, P,,) the length of Hom,(P,. PI) as an End,(P,) module. Then the h x n matrix C(n j, having knd.,cPi, Hom.,(P,, P,i) as the (i,j)th entry is called the Cartan matrix of rl. Let ,d be now an artin algebra of finite representation typethat is, having finitely many nonisomorphic indecomposable ‘4 modules. Let H= u End,(,tr) Hom.,(M, N)) be the matrix with integral entries, where M, N range through the set of the nonisomorphic indecomposable II modules. Donovan and Freislich [2], have shown that if .4 = kG is a group algebra of finite representation type, then det H = + 1. Then, using associated graded algebras, Wilson showed that det H = + 1 if A is a finite-dimensional algebra of finite-representation type over an algebraically closed field [ 5 IO Independently the same result was proved by Igusa and Todorov [4] without making any assumption on the ground field. Let P be the Auslander algebra of n (see [ 1 ] for further details). that is. r = End,,(M, 0 MT . . . @ M,JoD, where M, ,..., M, are the nonisomorphic indecomposable n modules. Then, it has been shown that H = C(T), where. C(T) denotes the Cartan matrix of the Auslander algebra r. It is known that gl dim r 2. [We recall that