On the finitistic dimension conjecture of Artin algebras.
Let \(A\) be an Artin algebra. The finitistic dimension of \(A\) is defined as the supremum of the projective dimensions of finitely generated left \(R\)-modules of finite projective dimension. The finitistic dimension conjecture, due to H. Bass in 1960, asserts that the finitistic dimension of any Artin algebra is finite. The representation dimension of \(A\) is defined as the smallest number \(d\) that is equal to the global dimension of the endomorphism ring of a left \(R\)-module which is both a generator and a cogenerator in the category of left \(A\)-modules. In this paper it is proved that if an Artin algebra \(A\) has representation dimension at most three, and \(e\) is an idempotent of \(A\), then the finitistic dimension of \(eAe\) is finite. As a corollary, it is deduced that if every quasi-hereditary algebra has representation dimension at most three, then the finitistic dimension conjecture holds.
scientific article; zbMATH DE number 5349051
scientific article