Global dimension of the endomorphism ring and [formula omitted]-modules

Abstract

We show that, if T is a selfsmall and selforthogonal module over a noetherian ring R of finite global dimension with the endomorphism ring A, then fd T A ⩽ gd A ⩽ id R T + fd T A . Applying the result we give answers to two questions left in [J. Wei et al., J. Algebra 168 (2) (2003) 404–418] concerning basic properties of * n -modules, by showing that the flat dimension of a * n -module with n ⩾ 3 over its endomorphism ring can even be arbitrarily far from the integer n while the flat dimension of a * 2 -module over its endomorphism ring is always bounded by the integer 2 and showing that * n -modules are not finitely generated in general, even in case n = 2 .