HOMOLOGICAL DIMENSIONS IN COTORSION PAIRS

Abstract. Two classes A and B of modules over a ring R are said to form a cotorsion pair (A,B) if A = Ker Ext1R(−,B) and B = Ker Ext1R(A,−). We investigate relative homological dimensions in cotorsion pairs. This can be applied to study the big and the little finitistic dimension of R. We show that FindimR < ∞ if and only if the following dimensions are finite for some cotorsion pair (A,B) in ModR: the relative projective dimension of A with respect to itself, and the A-resolution dimension of the category P of all R-modules of finite projective dimension. Moreover, we obtain an analogous result for findimR, and we characterize when FindimR = findimR. Introduction. The study of homological dimensions which are obtained by replacing the pro-jective or injective modules by certain subcategories was initiated by Auslander and Buchweitz in their seminal paper [5], which was one of the starting points for what is now called relative homological algebra