 
Summary: HOMOLOGICAL DIMENSIONS IN COTORSION PAIRS
LIDIA ANGELERI H¨UGEL AND OCTAVIO MENDOZA
Abstract. Two classes A and B of modules over a ring R are said to form
a cotorsion pair (A, B) if A = Ker Ext1
R(, B) and B = Ker Ext1
R(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 Findim R < if and only if the following dimensions are finite for some
cotorsion pair (A, B) in Mod R: the relative projective dimension of A with
respect to itself, and the Aresolution dimension of the category P of all R
modules of finite projective dimension. Moreover, we obtain an analogous
result for findim R, and we characterize when Findim R = findim R.
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.
Of course, the existence of approximations is the prerequisite for computing
relative dimensions. In recent years, a powerful machinery for producing approxi
