  In this paper, we first introduce stable functors with respect to a
preenveloping/precovering subcategory and investigate some of their properties.
Using that we then introduce and study a relative complete cohomology theory in
abelian categories. Some properties of the cohomology including vanishing are
given. As applications, we give some characterizations of objects of finite
homological dimensions including the flat dimension, cotorsion dimension,
Gorenstein injective/flat dimension and projectively coresolved Gorenstein flat
dimension.
