  Let R be a commutative noetherian local ring. As an analogue of the notion of
the dimension of a triangulated category defined by Rouquier, the notion of the
dimension of a subcategory of finitely generated R-modules is introduced in
this paper. We found evidence that certain categories over nice singularities
have small dimensions. When R is Cohen-Macaulay, under a mild assumption it is
proved that finiteness of the dimension of the full subcategory consisting of
maximal Cohen-Macaulay modules which are locally free on the punctured spectrum
is equivalent to saying that R is an isolated singularity. As an application,
the celebrated theorem of Auslander, Huneke, Leuschke and Wiegand is not only
recovered but also improved. The dimensions of stable categories of maximal
Cohen-Macaulay modules as triangulated categories are also investigated in the
case where R is Gorenstein, and special cases of the recent results of Aihara
and Takahashi, and Oppermann and Stovicek are recovered and improved. Our key
technique involves a careful study of annihilators and supports of Tor, Ext and
\underline{Hom} between two subcategories.
