  We investigate the notion of the C-projective dimension of a module, where C
is a semidualizing module. When C=R, this recovers the standard projective
dimension. We show that three natural definitions of finite C-projective
dimension agree, and investigate the relationship between relative cohomology
modules and absolute cohomology modules in this setting. Finally, we prove
several results that demonstrate the deep connections between modules of finite
projective dimension and modules of finite C-projective dimension. In parallel,
we develop the dual theory for injective dimension and C-injective dimension.
