Injective dimension of cofinite modules and local cohomology

Let (R, m) be a commutative Noetherian local ring, I an ideal of R and let M be a non-zero I -cofinite R-module. In this paper we show that if M has finite injective dimension, then dim R/I <= inj dim M <= depth R; and inj dim M = depth R, whenever mM not equal M. These generalize the classical Bass formulas for injective dimension. As an application we obtain some results on the injective dimension of local cohomology modules. In addition, we show that R is a Cohen-Macaulay ring if admits a Cohen-Macaulay R-module of finite projective dimension.