  For a noetherian ring R we call an R-module M cofinite if there exists an
ideal I of R such that M is I-cofinite; we show that every cofinite module M
satisfies dim_R(M)<=injdimR(M). As an application we study the question which
local cohomology modules H^i_I(R) satisfy injdim_R(H^i_I(R)) = dim_R(H^i_I(R)).
There are two situations where the answer is positive. On the other hand we
present two counter-examples, the failure in these two examples coming from
different reasons.
