A note on the injective dimension of local cohomology modules

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 $\dim _R(M)\leq \operatorname{injdim}_R(M)$. As an application we study the question which local cohomology modules $H^i_I(R)$ satisfy $\operatorname{injdim}_R(H^i_I(R))=\dim_R(H^i_I(R))$. There are two situations where the answer is positive. On the other hand, we present two counterexamples, the failure in these two examples coming from different reasons.