Let $R$ be a local (with maximal ideal $m$) commutative Gorenstein ring of dimension $d$. Then for any $0 \leq i \leq d$ there are isomorphisms for the local cohomology: $H_m^i(M) \cong D Ext_R^{d-i}(M,R)$ where $D$ is the Matlis duality and $M$ is finitely generated.
Question 1: Is there a more general version of this result that also includes (at least certain) infinitely generated modules $M$? What happens when $M$ is a general injective module?
When $M=E(R/p)$ is an indecomposable injective module we should have $H_m^i(M)=0$ for all $i >0$ and $H_m^0(M)) \cong M$ iff $p=m$ and zero else according to example 7.6. in the book "Twenty four hours of local cohomology" , but I think that we should have that $Ext_R^i(M,R)$ is non-zero for $i=projdim(M)$, so the formula can not be true in general.
Question 2: Is it true that $Ext_R^i(M,R)$ is non-zero in exactly one degree when $M$ is indecomposable injective?
