Let $(R, \mathfrak{m})$ be a local ring and $X = Spec(R)$. Let $Y = V(I)$ be a closed subscheme of $X$, defined by an ideal $I \subset R$, and let $P \in X$ (in fact, $P \in Y$) be the closed point. Let $(\hat{X}, \mathcal{O}_{\hat{X}})$ be the formal completion of $X$ along $Y$ ($\hat{X} = Y$ as a topological space, and the sheaf of rings $\mathcal{O}_{\hat{X}}$ is $\varprojlim \mathcal{O}_X/\tilde{I}^n$).

Can we write the cohomology of $\mathcal{O}_{\hat{X}}$ with support in the closed point $P$ in terms of local cohomology on $R$? Specifically, do we have isomorphisms $H^j_P(\hat{X}, \mathcal{O}_{\hat{X}}) \simeq \varprojlim H^j_{\mathfrak{m}}(R/I^n)$?

Without the "support at P" subscript, this is Corollaire 4.1.7 of EGA III, Grothendieck's comparison theorem. So essentially what I am asking is: is there an analogue of the comparison theorem for local cohomology?

One natural strategy for deducing a local comparison theorem from the global one would be to use the long exact sequence relating $H^j_P(\hat{X}, \mathcal{O}_{\hat{X}})$ to $H^j(\hat{X}, \mathcal{O}_{\hat{X}})$ and $H^j(\hat{X} \setminus P, \mathcal{O}_{\hat{X}})$, but this runs into trouble because the inclusion $X \setminus P \hookrightarrow X$ won't be proper, so the global comparison theorem will fail for $X \setminus P$.

SGA 2 would seem to be the natural place to look for results of this type, but its Expose IX (where formal completions are treated) doesn't address local cohomology, as far as I can tell.

artinian$R$-module, so the ML condition in the comment above is trivially verified (as any projective system of artinian $R$-modules is automatically ML). – anon Feb 28 '13 at 15:27