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.



Comparison for formal local cohomology
Dear Nick: If your local ring $R$ is Gorenstein, then $H^i\_p(\hat{X},\mathcal{O}\_{\hat{X}})\cong\varprojlim H^i\_{\mathfrak{m}}(R/I^n)$ for all $i$, as you want. This is proved, for instance, in Proposition 2.2, page 334 of A. Ogus' *[Local cohomological dimension of algebraic varieties](http://www.jstor.org/discover/10.2307/1970785?uid=3739832&uid=2129&uid=2&uid=70&uid=4&uid=3739256&sid=21101711289563)*, Annals of Mathematics, Vol. 98, No. 2 (1973). He attributes it to Peskine Szpiro. You should also check Proposition A8 and Corollary A9 on pages 362 and 363 of Ogus' paper, as well as Propositions 2.1 and 2.2 on pages 106-107 of Peskine-Szpiro's *[Dimension projective finie et cohomologie locale](http://link.springer.com/article/10.1007/BF02685877?LI=true)*.


I would like to add that as anon commented above, the proof given by Ogus is achieved by showing that when $R$ is Gorenstein the suggested (by anon) $\mathrm{lim}^1$ vanishes.


**Edit.** Following anon's comment, I don't think Gorenstein assumption is needed for this.


