Let $A$ be a ring, Noetherian or even of finite type over a field if necessary. Let $I$ be an ideal in $A$, $\widehat{A}$ the formal completion of $A$ along $I$, $M$ an $A$-module, finitely generated if necessary, and $\widehat{M}$ the formal completion. Then the sheaf cohomology $H^{i}(Spec(A),M)=0$ for $i > 0$.

Is the same true for $\widehat{M}$ on the formal completion? (Clarification based on the below comment: The underlying topological space of the formal completion is $Spec(A/I)$, *not* $Spec(\widehat{A}))$, thus there is something to prove.)

formal schemewhose underlying topological space is $Spec(A/I)$. In particular, $\widehat{M}$ isnotan $A/I$-module and so I can't just apply the usual vanishing on an affine scheme. $\endgroup$ – A. Pascal Aug 1 '12 at 8:11