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.)



Cohomology vanishing for formal completions of modules?
Yes --- this follows from [EGA3 I, Chapter 0, Proposition 13.3.1](http://www.numdam.org/numdam-bin/feuilleter?id=PMIHES_1961__11_) . This general result gives conditions under which it is possible to conclude that


$H^i( X, \lim\limits\_\longleftarrow \mathcal{F}\_k )$


is isomorphic to


$\lim\limits\_\longleftarrow H^i( X, \mathcal{F}\_k)$


for an inverse system $(\mathcal{F}\_k)$ of sheaves of abelian groups on a topological space $X$. 


