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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.)

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I think you have unnecessarily complicated the question. The cohomology vanishing is true for any (Noetherian) commutative ring and any module. –  Mohan Jul 31 '12 at 23:11
    
My question is not about $\widehat{M}$ on $Spec(\widehat{A})$, which is a quasi-coherent sheaf on an affine scheme, but about $\widehat{M}$ on the formal scheme whose underlying topological space is $Spec(A/I)$. In particular, $\widehat{M}$ is not an $A/I$-module and so I can't just apply the usual vanishing on an affine scheme. –  A. Pascal Aug 1 '12 at 8:11
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1 Answer

up vote 2 down vote accepted

Yes --- this follows from EGA3 I, Chapter 0, Proposition 13.3.1 . 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 $X$.

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