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.)

  • 2
    $\begingroup$ I think you have unnecessarily complicated the question. The cohomology vanishing is true for any (Noetherian) commutative ring and any module. $\endgroup$ – Mohan Jul 31 '12 at 23:11
  • $\begingroup$ 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. $\endgroup$ – A. Pascal Aug 1 '12 at 8:11

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 $X$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.