TITLE: Mayer-Vietoris for sheaf cohomology
QUESTION [6 upvotes]: Assume the following is given:

$X$  quasi-projective (smooth) variety,
$\overline X$ a projective variety, such that $X$ is the complement of a divisor $D$ in $X$, and
$\mathcal F$ a coherent sheaf on $\hat X$.

Suppose, we know the cohomology $H^\ast(X,{\mathcal F}|_X)$ of $\mathcal F$ on $X$.  Now, for sure, this is not enough to be able to compute $H^{\ast}({\overline X},{\mathcal F})$.
My question is what kind of additional data (localized near D) do we need to know in order to be able to compute $H^\ast({\overline X},{\mathcal F})$?

REPLY [3 votes]: You need the cohomology of the extension of $\mathcal{F}$ to the completion of $X$ along $D$ (a formal scheme). There is an equivalence of categories that you can look up in 
Moret-Bailly, Laurent 
Un problème de descente. Bulletin de la Société Mathématique de France, 124 no. 4 (1996), p. 559-585 (numdam)
see also references therein.
From this description, a way of computing cohomology should follow.