Let $Y$ be a closed smooth subvariety in a (smooth) affine variety $X$. What can one say about the etale cohomology of the generic points of the formal completion of $X$ along $Y$ i.e. about the Galois cohomology of the fraction fields of the completions of $X$ along the connected components of $Y$? Are there any methods for computing this?
Actually, I am rather interested in the following situation. Let $X$ be projective (!), and consider a Zarisky (or etale) affine hypercovering $Y.$ of $Y$ ($Y$ is again a closed smooth subvariety of $X$). Then each component $Y_I$ of $Y.$ could be considered as the intersection of $Y$ with some open affine subvariety $X_I$ of $X$ (resp. with an affine etale $X_I/X$); so we may consider the formal completion $\hat{X}_{Y_I}$ of $X_I$ along $Y_I$ (it seems that it depends only on $X$ and $Y_I$). So, we obtain a 'simplicial completion'; then one may consider the generic points and their (etale=Galois) cohomology. Are there any interesting examples when one can compute this, or any general results for this setting?
