# Sheaf cohomology of completion along a subvariety

Suppose we are given a subvariety $Y \subset X$ where $X$ is a projective non-singular variety. We also have a coherent sheaf $\mathcal{F}$ on $Y$. We look at the completion of $X$ along $Y$ and denote it by $\hat{X}$. We can view $\mathcal{F}$ as a sheaf on $\hat{X}$ under the pushforward map.

Now, topologically $\hat{X}$ and $Y$ are the same space. My question is:

How does the sheaf cohomologies $H^i(\hat{X}, \mathcal{F})$ and $H^i(Y, \mathcal{F})$ related to each other?

I came across this question while reading a proof from Ample subvarieties of algebraic varieties [Prop 1.3 Page 168] by Hartshorne.

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