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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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up vote 2 down vote accepted

By definition, sheaf cohomology depends only on the underlying topological space. (If you look in Hartshorne, for example, you will see that he is careful to define sheaf cohomology in terms of injective resolutions by sheaves of abelian groups, with no reference to the structure sheaf.) So the two cohomology spaces you ask about will coincide.

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