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Let $X$ be a smooth, projective surface and $T$ a finite set of points in $X$ i.e., of codimension $2$ in $X$. Is it true that $H^i(\mathcal{O}_X)=H^i(\mathcal{O}_{X\backslash T})$ for $i \ge 1$?

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    $\begingroup$ No, this is false. Look at SGA 2, in particular chap. III. $\endgroup$
    – abx
    Commented Sep 14, 2018 at 5:04

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