Let $X$ be a regular projective (complex) surface and $S$ be a finite set of closed points on $X$. Denote by $j:X\backslash S \to X$ the open immersion. Is $H^0(R^1j_*\mathcal{O}_X) \cong H^1(j_*\mathcal{O}_X)$?
Global sections of higher direct image sheaf
Ron
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