= Castelnuovo–Mumford regularity
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{title2=$H^i(\mathcal F(r-i))=0\ (i>0)$}
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A <coherent sheaf> $\mathcal F$ on $\mathbb P^N$ is $r$-regular if $H^i(\mathbb P^N,\mathcal F(r-i))=0$ for every $i>0$. It is then $(r+1)$-regular and $\mathcal F(r)$ is globally generated. Persistence follows by induction with a hyperplane avoiding the associated points and its restriction exact sequence. A sheaf on an embedded <projective scheme> is tested after pushforward to projective space.
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