Castelnuovo–Mumford regularity ($H^i(\mathcal{F}(r-i))=0(i>0)$)

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.