Let $X$ be a connected projective scheme. Let $U$ be a finite type open substack of the algebraic stack of coherent sheaves on $X$ with a fixed Hilbert polynomial. Can one take $p>0$ such that every sheaf in $U$ is CastelnuovoMumford $p$regular? It seems true and used in some papers (maybe with additional condition), but I cannot show this statement. Could anyone give me a sketch of a proof?
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Lemma 1.7.6, p. 28 of the book by Huybrechts and Lehn. 


$\mathcal{O}(n)\oplus \mathcal{O}(n)$
on$\mathbb{P}^1$
. – Jason Starr Nov 9 '12 at 16:06