Consider a short exact sequence $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ of coherent sheaves on $\mathbb{P}^n$. Assume that $\mathcal{F}''$ (resp. $\mathcal{F}$) is $m-1$ (resp. $m$)-regular. Why does this imply that $\mathcal{F}'$ is $m$-regular?

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wouldbe correct if the induced map $\Gamma_*(\mathbb{P}^n,\mathcal{F})\to \Gamma_*(\mathbb{P}^n,\mathcal{F}'')$ were surjective, i.e., if we still had a short exact sequence after we pass from coherent sheaves on $\mathbb{P}^n$ to graded modules over the polynomial ring. However, as written, the exercise is wrong. Also my first sheaf map sends the first generator to $x^d$ and the second generator to $y^d$. The second map comes by Koszul (and twisting). – Jason Starr Nov 24 '13 at 12:53