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Let $k$ be an algebraically closed field of characteristic zero, $X, Y$ be projective $k$-schemes. Fix projective immersions $i:X \hookrightarrow \mathbb{P}^n$ and $j:Y \hookrightarrow \mathbb{P}^m$ for some integers $n$ and $m$. This induces a natural closed immersion $i \times j:X \times_k Y \to \mathbb{P}^n \times_k \mathbb{P}^m \to \mathbb{P}^{nm+n+m}$, where the last morphism is the Segre embedding. Let $F$ be a coherent sheaf on $X \times_k Y$. Denote by $p:X \times_k Y \to X$ a natural projection map. Is it then true that $(p^*p_*F)(m) \cong p^*p_*(F(m))$ for $m$ large enough?

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    What do you mean by $F(m)$ when $F$ is a sheaf on $X \times_k Y$? Which line bundle are you twisting by? – 2017-01-19
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    @FredrikMeyer I have edited the question a little. In the current notation I am twisting $F$ by $(i \times j)^*\mathcal{O}_{\mathbb{P}^{nm+n+m}}(m)$. – 2017-01-19
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    Have you tried to see what happens when $X$ is a point? – 2017-01-19

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