The Stacks project

Remark 33.35.4. Let $k$ be an infinite field. Let $n \geq 1$. Given a finite number of coherent modules $\mathcal{F}_ i$ on $\mathbf{P}^ n_ k$ we can choose a single $s \in \Gamma (\mathbf{P}^ n_ k, \mathcal{O}(1))$ such that the statement of Lemma 33.35.3 works for each of them. To prove this, just apply the lemma to $\bigoplus \mathcal{F}_ i$.