The Stacks project

Proof. We prove this by induction on the integer $r = \sum r_ i$. If $r = 0$ we can take $\pi = \text{id}_ X$. If $r_ i = 1$ for all $i$, then we can also take $\pi = \text{id}_ X$. Assume that $r_{i_0} > 1$ for some $i_0$. Let $(\pi : P \to X, \mathcal{O}_ P(1))$ be the projective bundle associated to $\mathcal{E}_{i_0}$. The canonical map $\pi ^*\mathcal{E}_{i_0} \to \mathcal{O}_ P(1)$ is surjective and hence its kernel $\mathcal{E}'_{i_0}$ is finite locally free of rank $r_{i_0} - 1$. Observe that $\pi _ Y^*$ is injective for any morphism $f : Y \to X$ of good algebraic spaces over $B$, see Lemma 82.27.2. Thus it suffices to prove the lemma for $P$ and the locally free sheaves $\pi ^*\mathcal{E}_ i$. However, because we have the subbundle $\mathcal{E}_{i_0} \subset \pi ^*\mathcal{E}_{i_0}$ with invertible quotient, it now suffices to prove the lemma for the collection $\{ \mathcal{E}_ i\} _{i \not= i_0} \cup \{ \mathcal{E}'_{i_0}\} $. This decreases $r$ by $1$ and we win by induction hypothesis. $\square$