Lemma 59.62.1. Let $S$ be a scheme. Let $\mathcal{F}$ and $\mathcal{G}$ be finite locally free sheaves of $\mathcal{O}_ S$-modules of positive rank. If there exists an isomorphism $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{F}, \mathcal{F}) \cong \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{G}, \mathcal{G})$ of $\mathcal{O}_ S$-algebras, then there exists an invertible sheaf $\mathcal{L}$ on $S$ such that $\mathcal{F} \otimes _{\mathcal{O}_ S} \mathcal{L} \cong \mathcal{G}$ and such that this isomorphism induces the given isomorphism of endomorphism algebras.
Proof. Fix an isomorphism $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{F}, \mathcal{F}) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{G}, \mathcal{G})$. Consider the sheaf $\mathcal{L} \subset \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}, \mathcal{G})$ generated as an $\mathcal{O}_ S$-module by the local isomorphisms $\varphi : \mathcal{F} \to \mathcal{G}$ such that conjugation by $\varphi $ is the given isomorphism of endomorphism algebras. A local calculation (reducing to the case that $\mathcal{F}$ and $\mathcal{G}$ are finite free and $S$ is affine) shows that $\mathcal{L}$ is invertible. Another local calculation shows that the evaluation map
\[ \mathcal{F} \otimes _{\mathcal{O}_ S} \mathcal{L} \longrightarrow \mathcal{G} \]
is an isomorphism. $\square$
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