Lemma 18.44.2. In the situation above the map to the sheafification
\mathcal{F} \longrightarrow (\mathcal{S}^{-1}\mathcal{F})^\#
has the following universal property: for any homomorphism of \mathcal{O}-modules \mathcal{F} \to \mathcal{G} such that each local section of \mathcal{S} acts invertibly on \mathcal{G} there exists a unique factorization (\mathcal{S}^{-1}\mathcal{F})^\# \to \mathcal{G}. Moreover we have
(\mathcal{S}^{-1}\mathcal{F})^\# = (\mathcal{S}^{-1}\mathcal{O})^\# \otimes _\mathcal {O} \mathcal{F}
as sheaves of (\mathcal{S}^{-1}\mathcal{O})^\# -modules.
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