Lemma 19.8.3. Let $\mathcal{O}$ be a sheaf of rings. For every $\mathcal{O}$-module $\mathcal{F}$ the $\mathcal{O}$-module $J(\mathcal{F})$ is injective.

Proof. We have to show that the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{G}, J(\mathcal{F}))$ is exact. Note that

\begin{eqnarray*} \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{G}, J(\mathcal{F})) & = & \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{G}, (F(\mathcal{F}^\vee ))^\vee ) \\ & = & \mathop{\mathrm{Hom}}\nolimits _\mathcal {O} (\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (F(\mathcal{F}^\vee ), \mathcal{J})) \\ & = & \mathop{\mathrm{Hom}}\nolimits (\mathcal{G} \otimes _\mathcal {O} F(\mathcal{F}^\vee ), \mathcal{J}) \end{eqnarray*}

Thus what we want follows from the fact that $F(\mathcal{F}^\vee )$ is flat and $\mathcal{J}$ is injective. $\square$

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