Lemma 29.32.2 (01UR)—The Stacks project

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Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.32: Sheaf of differentials of a morphism
Lemma 29.32.2 (cite)

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Lemma 29.32.2. Let $f : X \to S$ be a morphism of schemes. The map

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/S}, \mathcal{F}) \longrightarrow \text{Der}_ S(\mathcal{O}_ X, \mathcal{F}),\quad \alpha \longmapsto \alpha \circ \text{d}_{X/S}$

is an isomorphism of functors $\textit{Mod}(\mathcal{O}_ X) \to \textit{Sets}$ .

Proof. This is just a restatement of the definition. $\square$

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