# The Stacks Project

## Tag: 01UR

This tag has label morphisms-lemma-universal-derivation-universal and it points to

The corresponding content:

Lemma 25.34.2. Let $f : X \to S$ be a morphism of schemes. The map $$\mathop{\rm Hom}\nolimits_{\mathcal{O}_X}(\Omega_{X/S}, \mathcal{F}) \longrightarrow \text{Der}_S(\mathcal{O}_X, \mathcal{F}), \ \alpha \longmapsto \alpha \circ \text{d}_{X/S}$$ is an isomorphism of functors $\textit{Mod}(\mathcal{O}_X) \to \textit{Sets}$.

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

\begin{lemma}
\label{lemma-universal-derivation-universal}
Let $f : X \to S$ be a morphism of schemes. The map
$$\Hom_{\mathcal{O}_X}(\Omega_{X/S}, \mathcal{F}) \longrightarrow \text{Der}_S(\mathcal{O}_X, \mathcal{F}), \ \ \alpha \longmapsto \alpha \circ \text{d}_{X/S}$$
is an isomorphism of functors $\textit{Mod}(\mathcal{O}_X) \to \textit{Sets}$.
\end{lemma}

\begin{proof}
This is just a restatement of the definition.
\end{proof}


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