The Stacks project

Lemma 39.6.3. Let $(G, m, e, i)$ be a group scheme over the scheme $S$. Denote $f : G \to S$ the structure morphism. Assume $f$ is flat. Then there exist canonical isomorphisms

\[ \Omega _{G/S} \cong f^*\mathcal{C}_{S/G} \cong f^*e^*\Omega _{G/S} \]

where $\mathcal{C}_{S/G}$ denotes the conormal sheaf of the immersion $e$. In particular, if $S$ is the spectrum of a field, then $\Omega _{G/S}$ is a free $\mathcal{O}_ G$-module.

Proof. In Morphisms, Lemma 29.32.7 we identified $\Omega _{G/S}$ with the conormal sheaf of the diagonal morphism $\Delta _{G/S}$. In the proof of Lemma 39.6.1 we showed that $\Delta _{G/S}$ is a base change of the immersion $e$ by the morphism $(g, g') \mapsto m(i(g), g')$. This morphism is isomorphic to the morphism $(g, g') \mapsto m(g, g')$ hence is flat by Lemma 39.6.2. Hence we get the first isomorphism by Morphisms, Lemma 29.31.4. By Morphisms, Lemma 29.32.16 we have $\mathcal{C}_{S/G} \cong e^*\Omega _{G/S}$.

If $S$ is the spectrum of a field, then $G \to S$ is flat, and any $\mathcal{O}_ S$-module on $S$ is free. $\square$


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