Lemma 39.6.2. Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Let $T$ be a scheme over $S$ and let $\psi : T \to G$ be a morphism over $S$. If $T$ is flat over $S$, then the morphism

\[ T \times _ S G \longrightarrow G, \quad (t, g) \longmapsto m(\psi (t), g) \]

is flat. In particular, if $G$ is flat over $S$, then $m : G \times _ S G \to G$ is flat.

**Proof.**
Consider the diagram

\[ \xymatrix{ T \times _ S G \ar[rrr]_{(t, g) \mapsto (t, m(\psi (t), g))} & & & T \times _ S G \ar[r]_{\text{pr}} \ar[d] & G \ar[d] \\ & & & T \ar[r] & S } \]

The left top horizontal arrow is an isomorphism and the square is cartesian. Hence the lemma follows from Morphisms, Lemma 29.25.8.
$\square$

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