Lemma 77.9.10. In Situation 77.9.1 assume $G$ locally of finite type. For all $g \in |G|$

$\dim (G) = \dim _ g(G)$,

if the transcendence degree of $g$ over $k$ is $0$, then $\dim (G) = \dim (\mathcal{O}_{G, \overline{g}})$.

Lemma 77.9.10. In Situation 77.9.1 assume $G$ locally of finite type. For all $g \in |G|$

$\dim (G) = \dim _ g(G)$,

if the transcendence degree of $g$ over $k$ is $0$, then $\dim (G) = \dim (\mathcal{O}_{G, \overline{g}})$.

**Proof.**
Immediate from Lemma 77.9.9 via (77.9.2.1).
$\square$

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