Remark 40.10.11. Warning: Lemma 40.10.10 is wrong without the condition that $s$ and $t$ are locally of finite type. An easy example is to start with the action
\[ \mathbf{G}_{m, \mathbf{Q}} \times _{\mathbf{Q}} \mathbf{A}^1_{\mathbf{Q}} \to \mathbf{A}^1_{\mathbf{Q}} \]
and restrict the corresponding groupoid scheme to the generic point of $\mathbf{A}^1_{\mathbf{Q}}$. In other words restrict via the morphism $\mathop{\mathrm{Spec}}(\mathbf{Q}(x)) \to \mathop{\mathrm{Spec}}(\mathbf{Q}[x]) = \mathbf{A}^1_{\mathbf{Q}}$. Then you get a groupoid scheme $(U, R, s, t, c)$ with $U = \mathop{\mathrm{Spec}}(\mathbf{Q}(x))$ and
\[ R = \mathop{\mathrm{Spec}}\left( \mathbf{Q}(x)[y]\left[ \frac{1}{P(xy)}, P \in \mathbf{Q}[T], P \not= 0 \right] \right) \]
In this case $\dim (R) = 1$ and $\dim (G) = 0$.
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