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

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

In this case $\dim (R) = 1$ and $\dim (G) = 0$.

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