Lemma 100.18.3. Let $\mathcal{X}$ be an algebraic stack. We have

where $U_0$ is the set of closed points of $U$. Here we may let $U$ range over all schemes smooth over $\mathcal{X}$ or over all affine schemes smooth over $\mathcal{X}$.

Lemma 100.18.3. Let $\mathcal{X}$ be an algebraic stack. We have

\[ \mathcal{X}_{\text{ft-pts}} = \bigcup \nolimits _{\varphi : U \to \mathcal{X}\text{ smooth}} |\varphi |(U_0) \]

where $U_0$ is the set of closed points of $U$. Here we may let $U$ range over all schemes smooth over $\mathcal{X}$ or over all affine schemes smooth over $\mathcal{X}$.

**Proof.**
Immediate from Lemma 100.18.1.
$\square$

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