The Stacks project

Lemma 106.11.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ finite type and $\mathcal{Y}$ locally Noetherian. Let $y \in |\mathcal{Y}|$ be a point in the closure of the image of $|f|$. Then there exists a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r] & \mathcal{Y} } \]

of algebraic stacks where $A$ is a discrete valuation ring and $K$ is its field of fractions mapping the closed point of $\mathop{\mathrm{Spec}}(A)$ to $y$.

Proof. Choose an affine scheme $V$, a point $v \in V$ and a smooth morphism $V \to \mathcal{Y}$ mapping $v$ to $y$. The map $|V| \to |\mathcal{Y}|$ is open and by Properties of Stacks, Lemma 100.4.3 the image of $|\mathcal{X} \times _\mathcal {Y} V| \to |V|$ is the inverse image of the image of $|f|$. We conclude that the point $v$ is in the closure of the image of $|\mathcal{X} \times _\mathcal {Y} V| \to |V|$. If we prove the lemma for $\mathcal{X} \times _\mathcal {Y} V \to V$ and the point $v$, then the lemma follows for $f$ and $y$. In this way we reduce to the situation described in the next paragraph.

Assume we have $f : \mathcal{X} \to Y$ and $y \in |Y|$ as in the lemma where $Y$ is a Noetherian affine scheme. Since $f$ is quasi-compact, we conclude that $\mathcal{X}$ is quasi-compact. Hence we can choose an affine scheme $W$ and a surjective smooth morphism $W \to \mathcal{X}$. Then the image of $|f|$ is the same as the image of $|W| \to |Y|$. In this way we reduce to the case of schemes which is Limits, Lemma 32.15.1. $\square$

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