The Stacks project

Lemma 99.9.15. Let $\mathcal X$ be an algebraic stack and $\mathcal X' \subset \mathcal X$ a quasi-compact open substack. Suppose that we have a collection of open substacks $\mathcal{X}_ i \subset \mathcal X$ indexed by $i \in I$ such that $\mathcal{X}' \subset \bigcup _{i \in I} \mathcal{X}_ i$, where we define the union as in Lemma 99.9.14. Then there exists a finite subset $I' \subset I$ such that $\mathcal{X}' \subset \bigcup _{i \in I'} \mathcal{X}_ i$.

Proof. Since $\mathcal X$ is algebraic, there exists a scheme $U$ with a surjective smooth morphism $U \to \mathcal X$. Let $U_ i \subset U$ be the open subscheme representing $\mathcal{X}_ i \times _{\mathcal X} U$ and $U' \subset U$ the open subscheme representing $\mathcal{X}' \times _{\mathcal X} U$. By hypothesis, $U'\subset \bigcup _{i\in I} U_ i$. From the proof of Lemma 99.6.2, there is a quasi-compact open $V \subset U'$ such that $V \to \mathcal{X}'$ is a surjective smooth morphism. Therefore there exists a finite subset $I' \subset I$ such that $V \subset \bigcup _{i \in I'} U_ i$. We claim that $\mathcal{X}' \subset \bigcup _{i \in I'} \mathcal{X}_ i$. Take $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}'_ T)$ for $T \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$. Since $\mathcal{X}' \to \mathcal{X}$ is a monomorphism, we have cartesian squares

\[ \xymatrix{ V \times _\mathcal {X} T \ar[r] \ar[d] & T \ar[d]^ x \ar@{=}[r] & T \ar[d]^ x \\ V \ar[r] & \mathcal{X}' \ar[r] & \mathcal X } \]

By base change, $V \times _{\mathcal X} T \to T$ is surjective. Therefore $\bigcup _{i \in I'} U_ i \times _{\mathcal X} T \to T$ is also surjective. Let $T_ i \subset T$ be the open subscheme representing $\mathcal{X}_ i \times _{\mathcal X} T$. By a formal argument, we have a Cartesian square

\[ \xymatrix{ U_ i \times _{\mathcal{X}_ i} T_ i \ar[r] \ar[d] & U \times _{\mathcal X} T \ar[d] \\ T_ i \ar[r] & T } \]

where the vertical arrows are surjective by base change. Since $U_ i \times _{\mathcal{X}_ i} T_ i \simeq U_ i \times _{\mathcal X} T$, we find that $\bigcup _{i \in I'} T_ i = T$. Hence $x$ is an object of $(\bigcup _{i\in I'} \mathcal{X}_ i)_ T$ by definition of the union. Observe that the inclusion $\mathcal{X}' \subset \bigcup _{i \in I'} \mathcal{X}_ i$ is automatically an open substack. $\square$


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