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The Stacks project

Lemma 100.6.2. Let \mathcal{X} be an algebraic stack. The following are equivalent:

  1. \mathcal{X} is quasi-compact,

  2. there exists a surjective smooth morphism U \to \mathcal{X} with U an affine scheme,

  3. there exists a surjective smooth morphism U \to \mathcal{X} with U a quasi-compact scheme,

  4. there exists a surjective smooth morphism U \to \mathcal{X} with U a quasi-compact algebraic space, and

  5. there exists a surjective morphism \mathcal{U} \to \mathcal{X} of algebraic stacks such that \mathcal{U} is quasi-compact.

Proof. We will use Lemma 100.4.4. Suppose \mathcal{U} and \mathcal{U} \to \mathcal{X} are as in (5). Then since |\mathcal{U}| \to |\mathcal{X}| is surjective and continuous we conclude that |\mathcal{X}| is quasi-compact. Thus (5) implies (1). The implications (2) \Rightarrow (3) \Rightarrow (4) \Rightarrow (5) are immediate. Assume (1), i.e., \mathcal{X} is quasi-compact, i.e., that |\mathcal{X}| is quasi-compact. Choose a scheme U and a surjective smooth morphism U \to \mathcal{X}. Then since |U| \to |\mathcal{X}| is open we see that there exists a quasi-compact open U' \subset U such that |U'| \to |X| is surjective (and still smooth). Choose a finite affine open covering U' = U_1 \cup \ldots \cup U_ n. Then U_1 \amalg \ldots \amalg U_ n \to \mathcal{X} is a surjective smooth morphism whose source is an affine scheme (Schemes, Lemma 26.6.8). Hence (2) holds. \square


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