The Stacks project

Lemma 105.13.3. Let the algebraic stack $\mathcal{X}$ be well-nigh affine.

  1. If $\mathcal{X}$ is an algebraic space, then it is affine.

  2. If $\mathcal{X}' \to \mathcal{X}$ is an affine morphism of algebraic stacks, then $\mathcal{X}'$ is well-nigh affine.

Proof. Part (1) follows from immediately from Limits of Spaces, Lemma 69.15.1. However, this is overkill, since (1) also follows from Lemma 105.13.2 combined with Groupoids, Proposition 39.23.9.

To prove (2) we choose an affine scheme $U$ and a surjective, flat, finite, and finitely presented morphism $U \to \mathcal{X}$. Then $U' = \mathcal{X}' \times _\mathcal {X} U$ admits an affine morphism to $U$ (Morphisms of Stacks, Lemma 100.9.2). Therefore $U'$ is an affine scheme. Of course $U' \to \mathcal{X}'$ is surjective, flat, finite, and finitely presented as a base change of $U \to \mathcal{X}$. $\square$


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