The Stacks project

Lemma 99.3.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Then

  1. $\Delta _ f$ is representable (by schemes),

  2. $\Delta _ f$ is locally of finite type,

  3. $\Delta _ f$ is a monomorphism,

  4. $\Delta _ f$ is separated, and

  5. $\Delta _ f$ is locally quasi-finite.

Proof. We have already seen in Lemma 99.3.3 that $\Delta _ f$ is representable by algebraic spaces. Hence the statements (2) – (5) make sense, see Properties of Stacks, Section 98.3. Also Lemma 99.3.3 guarantees (2) holds. Let $T \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ be a morphism and contemplate Diagram (99.3.3.1). By Algebraic Stacks, Lemma 92.9.2 the right vertical arrow is injective as a map of sheaves, i.e., a monomorphism of algebraic spaces. Hence also the morphism $T \times _{\mathcal{X} \times _\mathcal {Y} \mathcal{X}} \mathcal{X} \to T$ is a monomorphism. Thus (3) holds. We already know that $T \times _{\mathcal{X} \times _\mathcal {Y} \mathcal{X}} \mathcal{X} \to T$ is locally of finite type. Thus Morphisms of Spaces, Lemma 65.27.10 allows us to conclude that $T \times _{\mathcal{X} \times _\mathcal {Y} \mathcal{X}} \mathcal{X} \to T$ is locally quasi-finite and separated. This proves (4) and (5). Finally, Morphisms of Spaces, Proposition 65.50.2 implies that $T \times _{\mathcal{X} \times _\mathcal {Y} \mathcal{X}} \mathcal{X}$ is a scheme which proves (1). $\square$


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