The Stacks project

Proof. By Algebraic Stacks, Lemma 94.16.2 we may assume that $\mathcal{X} = [U/R]$ for some smooth groupoid in algebraic spaces. By Descent on Spaces, Lemma 74.11.9 it suffices to check the property fppf locally on $T$. Thus we may assume that $x, y$ come from morphisms $x', y' : T \to U$. By Groupoids in Spaces, Lemma 78.22.1 we see that in this case $\mathit{Isom}_\mathcal {X}(x, y) = T \times _{(y', x'), U \times _ S U} R$. Hence it suffices to prove that $R \to U \times _ S U$ is locally of finite type. This follows from the fact that the composition $s : R \to U \times _ S U \to U$ is smooth (hence locally of finite type, see Morphisms of Spaces, Lemmas 67.37.5 and 67.28.5) and Morphisms of Spaces, Lemma 67.23.6. $\square$

Proof. See Groupoids in Spaces, Definitions 78.5.1 and 78.9.1. The lemma follows immediately from the fact that $\mathcal{X}$ is a stack in groupoids. $\square$

Proof. Let $T$ be a scheme and let $a : T \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ be a morphism. By definition of the fibre product and the $2$-Yoneda lemma the morphism $a$ is given by a triple $a = (x, x', \alpha )$ where $x, x'$ are objects of $\mathcal{X}$ over $T$, and $\alpha : f(x) \to f(x')$ is a morphism in the fibre category of $\mathcal{Y}$ over $T$. By definition of an algebraic stack the sheaves $\mathit{Isom}_\mathcal {X}(x, x')$ and $\mathit{Isom}_\mathcal {Y}(f(x), f(x'))$ are algebraic spaces over $T$. In this language $\alpha $ defines a section of the morphism $\mathit{Isom}_\mathcal {Y}(f(x), f(x')) \to T$. A $T'$-valued point of $\mathcal{X} \times _{\mathcal{X} \times _\mathcal {Y} \mathcal{X}, a} T$ for $T' \to T$ a scheme over $T$ is the same thing as an isomorphism $x|_{T'} \to x'|_{T'}$ whose image under $f$ is $\alpha |_{T'}$. Thus we see that

is a fibre square of sheaves over $T$. In particular we see that $\mathcal{X} \times _{\mathcal{X} \times _\mathcal {Y} \mathcal{X}, a} T$ is an algebraic space which proves part (1) of the lemma.

To prove the second statement we have to show that the left vertical arrow of Diagram (101.3.3.1) is locally of finite type. By Lemma 101.3.1 the algebraic space $\mathit{Isom}_\mathcal {X}(x, x')$ and is locally of finite type over $T$. Hence the right vertical arrow of Diagram (101.3.3.1) is locally of finite type, see Morphisms of Spaces, Lemma 67.23.6. We conclude by Morphisms of Spaces, Lemma 67.23.3. $\square$

Proof. We have already seen in Lemma 101.3.3 that $\Delta _ f$ is representable by algebraic spaces. Hence the statements (2) – (5) make sense, see Properties of Stacks, Section 100.3. Also Lemma 101.3.3 guarantees (2) holds. Let $T \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ be a morphism and contemplate Diagram (101.3.3.1). By Algebraic Stacks, Lemma 94.9.2 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 67.27.10 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 67.50.2 implies that $T \times _{\mathcal{X} \times _\mathcal {Y} \mathcal{X}} \mathcal{X}$ is a scheme which proves (1). $\square$

Proof. The statements “$f$ is separated”, “$\Delta _ f$ is a closed immersion”, “$\Delta _ f$ is universally closed”, and “$\Delta _ f$ is proper” refer to the notions defined in Properties of Stacks, Section 100.3. Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. Set $U = \mathcal{X} \times _\mathcal {Y} V$ which is an algebraic space by assumption, and the morphism $U \to \mathcal{X}$ is surjective and smooth. By Categories, Lemma 4.31.14 and Properties of Stacks, Lemma 100.3.3 we see that for any property $P$ (as in that lemma) we have: $\Delta _ f$ has $P$ if and only if $\Delta _{U/V} : U \to U \times _ V U$ has $P$. Hence the equivalence of (2), (3) and (4) follows from Morphisms of Spaces, Lemma 67.40.9 applied to $U \to V$. Moreover, if (1) holds, then $U \to V$ is separated and we see that $\Delta _{U/V}$ is a closed immersion, i.e., (2) holds. Finally, assume (2) holds. Let $T$ be a scheme, and $a : T \to \mathcal{Y}$ a morphism. Set $T' = \mathcal{X} \times _\mathcal {Y} T$. To prove (1) we have to show that the morphism of algebraic spaces $T' \to T$ is separated. Using Categories, Lemma 4.31.14 once more we see that $\Delta _{T'/T}$ is the base change of $\Delta _ f$. Hence our assumption (2) implies that $\Delta _{T'/T}$ is a closed immersion, hence $T' \to T$ is separated as desired. $\square$

Proof. The statements “$f$ is quasi-separated”, “$\Delta _ f$ is quasi-compact”, and “$\Delta _ f$ is of finite type” refer to the notions defined in Properties of Stacks, Section 100.3. Note that (2) and (3) are equivalent in view of the fact that $\Delta _ f$ is locally of finite type by Lemma 101.3.4 (and Algebraic Stacks, Lemma 94.10.9). Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. Set $U = \mathcal{X} \times _\mathcal {Y} V$ which is an algebraic space by assumption, and the morphism $U \to \mathcal{X}$ is surjective and smooth. By Categories, Lemma 4.31.14 and Properties of Stacks, Lemma 100.3.3 we see that we have: $\Delta _ f$ is quasi-compact if and only if $\Delta _{U/V} : U \to U \times _ V U$ is quasi-compact. If (1) holds, then $U \to V$ is quasi-separated and we see that $\Delta _{U/V}$ is quasi-compact, i.e., (2) holds. Assume (2) holds. Let $T$ be a scheme, and $a : T \to \mathcal{Y}$ a morphism. Set $T' = \mathcal{X} \times _\mathcal {Y} T$. To prove (1) we have to show that the morphism of algebraic spaces $T' \to T$ is quasi-separated. Using Categories, Lemma 4.31.14 once more we see that $\Delta _{T'/T}$ is the base change of $\Delta _ f$. Hence our assumption (2) implies that $\Delta _{T'/T}$ is quasi-compact, hence $T' \to T$ is quasi-separated as desired. $\square$

Proof. The statements “$f$ is locally separated”, and “$\Delta _ f$ is an immersion” refer to the notions defined in Properties of Stacks, Section 100.3. Proof omitted. Hint: Argue as in the proofs of Lemmas 101.3.5 and 101.3.6. $\square$