Definition 101.9.1. A morphism of algebraic stacks is said to be affine if it is representable and affine in the sense of Properties of Stacks, Section 100.3.
101.9 Affine morphisms
Affine morphisms of algebraic stacks are defined as follows.
For us it is a little bit more convenient to think of an affine morphism of algebraic stacks as a morphism of algebraic stacks which is representable by algebraic spaces and affine in the sense of Properties of Stacks, Section 100.3. (Recall that the default for “representable” in the Stacks project is representable by schemes.) Since this is clearly equivalent to the notion just defined we shall use this characterization without further mention. We prove a few simple lemmas about this notion.
Lemma 101.9.2. Let $\mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Z} \to \mathcal{Y}$ be an affine morphism of algebraic stacks. Then $\mathcal{Z} \times _\mathcal {Y} \mathcal{X} \to \mathcal{X}$ is an affine morphism of algebraic stacks.
Proof. This follows from the discussion in Properties of Stacks, Section 100.3. $\square$
Lemma 101.9.3. Compositions of affine morphisms of algebraic stacks are affine.
Proof. This follows from the discussion in Properties of Stacks, Section 100.3 and Morphisms of Spaces, Lemma 67.20.4. $\square$
Lemma 101.9.4. Let be a commutative diagram of morphisms of algebraic stacks. If $a$ is affine and $\Delta _ b$ is affine, then $f$ is affine.
\[ \xymatrix{ \mathcal{X} \ar[rr]_ f \ar[rd]_ a & & \mathcal{Y} \ar[dl]^ b \\ & \mathcal{Z} } \]
Proof. The base change $\text{pr}_2 : \mathcal{X} \times _\mathcal {Z} \mathcal{Y} \to \mathcal{Y}$ of $a$ is affine by Lemma 101.9.2. The morphism $(1, f) : \mathcal{X} \to \mathcal{X} \times _\mathcal {Z} \mathcal{Y}$ is the base change of $\Delta _ b : \mathcal{Y} \to \mathcal{Y} \times _\mathcal {Z} \mathcal{Y}$ by the morphism $\mathcal{X} \times _\mathcal {Z} \mathcal{Y} \to \mathcal{Y} \times _\mathcal {Z} \mathcal{Y}$ (see material in Categories, Section 4.31). Hence it is affine by Lemma 101.9.2. The composition $f = \text{pr}_2 \circ (1, f)$ of affine morphisms is affine by Lemma 101.9.3 and the proof is done. $\square$
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