## 100.9 Affine morphisms

Affine morphisms of algebraic stacks are defined as follows.

Definition 100.9.1. A morphism of algebraic stacks is said to be affine if it is representable and affine in the sense of Properties of Stacks, Section 99.3.

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 99.3. (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 100.9.2. 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 99.3. $\square$

Lemma 100.9.3. Compositions of affine morphisms of algebraic stacks are affine.

Proof. This follows from the discussion in Properties of Stacks, Section 99.3 and Morphisms of Spaces, Lemma 66.20.4. $\square$

$\xymatrix{ \mathcal{X} \ar[rr]_ f \ar[rd]_ a & & \mathcal{Y} \ar[dl]^ b \\ & \mathcal{Z} }$

be a commutative diagram of morphisms of algebraic stacks. If $a$ is affine and $\Delta _ b$ is affine, then $f$ is affine.

Proof. The base change $\text{pr}_2 : \mathcal{X} \times _\mathcal {Z} \mathcal{Y} \to \mathcal{Y}$ of $a$ is affine by Lemma 100.9.2. 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 100.9.2. The composition $f = \text{pr}_2 \circ (1, f)$ of affine morphisms is affine by Lemma 100.9.3 and the proof is done. $\square$

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