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.

Lemma 101.9.4. Let

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

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 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.
$\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)