The Stacks project

Lemma 101.45.5. Let

\[ \xymatrix{ \mathcal{X}' \ar[r] \ar[d]_{f'} & \mathcal{X} \ar[d]^ f \\ \mathcal{Y}' \ar[r] & \mathcal{Y} } \]

be a cartesian diagram of algebraic stacks.

  1. Let $x' \in |\mathcal{X}'|$ with image $x \in |\mathcal{X}|$. If $f$ induces an isomorphism between automorphism groups at $x$ and $f(x)$ (Remark 101.19.5), then $f'$ induces an isomorphism between automorphism groups at $x'$ and $f(x')$.

  2. If $\mathcal{I}_\mathcal {X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{I}_\mathcal {Y}$ is an isomorphism, then $\mathcal{I}_{\mathcal{X}'} \to \mathcal{X}' \times _{\mathcal{Y}'} \mathcal{I}_{\mathcal{Y}'}$ is an isomorphism.

Proof. Omitted. $\square$

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