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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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