Lemma 78.6.3 (0DSJ)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 78: Groupoids in Algebraic Spaces
Section 78.6: Properties of group algebraic spaces
Lemma 78.6.3 (cite)

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Lemma 78.6.3. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Assume $G \to B$ is locally of finite type.

There exists a maximal open subspace $U \subset B$ such that $G_ U \to U$ is unramified and formation of $U$ commutes with base change.
There exists a maximal open subspace $U \subset B$ such that $G_ U \to U$ is locally quasi-finite and formation of $U$ commutes with base change.

Proof. By Morphisms of Spaces, Lemma 67.38.10 (resp. Morphisms of Spaces, Lemma 67.27.2) there is a maximal open subspace $W \subset G$ such that $W \to B$ is unramified (resp. locally quasi-finite). Moreover formation of $W$ commutes with base change. By Lemma 78.6.2 we see that $U = e^{-1}(W)$ in either case. $\square$

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