Lemma 100.9.18. Let \mathcal{P}, \mathcal{Q}, \mathcal{R} be properties of morphisms of algebraic spaces. Assume
\mathcal{P}, \mathcal{Q}, \mathcal{R} are fppf local on the target and stable under arbitrary base change,
\text{smooth} \Rightarrow \mathcal{R},
for any morphism f : X \to Y which has \mathcal{Q} there exists a largest open subspace W(\mathcal{P}, f) \subset X such that f|_{W(\mathcal{P}, f)} has \mathcal{P}, and
for any morphism f : X \to Y which has \mathcal{Q}, and any morphism Y' \to Y which has \mathcal{R} we have Y' \times _ Y W(\mathcal{P}, f) = W(\mathcal{P}, f'), where f' : X_{Y'} \to Y' is the base change of f.
Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks representable by algebraic spaces. Assume f has \mathcal{Q}. Then
there exists a largest open substack \mathcal{X}' \subset \mathcal{X} such that f|_{\mathcal{X}'} has \mathcal{P}, and
if \mathcal{Z} \to \mathcal{Y} is a morphism of algebraic stacks representable by algebraic spaces which has \mathcal{R} then \mathcal{Z} \times _\mathcal {Y} \mathcal{X}' is the largest open substack of \mathcal{Z} \times _\mathcal {Y} \mathcal{X} over which the base change \text{id}_\mathcal {Z} \times f has property \mathcal{P}.
Proof.
Choose a scheme V and a surjective smooth morphism V \to \mathcal{Y}. Set U = V \times _\mathcal {Y} \mathcal{X} and let f' : U \to V be the base change of f. The morphism of algebraic spaces f' : U \to V has property \mathcal{Q}. Thus we obtain the open W(\mathcal{P}, f') \subset U by assumption (3). Note that U \times _\mathcal {X} U = (V \times _\mathcal {Y} V) \times _\mathcal {Y} \mathcal{X} hence the morphism f'' : U \times _\mathcal {X} U \to V \times _\mathcal {Y} V is the base change of f via either projection V \times _\mathcal {Y} V \to V. By our choice of V these projections are smooth, hence have property \mathcal{R} by (2). Thus by (4) we see that the inverse images of W(\mathcal{P}, f') under the two projections \text{pr}_ i : U \times _\mathcal {X} U \to U agree. In other words, W(\mathcal{P}, f') is an R-invariant subspace of U (where R = U \times _\mathcal {X} U). Let \mathcal{X}' be the open substack of \mathcal{X} corresponding to W(\mathcal{P}, f) via Lemma 100.9.7. By construction W(\mathcal{P}, f') = \mathcal{X}' \times _\mathcal {Y} V hence f|_{\mathcal{X}'} has property \mathcal{P} by Lemma 100.3.3. Also, \mathcal{X}' is the largest open substack such that f|_{\mathcal{X}'} has \mathcal{P} as the same maximality holds for W(\mathcal{P}, f). This proves (A).
Finally, if \mathcal{Z} \to \mathcal{Y} is a morphism of algebraic stacks representable by algebraic spaces which has \mathcal{R} then we set T = V \times _\mathcal {Y} \mathcal{Z} and we see that T \to V is a morphism of algebraic spaces having property \mathcal{R}. Set f'_ T : T \times _ V U \to T the base change of f'. By (4) again we see that W(\mathcal{P}, f'_ T) is the inverse image of W(\mathcal{P}, f) in T \times _ V U. This implies (B); some details omitted.
\square
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