Lemma 40.14.6. Let (U, R, s, t, c) be a groupoid scheme over a scheme S with s, t integral. Let g : U' \to U be an integral morphism such that every R-orbit in U meets g(U'). Let (U', R', s', t', c') be the restriction of R to U'. If u' \in U' is contained in an R'-invariant affine open, then the image u \in U is contained in an R-invariant affine open of U.
Proof. Let W' \subset U' be an R'-invariant affine open. Set \tilde R = U' \times _{g, U, t} R with maps \text{pr}_0 : \tilde R \to U' and h = s \circ \text{pr}_1 : \tilde R \to U. Observe that \text{pr}_0 and h are integral. It follows that \tilde W = \text{pr}_0^{-1}(W') is affine. Since W' is R'-invariant, the image W = h(\tilde W) is set theoretically R-invariant and \tilde W = h^{-1}(W) set theoretically (details omitted). Thus, if we can show that W is open, then W is a scheme and the morphism \tilde W \to W is integral surjective which implies that W is affine by Limits, Proposition 32.11.2. However, our assumption on orbits meeting U' implies that h : \tilde R \to U is surjective. Since an integral surjective morphism is submersive (Topology, Lemma 5.6.5 and Morphisms, Lemma 29.44.7) it follows that W is open. \square
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