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The Stacks project

Lemma 79.15.6 (Existence of splitting). In Situation 79.15.3 there exists an algebraic space U', an étale morphism U' \to U, and a point u' : \mathop{\mathrm{Spec}}(\kappa (u)) \to U' lying over u : \mathop{\mathrm{Spec}}(\kappa (u)) \to U such that the restriction R' = R|_{U'} of R to U' is split over u'.

Proof. Let f : (U', Z_{univ}, s', t', c') \to (U, R, s, t, c) be as constructed in Lemma 79.14.1. Recall that R' = R \times _{(U \times _ S U)} (U' \times _ S U'). Thus we get a morphism (f, t', s') : Z_{univ} \to R' of groupoids in algebraic spaces

(U', Z_{univ}, s', t', c') \to (U', R', s', t', c')

(by abuse of notation we indicate the morphisms in the two groupoids by the same symbols). Now, as Z_{univ} \subset R \times _{s, U, g} U' is open and R' \to R \times _{s, U, g} U' is étale (as a base change of U' \to U) we see that Z_{univ} \to R' is an open immersion. By construction the morphisms s', t' : Z_{univ} \to U' are finite. It remains to find the point u' of U'.

We think of u as a morphism \mathop{\mathrm{Spec}}(\kappa (u)) \to U as in the statement of the lemma. Set F_ u = R \times _{s, U} \mathop{\mathrm{Spec}}(\kappa (u)). Let G_ u \subset F_ u be the scheme theoretic fibre of G \to U over u. By assumption G_ u is finite and F_ u \to \mathop{\mathrm{Spec}}(\kappa (u)) is quasi-finite at each point of G_ u by assumption. Hence we can find a decomposition into open and closed subschemes

F_ u = Z_ u \amalg Rest

for some scheme Z_ u finite over \kappa (u) whose support is G_ u. Note that e(u) \in Z_ u. Hence by the construction of U' in Section 79.14 (u, Z_ u) defines a \mathop{\mathrm{Spec}}(\kappa (u))-valued point u' of U'.

We still have to show that the set \{ g' \in |G'| : g'\text{ maps to }u'\} is contained in |Z_{univ}|. Pick any point g' in this set and represent it by a morphism z' : \mathop{\mathrm{Spec}}(k) \to G'. Denote z : \mathop{\mathrm{Spec}}(k) \to G the composition of z' with the map G' \to G. Clearly, z defines a point of G_ u. In fact, let us write \tilde u : \mathop{\mathrm{Spec}}(k) \to u \to U for the corresponding map to u or U. Consider the triple

(\tilde u, Z_ u \times _{u, \tilde u} \mathop{\mathrm{Spec}}(k), z)

where Z_ u is as above. This defines a \mathop{\mathrm{Spec}}(k)-valued point of Z_{univ} whose image via s', t' in U' is u' and whose image via Z_{univ} \to R' is the point z' (because the image in R is z). This finishes the proof. \square


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