Lemma 37.34.7. Let f : X \to S be a morphism of affine schemes, which is of finite presentation with geometrically irreducible fibres. Then there exists a diagram as in Lemma 37.34.1 such that in addition f_0 has geometrically irreducible fibres.
Proof. Apply Lemma 37.34.1 to get a cartesian diagram
of affine schemes with X_0 \to S_0 a finite type morphism of schemes of finite type over \mathbf{Z}. By Lemma 37.27.7 the set E \subset S_0 of points where the fibre of f_0 is geometrically irreducible is a constructible subset. By Lemma 37.27.2 we have h(S) \subset E. Write S_0 = \mathop{\mathrm{Spec}}(A_0) and S = \mathop{\mathrm{Spec}}(A). Write A = \mathop{\mathrm{colim}}\nolimits _ i A_ i as a direct colimit of finite type A_0-algebras. By Limits, Lemma 32.4.10 we see that \mathop{\mathrm{Spec}}(A_ i) \to S_0 has image contained in E for some i. After replacing S_0 by \mathop{\mathrm{Spec}}(A_ i) and X_0 by X_0 \times _{S_0} \mathop{\mathrm{Spec}}(A_ i) we see that all fibres of f_0 are geometrically irreducible. \square
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