Lemma 37.47.3. Assumption and notation as in Lemma 37.47.2. In addition to properties (1) – (6) we may also arrange it so that
S', Y', X' are affine.
Lemma 37.47.3. Assumption and notation as in Lemma 37.47.2. In addition to properties (1) – (6) we may also arrange it so that
S', Y', X' are affine.
Proof. Note that if Y' is affine, then X' is affine as \pi is finite. Choose an affine open neighbourhood U' \subset S' of s'. Choose an affine open neighbourhood V' \subset h^{-1}(U') of y'. Let W' = h(V'). This is an open neighbourhood of s' in S', see Morphisms, Lemma 29.34.10, contained in U'. Choose an affine open neighbourhood U'' \subset W' of s'. Then h^{-1}(U'') \cap V' is affine because it is equal to U'' \times _{U'} V'. By construction h^{-1}(U'') \cap V' \to U'' is a surjective smooth morphism whose fibres are (nonempty) open subschemes of geometrically integral fibres of Y' \to S', and hence geometrically integral. Thus we may replace S' by U'' and Y' by h^{-1}(U'') \cap V'. \square
Comments (0)