Lemma 99.5.7. In Situation 99.5.1. Let
be a pushout in the category of schemes over S where Z \to Z' is a thickening and Z \to Y is affine, see More on Morphisms, Lemma 37.14.3. Then the functor on fibre categories
is an equivalence.
Lemma 99.5.7. In Situation 99.5.1. Let
be a pushout in the category of schemes over S where Z \to Z' is a thickening and Z \to Y is affine, see More on Morphisms, Lemma 37.14.3. Then the functor on fibre categories
is an equivalence.
Proof. Observe that the corresponding map
is a bijection, see Pushouts of Spaces, Lemma 81.6.1. Thus using the commutative diagram
we see that we may assume that Y' is a scheme over B'. By Remark 99.5.5 we may replace B by Y' and X by X \times _ B Y'. Thus we may assume B = Y'. In this case the statement follows from Pushouts of Spaces, Lemma 81.6.6. \square
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