Lemma 71.18.2 (Étale localization and strict transform). In the situation of Definition 71.18.1. Let
be a commutative diagram of morphisms with U and V schemes and étale horizontal arrows. Let V' \to V be the blowup of V in Z \times _ B V. Then
V' = V \times _ B B' and the maps V' \to B' and U \times _ V V' \to X \times _ B B' are étale,
the strict transform U' of U relative to V' \to V is equal to X' \times _ X U where X' is the strict transform of X relative to B' \to B, and
for a quasi-coherent \mathcal{O}_ X-module \mathcal{F} the restriction of the strict transform \mathcal{F}' to U \times _ V V' is the strict transform of \mathcal{F}|_ U relative to V' \to V.
Comments (0)
There are also: