Lemma 71.18.2 (0863): Étale localization and strict transform

Lemma 71.18.2 (Étale localization and strict transform). In the situation of Definition 71.18.1. Let

\[ \xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ V \ar[r] & B } \]

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 $V' \to V$.

Proof. Part (1) follows from the fact that blowup commutes with flat base change (Lemma 71.17.3), the fact that étale morphisms are flat, and that the base change of an étale morphism is étale. Part (3) then follows from the fact that taking the sheaf of sections supported on a closed commutes with pullback by étale morphisms, see Limits of Spaces, Lemma 70.14.5. Part (2) follows from (3) applied to $\mathcal{F} = \mathcal{O}_ X$. $\square$

The Stacks project

Comments (0)

Post a comment