Lemma 61.11.7. Let $A$ be a ring. Let $Z \subset \mathop{\mathrm{Spec}}(A)$ be a closed subset of the form $Z = V(f_1, \ldots , f_ r)$. Set $B = A_ Z^\sim$, see Lemma 61.5.1. If $A$ is w-contractible, so is $B$.

Proof. Let $A_ Z^\sim \to B$ be a weakly étale faithfully flat ring map. Consider the ring map

$A \longrightarrow A_{f_1} \times \ldots \times A_{f_ r} \times B$

this is faithful flat and weakly étale. If $A$ is w-contractible, then there is a section $\sigma$. Consider the morphism

$\mathop{\mathrm{Spec}}(A_ Z^\sim ) \to \mathop{\mathrm{Spec}}(A) \xrightarrow {\mathop{\mathrm{Spec}}(\sigma )} \coprod \mathop{\mathrm{Spec}}(A_{f_ i}) \amalg \mathop{\mathrm{Spec}}(B)$

Every point of $Z \subset \mathop{\mathrm{Spec}}(A_ Z^\sim )$ maps into the component $\mathop{\mathrm{Spec}}(B)$. Since every point of $\mathop{\mathrm{Spec}}(A_ Z^\sim )$ specializes to a point of $Z$ we find a morphism $\mathop{\mathrm{Spec}}(A_ Z^\sim ) \to \mathop{\mathrm{Spec}}(B)$ as desired. $\square$

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