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 retraction \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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