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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)