The Stacks project

Proposition 61.11.3. For every ring $A$ there exists a faithfully flat, ind-étale ring map $A \to D$ such that $D$ is w-contractible.

Proof. Applying Lemma 61.8.7 to $\text{id}_ A : A \to A$ we find a faithfully flat, ind-étale ring map $A \to C$ such that $C$ is w-local and such that every local ring at a maximal ideal of $C$ is strictly henselian. Choose an extremally disconnected space $T$ and a surjective continuous map $T \to \pi _0(\mathop{\mathrm{Spec}}(C))$, see Topology, Lemma 5.26.9. Note that $T$ is profinite. Apply Lemma 61.6.2 to find an ind-Zariski ring map $C \to D$ such that $\pi _0(\mathop{\mathrm{Spec}}(D)) \to \pi _0(\mathop{\mathrm{Spec}}(C))$ realizes $T \to \pi _0(\mathop{\mathrm{Spec}}(C))$ and such that

\[ \xymatrix{ \mathop{\mathrm{Spec}}(D) \ar[r] \ar[d] & \pi _0(\mathop{\mathrm{Spec}}(D)) \ar[d] \\ \mathop{\mathrm{Spec}}(C) \ar[r] & \pi _0(\mathop{\mathrm{Spec}}(C)) } \]

is cartesian in the category of topological spaces. Note that $\mathop{\mathrm{Spec}}(D)$ is w-local, that $\mathop{\mathrm{Spec}}(D) \to \mathop{\mathrm{Spec}}(C)$ is w-local, and that the set of closed points of $\mathop{\mathrm{Spec}}(D)$ is the inverse image of the set of closed points of $\mathop{\mathrm{Spec}}(C)$, see Lemma 61.2.5. Thus it is still true that the local rings of $D$ at its maximal ideals are strictly henselian (as they are isomorphic to the local rings at the corresponding maximal ideals of $C$). It follows from Lemma 61.11.2 that $D$ is w-contractible. $\square$


Comments (0)


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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0983. Beware of the difference between the letter 'O' and the digit '0'.