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. $\square$

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