The Stacks project

Lemma 61.11.2. Let $A$ be a ring. The following are equivalent

  1. $A$ is w-contractible,

  2. every faithfully flat, ind-étale ring map $A \to B$ has a section, and

  3. $A$ satisfies

    1. $\mathop{\mathrm{Spec}}(A)$ is w-local,

    2. $\pi _0(\mathop{\mathrm{Spec}}(A))$ is extremally disconnected, and

    3. for every maximal ideal $\mathfrak m \subset A$ the local ring $A_\mathfrak m$ is strictly henselian.

Proof. The equivalence of (1) and (2) follows immediately from Proposition 61.9.1.

Assume (3)(a), (3)(b), and (3)(c). Let $A \to B$ be faithfully flat and ind-étale. We will use without further mention the fact that a flat map $A \to B$ is faithfully flat if and only if every closed point of $\mathop{\mathrm{Spec}}(A)$ is in the image of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ We will show that $A \to B$ has a section.

Let $I \subset A$ be an ideal such that $V(I) \subset \mathop{\mathrm{Spec}}(A)$ is the set of closed points of $\mathop{\mathrm{Spec}}(A)$. We may replace $B$ by the ring $C$ constructed in Lemma 61.5.8 for $A \to B$ and $I \subset A$. Thus we may assume $\mathop{\mathrm{Spec}}(B)$ is w-local such that the set of closed points of $\mathop{\mathrm{Spec}}(B)$ is $V(IB)$. In this case $A \to B$ identifies local rings by condition (3)(c) as it suffices to check this at maximal ideals of $B$ which lie over maximal ideals of $A$. Thus $A \to B$ has a section by Lemma 61.6.7.

Assume (1) or equivalently (2). We have (3)(c) by Lemma 61.8.5. Properties (3)(a) and (3)(b) follow from Lemma 61.6.7. $\square$


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