The Stacks project

61.9 Weakly étale versus pro-étale

Recall that a ring homomorphism $A \to B$ is weakly étale if $A \to B$ is flat and $B \otimes _ A B \to B$ is flat. We have proved some properties of such ring maps in More on Algebra, Section 15.104. In particular, if $A \to B$ is a local homomorphism, and $A$ is a strictly henselian local rings, then $A = B$, see More on Algebra, Theorem 15.104.24. Using this theorem and the work we've done above we obtain the following structure theorem for weakly étale ring maps.

Proposition 61.9.1. Let $A \to B$ be a weakly étale ring map. Then there exists a faithfully flat, ind-étale ring map $B \to B'$ such that $A \to B'$ is ind-étale.

Proof. The ring map $A \to B$ induces (separable) algebraic extensions of residue fields, see More on Algebra, Lemma 15.104.17. Thus we may apply Lemma 61.8.7 and choose a diagram

\[ \xymatrix{ B \ar[r] & D \\ A \ar[r] \ar[u] & C \ar[u] } \]

with the properties as listed in the lemma. Note that $C \to D$ is weakly étale by More on Algebra, Lemma 15.104.11. Pick a maximal ideal $\mathfrak m \subset D$. By construction this lies over a maximal ideal $\mathfrak m' \subset C$. By More on Algebra, Theorem 15.104.24 the ring map $C_{\mathfrak m'} \to D_\mathfrak m$ is an isomorphism. As every point of $\mathop{\mathrm{Spec}}(C)$ specializes to a closed point we conclude that $C \to D$ identifies local rings. Thus Proposition 61.6.6 applies to the ring map $C \to D$. Pick $D \to D'$ faithfully flat and ind-Zariski such that $C \to D'$ is ind-Zariski. Then $B \to D'$ is a solution to the problem posed in the proposition. $\square$

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