Lemma 10.144.2. Let R \to R[x]_ g/(f) be standard étale.
The ring map R \to R[x]_ g/(f) is étale.
For any ring map R \to R' the base change R' \to R'[x]_ g/(f) of the standard étale ring map R \to R[x]_ g/(f) is standard étale.
Any principal localization of R[x]_ g/(f) is standard étale over R.
A composition of standard étale maps is not standard étale in general.
Proof. Omitted. Here is an example for (4). The ring map \mathbf{F}_2 \to \mathbf{F}_{2^2} is standard étale. The ring map \mathbf{F}_{2^2} \to \mathbf{F}_{2^2} \times \mathbf{F}_{2^2} \times \mathbf{F}_{2^2} \times \mathbf{F}_{2^2} is standard étale. But the ring map \mathbf{F}_2 \to \mathbf{F}_{2^2} \times \mathbf{F}_{2^2} \times \mathbf{F}_{2^2} \times \mathbf{F}_{2^2} is not standard étale. \square
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