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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