The Stacks project

Lemma 10.144.2. Let $R \to R[x]_ g/(f)$ be standard étale.

  1. The ring map $R \to R[x]_ g/(f)$ is étale.

  2. 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.

  3. Any principal localization of $R[x]_ g/(f)$ is standard étale over $R$.

  4. 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. $\square$

Comments (2)

Comment #4606 by Rex on

I believe part (1) is actually proven in 03PA. It may be helpful to include a link to that here.

Comment #4776 by on

Pointing there would be a forward reference. I would be happy to accept a small patch from somebody providing a proof of this lemma.

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