Lemma 10.148.3 (04E8)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.148: Formally unramified maps
Lemma 10.148.3 (cite)

previous lemma
next lemma

numberstags

history
statistics
1 tag refers to this tag

Lemma 10.148.3. Let $R \to S$ be a ring map. The following are equivalent:

$R \to S$ is formally unramified,
$R \to S_{\mathfrak q}$ is formally unramified for all primes $\mathfrak q$ of $S$ , and
$R_{\mathfrak p} \to S_{\mathfrak q}$ is formally unramified for all primes $\mathfrak q$ of $S$ with $\mathfrak p = R \cap \mathfrak q$ .

Proof. We have seen in Lemma 10.148.2 that (1) is equivalent to $\Omega _{S/R} = 0$ . Similarly, by Lemma 10.131.8 we see that (2) and (3) are equivalent to $(\Omega _{S/R})_{\mathfrak q} = 0$ for all $\mathfrak q$ . Hence the equivalence follows from Lemma 10.23.1. $\square$

Comments (0)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment