Lemma 10.160.6 (0328)—The Stacks project

Lemma 10.160.6. Let $p$ be a prime number. Let $k$ be a field of characteristic $p$. There exists a Cohen ring $\Lambda $ with $\Lambda /p\Lambda \cong k$.

Proof. First note that the $p$-adic integers $\mathbf{Z}_ p$ form a Cohen ring for $\mathbf{F}_ p$. Let $k$ be an arbitrary field of characteristic $p$. Let $\mathbf{Z}_ p \to R$ be a flat local ring map such that $\mathfrak m_ R = pR$ and $R/pR = k$, see Lemma 10.159.1. By Lemma 10.97.5 the completion $\Lambda = R^\wedge $ is Noetherian. It is a complete Noetherian local ring with maximal ideal $(p)$ as $\Lambda /p\Lambda = R/pR$ is a field (use Lemma 10.96.3). Since $\mathbf{Z}_ p \to R \to \Lambda $ is flat (by Lemma 10.97.2) we see that $p$ is a nonzerodivisor in $\Lambda $. Hence $\Lambda $ has dimension $\geq 1$ (Lemma 10.60.13) and we conclude that $\Lambda $ is regular of dimension $1$, i.e., a discrete valuation ring by Lemma 10.119.7. We conclude $\Lambda $ is a Cohen ring for $k$. $\square$

Comments (5)

Comment #6049 by Mark on April 11, 2021 at 09:35

It might be better to add a citation for the sentence "Then clearly $R$ is a discrete valuation ring." I think it's regular+dimension 1 (Tag 00PD (3)). (At least this was not clear to me at first glance, I wondered quite a while around noetherianness.)

Comment #6186 by Johan on May 02, 2021 at 00:28

Thanks for catching this! No you can't get this immediately from Lemma 10.119.7 because indeed you don't know that it is Noetherian (a priori). And indeed right now I don't see why $\bigcap p^nR = 0$ (with $R$ as in the proof). I fixed it by completing first. Unfortunately now the proof is rather long and I feel it doesn't do justice to what's really going on. Anway, the changes can be viewed here.

Comment #7784 by Juhani on September 23, 2022 at 06:09

The original version (as far as I know) of Lemma 03C3 [EGA, ( $\mathbf{0}_{III}$ .10.3.1)] applies only to Noetherian base rings but shows that the extension is then also Noetherian. (That the Noether property is preserved in (the full version of) 03C3 is also embedded in [Bourbaki, Algèbre Commutative, Ch. X, Appendice].)

This gives immediately that R in the proof is a DVR (using also Lemma 031E). But of course the preservation of Noetherian property needs to go somewhere (in EGA it also goes via completion).

Comment #7832 by Juhani on October 10, 2022 at 11:32

Should have checked before posting #7784 above: the preservation of the Noetherian property (as well as being a DVR) is in [EGA I 2nd ed., ( $\mathbf{0}_{I}$ .6.8.3)]. The proof is a quick lemma ((6.8.3.1), loc.cit.).

Comment #8025 by Stacks Project on January 18, 2023 at 15:32

Dear Juhani, thank you for mentioning this.

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