Lemma 10.97.3. Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $I \subset \mathfrak m$ be an ideal. Denote $R^\wedge$ the completion of $R$ with respect to $I$. The ring map $R \to R^\wedge$ is faithfully flat. In particular the completion with respect to $\mathfrak m$, namely $\mathop{\mathrm{lim}}\nolimits _ n R/\mathfrak m^ n$ is faithfully flat.

Proof. By Lemma 10.97.2 it is flat. The composition $R \to R^\wedge \to R/\mathfrak m$ where the last map is the projection map $R^\wedge \to R/I$ combined with $R/I \to R/\mathfrak m$ shows that $\mathfrak m$ is in the image of $\mathop{\mathrm{Spec}}(R^\wedge ) \to \mathop{\mathrm{Spec}}(R)$. Hence the map is faithfully flat by Lemma 10.39.15. $\square$

Comment #8360 by Rankeya on

Tag 00MC doesn't need $(R,\mathfrak m)$ to be local. The result holds as long as $I$ is contained in the Jacobson radical of $R$. Perhaps it could be stated this way since the proof does not require any serious modifications?

There are also:

• 5 comment(s) on Section 10.97: Completion for Noetherian rings

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).