The Stacks project

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

Proof. By Lemma 10.97.2 it is flat. The composition $R \to R^\wedge \to R/I$ where the last map is the projection map $R^\wedge \to R/I$ shows that any maximal ideal of $R$ 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. $\square$


Comments (3)

Comment #8360 by Rankeya on

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

Comment #9476 by on

Alternative suggested proof: By Lemma 10.97.2 it is flat. Thus, if is some non-zero -module, to see that it suffices to see that , for finitely generated submodules . By Lemma 10.97.1, . Since , the map is injective by Krull's intersection theorem (Lemma 10.51.4), so implies .

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

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