Strengthen a lemma

Thanks to Rankeya
https://stacks.math.columbia.edu/tag/00MC#comment-8360

algebra.tex

@@ -22758,20 +22758,18 @@ \section{Completion for Noetherian rings}
 
 \begin{lemma}
 \label{lemma-completion-faithfully-flat}
-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 $\lim_n R/\mathfrak m^n$ is faithfully flat.
+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 $\lim_n R/\mathfrak m^n$ is faithfully flat.
 \end{lemma}
 
 \begin{proof}
 By Lemma \ref{lemma-completion-flat} 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 $\Spec(R^\wedge)
+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 $\Spec(R^\wedge)
 \to \Spec(R)$. Hence the map is faithfully
 flat by Lemma \ref{lemma-ff}.
 \end{proof}