# The Stacks Project

## Tag: 07PJ

This tag has label more-algebra-proposition-ubiquity-J-2 and it points to

The corresponding content:

Proposition 14.34.6. The following types of rings are J-2:
1. fields,
2. Noetherian complete local rings,
3. $\mathbf{Z}$,
4. Dedekind domains with fraction field of characteristic zero,
5. finite type ring extensions of any of the above.

Proof. For fields, $\mathbf{Z}$ and Dedekind domains of characteristic zero you just check condition (4) of Lemma 14.33.6. In the case of Noetherian complete local rings, note that if $R \to R'$ is finite and $R$ is a Noetherian complete local ring, then $R'$ is a product of Noetherian complete local rings, see Algebra, Lemma 9.147.2. Hence it suffices to prove that a Noetherian complete local ring which is a domain is J-0, which is Lemma 14.34.5. $\square$

\begin{proposition}
\label{proposition-ubiquity-J-2}
The following types of rings are J-2:
\begin{enumerate}
\item fields,
\item Noetherian complete local rings,
\item $\mathbf{Z}$,
\item Dedekind domains with fraction field of characteristic zero,
\item finite type ring extensions of any of the above.
\end{enumerate}
\end{proposition}

\begin{proof}
For fields, $\mathbf{Z}$ and Dedekind domains of characteristic zero
you just check condition (4) of Lemma \ref{lemma-J-2}.
In the case of Noetherian complete local rings, note that if
$R \to R'$ is finite and $R$ is a Noetherian complete local ring,
then $R'$ is a product of Noetherian complete local rings, see
Algebra, Lemma \ref{algebra-lemma-quotient-complete-local}.
Hence it suffices to prove that a Noetherian complete local ring
which is a domain is J-0, which is
Lemma \ref{lemma-complete-Noetherian-domain-J-0}.
\end{proof}


To cite this tag (see How to reference tags), use:

\cite[\href{http://stacks.math.columbia.edu/tag/07PJ}{Tag 07PJ}]{stacks-project}


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 lower-right corner).