# The Stacks Project

## Tag 01OW

Lemma 27.5.2. Let $X$ be a scheme. The following are equivalent:

1. The scheme $X$ is locally Noetherian.
2. For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is Noetherian.
3. There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Noetherian.
4. There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is locally Noetherian.

Moreover, if $X$ is locally Noetherian then every open subscheme is locally Noetherian.

Proof. To show this it suffices to show that being Noetherian is a local property of rings, see Lemma 27.4.3. Any localization of a Noetherian ring is Noetherian, see Algebra, Lemma 10.30.1. By Algebra, Lemma 10.23.2 we see the second property to Definition 27.4.1. $\square$

The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 400–414 (see updates for more information).

\begin{lemma}
\label{lemma-locally-Noetherian}
Let $X$ be a scheme. The following are equivalent:
\begin{enumerate}
\item The scheme $X$ is locally Noetherian.
\item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$
is Noetherian.
\item There exists an affine open covering $X = \bigcup U_i$ such that
each $\mathcal{O}_X(U_i)$ is Noetherian.
\item There exists an open covering $X = \bigcup X_j$
such that each open subscheme $X_j$ is locally Noetherian.
\end{enumerate}
Moreover, if $X$ is locally Noetherian then every open subscheme
is locally Noetherian.
\end{lemma}

\begin{proof}
To show this it suffices to show that being Noetherian is a local
property of rings, see Lemma \ref{lemma-locally-P}.
Any localization of a Noetherian ring is Noetherian, see
Algebra, Lemma \ref{algebra-lemma-Noetherian-permanence}.
By Algebra, Lemma \ref{algebra-lemma-cover} we see the second
property to Definition \ref{definition-property-local}.
\end{proof}

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