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Tag 01OW

Chapter 27: Properties of Schemes > Section 27.5: Noetherian schemes

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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