Definition 66.24.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say $X$ is *Noetherian* if $X$ is quasi-compact, quasi-separated and locally Noetherian.

## 66.24 Noetherian spaces

We have already defined locally Noetherian algebraic spaces in Section 66.7.

Note that a Noetherian algebraic space $X$ is not just quasi-compact and locally Noetherian, but also quasi-separated. This does not conflict with the definition of a Noetherian scheme, as a locally Noetherian scheme is quasi-separated, see Properties, Lemma 28.5.4. This does not hold for algebraic spaces. Namely, $X = \mathbf{A}^1_ k/\mathbf{Z}$, see Spaces, Example 65.14.8 is locally Noetherian and quasi-compact but not quasi-separated (hence not Noetherian according to our definitions).

A consequence of the choice made above is that an algebraic space of finite type over a Noetherian algebraic space is not automatically Noetherian, i.e., the analogue of Morphisms, Lemma 29.15.6 does not hold. The correct statement is that an algebraic space of finite presentation over a Noetherian algebraic space is Noetherian (see Morphisms of Spaces, Lemma 67.28.6).

A Noetherian algebraic space $X$ is very close to being a scheme. In the rest of this section we collect some lemmas to illustrate this.

Lemma 66.24.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

If $X$ is locally Noetherian then $|X|$ is a locally Noetherian topological space.

If $X$ is quasi-compact and locally Noetherian, then $|X|$ is a Noetherian topological space.

**Proof.**
Assume $X$ is locally Noetherian. Choose a scheme $U$ and a surjective étale morphism $U \to X$. As $X$ is locally Noetherian we see that $U$ is locally Noetherian. By Properties, Lemma 28.5.5 this means that $|U|$ is a locally Noetherian topological space. Since $|U| \to |X|$ is open and surjective we conclude that $|X|$ is locally Noetherian by Topology, Lemma 5.9.3. This proves (1). If $X$ is quasi-compact and locally Noetherian, then $|X|$ is quasi-compact and locally Noetherian. Hence $|X|$ is Noetherian by Topology, Lemma 5.12.14.
$\square$

Lemma 66.24.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is Noetherian, then $|X|$ is a sober Noetherian topological space.

**Proof.**
A quasi-separated algebraic space has an underlying sober topological space, see Lemma 66.15.1. It is Noetherian by Lemma 66.24.2.
$\square$

Lemma 66.24.4. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. Then $\mathcal{O}_{X, \overline{x}}$ is a Noetherian local ring.

**Proof.**
Choose an étale neighbourhood $(U, \overline{u})$ of $\overline{x}$ where $U$ is a scheme. Then $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of the local ring of $U$ at $u$, see Lemma 66.22.1. By our definition of Noetherian spaces the scheme $U$ is locally Noetherian. Hence we conclude by More on Algebra, Lemma 15.45.3.
$\square$

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