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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
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 toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)