The Stacks project

66.23 Jacobson spaces

We have defined the Jacobson property for algebraic spaces in Properties of Spaces, Remark 64.7.3. For representable algebraic spaces it agrees with the property discussed in Properties, Section 28.6. The relationship between the Jacobson property and the behaviour of the topological space $|X|$ is not evident for general algebraic spaces $|X|$. However, a decent (for example quasi-separated or locally separated) algebraic space $X$ is Jacobson if and only if $|X|$ is Jacobson (see Lemma 66.23.4).

Lemma 66.23.1. Let $S$ be a scheme. Let $X$ be a Jacobson algebraic space over $S$. Any algebraic space locally of finite type over $X$ is Jacobson.

Proof. Let $U \to X$ be a surjective étale morphism where $U$ is a scheme. Then $U$ is Jacobson (by definition) and for a morphism of schemes $V \to U$ which is locally of finite type we see that $V$ is Jacobson by the corresponding result for schemes (Morphisms, Lemma 29.16.9). Thus if $Y \to X$ is a morphism of algebraic spaces which is locally of finite type, then setting $V = U \times _ X Y$ we see that $Y$ is Jacobson by definition. $\square$

Lemma 66.23.2. Let $S$ be a scheme. Let $X$ be a Jacobson algebraic space over $S$. For $x \in X_{\text{ft-pts}}$ and $g : W \to X$ locally of finite type with $W$ a scheme, if $x \in \mathop{\mathrm{Im}}(|g|)$, then there exists a closed point of $W$ mapping to $x$.

Proof. Let $U \to X$ be an étale morphism with $U$ a scheme and with $u \in U$ closed mapping to $x$, see Morphisms of Spaces, Lemma 65.25.3. Observe that $W$, $W \times _ X U$, and $U$ are Jacobson schemes by Lemma 66.23.1. Hence finite type points on these schemes are the same thing as closed points by Morphisms, Lemma 29.16.8. The inverse image $T \subset W \times _ X U$ of $u$ is a nonempty (as $x$ in the image of $W \to X$) closed subset. By Morphisms, Lemma 29.16.7 there is a closed point $t$ of $W \times _ X U$ which maps to $u$. As $W \times _ X U \to W$ is locally of finite type the image of $t$ in $W$ is closed by Morphisms, Lemma 29.16.8. $\square$

Lemma 66.23.3. Let $S$ be a scheme. Let $X$ be a decent Jacobson algebraic space over $S$. Then $X_{\text{ft-pts}} \subset |X|$ is the set of closed points.

Proof. If $x \in |X|$ is closed, then we can represent $x$ by a closed immersion $\mathop{\mathrm{Spec}}(k) \to X$, see Lemma 66.14.6. Hence $x$ is certainly a finite type point.

Conversely, let $x \in |X|$ be a finite type point. We know that $x$ can be represented by a quasi-compact monomorphism $\mathop{\mathrm{Spec}}(k) \to X$ where $k$ is a field (Definition 66.6.1). On the other hand, by definition, there exists a morphism $\mathop{\mathrm{Spec}}(k') \to X$ which is locally of finite type and represents $x$ (Morphisms, Definition 29.16.3). We obtain a factorization $\mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k) \to X$. Let $U \to X$ be any étale morphism with $U$ affine and consider the morphisms

\[ \mathop{\mathrm{Spec}}(k') \times _ X U \to \mathop{\mathrm{Spec}}(k) \times _ X U \to U \]

The quasi-compact scheme $\mathop{\mathrm{Spec}}(k) \times _ X U$ is étale over $\mathop{\mathrm{Spec}}(k)$ hence is a finite disjoint union of spectra of fields (Remark 66.4.1). Moreover, the first morphism is surjective and locally of finite type (Morphisms, Lemma 29.15.8) hence surjective on finite type points (Morphisms, Lemma 29.16.6) and the composition (which is locally of finite type) sends finite type points to closed points as $U$ is Jacobson (Morphisms, Lemma 29.16.8). Thus the image of $\mathop{\mathrm{Spec}}(k) \times _ X U \to U$ is a finite set of closed points hence closed. Since this is true for every affine $U$ and étale morphism $U \to X$, we conclude that $x \in |X|$ is closed. $\square$

Lemma 66.23.4. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Then $X$ is Jacobson if and only if $|X|$ is Jacobson.

Proof. Assume $X$ is Jacobson and that $T \subset |X|$ is a closed subset. By Morphisms of Spaces, Lemma 65.25.6 we see that $T \cap X_{\text{ft-pts}}$ is dense in $T$. By Lemma 66.23.3 we see that $X_{\text{ft-pts}}$ are the closed points of $|X|$. Thus $|X|$ is indeed Jacobson.

Assume $|X|$ is Jacobson. Let $f : U \to X$ be an étale morphism with $U$ an affine scheme. We have to show that $U$ is Jacobson. If $x \in |X|$ is closed, then the fibre $F = f^{-1}(\{ x\} )$ is a finite (by definition of decent) closed (by construction of the topology on $|X|$) subset of $U$. Since there are no specializations between points of $F$ (Lemma 66.12.1) we conclude that every point of $F$ is closed in $U$. If $U$ is not Jacobson, then there exists a non-closed point $u \in U$ such that $\{ u\} $ is locally closed (Topology, Lemma 5.18.3). We will show that $f(u) \in |X|$ is closed; by the above $u$ is closed in $U$ which is a contradiction and finishes the proof. To prove this we may replace $U$ by an affine open neighbourhood of $u$. Thus we may assume that $\{ u\} $ is closed in $U$. Let $R = U \times _ X U$ with projections $s, t : R \to U$. Then $s^{-1}(\{ u\} ) = \{ r_1, \ldots , r_ m\} $ is finite (by definition of decent spaces). After replacing $U$ by a smaller affine open neighbourhood of $u$ we may assume that $t(r_ j) = u$ for $j = 1, \ldots , m$. It follows that $\{ u\} $ is an $R$-invariant closed subset of $U$. Hence $\{ f(u)\} $ is a locally closed subset of $X$ as it is closed in the open $|f|(|U|)$ of $|X|$. Since $|X|$ is Jacobson we conclude that $f(u)$ is closed in $|X|$ as desired. $\square$

Lemma 66.23.5. Let $S$ be a scheme. Let $X$ be a decent locally Noetherian algebraic space over $S$. Let $x \in |X|$. Then

\[ W = \{ x' \in |X| : x' \leadsto x,\ x' \not= x\} \]

is a Noetherian, spectral, sober, Jacobson topological space.

Proof. We may replace by any open subspace containing $x$. Thus we may assume that $X$ is quasi-compact. Then $|X|$ is a Noetherian topological space (Properties of Spaces, Lemma 64.24.2). Thus $W$ is a Noetherian topological space (Topology, Lemma 5.9.2).

Combining Lemma 66.14.1 with Properties of Spaces, Lemma 64.15.2 we see that $|X|$ is a spectral toplogical space. By Topology, Lemma 5.24.7 we see that $W \cup \{ x\} $ is a spectral topological space. Now $W$ is a quasi-compact open of $W \cup \{ x\} $ and hence $W$ is spectral by Topology, Lemma 5.23.5.

Let $E \subset W$ be an irreducible closed subset. Then if $Z \subset |X|$ is the closure of $E$ we see that $x \in Z$. There is a unique generic point $\eta \in Z$ by Proposition 66.12.4. Of course $\eta \in W$ and hence $\eta \in E$. We conclude that $E$ has a unique generic point, i.e., $W$ is sober.

Let $x' \in W$ be a point such that $\{ x'\} $ is locally closed in $W$. To finish the proof we have to show that $x'$ is a closed point of $W$. If not, then there exists a nontrivial specialization $x' \leadsto x'_1$ in $W$. Let $U$ be an affine scheme, $u \in U$ a point, and let $U \to X$ be an étale morphism mapping $u$ to $x$. By Lemma 66.12.2 we can choose specializations $u' \leadsto u'_1 \leadsto u$ mapping to $x' \leadsto x'_1 \leadsto x$. Let $\mathfrak p' \subset \mathcal{O}_{U, u}$ be the prime ideal corresponding to $u'$. The existence of the specializations implies that $\dim (\mathcal{O}_{U, u}/\mathfrak p') \geq 2$. Hence every nonempty open of $\mathop{\mathrm{Spec}}(\mathcal{O}_{U, u}/\mathfrak p')$ is infinite by Algebra, Lemma 10.60.1. By Lemma 66.12.1 we obtain a continuous map

\[ \mathop{\mathrm{Spec}}(\mathcal{O}_{U, u}/\mathfrak p') \setminus \{ \mathfrak m_ u/\mathfrak p'\} \longrightarrow W \]

Since the generic point of the LHS maps to $x'$ the image is contained in $\overline{\{ x'\} }$. We conclude the inverse image of $\{ x'\} $ under the displayed arrow is nonempty open hence infinite. However, the fibres of $U \to X$ are finite as $X$ is decent and we conclude that $\{ x'\} $ is infinite. This contradiction finishes the proof. $\square$


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