## 5.19 Specialization

Definition 5.19.1. Let $X$ be a topological space.

1. If $x, x' \in X$ then we say $x$ is a specialization of $x'$, or $x'$ is a generalization of $x$ if $x \in \overline{\{ x'\} }$. Notation: $x' \leadsto x$.

2. A subset $T \subset X$ is stable under specialization if for all $x' \in T$ and every specialization $x' \leadsto x$ we have $x \in T$.

3. A subset $T \subset X$ is stable under generalization if for all $x \in T$ and every generalization $x' \leadsto x$ we have $x' \in T$.

Lemma 5.19.2. Let $X$ be a topological space.

1. Any closed subset of $X$ is stable under specialization.

2. Any open subset of $X$ is stable under generalization.

3. A subset $T \subset X$ is stable under specialization if and only if the complement $T^ c$ is stable under generalization.

Proof. Omitted. $\square$

Lemma 5.19.3. Let $T \subset X$ be a subset of a topological space $X$. The following are equivalent

1. $T$ is stable under specialization, and

2. $T$ is a (directed) union of closed subsets of $X$.

Proof. Omitted. $\square$

Definition 5.19.4. Let $f : X \to Y$ be a continuous map of topological spaces.

1. We say that specializations lift along $f$ or that $f$ is specializing if given $y' \leadsto y$ in $Y$ and any $x'\in X$ with $f(x') = y'$ there exists a specialization $x' \leadsto x$ of $x'$ in $X$ such that $f(x) = y$.

2. We say that generalizations lift along $f$ or that $f$ is generalizing if given $y' \leadsto y$ in $Y$ and any $x\in X$ with $f(x) = y$ there exists a generalization $x' \leadsto x$ of $x$ in $X$ such that $f(x') = y'$.

Lemma 5.19.5. Suppose $f : X \to Y$ and $g : Y \to Z$ are continuous maps of topological spaces. If specializations lift along both $f$ and $g$ then specializations lift along $g \circ f$. Similarly for “generalizations lift along”.

Proof. Omitted. $\square$

Lemma 5.19.6. Let $f : X \to Y$ be a continuous map of topological spaces.

1. If specializations lift along $f$, and if $T \subset X$ is stable under specialization, then $f(T) \subset Y$ is stable under specialization.

2. If generalizations lift along $f$, and if $T \subset X$ is stable under generalization, then $f(T) \subset Y$ is stable under generalization.

Proof. Omitted. $\square$

Lemma 5.19.7. Let $f : X \to Y$ be a continuous map of topological spaces.

1. If $f$ is closed then specializations lift along $f$.

2. If $f$ is open, $X$ is a Noetherian topological space, each irreducible closed subset of $X$ has a generic point, and $Y$ is Kolmogorov then generalizations lift along $f$.

Proof. Assume $f$ is closed. Let $y' \leadsto y$ in $Y$ and any $x'\in X$ with $f(x') = y'$ be given. Consider the closed subset $T = \overline{\{ x'\} }$ of $X$. Then $f(T) \subset Y$ is a closed subset, and $y' \in f(T)$. Hence also $y \in f(T)$. Hence $y = f(x)$ with $x \in T$, i.e., $x' \leadsto x$.

Assume $f$ is open, $X$ Noetherian, every irreducible closed subset of $X$ has a generic point, and $Y$ is Kolmogorov. Let $y' \leadsto y$ in $Y$ and any $x \in X$ with $f(x) = y$ be given. Consider $T = f^{-1}(\{ y'\} ) \subset X$. Take an open neighbourhood $x \in U \subset X$ of $x$. Then $f(U) \subset Y$ is open and $y \in f(U)$. Hence also $y' \in f(U)$. In other words, $T \cap U \not= \emptyset$. This proves that $x \in \overline{T}$. Since $X$ is Noetherian, $T$ is Noetherian (Lemma 5.9.2). Hence it has a decomposition $T = T_1 \cup \ldots \cup T_ n$ into irreducible components. Then correspondingly $\overline{T} = \overline{T_1} \cup \ldots \cup \overline{T_ n}$. By the above $x \in \overline{T_ i}$ for some $i$. By assumption there exists a generic point $x' \in \overline{T_ i}$, and we see that $x' \leadsto x$. As $x' \in \overline{T}$ we see that $f(x') \in \overline{\{ y'\} }$. Note that $f(\overline{T_ i}) = f(\overline{\{ x'\} }) \subset \overline{\{ f(x')\} }$. If $f(x') \not= y'$, then since $Y$ is Kolmogorov $f(x')$ is not a generic point of the irreducible closed subset $\overline{\{ y'\} }$ and the inclusion $\overline{\{ f(x')\} } \subset \overline{\{ y'\} }$ is strict, i.e., $y' \not\in f(\overline{T_ i})$. This contradicts the fact that $f(T_ i) = \{ y'\}$. Hence $f(x') = y'$ and we win. $\square$

Lemma 5.19.8. Suppose that $s, t : R \to U$ and $\pi : U \to X$ are continuous maps of topological spaces such that

1. $\pi$ is open,

2. $U$ is sober,

3. $s, t$ have finite fibres,

4. generalizations lift along $s, t$,

5. $(t, s)(R) \subset U \times U$ is an equivalence relation on $U$ and $X$ is the quotient of $U$ by this equivalence relation (as a set).

Then $X$ is Kolmogorov.

Proof. Properties (3) and (5) imply that a point $x$ corresponds to an finite equivalence class $\{ u_1, \ldots , u_ n\} \subset U$ of the equivalence relation. Suppose that $x' \in X$ is a second point corresponding to the equivalence class $\{ u'_1, \ldots , u'_ m\} \subset U$. Suppose that $u_ i \leadsto u'_ j$ for some $i, j$. Then for any $r' \in R$ with $s(r') = u'_ j$ by (4) we can find $r \leadsto r'$ with $s(r) = u_ i$. Hence $t(r) \leadsto t(r')$. Since $\{ u'_1, \ldots , u'_ m\} = t(s^{-1}(\{ u'_ j\} ))$ we conclude that every element of $\{ u'_1, \ldots , u'_ m\}$ is the specialization of an element of $\{ u_1, \ldots , u_ n\}$. Thus $\overline{\{ u_1\} } \cup \ldots \cup \overline{\{ u_ n\} }$ is a union of equivalence classes, hence of the form $\pi ^{-1}(Z)$ for some subset $Z \subset X$. By (1) we see that $Z$ is closed in $X$ and in fact $Z = \overline{\{ x\} }$ because $\pi (\overline{\{ u_ i\} }) \subset \overline{\{ x\} }$ for each $i$. In other words, $x \leadsto x'$ if and only if some lift of $x$ in $U$ specializes to some lift of $x'$ in $U$, if and only if every lift of $x'$ in $U$ is a specialization of some lift of $x$ in $U$.

Suppose that both $x \leadsto x'$ and $x' \leadsto x$. Say $x$ corresponds to $\{ u_1, \ldots , u_ n\}$ and $x'$ corresponds to $\{ u'_1, \ldots , u'_ m\}$ as above. Then, by the results of the preceding paragraph, we can find a sequence

$\ldots \leadsto u'_{j_3} \leadsto u_{i_3} \leadsto u'_{j_2} \leadsto u_{i_2} \leadsto u'_{j_1} \leadsto u_{i_1}$

which must repeat, hence by (2) we conclude that $\{ u_1, \ldots , u_ n\} = \{ u'_1, \ldots , u'_ m\}$, i.e., $x = x'$. Thus $X$ is Kolmogorov. $\square$

Lemma 5.19.9. Let $f : X \to Y$ be a morphism of topological spaces. Suppose that $Y$ is a sober topological space, and $f$ is surjective. If either specializations or generalizations lift along $f$, then $\dim (X) \geq \dim (Y)$.

Proof. Assume specializations lift along $f$. Let $Z_0 \subset Z_1 \subset \ldots Z_ e \subset Y$ be a chain of irreducible closed subsets of $X$. Let $\xi _ e \in X$ be a point mapping to the generic point of $Z_ e$. By assumption there exists a specialization $\xi _ e \leadsto \xi _{e - 1}$ in $X$ such that $\xi _{e - 1}$ maps to the generic point of $Z_{e - 1}$. Continuing in this manner we find a sequence of specializations

$\xi _ e \leadsto \xi _{e - 1} \leadsto \ldots \leadsto \xi _0$

with $\xi _ i$ mapping to the generic point of $Z_ i$. This clearly implies the sequence of irreducible closed subsets

$\overline{\{ \xi _0\} } \subset \overline{\{ \xi _1\} } \subset \ldots \overline{\{ \xi _ e\} }$

is a chain of length $e$ in $X$. The case when generalizations lift along $f$ is similar. $\square$

Lemma 5.19.10. Let $X$ be a Noetherian sober topological space. Let $E \subset X$ be a subset of $X$.

1. If $E$ is constructible and stable under specialization, then $E$ is closed.

2. If $E$ is constructible and stable under generalization, then $E$ is open.

Proof. Let $E$ be constructible and stable under generalization. Let $Y \subset X$ be an irreducible closed subset with generic point $\xi \in Y$. If $E \cap Y$ is nonempty, then it contains $\xi$ (by stability under generalization) and hence is dense in $Y$, hence it contains a nonempty open of $Y$, see Lemma 5.16.3. Thus $E$ is open by Lemma 5.16.5. This proves (2). To prove (1) apply (2) to the complement of $E$ in $X$. $\square$

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