Lemma 5.23.2. Let $X$ be a spectral space. The constructible topology is Hausdorff and quasi-compact.

Proof. Since the collection of all quasi-compact opens forms a basis for the topology on $X$ and $X$ is sober, it is clear that $X$ is Hausdorff in the constructible topology.

Let $\mathcal{B}$ be the collection of subsets $B \subset X$ with $B$ either quasi-compact open or closed with quasi-compact complement. If $B \in \mathcal{B}$ then $B^ c \in \mathcal{B}$. It suffices to show every covering $X = \bigcup _{i \in I} B_ i$ with $B_ i \in \mathcal{B}$ has a finite refinement, see Lemma 5.12.15. Taking complements we see that we have to show that any family $\{ B_ i\} _{i \in I}$ of elements of $\mathcal{B}$ such that $B_{i_1} \cap \ldots \cap B_{i_ n} \not= \emptyset$ for all $n$ and all $i_1, \ldots , i_ n \in I$ has a common point of intersection. We may and do assume $B_ i \not= B_{i'}$ for $i \not= i'$.

To get a contradiction assume $\{ B_ i\} _{i \in I}$ is a family of elements of $\mathcal{B}$ having the finite intersection property but empty intersection. An application of Zorn's lemma shows that we may assume our family is maximal (details omitted). Let $I' \subset I$ be those indices such that $B_ i$ is closed and set $Z = \bigcap _{i \in I'} B_ i$. This is a closed subset of $X$. If $Z$ is reducible, then we can write $Z = Z' \cup Z''$ as a union of two closed subsets, neither equal to $Z$. This means in particular that we can find a quasi-compact open $U' \subset X$ meeting $Z'$ but not $Z''$. Similarly, we can find a quasi-compact open $U'' \subset X$ meeting $Z''$ but not $Z'$. Set $B' = X \setminus U'$ and $B'' = X \setminus U''$. Note that $Z'' \subset B'$ and $Z' \subset B''$. If there exist a finite number of indices $i_1, \ldots , i_ n \in I$ such that $B' \cap B_{i_1} \cap \ldots \cap B_{i_ n} = \emptyset$ as well as a finite number of indices $j_1, \ldots , j_ m \in I$ such that $B'' \cap B_{j_1} \cap \ldots \cap B_{j_ m} = \emptyset$ then we find that $Z \cap B_{i_1} \cap \ldots \cap B_{i_ n} \cap B_{j_1} \cap \ldots \cap B_{j_ m} = \emptyset$. However, the set $B_{i_1} \cap \ldots \cap B_{i_ n} \cap B_{j_1} \cap \ldots \cap B_{j_ m}$ is quasi-compact hence we would find a finite number of indices $i'_1, \ldots , i'_ l \in I'$ with $B_{i_1} \cap \ldots \cap B_{i_ n} \cap B_{j_1} \cap \ldots \cap B_{j_ m} \cap B_{i'_1} \cap \ldots \cap B_{i'_ l} = \emptyset$, a contradiction. Thus we see that we may add either $B'$ or $B''$ to the given family contradicting maximality. We conclude that $Z$ is irreducible. However, this leads to a contradiction as well, as now every nonempty (by the same argument as above) open $Z \cap B_ i$ for $i \in I \setminus I'$ contains the unique generic point of $Z$. This contradiction proves the lemma. $\square$

Comment #4224 by Laurent Moret-Bailly on

I suggest to add the property that the constructible topology is totally disconnected. (Proof: for $x$ and $y$ in $X$ and (say) $y$ not a specialization of $x$, there is a quasicompact open $U$ containing $y$ but not $x$. Now observe that $U$ is clopen in the constructible topology.) With the other properties this shows that the constructible topology is spectral.

Comment #4226 by Laurent Moret-Bailly on

Suggestion for a cleaner proof of quasi-compactness: let $\mathscr{P}$ be the set of finite partitions of $X$ into quasicompact, locally closed subspaces. For $P\in\mathscr{P}$, put $X_P:=\coprod_{Z\in P}Z$. Then observe that the constructible topology is $\varprojlim_{P\in\mathscr{P}}X_P$. This construction carries over nicely to schemes, showing that for a scheme $X$, $X^{\mathrm{cons}}$ has a natural scheme structure, giving a right adjoint to the inclusion (absolutely flat schemes)$\to$(schemes). Explicitly, let $X=\mathrm{Spec}(R)$. For $f\in R$, put $R\langle f\rangle=R_f\times R/f$. Then $X^{\mathrm{cons}}=\mathrm{Spec}(R^{\mathrm{cons}})$ where $R^{\mathrm{cons}}=\bigotimes_{f\in R}R\langle f\rangle$. The construction localizes well, giving rise to an affine morphism $X^{\mathrm{cons}}\to X$ for general $X$. It loooks simpler than the constructions given by Olivier and Hochster.

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