Lemma 28.2.1. Let $X$ be a scheme. A subset $E$ of $X$ is locally constructible in $X$ if and only if $E \cap U$ is constructible in $U$ for every affine open $U$ of $X$.

## 28.2 Constructible sets

Constructible and locally constructible sets are introduced in Topology, Section 5.15. We may characterize locally constructible subsets of schemes as follows.

**Proof.**
Assume $E$ is locally constructible. Then there exists an open covering $X = \bigcup U_ i$ such that $E \cap U_ i$ is constructible in $U_ i$ for each $i$. Let $V \subset X$ be any affine open. We can find a finite open affine covering $V = V_1 \cup \ldots \cup V_ m$ such that for each $j$ we have $V_ j \subset U_ i$ for some $i = i(j)$. By Topology, Lemma 5.15.4 we see that each $E \cap V_ j$ is constructible in $V_ j$. Since the inclusions $V_ j \to V$ are quasi-compact (see Schemes, Lemma 26.19.2) we conclude that $E \cap V$ is constructible in $V$ by Topology, Lemma 5.15.6. The converse implication is immediate.
$\square$

Lemma 28.2.2. Let $X$ be a scheme and let $E \subset X$ be a locally constructible subset. Let $\xi \in X$ be a generic point of an irreducible component of $X$.

If $\xi \in E$, then an open neighbourhood of $\xi $ is contained in $E$.

If $\xi \not\in E$, then an open neighbourhood of $\xi $ is disjoint from $E$.

**Proof.**
As the complement of a locally constructible subset is locally constructible it suffices to show (2). We may assume $X$ is affine and hence $E$ constructible (Lemma 28.2.1). In this case $X$ is a spectral space (Algebra, Lemma 10.26.2). Then $\xi \not\in E$ implies $\xi \not\in \overline{E}$ by Topology, Lemma 5.23.6 and the fact that there are no points of $X$ different from $\xi $ which specialize to $\xi $.
$\square$

Lemma 28.2.3. Let $X$ be a quasi-separated scheme. The intersection of any two quasi-compact opens of $X$ is a quasi-compact open of $X$. Every quasi-compact open of $X$ is retrocompact in $X$.

**Proof.**
If $U$ and $V$ are quasi-compact open then $U \cap V = \Delta ^{-1}(U \times V)$, where $\Delta : X \to X \times X$ is the diagonal. As $X$ is quasi-separated we see that $\Delta $ is quasi-compact. Hence we see that $U \cap V$ is quasi-compact as $U \times V$ is quasi-compact (details omitted; use Schemes, Lemma 26.17.4 to see $U \times V$ is a finite union of affines). The other assertions follow from the first and Topology, Lemma 5.27.1.
$\square$

Lemma 28.2.4. Let $X$ be a quasi-compact and quasi-separated scheme. Then the underlying topological space of $X$ is a spectral space.

**Proof.**
By Topology, Definition 5.23.1 we have to check that $X$ is sober, quasi-compact, has a basis of quasi-compact opens, and the intersection of any two quasi-compact opens is quasi-compact. This follows from Schemes, Lemma 26.11.1 and 26.11.2 and Lemma 28.2.3 above.
$\square$

Lemma 28.2.5. Let $X$ be a quasi-compact and quasi-separated scheme. Any locally constructible subset of $X$ is constructible.

**Proof.**
As $X$ is quasi-compact we can choose a finite affine open covering $X = V_1 \cup \ldots \cup V_ m$. As $X$ is quasi-separated each $V_ i$ is retrocompact in $X$ by Lemma 28.2.3. Hence by Topology, Lemma 5.15.6 we see that $E \subset X$ is constructible in $X$ if and only if $E \cap V_ j$ is constructible in $V_ j$. Thus we win by Lemma 28.2.1.
$\square$

Lemma 28.2.6. Let $X$ be a scheme. A subset $E$ of $X$ is retrocompact in $X$ if and only if $E \cap U$ is quasi-compact for every affine open $U$ of $X$.

**Proof.**
Immediate from the fact that every quasi-compact open of $X$ is a finite union of affine opens.
$\square$

Lemma 28.2.7. A partition $X = \coprod _{i \in I} X_ i$ of a scheme $X$ with retrocompact parts is locally finite if and only if the parts are locally constructible.

**Proof.**
See Topology, Definitions 5.12.1, 5.28.1, and 5.28.4 for the definitions of retrocompact, partition, and locally finite.

If the partition is locally finite and $U \subset X$ is an affine open, then we see that $U = \coprod _{i \in I} U \cap X_ i$ is a finite partition (more precisely, all but a finite number of its parts are empty). Hence $U \cap X_ i$ is quasi-compact and its complement is retrocompact in $U$ as a finite union of retrocompact parts. Thus $U \cap X_ i$ is constructible by Topology, Lemma 5.15.13. It follows that $X_ i$ is locally constructible by Lemma 28.2.1.

Assume the parts are locally constructible. Then for any affine open $U \subset X$ we obtain a covering $U = \coprod X_ i \cap U$ by constructible subsets. Since the constructible topology is quasi-compact, see Topology, Lemma 5.23.2, this covering has a finite refinement, i.e., the partition is locally finite. $\square$

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