The Stacks project

110.26 Nonexistence of suitable opens

This section complements the results of Properties, Section 28.29.

Let $k$ be a field and let $A = k[z_1, z_2, z_3, \ldots ]/I$ where $I$ is the ideal generated by all pairwise products $z_ iz_ j$, $i \not= j$, $i, j \in \mathbf{N}$. Set $S = \mathop{\mathrm{Spec}}(A)$. Let $s \in S$ be the closed point corresponding to the maximal ideal $(z_ i)$. We claim there is no quasi-compact open $V \subset S \setminus \{ s\} $ which is dense in $S \setminus \{ s\} $. Note that $S \setminus \{ s\} = \bigcup D(z_ i)$. Each $D(z_ i)$ is open and irreducible with generic point $\eta _ i$. We conclude that $\eta _ i \in V$ for all $i$. However, a principal affine open of $S \setminus \{ s\} $ is of the form $D(f)$ where $f \in (z_1, z_2, \ldots )$. Then $f \in (z_1, \ldots , z_ n)$ for some $n$ and we see that $D(f)$ contains only finitely many of the points $\eta _ i$. Thus $V$ cannot be quasi-compact.

Let $k$ be a field and let $B = k[x, z_1, z_2, z_3, \ldots ]/J$ where $J$ is the ideal generated by the products $xz_ i$, $i \in \mathbf{N}$ and by all pairwise products $z_ iz_ j$, $i \not= j$, $i, j \in \mathbf{N}$. Set $T = \mathop{\mathrm{Spec}}(B)$. Consider the principal open $U = D(x)$. We claim there is no quasi-compact open $V \subset S$ such that $V \cap U = \emptyset $ and $V \cup U$ is dense in $S$. Let $t \in T$ be the closed point corresponding to the maximal ideal $(x, z_ i)$. The closure of $U$ in $T$ is $\overline{U} = U \cup \{ t\} $. Hence $V \subset \bigcup _ i D(z_ i)$ is a quasi-compact open. By the arguments of the previous paragraph we see that $V$ cannot be dense in $\bigcup D(z_ i)$.

Lemma 110.26.1. Nonexistence quasi-compact opens of affines:

1. There exist an affine scheme $S$ and affine open $U \subset S$ such that there is no quasi-compact open $V \subset S$ with $U \cap V = \emptyset $ and $U \cup V$ dense in $S$.

2. There exists an affine scheme $S$ and a closed point $s \in S$ such that $S \setminus \{ s\} $ does not contain a quasi-compact dense open.

Proof. See discussion above. $\square$

Let $X$ be the glueing of two copies of the affine scheme $T$ (see above) along the affine open $U$. Thus there is a morphism $\pi : X \to T$ and $X = U_1 \cup U_2$ such that $\pi $ maps $U_ i$ isomorphically to $T$ and $U_1 \cap U_2$ isomorphically to $U$. Note that $X$ is quasi-separated (by Schemes, Lemma 26.21.6) and quasi-compact. We claim there does not exist a separated, dense, quasi-compact open $W \subset X$. Namely, consider the two closed points $x_1 \in U_1$, $x_2 \in U_2$ mapping to the closed point $t \in T$ introduced above. Let $\tilde\eta \in U_1 \cap U_2$ be the generic point mapping to the (unique) generic point $\eta $ of $U$. Note that $\tilde\eta \leadsto x_1$ and $\tilde\eta \leadsto x_2$ lying over the specialization $\eta \leadsto s$. Since $\pi |_ W : W \to T$ is separated we conclude that we cannot have both $x_1$ and $x_2 \in W$ (by the valuative criterion of separatedness Schemes, Lemma 26.22.2). Say $x_1 \not\in W$. Then $W \cap U_1$ is a quasi-compact (as $X$ is quasi-separated) dense open of $U_1$ which does not contain $x_1$. Now observe that there exists an isomorphism $(T, t) \cong (S, s)$ of schemes (by sending $x$ to $z_1$ and $z_ i$ to $z_{i + 1}$). Hence by the first paragraph of this section we arrive at a contradiction.

Lemma 110.26.2. There exists a quasi-compact and quasi-separated scheme $X$ which does not contain a separated quasi-compact dense open.

Proof. See discussion above. $\square$