110.27 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 110.27.1. Nonexistence quasi-compact opens of affines:
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.
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 110.27.2. 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
Comments (3)
Comment #3232 by Laurent Moret-Bailly on
Comment #3331 by Johan on
Comment #9567 by Branislav Sobot on