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Rational points of smooth finite-type schemes over a separably closed field are schematically dense

Statement

Assume the Axiom of Choice. Let $k$ be a field with no nontrivial finite separable extension (for example a separably closed or algebraically closed field), let $X$ be a reduced finite-type $k$ -scheme, and let $S \subseteq X (k)$ be a subset. If $S$ is dense in the underlying topological space of $X$ , then every closed subscheme $Z \subseteq X$ with $Z (k) \supseteq S$ equals $X$ ; in other words, $S$ is schematically dense in $X$ .

In particular, if $X$ is smooth over $k$ , then $X (k)$ is dense in $X$ and hence schematically dense in $X$ . The reducedness hypothesis cannot be dropped: for $X = Spec k [ε] / (ε^{2})$ the closed subscheme $Z = Spec k$ has the same underlying space and satisfies $Z (k) = X (k)$ , but $Z \neq X$ .

The Axiom of Choice is used through the finite-separable-point lemma and the affine description of closed immersions.

Facts & Assumptions

Given: The Axiom of Choice, a field $k$ with no nontrivial finite separable extension, a reduced finite-type $k$ -scheme $X$ , a dense subset $S \subseteq X (k)$ , and a closed subscheme $Z \subseteq X$ with $Z (k) \supseteq S$ .

[F1]

A closed immersion $i : Z \to X$ has underlying map a homeomorphism onto a closed subset, and for every affine open $U = Spec A \subseteq X$ there is a unique ideal $I \subseteq A$ with $i^{- 1} (U) ≅ Spec (A / I)$ over $U$ . (Closed immersions of schemes, Closed immersions are affine quotients and survive base change)

[F2]

The reduction $X_{r e d}$ is the closed subscheme defined by the ideal sheaf of nilpotents; $X$ is reduced exactly when $N_{X} = 0$ , equivalently when every affine chart ring is reduced. (The reduction of a scheme)

[F4]

Smoothness is preserved by restricting the source to an open subscheme. A smooth scheme over a field is reduced: its local rings are regular by the geometric-regularity clause of smoothness, hence domains and therefore reduced. (Smooth morphism of schemes, regular local domain induction)

[F5]

Assume AC. Every nonempty smooth finite-type $k$ -scheme $U$ has a closed point $P$ with $κ (P)$ finite and separable over $k$ . (A nonempty smooth scheme has a finite separable point)

[F6]

For a field $K$ and scheme $X$ , morphisms $Spec K \to X$ correspond bijectively to pairs $(x, ι)$ with $x \in X$ and a field embedding $ι : κ (x) \to K$ ; for $K = k$ this identifies $X (k)$ with the points of residue field $k$ . (Field-valued points and local-ring points)

Proof

1.1F1given

By [F1] the underlying space $∣ Z ∣$ is closed in $X$ and contains $Z (k) \supseteq S$ ; since $S$ is dense in $∣ X ∣$ , every closed subset containing $S$ equals $∣ X ∣$ , so $∣ Z ∣ = ∣ X ∣$ .

1.2F4F5F6

Assume now that $X$ is smooth over $k$ , and let $U \subseteq X$ be a nonempty open subscheme. Then $U$ is smooth over $k$ by [F4], nonempty and of finite type, so by [F5] it has a closed point $P$ with $κ (P)$ finite and separable over $k$ . By hypothesis on $k$ , $κ (P) = k$ , and [F6] identifies $P$ with a $k$ -point of $U$ . Hence every nonempty open subscheme of $X$ meets $X (k)$ , so $X (k)$ is dense in $X$ .

2.1F1F2step 1.1

I claim that $Z = X$ . Let $U = Spec A \subseteq X$ be an affine open; by [F1] write $Z \cap U = Spec (A / I)$ for a unique ideal $I \subseteq A$ , and by [step 1.1] its underlying space is all of $U$ , so $V (I) = Spec A$ and hence every $f \in I$ lies in every prime ideal of $A$ , i.e. $I \subseteq \sqrt{(0)}$ . Since $X$ is reduced, $A$ is reduced by [F2], so $\sqrt{(0)} = 0$ and $I = 0$ , giving $Z \cap U = U$ . As the affine opens cover $X$ and two closed subschemes of $X$ that agree on an open cover agree, $Z = X$ .

3.1F4step 1.1step 2.1step 1.2∎

Combining: the first assertion is [step 1.1] with [step 2.1]; applying it to the reduced smooth scheme $X$ of [F4] with $S = X (k)$ , which is dense by [step 1.2], shows that every closed subscheme $Z \subseteq X$ with $Z (k) \supseteq X (k)$ equals $X$ , that is, $X (k)$ is schematically dense in $X$ .

Depends on

Dependency tree · 2 levels

Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press)