Lemma 45.8.1. Let $k$ be a field. Let $X$ be a smooth projective scheme over $k$. Then $\mathop{\mathrm{CH}}\nolimits _0(X)$ is generated by classes of closed points whose residue fields are separable over $k$.
Proof. The lemma is immediate if $k$ has characteristic $0$ or is perfect. Thus we may assume $k$ is an infinite field of characteristic $p > 0$.
We may assume $X$ is irreducible of dimension $d$. Then $k' = H^0(X, \mathcal{O}_ X)$ is a finite separable field extension of $k$ and that $X$ is geometrically integral over $k'$. See Varieties, Lemmas 33.25.4, 33.9.3, and 33.9.4. We may and do replace $k$ by $k'$ and assume that $X$ is geometrically integral.
Let $x \in X$ be a closed point. To prove the lemma we are going to show that $[x] \in \mathop{\mathrm{CH}}\nolimits _0(X)$ is rationally equivalent to an integer linear combination of classes of closed points whose residue fields are separable over $k$. Choose an ample invertible $\mathcal{O}_ X$-module $\mathcal{L}$. Set
\[ V = \{ s \in H^0(X, \mathcal{L}) \mid s(x) = 0 \} \]
After replacing $\mathcal{L}$ by a power we may assume (a) $\mathcal{L}$ is very ample, (b) $V$ generates $\mathcal{L}$ over $X \setminus x$, (c) the morphism $X \setminus x \to \mathbf{P}(V)$ is an immersion, (d) the map $V \to \mathfrak m_ x\mathcal{L}_ x/\mathfrak m_ x^2\mathcal{L}_ x$ is surjective, see Morphisms, Lemma 29.39.5, Varieties, Lemma 33.47.1, and Properties, Proposition 28.26.13. Consider the set
\[ V^ d \supset U = \{ (s_1, \ldots , s_ d) \in V^ d \mid s_1, \ldots , s_ d \text{ generate } \mathfrak m_ x\mathcal{L}_ x/\mathfrak m_ x^2\mathcal{L}_ x \text{ over }\kappa (x) \} \]
Since $\mathcal{O}_{X, x}$ is a regular local ring of dimension $d$ we have $\dim _{\kappa (x)}(\mathfrak m_ x/\mathfrak m_ x^2) = d$ and hence we see that $U$ is a nonempty (Zariski) open of $V^ d$. For $(s_1, \ldots , s_ d) \in U$ set $H_ i = Z(s_ i)$. Since $s_1, \ldots , s_ d$ generate $\mathfrak m_ x\mathcal{L}_ x$ we see that
\[ H_1 \cap \ldots \cap H_ d = x \amalg Z \]
scheme theoretically for some closed subscheme $Z \subset X$. By Bertini (in the form of Varieties, Lemma 33.47.3) for a general element $s_1 \in V$ the scheme $H_1 \cap (X \setminus x)$ is smooth over $k$ of dimension $d - 1$. Having chosen $s_1$, for a general element $s_2 \in V$ the scheme $H_1 \cap H_2 \cap (X \setminus x)$ is smooth over $k$ of dimension $d - 2$. And so on. We conclude that for sufficiently general $(s_1, \ldots , s_ d) \in U$ the scheme $Z$ is étale over $\mathop{\mathrm{Spec}}(k)$. In particular $H_1 \cap \ldots \cap H_ d$ has dimension $0$ and hence
\[ [H_1] \cdot \ldots \cdot [H_ d] = [x] + [Z] \]
in $\mathop{\mathrm{CH}}\nolimits _0(X)$ by repeated application of Chow Homology, Lemma 42.62.5 (details omitted). This finishes the proof as it shows that $[x] \sim _{rat} - [Z] + [Z']$ where $Z' = H'_1 \cap \ldots \cap H'_ d$ is a general complete intersection of vanishing loci of sufficiently general sections of $\mathcal{L}$ which will be étale over $k$ by the same argument as before. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)