The Stacks project

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 ├ętale 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 ├ętale over $k$ by the same argument as before. $\square$


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