## 45.8 Cycles over non-closed fields

Some lemmas which will help us in our study of motives over base fields which are not algebraically closed.

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. $\square$

Lemma 45.8.2. Let $K/k$ be an algebraic field extension. Let $X$ be a finite type scheme over $k$. Then $\mathop{\mathrm{CH}}\nolimits _ i(X_ K) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{CH}}\nolimits _ i(X_{k'})$ where the colimit is over the subextensions $K/k'/k$ with $k'/k$ finite.

Proof. This is a special case of Chow Homology, Lemma 42.67.10. $\square$

Lemma 45.8.3. Let $k$ be a field. Let $X$ be a geometrically irreducible smooth projective scheme over $k$. Let $x, x' \in X$ be $k$-rational points. Let $n$ be an integer invertible in $k$. Then there exists a finite separable extension $k'/k$ such that the pullback of $[x] - [x']$ to $X_{k'}$ is divisible by $n$ in $\mathop{\mathrm{CH}}\nolimits _0(X_{k'})$.

Proof. Let $k'$ be a separable algebraic closure of $k$. Suppose that we can show the the pullback of $[x] - [x']$ to $X_{k'}$ is divisible by $n$ in $\mathop{\mathrm{CH}}\nolimits _0(X_{k'})$. Then we conclude by Lemma 45.8.2. Thus we may and do assume $k$ is separably algebraically closed.

Suppose $\dim (X) > 1$. Let $\mathcal{L}$ be an ample invertible sheaf on $X$. Set

$V = \{ s \in H^0(X, \mathcal{L}) \mid s(x) = 0\text{ and }s(x') = 0 \}$

After replacing $\mathcal{L}$ by a power we see that for a general $v \in V$ the corresponding divisor $H_ v \subset X$ is smooth away from $x$ and $x'$, see Varieties, Lemmas 33.47.1 and 33.47.3. To find $v$ we use that $k$ is infinite (being separably algebraically closed). If we choose $s$ general, then the image of $s$ in $\mathfrak m_ x\mathcal{L}_ x/\mathfrak m_ x^2\mathcal{L}_ x$ will be nonzero, which implies that $H_ v$ is smooth at $x$ (details omitted). Similarly for $x'$. Thus $H_ v$ is smooth. By Varieties, Lemma 33.48.3 (applied to the base change of everything to the algebraic closure of $k$) we see that $H_ v$ is geometrically connected. It suffices to prove the result for $[x] - [x']$ seen as an element of $\mathop{\mathrm{CH}}\nolimits _0(H_ v)$. In this way we reduce to the case of a curve.

Assume $X$ is a curve. Then we see that $\mathcal{O}_ X(x - x')$ defines a $k$-rational point $g$ of $J = \underline{\mathop{\mathrm{Pic}}\nolimits }^0_{X/k}$, see Picard Schemes of Curves, Lemma 44.6.7. Recall that $J$ is a proper smooth variety over $k$ which is also a group scheme over $k$ (same reference). Hence $J$ is geometrically integral (see Varieties, Lemma 33.7.13 and 33.25.4). In other words, $J$ is an abelian variety, see Groupoids, Definition 39.9.1. Thus $[n] : J \to J$ is finite étale by Groupoids, Proposition 39.9.11 (this is where we use $n$ is invertible in $k$). Since $k$ is separably closed we conclude that $g = [n](g')$ for some $g' \in J(k)$. If $\mathcal{L}$ is the degree $0$ invertible module on $X$ corresponding to $g'$, then we conclude that $\mathcal{O}_ X(x - x') \cong \mathcal{L}^{\otimes n}$ as desired. $\square$

Lemma 45.8.4. Let $K/k$ be an algebraic extension of fields. Let $X$ be a finite type scheme over $k$. The kernel of the map $\mathop{\mathrm{CH}}\nolimits _ i(X) \to \mathop{\mathrm{CH}}\nolimits _ i(X_ K)$ constructed in Lemma 45.8.2 is torsion.

Proof. It clearly suffices to show that the kernel of flat pullback $\mathop{\mathrm{CH}}\nolimits _ i(X) \to \mathop{\mathrm{CH}}\nolimits _ i(X_{k'})$ by $\pi : X_{k'} \to X$ is torsion for any finite extension $k'/k$. This is clear because $\pi _* \pi ^* \alpha = [k' : k] \alpha$ by Chow Homology, Lemma 42.15.2. $\square$

Lemma 45.8.5 (Voevodsky). Let $k$ be a field. Let $X$ be a geometrically irreducible smooth projective scheme over $k$. Let $x, x' \in X$ be $k$-rational points. For $n$ large enough the class of the zero cycle

$([x] - [x']) \times \ldots \times ([x] - [x']) \in \mathop{\mathrm{CH}}\nolimits _0(X^ n)$

is torsion.

Proof. If we can show this after base change to the algebraic closure of $k$, then the result follows over $k$ because the kernel of pullback is torsion by Lemma 45.8.4. Hence we may and do assume $k$ is algebraically closed.

Using Bertini we can choose a smooth curve $C \subset X$ passing through $x$ and $x'$. See proof of Lemma 45.8.3. Hence we may assume $X$ is a curve.

Assume $X$ is a curve and $k$ is algebraically closed. Write $S^ n(X) = \underline{\mathrm{Hilb}}^ n_{X/k}$ with notation as in Picard Schemes of Curves, Sections 44.2 and 44.3. There is a canonical morphism

$\pi : X^ n \longrightarrow S^ n(X)$

which sends the $k$-rational point $(x_1, \ldots , x_ n)$ to the $k$-rational point corresponding to the divisor $[x_1] + \ldots + [x_ n]$ on $X$. There is a faithful action of the symmetric group $S_ n$ on $X^ n$. The morphism $\pi$ is $S_ n$-invariant and the fibres of $\pi$ are $S_ n$-orbits (set theoretically). Finally, $\pi$ is finite flat of degree $n!$, see Picard Schemes of Curves, Lemma 44.3.4.

Let $\alpha _ n$ be the zero cycle on $X^ n$ given by the formula in the statement of the lemma. Let $\mathcal{L} = \mathcal{O}_ X(x - x')$. Then $c_1(\mathcal{L}) \cap [X] = [x] - [x']$. Thus

$\alpha _ n = c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ n) \cap [X^ n]$

where $\mathcal{L}_ i = \text{pr}_ i^*\mathcal{L}$ and $\text{pr}_ i : X^ n \to X$ is the $i$th projection. By either Divisors, Lemma 31.17.6 or Divisors, Lemma 31.17.7 there is a norm for $\pi$. Set $\mathcal{N} = \text{Norm}_\pi (\mathcal{L}_1)$, see Divisors, Lemma 31.17.2. We have

$\pi ^*\mathcal{N} = (\mathcal{L}_1 \otimes \ldots \otimes \mathcal{L}_ n)^{\otimes (n - 1)!}$

in $\mathop{\mathrm{Pic}}\nolimits (X^ n)$ by a calculation. Deails omitted; hint: this follows from the fact that $\text{Norm}_\pi : \pi _*\mathcal{O}_{X^ n} \to \mathcal{O}_{S^ n(X)}$ composed with the natural map $\pi _*\mathcal{O}_{S^ n(X)} \to \mathcal{O}_{X^ n}$ is equal to the product over all $\sigma \in S_ n$ of the action of $\sigma$ on $\pi _*\mathcal{O}_{X^ n}$. Consider

$\beta _ n = c_1(\mathcal{N})^ n \cap [S^ n(X)]$

in $\mathop{\mathrm{CH}}\nolimits _0(S^ n(X))$. Observe that $c_1(\mathcal{L}_ i) \cap c_1(\mathcal{L}_ i) = 0$ because $\mathcal{L}_ i$ is pulled back from a curve, see Chow Homology, Lemma 42.34.6. Thus we see that

\begin{align*} \pi ^*\beta _ n & = ((n - 1)!)^ n (\sum \nolimits _{i = 1, \ldots , n} c_1(\mathcal{L}_ i))^ n \cap [X^ n] \\ & = ((n - 1)!)^ n n^ n c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ n) \cap [X^ n] \\ & = (n!)^ n \alpha _ n \end{align*}

Thus it suffices to show that $\beta _ n$ is torsion.

There is a canonical morphism

$f : S^ n(X) \longrightarrow \underline{\mathrm{Pic}}^ n_{X/k}$

See Picard Schemes of Curves, Lemma 44.6.7. For $n \geq 2g - 1$ this morphism is a projective space bundle (details omitted; compare with the proof of Picard Schemes of Curves, Lemma 44.6.7). The invertible sheaf $\mathcal{N}$ is trivial on the fibres of $f$, see below. Thus by the projective space bundle formula (Chow Homology, Lemma 42.36.2) we see that $\mathcal{N} = f^*\mathcal{M}$ for some invertible module $\mathcal{M}$ on $\underline{\mathrm{Pic}}^ n_{X/k}$. Of course, then we see that

$c_1(\mathcal{N})^ n = f^*(c_1(\mathcal{M})^ n)$

is zero because $n > g = \dim (\underline{\mathrm{Pic}}^ n_{X/k})$ and we can use Chow Homology, Lemma 42.34.6 as before.

We still have to show that $\mathcal{N}$ is trivial on a fibre $F$ of $f$. Since the fibres of $f$ are projective spaces and since $\mathop{\mathrm{Pic}}\nolimits (\mathbf{P}^ m_ k) = \mathbf{Z}$ (Divisors, Lemma 31.28.5), this can be shown by computing the degree of $\mathcal{N}$ on a line contained in the fibre. Instead we will prove it by proving that $\mathcal{N}$ is algebraically equivalent to zero. First we claim there is a connected finite type scheme $T$ over $k$, an invertible module $\mathcal{L}'$ on $T \times X$ and $k$-rational points $p, q \in T$ such that $\mathcal{M}_ p \cong \mathcal{O}_ X$ and $\mathcal{M}_ q = \mathcal{L}$. Namely, since $\mathcal{L} = \mathcal{O}_ X(x - x')$ we can take $T = X$, $p = x'$, $q = x$, and $\mathcal{L}' = \mathcal{O}_{X \times X}(\Delta ) \otimes \text{pr}_2^*\mathcal{O}_ X(-x')$. Then we let $\mathcal{L}'_ i$ on $T \times X^ n$ for $i = 1, \ldots , n$ be the pullback of $\mathcal{L}'$ by $\text{id}_ T \times \text{pr}_ i : T \times X^ n \to T \times X$. Finally, we let $\mathcal{N}' = \text{Norm}_{\text{id}_ T \times \pi }(\mathcal{L}'_1)$ on $T \times S^ n(X)$. By construction we have $\mathcal{N}'_ p = \mathcal{O}_{S^ n(X)}$ and $\mathcal{N}'_ q = \mathcal{N}$. We conclude that

$\mathcal{N}'|_{T \times F}$

is an invertible module on $T \times F \cong T \times \mathbf{P}^ m_ k$ whose fibre over $p$ is the trivial invertible module and whose fibre over $q$ is $\mathcal{N}|_ F$. Since the euler characteristic of the trivial bundle is $1$ and since this euler characteristic is locally constant in families (Derived Categories of Schemes, Lemma 36.32.2) we conclude $\chi (F, \mathcal{N}^{\otimes s}|_ F) = 1$ for all $s \in \mathbf{Z}$. This can happen only if $\mathcal{N}|_ F \cong \mathcal{O}_ F$ (see Cohomology of Schemes, Lemma 30.8.1) and the proof is complete. Some details omitted. $\square$

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