## 62.8 Effective relative cycles

Here is the definition.

Definition 62.8.1. Let $f : X \to S$ be a morphism of schemes. Assume $S$ is locally Noetherian and $f$ is locally of finite type. Let $r \geq 0$ be an integer. We say a relative $r$-cycle $\alpha$ on $X/S$ effective if $\alpha _ s$ is an effective cycle (Chow Homology, Definition 42.8.4) for all $s \in S$. The monoid of all effective relative $r$-cycles on $X/S$ is denoted $z^{eff}(X/S, r)$.

Below we will show that an effective relative cycle is equidimensional, see Lemma 62.8.7.

Lemma 62.8.2. Let $f : X \to S$ be a morphism of schemes. Assume $S$ is locally Noetherian and $f$ is locally of finite type. Let $r \geq 0$ be an integer. Let $\alpha$ be a relative $r$-cycle on $X/S$. If $\alpha$ is effective, then any restriction, base change, flat pullback, or proper pushforward of $\alpha$ is effective.

Proof. Omitted. $\square$

Lemma 62.8.3. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r \geq 0$ be an integer. Let $\alpha$ be a relative $r$-cycle on $X/S$. Then to check that $\alpha$ is effective we may work Zariski locally on $X$ and $S$.

Proof. Omitted. $\square$

Lemma 62.8.4. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r \geq 0$ be an integer. Let $\alpha$ be a relative $r$-cycle on $X/S$. Let $g : S' \to S$ be a surjective morphism. Then $\alpha$ is effective if and only if the base change $g^*\alpha$ is effective.

Proof. Omitted. $\square$

Lemma 62.8.5. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r, e \geq 0$ be integers. Let $\alpha$ be a relative $r$-cycle on $X/S$. Let $\{ f_ i : X_ i \to X\}$ be a jointly surjective family of flat morphisms, locally of finite type, and of relative dimension $e$. Then $\alpha$ is effective if and only if each flat pullback $f_ i^*\alpha$ is effective.

Proof. Omitted. $\square$

Lemma 62.8.6. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r, e \geq 0$ be integers. Let $\alpha$ be a relative $r$-cycle on $X/S$. If $\alpha$ is effective, then $\text{Supp}(\alpha )$ is closed in $X$.

Proof. Let $g : S' \to S$ be the inclusion of an irreducible component viewed as an integral closed subscheme. By Lemmas 62.8.2 and 62.5.7 it suffices to show that the support of the base change $g^*\alpha$ is closed in $S' \times _ S S$. Thus we may assume $S$ is an integral scheme with generic point $\eta$. We will show that $\text{Supp}(\alpha )$ is the closure of $\text{Supp}(\alpha _\eta )$. To do this, pick any $s \in S$. We can find a morphism $g : S' \to S$ where $S'$ is the spectrum of a discrete valuation ring mapping the generic point $\eta ' \in S'$ to $\eta$ and the closed point $0 \in S'$ to $s$, see Properties, Lemma 28.5.10. Then it suffices to prove that the support of $g^*\alpha$ is equal to the closure of $\text{Supp}((g^\alpha )_{\eta '})$. This reduces us to the case discussed in the next paragraph.

Here $S$ is the spectrum of a discrete valuation ring with generic point $\eta$ and closed point $0$. We have to show that $\text{Supp}(\alpha )$ is the closure of $\text{Supp}(\alpha _\eta )$. Since $\alpha$ is effective we may write $\alpha _\eta = \sum n_ i[Z_ i]$ with $n_ i > 0$ and $Z_ i \subset X_\eta$ integral closed of dimension $r$. Since $\alpha _0 = sp_{X/S}(\alpha _\eta )$ we know that $\alpha _0 = \sum n_ i [\overline{Z}_{i, 0}]_ r$ where $\overline{Z}_ i$ is the closure of $Z_ i$. By Varieties, Lemma 33.19.2 we see that $\overline{Z}_{i, 0}$ is equidimensional of dimension $r$. Since $n_ i > 0$ we conclude that $\text{Supp}(\alpha _0)$ is equal to the union of the $\overline{Z}_{i, 0}$ which is the fibre over $0$ of $\bigcup \overline{Z}_ i$ which in turn is the closure of $\bigcup Z_ i$ as desired. $\square$

Lemma 62.8.7. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r, e \geq 0$ be integers. Let $\alpha$ be a relative $r$-cycle on $X/S$. If $\alpha$ is effective, then $\alpha$ is equidimensional.

Proof. Assume $\alpha$ is effective. By Lemma 62.8.6 the support $\text{Supp}(\alpha )$ is closed in $X$. Thus $\alpha$ is equidimensional as the fibres of $\text{Supp}(\alpha ) \to S$ are the supports of the cycles $\alpha _ s$ and hence have dimension $r$. $\square$

Remark 62.8.8. Let $f : X \to S$ be a morphism of schemes with $S$ locally Noetherian and $f$ locally of finite type. We can ask if the contravariant functor

$\begin{matrix} \text{schemes }S'\text{ locally} \\ \text{of finite type over }S \end{matrix} \longrightarrow z^{eff}(X'/S', r)\text{ where }X' = S' \times _ S X$

is representable. Since $z(X'/S', r) = z(X'_{red}/S'_{red}, r)$ this cannot be true (we leave it to the reader to make an actual counter example). A better question would be if we can find a subcategory of the left hand side on which the functor is representable. Lemma 62.6.16 suggests we should restrict at least to the category of seminormal schemes over $S$.

If $S/\mathop{\mathrm{Spec}}(\mathbf{Q})$ is Nagata and $f$ is a projective morphism, then it turns out that $S' \mapsto z^{eff}(X'/S', r)$ is representable on the category of seminormal $S'$. Roughly speaking this is the content of [Theorem 3.21, KRC].

If $S$ has points of positive characteristic, then this no longer works even if we replace seminormality with weak normality; a locally Noetherian scheme $T$ is weakly normal if any birational universal homeomorphism $T' \to T$ has a section. An example is to consider $0$-cycles of degree $2$ on $X = \mathbf{A}^2_ k$ over $S = \mathop{\mathrm{Spec}}(k)$ where $k$ is a field of characteristic $2$. Namely, over $W = X \times _ S X$ we have a canonical relative $0$-cycle $\alpha \in z^{eff}(X_ W/W, 0)$: for $w = (x_1, x_2) \in W = X^2$ we have the cycle $\alpha _ w = [x_1] + [x_2]$. This cycle is invariant under the involution $\sigma : W \to W$ switching the factors. Since $W$ is smooth (hence normal, hence weakly normal), if $z(-/-, r)$ was representable by $M$ on the category of weakly normal schemes of finite type over $k$ we would get a $\sigma$-invariant morphism from $W$ to $M$. This in turn would define a morphism from the quotient scheme $\text{Sym}^2_ S(X) = W/\langle \sigma \rangle$ to $M$. Since $\text{Sym}^2_ S(X)$ is normal, we would by the moduli property of $M$ obtain a relative $0$-cycle $\beta$ on $X \times _ S \text{Sym}^2_ S(X) / \text{Sym}^2_ S(X)$ whose pullback to $W$ is $\alpha$. However, there is no such cycle $\beta$. Namely, writing $X = \mathop{\mathrm{Spec}}(k[u, v])$ the scheme $\text{Sym}^2_ S(X)$ is the spectrum of

$k[u_1 + u_2, u_1u_2, v_1 + v_2, v_1v_2, u_1v_1 + u_2v_2] \subset k[u_1, u_2, v_1, v_2]$

The image of the diagonal $u_1 = u_2, v_1 = v_2$ in $\text{Sym}^2_ S(X)$ is the closed subscheme $V = \mathop{\mathrm{Spec}}(k[u_1^2, v_1^2])$; here we use that the characteristic of $k$ is $2$. Looking at the generic point $\eta$ of $V$, the cycle $\beta _\eta$ would be a zero cycle of degree $2$ on $\mathbf{A}^2_{k(u_1^2, v_1^2)}$ whose pullback to $\mathbf{A}^2_{k(u_1, u_2)}$ whould be $2[\text{the point with coordinates} (u_1, v_2)]$. This is clearly impossible.

The discussion above does not contradict [Theorem 4.13, KRC] as the Chow variety in that theorem only coarsely represents a functor (in fact 2 distinct functors, only one of which agrees with ours for projective $X$ as one can see with some work). Similarly, in [Section 4.4, SV] it is shown that for projective $X/S$ the $h$-sheafification of the presheaf $S' \mapsto z^{eff}(S' \times _ S X/S', r)$ is equal to the $h$-sheafification of a representable functor.

Remark 62.8.9. Let $f : X \to S$ be a morphism of schemes. Let $r \geq 0$. Let $Z \subset X$ be a closed subscheme. Assume

1. $S$ is Noetherian and geometrically unibranch,

2. $f$ is of finite type, and

3. $Z \to S$ has relative dimension $\leq r$.

Then for all sufficiently divisible integers $n \geq 1$ there exists a unique effective relative $r$-cycle $\alpha$ on $X/S$ such that $\alpha _\eta = n[Z_\eta ]_ r$ for every generic point $\eta$ of $S$. This is a reformulation of [Theorem 3.4.2, SV]. If we ever need this result, we will precisely state and prove it here.

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