The Stacks project

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.

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