The Stacks project

Lemma 81.26.9. In Situation 81.2.1 let $X/B$ be good. Let $c \in A^ p(X)$. Then $c$ is zero if and only if $c \cap [Y] = 0$ in $\mathop{\mathrm{CH}}\nolimits _*(Y)$ for every integral algebraic space $Y$ locally of finite type over $X$.

Proof. The if direction is clear. For the converse, assume that $c \cap [Y] = 0$ in $\mathop{\mathrm{CH}}\nolimits _*(Y)$ for every integral algebraic space $Y$ locally of finite type over $X$. Let $X' \to X$ be locally of finite type. Let $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X')$. Write $\alpha = \sum n_ i [Y_ i]$ with $Y_ i \subset X'$ a locally finite collection of integral closed subschemes of $\delta $-dimension $k$. Then we see that $\alpha $ is pushforward of the cycle $\alpha ' = \sum n_ i[Y_ i]$ on $X'' = \coprod Y_ i$ under the proper morphism $X'' \to X'$. By the properties of bivariant classes it suffices to prove that $c \cap \alpha ' = 0$ in $\mathop{\mathrm{CH}}\nolimits _{k - p}(X'')$. We have $\mathop{\mathrm{CH}}\nolimits _{k - p}(X'') = \prod \mathop{\mathrm{CH}}\nolimits _{k - p}(Y_ i)$ as follows immediately from the definitions. The projection maps $\mathop{\mathrm{CH}}\nolimits _{k - p}(X'') \to \mathop{\mathrm{CH}}\nolimits _{k - p}(Y_ i)$ are given by flat pullback. Since capping with $c$ commutes with flat pullback, we see that it suffices to show that $c \cap [Y_ i]$ is zero in $\mathop{\mathrm{CH}}\nolimits _{k - p}(Y_ i)$ which is true by assumption. $\square$

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