The Stacks project

Lemma 42.35.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $c \in A^ p(X \to Y)$. For $Y'' \to Y' \to Y$ set $X'' = Y'' \times _ Y X$ and $X' = Y' \times _ Y X$. The following are equivalent

  1. $c$ is zero,

  2. $c \cap [Y'] = 0$ in $\mathop{\mathrm{CH}}\nolimits _*(X')$ for every integral scheme $Y'$ locally of finite type over $Y$, and

  3. for every integral scheme $Y'$ locally of finite type over $Y$, there exists a proper birational morphism $Y'' \to Y'$ such that $c \cap [Y''] = 0$ in $\mathop{\mathrm{CH}}\nolimits _*(X'')$.

Proof. The implications (1) $\Rightarrow $ (2) $\Rightarrow $ (3) are clear. Assumption (3) implies (2) because $(Y'' \to Y')_*[Y''] = [Y']$ and hence $c \cap [Y'] = (X'' \to X')_*(c \cap [Y''])$ as $c$ is a bivariant class. Assume (2). Let $Y' \to Y$ be locally of finite type. Let $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(Y')$. Write $\alpha = \sum n_ i [Y'_ i]$ with $Y'_ i \subset Y'$ 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 $Y'' = \coprod Y'_ i$ under the proper morphism $Y'' \to Y'$. 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 $\mathop{\mathrm{CH}}\nolimits _{k - p}(X'') = \prod \mathop{\mathrm{CH}}\nolimits _{k - p}(X'_ i)$ where $X'_ i = Y'_ i \times _ Y X$. This follows immediately from the definitions. The projection maps $\mathop{\mathrm{CH}}\nolimits _{k - p}(X'') \to \mathop{\mathrm{CH}}\nolimits _{k - p}(X'_ i)$ 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 $\mathop{\mathrm{CH}}\nolimits _{k - p}(X'_ i)$ which is true by assumption. $\square$

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