The Stacks project

Lemma 42.20.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be schemes locally of finite type over $S$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\alpha \sim _{rat} \beta $ be rationally equivalent $k$-cycles on $Y$. Then $f^*\alpha \sim _{rat} f^*\beta $ as $(k + r)$-cycles on $X$.

Proof. What do we have to show? Well, suppose we are given a collection

\[ i_ j : W_ j \longrightarrow Y \]

of closed immersions, with each $W_ j$ integral of $\delta $-dimension $k + 1$ and rational functions $g_ j \in R(W_ j)^*$. Moreover, assume that the collection $\{ i_ j(W_ j)\} _{j \in J}$ is locally finite on $Y$. Then we have to show that

\[ f^*(\sum i_{j, *}\text{div}(g_ j)) = \sum f^*i_{j, *}\text{div}(g_ j) \]

is rationally equivalent to zero on $X$. The sum on the right makes sense as $\{ W_ j\} $ is locally finite in $X$ by Lemma 42.13.2.

Consider the fibre products

\[ i'_ j : W'_ j = W_ j \times _ Y X \longrightarrow X. \]

and denote $f_ j : W'_ j \to W_ j$ the first projection. By Lemma 42.15.1 we can write the sum above as

\[ \sum i'_{j, *}(f_ j^*\text{div}(g_ j)) \]

By Lemma 42.20.1 we see that each $f_ j^*\text{div}(g_ j)$ is rationally equivalent to zero on $W'_ j$. Hence each $i'_{j, *}(f_ j^*\text{div}(g_ j))$ is rationally equivalent to zero. Then the same is true for the displayed sum by the discussion in Remark 42.19.6. $\square$

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