The Stacks project

Lemma 81.23.2. In Situation 81.2.1 let $X/B$ be good. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 81.22.1. The Gysin homomorphism factors through rational equivalence to give a map $i^* : \mathop{\mathrm{CH}}\nolimits _{k + 1}(X) \to \mathop{\mathrm{CH}}\nolimits _ k(D)$.

Proof. Let $\alpha \in Z_{k + 1}(X)$ and assume that $\alpha \sim _{rat} 0$. This means there exists a locally finite collection of integral closed subspaces $W_ j \subset X$ of $\delta $-dimension $k + 2$ and $f_ j \in R(W_ j)^*$ such that $\alpha = \sum i_{j, *}\text{div}_{W_ j}(f_ j)$. Set $X' = \coprod W_ i$ and consider the diagram

\[ \xymatrix{ D' \ar[d]_ q \ar[r]_{i'} & X' \ar[d]^ p \\ D \ar[r]^ i & X } \]

of Remark 81.22.3. Since $X' \to X$ is proper we see that $i^*p_* = q_*(i')^*$ by Lemma 81.22.5. As we know that $q_*$ factors through rational equivalence (Lemma 81.16.3), it suffices to prove the result for $\alpha ' = \sum \text{div}_{W_ j}(f_ j)$ on $X'$. Clearly this reduces us to the case where $X$ is integral and $\alpha = \text{div}(f)$ for some $f \in R(X)^*$.

Assume $X$ is integral and $\alpha = \text{div}(f)$ for some $f \in R(X)^*$. If $X = D$, then we see that $i^*\alpha $ is equal to $c_1(\mathcal{L}) \cap \alpha $. This is rationally equivalent to zero by Lemma 81.21.2. If $D \not= X$, then we see that $i^*\text{div}_ X(f)$ is equal to $c_1(\mathcal{O}_ D) \cap [D]_{n - 1}$ in $\mathop{\mathrm{CH}}\nolimits _ k(D)$ by Lemma 81.23.1. Of course capping with $c_1(\mathcal{O}_ D)$ is the zero map. $\square$

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