The Stacks project

Lemma 42.31.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $p : X \to Y$ be a flat morphism of relative dimension $r$. Let $i : D \to X$ be a relative effective Cartier divisor (Divisors, Definition 31.18.2). Let $\mathcal{L} = \mathcal{O}_ X(D)$. For any $\alpha \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(Y)$ we have

\[ i^*p^*\alpha = (p|_ D)^*\alpha \]

in $\mathop{\mathrm{CH}}\nolimits _{k + r}(D)$ and

\[ c_1(\mathcal{L}) \cap p^*\alpha = i_* ((p|_ D)^*\alpha ) \]

in $\mathop{\mathrm{CH}}\nolimits _{k + r}(X)$.

Proof. Let $W \subset Y$ be an integral closed subscheme of $\delta $-dimension $k + 1$. By Divisors, Lemma 31.18.1 we see that $D \cap p^{-1}W$ is an effective Cartier divisor on $p^{-1}W$. By Lemma 42.29.5 we get the first equality in

\[ i^*[p^{-1}W]_{k + r + 1} = [D \cap p^{-1}W]_{k + r} = [(p|_ D)^{-1}(W)]_{k + r}. \]

and the second because $D \cap p^{-1}(W) = (p|_ D)^{-1}(W)$ as schemes. Since by definition $p^*[W] = [p^{-1}W]_{k + r + 1}$ we see that $i^*p^*[W] = (p|_ D)^*[W]$ as cycles. If $\alpha = \sum m_ j[W_ j]$ is a general $k + 1$ cycle, then we get $i^*\alpha = \sum m_ j i^*p^*[W_ j] = \sum m_ j(p|_ D)^*[W_ j]$ as cycles. This proves then first equality. To deduce the second from the first apply Lemma 42.29.4. $\square$

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