The Stacks project

81.24 Relative effective Cartier divisors

This section is the analogue of Chow Homology, Section 42.31. Relative effective Cartier divisors are defined in Divisors on Spaces, Section 70.9. To develop the basic results on Chern classes of vector bundles we only need the case where both the ambient scheme and the effective Cartier divisor are flat over the base.

Lemma 81.24.1. In Situation 81.2.1. Let $X, Y/B$ be good. 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 on Spaces, Definition 70.9.2). 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 subspace of $\delta $-dimension $k + 1$. By Divisors on Spaces, Lemma 70.9.1 we see that $D \cap p^{-1}W$ is an effective Cartier divisor on $p^{-1}W$. By Lemma 81.22.7 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 algebraic spaces. 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 81.22.4. $\square$

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