This section is the analogue of Chow Homology, Section 42.28. Applying the key lemma we obtain the fundamental properties of intersecting with invertible sheaves. In particular, we will see that $c_1(\mathcal{L}) \cap -$ factors through rational equivalence and that these operations for different invertible sheaves commute.
Lemma 82.21.1. In Situation 82.2.1 let $X/B$ be good. Assume $X$ integral and $\dim _\delta (X) = n$. Let $\mathcal{L}$, $\mathcal{N}$ be invertible on $X$. Choose a nonzero meromorphic section $s$ of $\mathcal{L}$ and a nonzero meromorphic section $t$ of $\mathcal{N}$. Set $\alpha = \text{div}_\mathcal {L}(s)$ and $\beta = \text{div}_\mathcal {N}(t)$. Then in $\mathop{\mathrm{CH}}\nolimits _{n - 2}(X)$.
\[ c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{L}) \cap \beta \]
Proof. Immediate from the key Lemma 82.20.1 and the discussion preceding it. $\square$
Lemma 82.21.2. In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{L}$ be invertible on $X$. The operation $\alpha \mapsto c_1(\mathcal{L}) \cap \alpha $ factors through rational equivalence to give an operation
\[ c_1(\mathcal{L}) \cap - : \mathop{\mathrm{CH}}\nolimits _{k + 1}(X) \to \mathop{\mathrm{CH}}\nolimits _ k(X) \]
Proof. Let $\alpha \in Z_{k + 1}(X)$, and $\alpha \sim _{rat} 0$. We have to show that $c_1(\mathcal{L}) \cap \alpha $ as defined in Definition 82.18.1 is zero. By Definition 82.15.1 there exists a locally finite family $\{ W_ j\} $ of integral closed subspaces with $\dim _\delta (W_ j) = k + 2$ and rational functions $f_ j \in R(W_ j)^*$ such that
Note that $p : \coprod W_ j \to X$ is a proper morphism, and hence $\alpha = p_*\alpha '$ where $\alpha ' \in Z_{k + 1}(\coprod W_ j)$ is the sum of the principal divisors $\text{div}_{W_ j}(f_ j)$. By Lemma 82.19.4 we have $c_1(\mathcal{L}) \cap \alpha = p_*(c_1(p^*\mathcal{L}) \cap \alpha ')$. Hence it suffices to show that each $c_1(\mathcal{L}|_{W_ j}) \cap \text{div}_{W_ j}(f_ j)$ is zero. In other words we may assume that $X$ is integral and $\alpha = \text{div}_ X(f)$ for some $f \in R(X)^*$.
Assume $X$ is integral and $\alpha = \text{div}_ X(f)$ for some $f \in R(X)^*$. We can think of $f$ as a regular meromorphic section of the invertible sheaf $\mathcal{N} = \mathcal{O}_ X$. Choose a meromorphic section $s$ of $\mathcal{L}$ and denote $\beta = \text{div}_\mathcal {L}(s)$. By Lemma 82.21.1 we conclude that
However, by Lemma 82.18.2 we see that the right hand side is zero in $\mathop{\mathrm{CH}}\nolimits _ k(X)$ as desired. $\square$
In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{L}$ be invertible on $X$. We will denote
the operation $c_1(\mathcal{L}) \cap - $. This makes sense by Lemma 82.21.2. We will denote $c_1(\mathcal{L}^ s \cap -$ the $s$-fold iterate of this operation for all $s \geq 0$.
Lemma 82.21.3. In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{L}$, $\mathcal{N}$ be invertible on $X$. For any $\alpha \in \mathop{\mathrm{CH}}\nolimits _{k + 2}(X)$ we have as elements of $\mathop{\mathrm{CH}}\nolimits _ k(X)$.
\[ c_1(\mathcal{L}) \cap c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{N}) \cap c_1(\mathcal{L}) \cap \alpha \]
Proof. Write $\alpha = \sum m_ j[Z_ j]$ for some locally finite collection of integral closed subspaces $Z_ j \subset X$ with $\dim _\delta (Z_ j) = k + 2$. Consider the proper morphism $p : \coprod Z_ j \to X$. Set $\alpha ' = \sum m_ j[Z_ j]$ as a $(k + 2)$-cycle on $\coprod Z_ j$. By several applications of Lemma 82.19.4 we see that $c_1(\mathcal{L}) \cap c_1(\mathcal{N}) \cap \alpha = p_*(c_1(p^*\mathcal{L}) \cap c_1(p^*\mathcal{N}) \cap \alpha ')$ and $c_1(\mathcal{N}) \cap c_1(\mathcal{L}) \cap \alpha = p_*(c_1(p^*\mathcal{N}) \cap c_1(p^*\mathcal{L}) \cap \alpha ')$. Hence it suffices to prove the formula in case $X$ is integral and $\alpha = [X]$. In this case the result follows from Lemma 82.21.1 and the definitions. $\square$
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