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).
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
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).
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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