42.26 Intersecting with an invertible sheaf and push and pull
In this section we prove that the operation c_1(\mathcal{L}) \cap - commutes with flat pullback and proper pushforward.
Lemma 42.26.1. Let (S, \delta ) be as in Situation 42.7.1. Let X, Y be locally of finite type over S. Let f : X \to Y be a flat morphism of relative dimension r. Let \mathcal{L} be an invertible sheaf on Y. Assume Y is integral and n = \dim _\delta (Y). Let s be a nonzero meromorphic section of \mathcal{L}. Then we have
f^*\text{div}_\mathcal {L}(s) = \sum n_ i\text{div}_{f^*\mathcal{L}|_{X_ i}}(s_ i)
in Z_{n + r - 1}(X). Here the sum is over the irreducible components X_ i \subset X of \delta -dimension n + r, the section s_ i = f|_{X_ i}^*(s) is the pullback of s, and n_ i = m_{X_ i, X} is the multiplicity of X_ i in X.
Proof.
To prove this equality of cycles, we may work locally on Y. Hence we may assume Y is affine and s = p/q for some nonzero sections p \in \Gamma (Y, \mathcal{L}) and q \in \Gamma (Y, \mathcal{O}). If we can show both
f^*\text{div}_\mathcal {L}(p) = \sum n_ i\text{div}_{f^*\mathcal{L}|_{X_ i}}(p_ i) \quad \text{and}\quad f^*\text{div}_\mathcal {O}(q) = \sum n_ i\text{div}_{\mathcal{O}_{X_ i}}(q_ i)
(with obvious notations) then we win by the additivity, see Divisors, Lemma 31.27.5. Thus we may assume that s \in \Gamma (Y, \mathcal{L}). In this case we may apply the equality (42.25.3.1) to see that
[Z(f^*(s))]_{k + r - 1} = \sum n_ i\text{div}_{f^*\mathcal{L}|_{X_ i}}(s_ i)
where f^*(s) \in f^*\mathcal{L} denotes the pullback of s to X. On the other hand we have
f^*\text{div}_\mathcal {L}(s) = f^*[Z(s)]_{k - 1} = [f^{-1}(Z(s))]_{k + r - 1},
by Lemmas 42.24.2 and 42.14.4. Since Z(f^*(s)) = f^{-1}(Z(s)) we win.
\square
Lemma 42.26.2. Let (S, \delta ) be as in Situation 42.7.1. Let X, Y be locally of finite type over S. Let f : X \to Y be a flat morphism of relative dimension r. Let \mathcal{L} be an invertible sheaf on Y. Let \alpha be a k-cycle on Y. Then
f^*(c_1(\mathcal{L}) \cap \alpha ) = c_1(f^*\mathcal{L}) \cap f^*\alpha
in \mathop{\mathrm{CH}}\nolimits _{k + r - 1}(X).
Proof.
Write \alpha = \sum n_ i[W_ i]. We will show that
f^*(c_1(\mathcal{L}) \cap [W_ i]) = c_1(f^*\mathcal{L}) \cap f^*[W_ i]
in \mathop{\mathrm{CH}}\nolimits _{k + r - 1}(X) by producing a rational equivalence on the closed subscheme f^{-1}(W_ i) of X. By the discussion in Remark 42.19.6 this will prove the equality of the lemma is true.
Let W \subset Y be an integral closed subscheme of \delta -dimension k. Consider the closed subscheme W' = f^{-1}(W) = W \times _ Y X so that we have the fibre product diagram
\xymatrix{ W' \ar[r] \ar[d]_ h & X \ar[d]^ f \\ W \ar[r] & Y }
We have to show that f^*(c_1(\mathcal{L}) \cap [W]) = c_1(f^*\mathcal{L}) \cap f^*[W]. Choose a nonzero meromorphic section s of \mathcal{L}|_ W. Let W'_ i \subset W' be the irreducible components of \delta -dimension k + r. Write [W']_{k + r} = \sum n_ i[W'_ i] with n_ i the multiplicity of W'_ i in W' as per definition. So f^*[W] = \sum n_ i[W'_ i] in Z_{k + r}(X). Since each W'_ i \to W is dominant we see that s_ i = s|_{W'_ i} is a nonzero meromorphic section for each i. By Lemma 42.26.1 we have the following equality of cycles
h^*\text{div}_{\mathcal{L}|_ W}(s) = \sum n_ i\text{div}_{f^*\mathcal{L}|_{W'_ i}}(s_ i)
in Z_{k + r - 1}(W'). This finishes the proof since the left hand side is a cycle on W' which pushes to f^*(c_1(\mathcal{L}) \cap [W]) in \mathop{\mathrm{CH}}\nolimits _{k + r - 1}(X) and the right hand side is a cycle on W' which pushes to c_1(f^*\mathcal{L}) \cap f^*[W] in \mathop{\mathrm{CH}}\nolimits _{k + r - 1}(X).
\square
Lemma 42.26.3. Let (S, \delta ) be as in Situation 42.7.1. Let X, Y be locally of finite type over S. Let f : X \to Y be a proper morphism. Let \mathcal{L} be an invertible sheaf on Y. Let s be a nonzero meromorphic section s of \mathcal{L} on Y. Assume X, Y integral, f dominant, and \dim _\delta (X) = \dim _\delta (Y). Then
f_*\left(\text{div}_{f^*\mathcal{L}}(f^*s)\right) = [R(X) : R(Y)]\text{div}_\mathcal {L}(s).
as cycles on Y. In particular
f_*(c_1(f^*\mathcal{L}) \cap [X]) = [R(X) : R(Y)] c_1(\mathcal{L}) \cap [Y] = c_1(\mathcal{L}) \cap f_*[X]
Proof.
The last equation follows from the first since f_*[X] = [R(X) : R(Y)][Y] by definition. It turns out that we can re-use Lemma 42.18.1 to prove this. Namely, since we are trying to prove an equality of cycles, we may work locally on Y. Hence we may assume that \mathcal{L} = \mathcal{O}_ Y. In this case s corresponds to a rational function g \in R(Y), and we are simply trying to prove
f_*\left(\text{div}_ X(g)\right) = [R(X) : R(Y)]\text{div}_ Y(g).
Comparing with the result of the aforementioned Lemma 42.18.1 we see this true since \text{Nm}_{R(X)/R(Y)}(g) = g^{[R(X) : R(Y)]} as g \in R(Y)^*.
\square
Lemma 42.26.4. 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 proper morphism. Let \alpha \in Z_{k + 1}(X). Let \mathcal{L} be an invertible sheaf on Y. Then
p_*(c_1(p^*\mathcal{L}) \cap \alpha ) = c_1(\mathcal{L}) \cap p_*\alpha
in \mathop{\mathrm{CH}}\nolimits _ k(Y).
Proof.
Suppose that p has the property that for every integral closed subscheme W \subset X the map p|_ W : W \to Y is a closed immersion. Then, by definition of capping with c_1(\mathcal{L}) the lemma holds.
We will use this remark to reduce to a special case. Namely, write \alpha = \sum n_ i[W_ i] with n_ i \not= 0 and W_ i pairwise distinct. Let W'_ i \subset Y be the image of W_ i (as an integral closed subscheme). Consider the diagram
\xymatrix{ X' = \coprod W_ i \ar[r]_-q \ar[d]_{p'} & X \ar[d]^ p \\ Y' = \coprod W'_ i \ar[r]^-{q'} & Y. }
Since \{ W_ i\} is locally finite on X, and p is proper we see that \{ W'_ i\} is locally finite on Y and that q, q', p' are also proper morphisms. We may think of \sum n_ i[W_ i] also as a k-cycle \alpha ' \in Z_ k(X'). Clearly q_*\alpha ' = \alpha . We have q_*(c_1(q^*p^*\mathcal{L}) \cap \alpha ') = c_1(p^*\mathcal{L}) \cap q_*\alpha ' and (q')_*(c_1((q')^*\mathcal{L}) \cap p'_*\alpha ') = c_1(\mathcal{L}) \cap q'_*p'_*\alpha ' by the initial remark of the proof. Hence it suffices to prove the lemma for the morphism p' and the cycle \sum n_ i[W_ i]. Clearly, this means we may assume X, Y integral, f : X \to Y dominant and \alpha = [X]. In this case the result follows from Lemma 42.26.3.
\square
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