Lemma 82.23.4. In Situation 82.2.1 let X/B be good. Let (\mathcal{L}, s, i : D \to X) and (\mathcal{L}', s', i' : D' \to X) be two triples as in Definition 82.22.1. Then the diagram
commutes where each of the maps is a gysin map.
Lemma 82.23.4. In Situation 82.2.1 let X/B be good. Let (\mathcal{L}, s, i : D \to X) and (\mathcal{L}', s', i' : D' \to X) be two triples as in Definition 82.22.1. Then the diagram
commutes where each of the maps is a gysin map.
Proof. Denote j : D \cap D' \to D and j' : D \cap D' \to D' the closed immersions corresponding to (\mathcal{L}|_{D'}, s|_{D'} and (\mathcal{L}'_ D, s|_ D). We have to show that (j')^*i^*\alpha = j^* (i')^*\alpha for all \alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X). Let W \subset X be an integral closed subscheme of dimension k. We will prove the equality in case \alpha = [W]. The general case will then follow from the observation in Remark 82.15.3 (and the specific shape of our rational equivalence produced below). We will deduce the equality for \alpha = [W] from the key formula.
We let \sigma be a nonzero meromorphic section of \mathcal{L}|_ W which we require to be equal to s|_ W if W \not\subset D. We let \sigma ' be a nonzero meromorphic section of \mathcal{L}'|_ W which we require to be equal to s'|_ W if W \not\subset D'. Write
and similarly
as in the discussion in Section 82.20. Then we see that Z_ i \subset D if n_ i \not= 0 and Z'_ i \subset D' if n'_ i \not= 0. For each i, let \xi _ i \in |Z_ i| be the generic point. As in Section 82.20 we choose for each i an element \sigma _ i \in \mathcal{L}_{\xi _ i}, resp. \sigma '_ i \in \mathcal{L}'_{\xi _ i} which generates over B_ i = \mathcal{O}_{W, \xi _ i}^ h and which is equal to the image of s, resp. s' if Z_ i \not\subset D, resp. Z_ i \not\subset D'. Write \sigma = f_ i \sigma _ i and \sigma ' = f'_ i\sigma '_ i so that n_ i = \text{ord}_{B_ i}(f_ i) and n'_ i = \text{ord}_{B_ i}(f'_ i). From our definitions it follows that
as cycles and
The key formula (Lemma 82.20.1) now gives the equality
of cycles. Note that \text{div}_{Z_ i}(\partial _{B_ i}(f_ i, f'_ i)) = 0 if Z_ i \not\subset D \cap D' because in this case either f_ i = 1 or f'_ i = 1. Thus we get a rational equivalence between our specific cycles representing (j')^*i^*[W] and j^*(i')^*[W] on D \cap D' \cap W. \square
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