Lemma 42.5.3. Let (A, \mathfrak m) be a Noetherian local ring of dimension 1. Let A \subset B be a finite ring extension with B/A annihilated by a power of \mathfrak m and \mathfrak m not an associated prime of B. For a, b \in A nonzerodivisors we have
\partial _ A(a, b) = \prod \text{Norm}_{\kappa (\mathfrak m_ j)/\kappa (\mathfrak m)}(\partial _{B_ j}(a, b))
where the product is over the maximal ideals \mathfrak m_ j of B and B_ j = B_{\mathfrak m_ j}.
Proof.
Choose B_ j \subset C_ j as in Lemma 42.4.4 for a, b. By Lemma 42.4.1 we can choose a finite ring extension B \subset C with C_ j \cong C_{\mathfrak m_ j} for all j. Let \mathfrak m_{j, k} \subset C be the maximal ideals of C lying over \mathfrak m_ j. Let
a = u_{j, k}\pi _{j, k}^{f_{j, k}},\quad b = v_{j, k}\pi _{j, k}^{g_{j, k}}
be the local factorizations which exist by our choice of C_ j \cong C_{\mathfrak m_ j}. By definition we have
\partial _ A(a, b) = \prod \nolimits _{j, k} \text{Norm}_{\kappa (\mathfrak m_{j, k})/\kappa (\mathfrak m)} ((-1)^{f_{j, k}g_{j, k}}u_{j, k}^{g_{j, k}}v_{j, k}^{-f_{j, k}} \bmod \mathfrak m_{j, k})^{m_{j, k}}
and
\partial _{B_ j}(a, b) = \prod \nolimits _ k \text{Norm}_{\kappa (\mathfrak m_{j, k})/\kappa (\mathfrak m_ j)} ((-1)^{f_{j, k}g_{j, k}}u_{j, k}^{g_{j, k}}v_{j, k}^{-f_{j, k}} \bmod \mathfrak m_{j, k})^{m_{j, k}}
The result follows by transitivity of norms for \kappa (\mathfrak m_{j, k})/\kappa (\mathfrak m_ j)/\kappa (\mathfrak m), see Fields, Lemma 9.20.5.
\square
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