The Stacks project

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

\[ \partial _ A(a, b) = \prod \nolimits _{j, k} \text{Norm}_{\kappa (\mathfrak m_{j, k})/\kappa (\mathfrak m)} ((-1)^{f_{j, k}g_{j, k}}u_{j, k}^{g_{j, k}}v_{j, k}^{-f_{j, k}} \bmod \mathfrak m_{j, k})^{m_{j, k}} \]


\[ \partial _{B_ j}(a, b) = \prod \nolimits _ k \text{Norm}_{\kappa (\mathfrak m_{j, k})/\kappa (\mathfrak m_ j)} ((-1)^{f_{j, k}g_{j, k}}u_{j, k}^{g_{j, k}}v_{j, k}^{-f_{j, k}} \bmod \mathfrak m_{j, k})^{m_{j, k}} \]

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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