Remark 49.6.4. Let $A \to B$ be a quasi-finite homomorphism of Noetherian rings. Let $J$ be the annihilator of $\mathop{\mathrm{Ker}}(B \otimes _ A B \to B)$. There is a canonical $B$-bilinear pairing

49.6.4.1
$$\label{discriminant-equation-pairing-noether} \omega _{B/A} \times J \longrightarrow B$$

defined as follows. Choose a factorization $A \to B' \to B$ with $A \to B'$ finite and $B' \to B$ inducing an open immersion of spectra. Let $J'$ be the annihilator of $\mathop{\mathrm{Ker}}(B' \otimes _ A B' \to B')$. We first define

$\mathop{\mathrm{Hom}}\nolimits _ A(B', A) \times J' \longrightarrow B',\quad (\lambda , \sum b_ i \otimes c_ i) \longmapsto \sum \lambda (b_ i)c_ i$

This is $B'$-bilinear exactly because for $\xi \in J'$ and $b \in B'$ we have $(b \otimes 1)\xi = (1 \otimes b)\xi$. By Lemma 49.6.3 and the fact that $\omega _{B/A} = \mathop{\mathrm{Hom}}\nolimits _ A(B', A) \otimes _{B'} B$ we can extend this to a $B$-bilinear pairing as displayed above.

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