The Stacks project

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

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.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BVP. Beware of the difference between the letter 'O' and the digit '0'.