The Stacks project

Definition 49.4.1. Let $A \to B$ be a flat quasi-finite map of Noetherian rings. The trace element is the unique1 element $\tau _{B/A} \in \omega _{B/A}$ with the following property: for any Noetherian $A$-algebra $A_1$ such that $B_1 = B \otimes _ A A_1$ comes with a product decomposition $B_1 = C \times D$ with $A_1 \to C$ finite the image of $\tau _{B/A}$ in $\omega _{C/A_1}$ is $\text{Trace}_{C/A_1}$. Here we use the base change map ( and Lemma 49.2.7 to get $\omega _{B/A} \to \omega _{B_1/A_1} \to \omega _{C/A_1}$.

[1] Uniqueness and existence will be justified in Lemmas 49.4.2 and 49.4.6.

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Your email address will not be published.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 0BT6. Beware of the difference between the letter 'O' and the digit '0'.