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 (49.2.3.1) 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.

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).