Lemma 49.4.3. Let $A \to B$ be a finite flat map of Noetherian rings. Then $\text{Trace}_{B/A} \in \omega _{B/A}$ is the trace element.

**Proof.**
Suppose we have $A \to A_1$ with $A_1$ Noetherian and a product decomposition $B \otimes _ A A_1 = C \times D$ with $A_1 \to C$ finite. Of course in this case $A_1 \to D$ is also finite. Set $B_1 = B \otimes _ A A_1$. Since the construction of traces commutes with base change we see that $\text{Trace}_{B/A}$ maps to $\text{Trace}_{B_1/A_1}$. Thus the proof is finished by noticing that $\text{Trace}_{B_1/A_1} = (\text{Trace}_{C/A_1}, \text{Trace}_{D/A_1})$ under the isomorphism $\omega _{B_1/A_1} = \omega _{C/A_1} \times \omega _{D/A_1}$ of Lemma 49.2.7.
$\square$

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