The Stacks project

Lemma 49.4.2. Let $A \to B$ be a flat quasi-finite map of Noetherian rings. Then there is at most one trace element in $\omega _{B/A}$.

Proof. Let $\mathfrak q \subset B$ be a prime ideal lying over the prime $\mathfrak p \subset A$. By Algebra, Lemma 10.145.2 we can find an ├ętale ring map $A \to A_1$ and a prime ideal $\mathfrak p_1 \subset A_1$ lying over $\mathfrak p$ such that $\kappa (\mathfrak p_1) = \kappa (\mathfrak p)$ and such that

\[ B_1 = B \otimes _ A A_1 = C \times D \]

with $A_1 \to C$ finite and such that the unique prime $\mathfrak q_1$ of $B \otimes _ A A_1$ lying over $\mathfrak q$ and $\mathfrak p_1$ corresponds to a prime of $C$. Observe that $\omega _{C/A_1} = \omega _{B/A} \otimes _ B C$ (combine Lemmas 49.2.5 and 49.2.7). Since the collection of ring maps $B \to C$ obtained in this manner is a jointly injective family of flat maps and since the image of $\tau _{B/A}$ in $\omega _{C/A_1}$ is prescribed the uniqueness follows. $\square$

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