Lemma 49.7.4. Let $A$ be a ring. Let $n \geq 1$ and $f_1, \ldots , f_ n \in A[x_1, \ldots , x_ n]$. Set $B = A[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n)$. The Kähler different of $B$ over $A$ is the ideal of $B$ generated by $\det (\partial f_ i/\partial x_ j)$.

Proof. This is true because $\Omega _{B/A}$ has a presentation

$\bigoplus \nolimits _{i = 1, \ldots , n} B f_ i \xrightarrow {\text{d}} \bigoplus \nolimits _{j = 1, \ldots , n} B \text{d}x_ j \rightarrow \Omega _{B/A} \rightarrow 0$

by Algebra, Lemma 10.131.9. $\square$

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