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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