The Stacks project

Example 49.12.5. Let $A$ be a Noetherian ring. Let $f, h \in A[x]$ such that

\[ B = (A[x]/(f))_ h = A[x, 1/h]/(f) \]

is quasi-finite over $A$. Let $f' \in A[x]$ be the derivative of $f$ with respect to $x$. The ideal $\mathfrak {D} = (f') \subset B$ is the Noether different of $B$ over $A$, is the Kähler different of $B$ over $A$, and is the ideal whose associated quasi-coherent sheaf of ideals is the different of $\mathop{\mathrm{Spec}}(B)$ over $\mathop{\mathrm{Spec}}(A)$.

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View Example 49.12.5 as pdf