Definition 49.7.1. Let f : Y \to X be a morphism of schemes which is locally of finite type. The Kähler different is the 0th fitting ideal of \Omega _{Y/X}.
49.7 The Kähler different
Let A \to B be a finite type ring map. The Kähler different is the zeroth fitting ideal of \Omega _{B/A} as a B-module. We globalize the definition as follows.
The Kähler different is a quasi-coherent sheaf of ideals on Y.
Lemma 49.7.2. Consider a cartesian diagram of schemes
\xymatrix{ Y' \ar[d]_{f'} \ar[r] & Y \ar[d]^ f \\ X' \ar[r]^ g & X }
with f locally of finite type. Let R \subset Y, resp. R' \subset Y' be the closed subscheme cut out by the Kähler different of f, resp. f'. Then Y' \to Y induces an isomorphism R' \to R \times _ Y Y'.
Proof. This is true because \Omega _{Y'/X'} is the pullback of \Omega _{Y/X} (Morphisms, Lemma 29.32.10) and then we can apply More on Algebra, Lemma 15.8.4. \square
Lemma 49.7.3. Let f : Y \to X be a morphism of schemes which is locally of finite type. Let R \subset Y be the closed subscheme defined by the Kähler different. Then R \subset Y is exactly the set of points where f is not unramified.
Proof. This is a copy of Divisors, Lemma 31.10.2. \square
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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