The Stacks project

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.

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 $0$th fitting ideal of $\Omega _{Y/X}$.

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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