Lemma 49.12.6. Let $S$ be a Noetherian scheme. Let $X$, $Y$ be smooth schemes of relative dimension $n$ over $S$. Let $f : Y \to X$ be a locally quasi-finite morphism over $S$. Then $f$ is flat and the closed subscheme $R \subset Y$ cut out by the different of $f$ is the locally principal closed subscheme cut out by

$\wedge ^ n(\text{d}f) \in \Gamma (Y, (f^*\Omega ^ n_{X/S})^{\otimes -1} \otimes _{\mathcal{O}_ Y} \Omega ^ n_{Y/S})$

If $f$ is étale at the associated points of $Y$, then $R$ is an effective Cartier divisor and

$f^*\Omega ^ n_{X/S} \otimes _{\mathcal{O}_ Y} \mathcal{O}(R) = \Omega ^ n_{Y/S}$

as invertible sheaves on $Y$.

Proof. To prove that $f$ is flat, it suffices to prove $Y_ s \to X_ s$ is flat for all $s \in S$ (More on Morphisms, Lemma 37.16.3). Flatness of $Y_ s \to X_ s$ follows from Algebra, Lemma 10.128.1. By More on Morphisms, Lemma 37.62.10 the morphism $f$ is a local complete intersection morphism. Thus the statement on the different follows from the corresponding statement on the Kähler different by Lemma 49.12.3. Finally, since we have the exact sequence

$f^*\Omega _{X/S} \xrightarrow {\text{d}f} \Omega _{Y/S} \to \Omega _{Y/X} \to 0$

by Morphisms, Lemma 29.32.9 and since $\Omega _{X/S}$ and $\Omega _{Y/S}$ are finite locally free of rank $n$ (Morphisms, Lemma 29.34.12), the statement for the Kähler different is clear from the definition of the zeroth fitting ideal. If $f$ is étale at the associated points of $Y$, then $\wedge ^ n\text{d}f$ does not vanish in the associated points of $Y$, which implies that the local equation of $R$ is a nonzerodivisor. Hence $R$ is an effective Cartier divisor. The canonical isomorphism sends $1$ to $\wedge ^ n\text{d}f$, see Divisors, Lemma 31.14.10. $\square$

Comment #7485 by Hao Peng on

The central term of the exact sequence should by $\Omega_{Y/S}$

Comment #7486 by Hao Peng on

Borrowing drom differential geometry, would it be better to use $df^*$ instead if $df$?

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