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
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