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The Stacks project

Lemma 49.9.3. Let f : Y \to X be a flat quasi-finite morphism of Noetherian schemes. Let V = \mathop{\mathrm{Spec}}(B) \subset Y, U = \mathop{\mathrm{Spec}}(A) \subset X be affine open subschemes with f(V) \subset U. If \omega _{Y/X}|_ V is invertible, i.e., if \omega _{B/A} is an invertible B-module, then

\mathfrak {D}_ f|_ V = \widetilde{\mathfrak {D}}

as coherent ideal sheaves on V where \mathfrak {D} \subset B is the Noether different of B over A.

Proof. Consider the map

\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(\omega _{Y/X}, \mathcal{O}_ Y) \longrightarrow \mathcal{O}_ Y,\quad \varphi \longmapsto \varphi (\tau _{Y/X})

The image of this map corresponds to the Noether different on affine opens, see Lemma 49.6.7. Hence the result follows from the elementary fact that given an invertible module \omega and a global section \tau the image of \tau : \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\omega , \mathcal{O}) = \omega ^{\otimes -1} \to \mathcal{O} is the same as the annihilator of \mathop{\mathrm{Coker}}(\tau : \mathcal{O} \to \omega ). \square


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