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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