The Stacks project

Definition 49.9.1. Let $f : Y \to X$ be a flat locally quasi-finite morphism of locally Noetherian schemes. Let $\omega _{Y/X}$ be the relative dualizing module and let $\tau _{Y/X} \in \Gamma (Y, \omega _{Y/X})$ be the trace element (Remarks 49.2.11 and 49.4.7). The annihilator of

\[ \mathop{\mathrm{Coker}}(\mathcal{O}_ Y \xrightarrow {\tau _{Y/X}} \omega _{Y/X}) \]

is the different of $Y/X$. It is a coherent ideal $\mathfrak {D}_ f \subset \mathcal{O}_ Y$.

Comments (3)

Comment #7483 by Hao Peng on

Should it be that all the propositions are true for Noetherian repalce by locally Noetherian? I don't see why we need Noetherian.

Comment #7484 by Hao Peng on

in fact in tag0BWJ, may not by Noetherian

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View Definition 49.9.1 as pdf