Definition 49.9.1 (0BW4)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 49: Discriminants and Differents
Section 49.9: The different
Definition 49.9.1 (cite)

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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 July 14, 2022 at 09:18

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 July 14, 2022 at 09:37

in fact in tag0BWJ, $f: Y\to X$ may not by Noetherian

Comment #7631 by Stacks Project on August 14, 2022 at 11:18

Indeed! Thanks very much. Fix is here.

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