The Stacks project

49.1 Introduction

In this chapter we study the different and discriminant of locally quasi-finite morphisms of schemes. A good reference for some of this material is [Kunz].

Given a quasi-finite morphism $f : Y \to X$ of Noetherian schemes there is a relative dualizing module $\omega _{Y/X}$. In Section 49.2 we construct this module from scratch, using Zariski's main theorem and étale localization methods. The key property is that given a diagram

\[ \xymatrix{ Y' \ar[d]_{f'} \ar[r]_{g'} & Y \ar[d]^ f \\ X' \ar[r]^ g & X } \]

with $g : X' \to X$ flat, $Y' \subset X' \times _ X Y$ open, and $f' : Y' \to X'$ finite, then there is a canonical isomorphism

\[ f'_*(g')^*\omega _{Y/X} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{X'}}(f'_*\mathcal{O}_{Y'}, \mathcal{O}_{X'}) \]

as sheaves of $f'_*\mathcal{O}_{Y'}$-modules. In Section 49.4 we prove that if $f$ is flat, then there is a canonical global section $\tau _{Y/X} \in H^0(Y, \omega _{Y/X})$ which for every commutative diagram as above maps $(g')^*\tau _{Y/X}$ to the trace map of Section 49.3 for the finite locally free morphism $f'$. In Section 49.9 we define the different for a flat quasi-finite morphism of Noetherian schemes as the annihilator of the cokernel of $\tau _{Y/X} : \mathcal{O}_ X \to \omega _{Y/X}$.

The main goal of this chapter is to prove that for quasi-finite syntomic1 $f$ the different agrees with the Kähler different. The Kähler different is the zeroth fitting ideal of $\Omega _{Y/X}$, see Section 49.7. This agreement is not obvious; we use a slick argument due to Tate, see Section 49.12. On the way we also discuss the Noether different and the Dedekind different.

Only in the end of this chapter, see Sections 49.15 and 49.16, do we make the link with the more advanced material on duality for schemes.

[1] AKA flat and lci.

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