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First some context. In most algebraic number theory textbooks, the notion of discriminant and different of an extension of number fields $L/K$, or rather, of the corresponding extension $B/A$ of their rings of algebraic integers is defined. The discriminant, an ideal of $A$, is the ideal generated by the discriminant of the quadratic form $\text{tr}(xy)$ on $B$. The different, an ideal of $B$, is the inverse of the fractional ideal $c$ of $L$ defined by $c=\{x \in L, \text{tr}(xy) \in A \ \forall y \in B\}$. The norm of the different is the discriminant.

Now the discriminant makes sense in a much more general context, say for any extension of (commutative) rings $B/A$ that is finite projective, since the trace map $\operatorname{tr}$ makes sense in this context. My question is: is there a standard definition of the different in this context? if so, where can I find it in the literature, if possible with the basic results about it?

I am pretty sure the answer to the first question is yes, but I have not been able to find a reference. The problem when I try to use google or MathSciNet seems to be that "different" is not a very discriminant name: almost every paper in mathematics contains it.

Let me propose an answer to my own question: we could define the different of $B/A$ by the Fitting ideal of the universal $B$-modules of differentials $\Omega_{B/A}$. The fact that it gives the correct definition in the number field cases is [Serre, Local Fields, chapter III, Prop. 14], and moreover it behaves well under base change. This definition may very possibly be a remembrance of something I had heard in an earlier life. But even if it is the correct definition, I'd like to know a reference where it is stated.

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Welcome to MO, Joël. It is a very nice place and allows people to use the trémas in their names if they have any. –  Chandan Singh Dalawat Sep 16 '10 at 14:24
Your proposed definition can also be found (in a geometric context) in "The Geometry of Schemes" by Eisenbud-Harris (see Chapter V), although the fact that it agrees with the notion in number theory is not discussed there. –  Sam Lichtenstein Sep 16 '10 at 15:25
+1 for "different is not a very discriminant name" –  Cam McLeman Sep 16 '10 at 19:25

2 Answers 2

up vote 14 down vote accepted

In chapter 8 (entitled "Traces, Complementary Modules, and Differents") of the book Residues and Duality for Projective Algebraic Varieties by Kunz, he gives exactly the definition you propose and proves some basic results about its properties.

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Thanks. The book seems exactly what I need. It is not at my library but I will receive it soon. –  Joël Sep 17 '10 at 13:33
Three months after, an update: I have found everything I wanted in the book you need. In particular the notion of Noether's different, which fits exactly my need. Thanks again. –  Joël Dec 16 '10 at 4:43

Tu peux regarder la section 1 de mon article "La filtration de Harder-Narasimhan des schémas en groupes finis et plats":


Tout ce dont tu rêves y est démontré.

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"Tout ce dont tu rêves y est démontré." Quelle belle phrase... –  Olivier Sep 17 '10 at 11:57
But it is in French. I can't read this language. Non sérieusement, c'est très bien, merci, mais pour qppliquer tes resultats il me reste à savoir si dans mon cas B/A est syntomique. A part celle que tu dis (avec A et B regulier, ce qui est un peu trop fort pour moi), il y a d'autres conditions suffisantes simples de syntomicité ? –  Joël Sep 17 '10 at 13:31
Joel: I have an (unpublished, I think) survey article by "B.C. and M.L." (and the B.C. is Brian Conrad) on Tate's p-divisible groups paper, and in this article they work out the theory of differents for extensions which are finite locally free and Gorenstein, and prove norm of different equals discriminant etc. Let me know if you need more details (I suspect that Laurent's answer may be enough for you). –  Kevin Buzzard Sep 17 '10 at 14:33

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