Idea behind the definition of different ideal

Question

Let $L/K$ be an extension of number fields. Let $I$ be a fractional ideal in $L$ and $$I^*:=\{x\in L \mid \text{Tr}_{L/K}(xI)\subset \mathcal{O}_K\}.$$ The different of $I$ is the following fractional ideal $$\mathcal{D}_{L/K}(I):=(I^*)^{-1}.$$ I understand the importance of the different ideal in the study of ramification. For example, we know that if $P$ is a prime in $\mathcal{O}_K$ and $P\mathcal{O}_L=Q^eI$, with $(Q, I)=1$, then $Q^{e-1}\mid \mathcal{D}_{L/K}(\mathcal{O}_L)$. But I don't understand what is the idea behind its definition. What (historically) led to that definition?

Answer 1

I don’t think that the analogy with lattices in R^n is just a « similarity » as you say. On the contrary, since the trace pairing is a non degenerate bilinear form when the field extension E/F (not just K/Q) is separable, the definition of the « dual » of a lattice (extending naturally the one given in K. Conrad’s notes) is a general one, which is available as soon as one is given a non degenerate bilinear form together with « integral structures » inside the fields involved (here, the existence of the rings of integers). From the point of view of geometry of lattices,« the different ideal could be considered as a measure of how much O_E fails to be self-dual as a lattice in E » (op. cit.). As you point out, the interest of the different in number theory proper lies in Dedekind’s theorem (= « central theorem » of loc. cit.) which characterizes ramification. The classical discriminant does the same job, but the single fact that the discriminant ideal is the norm of the different ideal shows that the second invariant is a finer one. For example, as you notice, it gives the ramification index in the case of tame ramification (the theory of wild ramification is more complicated, it requires to put into the game the so called Hasse-Herbrand functions, see e.g. chapter IV of Serre’s book « Corps Locaux »). Concerning the origin of the notion, K. Conrad suggests that it could be related to differentiation, and this may be actually the case if you think of Kähler differentials, which provide an adaptation of differential forms to arbitrary commutative rings or schemes. In chapter III, §7 of his book op. cit., Serre shows that the universal ring Omega(O_F,O_E) of the O_F-differentials of the ring O_E is a cyclic O_E-module, the annihilator of which is the different of E/F. But since Kähler comes much later than Dedekind, I still wonder about the origin of the name « different ».