Canonical Isomorphisms In More Advanced Mathematics

Table Of Links

Abstract

1. Acknowledgements & Introduction

2. Universal properties

3. Products in practice

4. Universal properties in algebraic geometry

5. The problem with Grothendieck’s use of equality.

6. More on “canonical” maps

7. Canonical isomorphisms in more advanced mathematics

8. Summary And References

Canonical Isomorphisms In More Advanced Mathematics

We finish this paper by taking a look at some more advanced topics. We start with the Langlands program. In Langlands’ 1968 paper [Lan97] he proves a theorem which is now referred to as “the proof of the local and global Langlands conjectures for abelian algebraic groups”, although as Langlands himself explains, the result is an exercise using standard results from local and global class field theory. The set-up is that K/F is a finite Galois extension of fields, T is a torus over F which splits over K, L is the character group of T with its action of Gal(K/F), and Tb is the dual complex torus of T equipped with the induced action of Gal(K/F). We directly quote part of the first theorem in the paper.

\ “Theorem 1. If K is a global or local field there is a canonical isomorphism of H1 c (WK/F , Tb) with the group of generalized characters of HomGal(K/F )(L, CK).” Here, if K is local or global, then the Weil group WK/F is a certain topological group equipped with a map to Gal(K/F), and H1 c denotes continuous cocycles over coboundaries.

The issue with this “theorem” is that, just like the first isomorphism theorem, it is not a theorem – it is more than this. This theorem cannot be applied without reading the proof: this is in fact a sure fire sign that it is not actually a theorem. From a formalist point of view, theorems are mathematical objects which are completely encapsulated by their statements. Here is an argument to show that there must be more going on in this “theorem”.

\ The claim is that one group (a cohomology group) is canonically isomorphic to another group (a character group). Both groups in question are abelian, and are coming from completely different worlds; the cohomology group H1 c (WK/F , Tb) is a Galois-theoretic object and hence “algebraic”; the generalized characters are an automorphic object and hence “analytic”.

\ What Langlands gives in the proof, of course, is the explicit construction of a map from the cohomology group to the character group; part of his contribution is a definition of this map via an “explicit formula”; it is not completely constructive (in the sense of constructive mathematics) because it involves the inverse of a map which is only shown to be a bijection via a nonconstructive cohomological argument, and the inverse of a constructive bijection might not be constructive (see [Buz19]). However, it is an explicit definition. Let’s call it d. Now consider the function d ∗ obtained by composing d with the inversion map on the target abelian group.

\ This is still a bijection. Is it also “canonical”? This question cannot be answered, because the word “canonical” has no formal mathematical definition. However, unlike the example of the first isomorphism theorem, here the problem is genuinely worse. The issue in fact goes back to class field theory, where one of the main theorems is a “canonical” isomorphism between an abelianised global Galois group and an adelic group. Again both of these topological groups are abelian, so given one “canonical” isomorphism we can compose with inversion and get another one. Both of these “canonical” isomorphisms are known to mathematicians – indeed they have different names!

\ One is them is “the isomorphism of class field theory sending an arithmetic Frobenius element to a uniformiser” and the other is “the isomorphism of class field theory sending a geometric Frobenius element to a uniformiser”; a “geometric Frobenius element” is just defined to be the inverse of an arithmetic Frobenius element. Which of these maps is the “canonical” one? In my experience, both of these normalisations are used.

\ For example people working in the area of the cohomology of Shimura varieties use the geometric normalisation, but people working in the theory of Heegner points (and hence with Tate modules, which are homology groups) use the arithmetic one, in both cases because it locally minimises the number of minus signs in the results. There seems to be no good answer to the question of which isomorphism is “preferred” in mathematics – different mathematicians working in different areas prefer different choices, and have coherent reasons to prefer their choice over the other.

\ Another place where sign issues pervade mathematics is in the theory of homological algebra, also now a key tool being used in the Langlands program. Let us consider a very simple case, namely group cohomology. If G is a group and we have a short exact sequence 0 → A → B → C → 0 of G-modules, what is “the” definition of “the” boundary map H0 (G, C) → H1 (G, A)? One starts with a G-invariant element c ∈ C, lifts it to b ∈ B, and now given g ∈ G we observe that b and gb both map to c, so their difference maps to zero and is hence in A.

\ The 1-cocycle representing the cohomology class sends g to this difference. But is this difference b − gb or gb − b? Which is the “canonical” choice? It has taken me a long time to realise that this question does not have a preferred answer. Grothendieck was well aware of this; his concept of a universal δ-functor solves this problem. As part of the data of a universal δ-functor one has to supply the boundary homomorphisms (in all degrees); they do not come to us by magic.

\ As Grothendieck knew, it is not enough to define a cohomology theory by defining the cohomology groups; the extra data of the boundary homomorphisms must also be supplied: these are only “canonical up to a unit”, and −1 is a unit in the integers. Signs are also a nightmare whenever one is collapsing a double complex to a single complex; there is no “canonical” way to do this, and a choice must be made. Is the same choice made throughout the literature?

\ The answer, unfortunately, is “no”. This means that extreme care must be taken when quoting results from more than one reference in this area. Again I refer the reader to Conrad’s book [Con00], where in the introduction great lengths are taken to explain incompatibilities in the references he wants to cite. Strictly speaking, this is how all mathematics should look, unpleasant as it seems. And when it comes to formalisation of this mathematics, as it one day will, these things really will matter.

:::info Author: KEVIN BUZZARD

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:::info This paper is available on arxiv under CC BY 4.0 DEED license.

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