Table Of Links
4. Universal properties in algebraic geometry
5. The problem with Grothendieck’s use of equality.
7. Canonical isomorphisms in more advanced mathematics
The Problem With Grothendieck’s Use Of Equality
The above story is evidence that there is a missing argument in the literature, and that the statement of [Sta18, Tag 00EJ] in the stacks project is, strictly speaking, not strong enough deduce the claim that the structure presheaf is a sheaf just before [Sta18, Tag 01HU]. However the algebraic geometry community does not regard this issue as problematic, and indeed the way the theory is presented it is extremely difficult to even notice that this is an issue. I believe that one major reason for this can be traced back to Grothendieck’s seminal work EGA ([Gro60]), where he and Dieudonne develop the foundations of modern algebraic geometry. The word “canonique” appears hundreds of times in EGA1, with no definition ever supplied.
The argument which caused all our trouble is in section 1.3. Grothendieck claims that if R is a commutative ring and f, g are two elements contained in the same prime ideals (for example we could have R = C[X], f = X2 and g = X3 ) and if S is the multiplicative subset of R containing the s which divide some power f n of f (or equivalently divide some power of g) then R[1/f] and R[1/g] both “s’identifient canoniquement” with ring R[1/S] and hence R[1/f] = R[1/g] (see for example section 1.3.3 of [Gro60] where the stronger statement M[1/f] = M[1/g] is claimed for any R-module M).
Of course we certainly know what Grothendieck means – R[1/f] and R[1/g] are uniquely isomorphic as R-algebras, so we will identify them via this isomorphism and then call it an equality. Lean would tell Grothendieck that this equality simply isn’t true and would stubbornly point out any place where it was used. Let me emphasize once more: Grothendieck was well aware of what he was saying, but Lean would argue that he was confusing = and ∼=.
The idea that objects could be “canonically” isomorphic seems to have been taken on with some enthusiasm by many in the mathematical community 3 . By the 1970s it was clear that Grothendieck’s ideas were here to stay: his discovery of ´etale cohomology had led to a proof of the Weil conjectures, fundamental statements about the number of solutions to polynomial equations over finite fields which could be made without any reference to the theory of schemes, but which apparently could only be proved using them.
The book [Mil80] from 1980, one of the first textbook treatments of etale cohomology, contains in its “Terminology and conventions” section, the convention that “a canonical isomorphism [is denoted by] =”. Nowhere in any of these texts is any definition of the word “canonical”. Gordon James told me that he once asked John Conway what the word meant, and Conway’s reply was that if you and the person in the office next to yours both write down a map from A to B, and it’s the same map, then this map is canonical. This might be a good joke, but it is not a good definition.
In Milne’s book we do not just see localisations – we see pullbacks and pushforwards of sheaves, maps defined as adjoint functors, we see limits, colimits, quotients by equivalence relations, tensor products of modules, and constructions coming from Grothendieck’s six functor formalism. All of these constructions are universal and no doubt any maps produced by these universal properties are “canonical”. But whenever Milne is (ab)using the equality symbol, there should in theory be a check that whatever theory is being developed is valid for any object satisfying the universal property in question.
The reason that this is not happening is the devious technique of arguing that two objects which satisfy the same universal property are “canonically” isomorphic and hence “can be identified” and hence “are equal”. To give a random example from [Mil80]: in Section II.3, Remark 3.1(f) Milne talks about the direct and inverse image of a sheaf under a morphism of sites, and claims that (π ′π)∗ = π ′ ∗π∗ and π ∗π ′∗ = (π ′π) ∗ . Equality of functors is in some sense not a sensible mathematical notion, as it boils down to many statements about equality of objects in a category, and equality of objects is not invariant under equivalence of categories – it is hence sometimes referred to as an “evil” concept for this reason.
Moreover, as well as being evil, the claim is not actually correct for pullbacks, because “the” pullback of a sheaf involves making a choice of an explicit construction of sheafification of a presheaf, and the set-theoretic equality π ∗π ′∗F = (π ′π) ∗F fails for essentially the same reason as the equality R[1/f][1/g] = R[1/fg] fails. What is actually going on is a functorial identification which satisfies some unwritten compatibilities – the details are “left to the reader”. I thank the referee for pointing out that the implicit use of natural isomorphisms in sheaf cohomology when dealing with pullback functors goes back to Godement’s book on sheaf theory.
Before one formalises this kind of mathematics, one will have to think carefully about precisely which of the properties which uniquely define pullbacks of sheaves up to unique isomorphism should be used as the definition of a pullback, and, just as in the case of localisation of rings, it might not be the one which first springs to mind: the definition of pullback as being adjoint to pushforward involves quantification over all sheaves on a site and hence might not be the most ergonomic characterisation.
Author: KEVIN BUZZARD
This paper is