FOM: Another gap in Wiles's proof?

[Colin McLarty]

Joe Shipman <shipman at savera.com> has looked for references to Grothendieck
universes in the book "Modular Forms and Fermat's Last Theorem". He is
especially interested in the claim that the Universe assumption can always
be eliminated from this kind of proof.

It is actually much simpler than his concerns, and much more to do with the
particular proofs. We know that in fact the constructions used in derived
functor cohomology can be "dropped down" below the level of the categories
and functors one actually thinks about, and this in a coherent way, so that
the proofs can still be given without assuming universes. The very "large
scale" steps are fairly explicitly constructive. We can get around them by
just doing the constructions without ever saying what they construct.

As a trivial analogy: You and I know there is a bijection from the natural
numbers N to the set NxN of pairs of naturals. But we can also paraphrase
this to eliminate the infinite sets and the function, by specifying a
formula f(n,m) such that every natural is a value of f(n,m); and if
f(n,m)=f(p,q) then n=p and m=q.

As an example Joe asked me to explicate a statement I quoted from Tate (and
understand that this particular statement nothing special for our purposes,
I just looked for a concise quotable passage).

        "(For any ring R) the category of affine R-group algebras is 
        anti-equivalent to the category of commutative Hopf algebras
        over R"

The problem is merely that, looked at naively, the category of affine
R-group algebras is a proper class. First let me explicate it using
universes. Actually, as I will explicate it here, it is provable in ZFC but
none of its applications are.

"Let R be any ring and U any universe with R as a member. The category of
all affine R-group algebras lying in U (and R-group homomorphisms between
them) has a contravariant equivalence functor to the category of all
commutative Hopf algebras over R lying in U (and Hopf algebra homomorphisms
between them)."

Of course this implies nothing whatever about any affine R-group algebra, or
commutative Hopf algebra over R, unless R is contained in some Grothendieck
universe.

The paraphrase usable without universes is possible only because there are
ZF formulas Corr(R,A,H) and HCorr(R,f,g) explicitly defining the functorial
relation described above. Intuitively, Corr(R,A,H) says A is an affine
R-group algebra corresponding to the commutative Hopf algebra H over R.
Corr(R,f,g) intuitively says f is a homomorphism of affine R-group algebras
and g a homomorphism of Hopf algebras over R, and f corresponds to g. I
myself do not know the formulas Corr or HCorr even informally--I do not know
the correspondence Tate refers to. I do know such formulas exist, indeed
they can be varied in numerous ways fixing formal details of no real
significance (exactly like deciding which set to choose as "the real numbers").

The paraphrase reads:

"Let R be any ring. For each affine R-group algebra A there exists a unique 
commutative Hopf algebra H over R such that Corr(R,A,H), and for each affine
R-group homomorphism f there exists a unique Hopf algebra morphism g such
that HCorr(R,f,g). Further, if Corr(R,A,H) and Corr (R,A',H') and
HCorr(R,f,g) and we have f:A-->A' then we also have g:H'-->H.
    .......  [omit details saying Corr and HCorr also preserve composition 
        and identity homomorphisms]
    .......  [omit details saying HCorr is one-one for homomorphism with any
        given domain and codomain, and that it is onto in the same sense]
And each commutative Hopf algebra H is isomorphic to some H' such that there
is some A with Corr(R,A,H')."