[FOM] Grothendieck foundations: injectives without transfinite induction

[Colin McLarty]

One project discussed on FOM is to take various specific results in
cohomological number theory and seek weak adequate set theoretic or
arithmetic foundations for them.

A related project is to re-organize the general Grothendieck framework
for number theory without the assumption of universes.  One step in
this which I did not know how to take before turns out to be very
easy.  Indeed it is the only step which looks problematic to me -- but
the Grothendieck framework is a large thing and I do not yet claim to
know there are no other problems.

The simplest version of cohomological methods requires that for every
ring R, every R-module M embeds in an injective R-module.  This is a
classic theorem of Reinhold Baer, insofar as it applies to rings and
modules of sets.  See Baer "Abelian groups that are direct summands of
every containing abelian group" BAMS 1940.  Baer's proof uses
transfinite induction and the relevant cases apparently cannot be
bounded without using the axiom scheme of replacement.  If we want to
give Baer's proof inside some set as "universe" (as we do on
Grothendieck's framework) then that set must model ZFC -- it must be a
Gothendieck universe -- and ZFC does not prove such a set exists.

Further, Grothendieck wants this result not just for rings and modules
of sets but for every sheaf of modules M, over any sheaf of rings R,
in any Grothendieck  topos E.  The usually cited proof of this rests
on Grothendieck's 1957 Tohoku paper, where Baer's transfinite
construction is beautifully lifted to a general categorical context.

I lost some time looking for ways to bound the transfinite induction
so as to eliminate use of the axiom scheme of replacement -- and I
report this as an open problem in my BSL article of last month.  But
in fact a much simpler proof avoids the inductions.  The ingredients
for this proof have been in place since 1974 but it has apparently
never been published.

Eisenbud COMMUTATIVE ALGEBRA p. 621 proves the result for rings and
modules of sets in two steps:  Prove every divisible Abelian group is
injective as an Abelian group, then use simple formalities to prove
this implies every R-module embeds in an injective R-module.  (This
approach has been known at least since 1960.)

The first step is the hard one but it does not require transfinite
induction.  It requires the axiom of choice.  See Andreas Blass
"Injectivity, projectivity, and the axiom of choice" (Trans. AMS
Volume 255, November 1979).  So it will not transfer directly into
every Grothendieck topos.   But Barr showed in 1974 that if we assume
choice in our basic set theory then every Grothendieck topos is
"covered" by one that also satisfies choice, and Johnstone TOPOS
THEORY p. 261 uses Barr's theorem to show every divisible Abelian
group in any Grothendieck topos is injective (as an Abelian group in
that topos).  Barr's theorem does not use the axiom scheme of
separation.

The second step is sheer formalities.  You show that the functor which
takes each R-module M to its underlying (additive) Abelian group M has
both left and right adjoints.  Eisenbud does this only over sets, but
the exact same definitions work in any elementary topos with natural
number object, and a fortiori in any Grothendieck topos.

I will not lay out the topos theory here.  It is in Johnstone.  But I
will explain the adjoints to the underlying Abelian group functor
since Eisenbud does not actually call them that.  Think of an Abelian
group as a Z module, for the ring of integers Z.  For any ring R, the
underlying Abelian group functor from R-modules is just restriction of
scalars along the unique ring morphism Z-->R.  It has a left adjoint
called change of base, which is just tensor with R over Z.  Then
consider the functor taking each Abelian group A to the Abelian group
of all additive functions from R to A.  This group is Eisenbud's
Hom_Z(R,A) and it is in fact an R-module, not just an Abelian group,
in a way that Eisenbud describes.   Eisenbud's Lemma A3.8 amounts to a
proof that this is right adjoint to the underlying Abelian group
functor from R-modules to Abelian groups.

Generalities known at least since Grothendieck's SGA 4 show that
whenever a functor F:C-->C' has a left exact left adjoint, with monic
unit, and C' has enough injectives, then C also has enough injectives.
 This is our case, where C is the category of Abelian groups in any
topos E, and C' the category of R-modules for some ring R in E, and F
is the functor Hom_Z(R,_).

This proof requires at most ZC (Zermelo set theory with choice, and
separation but not replacement).  So it can be given inside any "small
universe" that is any set V_b for b a limit ordinal.  Since ZFC does
prove every set is a member of a small universe (and so of proper
class many), at least this part of Grothendieck's framework can be
given in ZFC without Grothendieck universes.

best, Colin