Proving enough injectives for modules over a Grothendieck topos

[Colin McLarty <colin.mclarty@case.edu>]

It seems to me there is a nearly elementary proof that the category of
sheaves of modules over any sheaf of rings in a Grothendieck topos has
enough injectives.  I want to check it here because I do not see it
published anywhere.  People normally cite Grothendieck's Tohoku to
prove the result.

To give the outline:  Johnstone TOPOS THEORY (p. 261) proves the
category of Abelian groups in any Grothendieck topos has enough
injectives.  Eisenbud COMMUTATIVE ALGEBRA (p. 621) uses adjunctions
between module categories (over sets) to turn injectives in Abelian
groups into enough injectives in any module category (over sets).
These adjunctions lift to the categories of Modules in ringed toposes
(E,R) over a fixed base topos E -- and so the category of R-Modules in
E has enough injectives.

In more detail, Eisenbud does not put it in terms of adjunctions.  But
in effect he shows the underlying group functor R-Mod-->AbGrp from
R-modules to Abelian groups (over Set) has a right adjoint taking each
Abelian group A to the R-module of additive functions from R to A
(i.e. the R-module of Abelian group morphisms from the underlying
group of R to A).

Of course the underlying group functor is also left exact (indeed has
a left adjoint, change of base from the integers Z to R).  And a right
adjoint to any exact functor preserves injectives.  So take any
R-module M, embed it additively into an injective Abelian group M>-->Q
and map this back to R-modules by the right adjoint.  It is easy to
see that M has a monic to the domain of the morphism and so embeds in
an injective R-module.

Grothendieck in SGA 4 casts this in terms of ringed toposes.  So far
as I can see he does not talk about the right adjoint to the
underlying group functor, and does not connect this to injectives, but
it lifts routinely unless I am missing something.

Every ring R in a topos E is also an additive Abelian group in E.  On
the other hand, for any Abelian group A in E we can form the group
HOM(R,A) where I write HOM to indicate this is an E object and not the
set of arrows.    But HOM(R,A) is also an R-module by the action of R
on arguments: any r \in R times a function f \in HOM(R,A) is the
function that multiplies by r first and then applies f.

So HOM(R,_) is a functor from Abelian groups in E to R-modules in E,
and the same straightforward calculation as we use  in Set shows it is
right adjoint to the underlying group functor.  So it preserves
injectives, and the same step as in Eisenbud shows we can use it to
embed every R-module into an injective R-module.

Is that reasonably clear?  Is it right?

best, Colin


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