Re: Proving enough injectives for modules over a Grothendieck topos

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

Dear Colin,

Yes, the argument is correct, and it'll be in volume 3 of the Elephant.
I don't know why it hasn't been published elsewhere.

Peter

On Sun, 10 Oct 2010, Colin McLarty wrote:

> 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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