Proving enough injectives for modules over a Grothendieck topos

Proving enough injectives for modules over a Grothendieck topos

[Colin McLarty]

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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Re: Proving enough injectives for modules over a Grothendieck topos

[Prof. Peter Johnstone]

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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Re: Proving enough injectives for modules over a Grothendieck topos

[Colin McLarty]

This proof really fell through the cracks.  The argument from
injective Abelian groups to injective R-modules was known by 1960.
The last piece was in place in 1974 with Barr's theorem on coverings
with AC.  The proof may be published somewhere but I can't find it and
Peter suggests it is not.

Its absence works mischief.  Eisenbud COMMUTATIVE ALGEBRA (p. 621)
proves modules (n Set) have enough injectives and then sends readers
off to Hartshorne for the Godement construction to prove modules over
topological spaces have enough injectives.   But he has in effect
already proved it for all Grothendieck toposes!  He merely has to send
readers off to Mike's paper or Peter's (1977) book for the AC result.
While Eisenbud states results explicitly for module categories over
Sets, he frames them to hold much more generally and he sends readers
to sources for that generality.

I am glad to hear it will be in the Elephant.

best, Colin

PS thanks to Carsten Butz for reminding me of Andreas Blass
"Injectivity, projectivity, and the axiom of choice" (Trans. AMS
Volume 255, November 1979) for both history and a proof that choice
will be required.


2010/10/11 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
>

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Re: Proving enough injectives for modules over a Grothendieck topos

[dalizan]

Dear Peter and Colin,

The result (and more general one asserting that if a variety V of
universal algebras has enough injectives then the  category of
V-algebras in a Grothendieck topos also has enough injectives) is
published in M.M. Ebrahimi, M. M., Algebra in a topos of sheaves:
Injectivity in quasi- equational classes, J. Pure and Appl. Alg., 26
(1982), 269–280. The different proof of the same result is given in
D.
Zangurashvili, Some categorical algebraic properties in quasi-varieties
of algebras in a Grothendieck topos, Bull. Acad. Sci. Georgian SSR,
139,
N1, 1990, 25-28. The second paper is mentioned in the Elephant.
Together
with the property to have enough injectives, the  amalgamation,
congruence extension, transferability properties and the property to
have enough absolute retracts in the categories of V- algebras in a
Grothendieck topos are studied in these papers.

Best regards,
Dali Zangurashvili





On Mon, 11 Oct 2010 10:19:12 -0400, Colin McLarty
<colin.mclarty@case.edu> wrote:
> This proof really fell through the cracks.  The argument from
> injective Abelian groups to injective R-modules was known by 1960.
> The last piece was in place in 1974 with Barr's theorem on coverings
> with AC.  The proof may be published somewhere but I can't find it and
> Peter suggests it is not.
> 
> Its absence works mischief.  Eisenbud COMMUTATIVE ALGEBRA (p. 621)
> proves modules (n Set) have enough injectives and then sends readers
> off to Hartshorne for the Godement construction to prove modules over
> topological spaces have enough injectives.   But he has in effect
> already proved it for all Grothendieck toposes!  He merely has to send
> readers off to Mike's paper or Peter's (1977) book for the AC result.
> While Eisenbud states results explicitly for module categories over
> Sets, he frames them to hold much more generally and he sends readers
> to sources for that generality.
> 
> I am glad to hear it will be in the Elephant.
> 
> best, Colin
> 
> PS thanks to Carsten Butz for reminding me of Andreas Blass
> "Injectivity, projectivity, and the axiom of choice" (Trans. AMS
> Volume 255, November 1979) for both history and a proof that choice
> will be required.
> 
> 
> 2010/10/11 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
>>
> 
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]



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