Re: Proving enough injectives for modules over a Grothendieck topos

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

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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