Re: Proving enough injectives for modules over a Grothendieck topos

[dalizan <dalizan@rmi.ge>]

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
>>
> 
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