[FOM] Proving FLT in PA

Colin McLarty colin.mclarty at case.edu
Tue Feb 21 23:50:40 EST 2006

>----- Original Message ----- 
>From  John Baldwin <jbaldwin at uic.edu> 
>Date  Tue, 21 Feb 2006 21:52:33 -0600 (Central Standard Time) 
>Can anyone provide some examples of the kinds of results in Algebraic 
>geometry which do use Grothendieck universes?

Of course the obvious ones are those that refer to universes.  So all 
the statements of SGA 4 that talk about U-small categories, where U is 
a universe, use universes. 

Contrary to the expectations of at least one anonymous expert, the 
theorems on cohomology in chapter III of Hartshorne ALGEBRAIC GEOMETRY 
involve quantifying over all functors from various categories the size 
of the category of all Abelian groups to the category of all Abelian 
groups.  These at least quantify over proper classes, and really the 
best way to understand them is as referring to e.g. the category of all 
U-small Abelian groups for some universe U.

If you mean results on number theory or classical varieties which 
actually require universes, there are probably none although, as Joe 
Shipman notes, no one seems to have really proved this. 

If you mean results on number theory or classical varieties whose known 
proofs in fact use universes, then that is everything that uses results 
from the Grothendieck school.  For example, Wiles' proof of FLT refers 
to Altman and Kleiman on Grothendieck duality.  And this includes all 
proofs that use results from Hartshorne chapter III.

best, Colin

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