Re: Grothendieck's 1973 Buffalo Colloquium

[Colin McLarty <cxm7@po.cwru.edu>]

I was just looking at Grothendieck's statement about schemes in EGA 3:

Pour obtenir un langage qui ``colle" sans effort à l'intuition 
g\'{e}om\'{e}trique, et \'{e}viter des circonlocutions
insupportables \`{a} la longue, nous identifions toujours un 
pr\'{e}sch\'{e}ma $X$ sur un autre $S$ au foncteur
\mbox{\clarrow{(\mathrm{Sch}/S)^\mathrm{o}}{\mathrm{Ens}}} qu'il 
repr\'{e}sente,

"To make the language stick to geometric intuition, and to avoid finally 
unbearable circumlocutions, we will always identify a scheme X over another 
S, with the functor from Sch/S to Set that it represents."

This quote is from the Springer Verlag edition page VI. This edition was 
printed in 1970. I do not yet know if it is printed in the earlier IHES 
edition.

The IHES edition of EGA chapter 0, printed in  1960, does urge the 
functorial rather than topological space conception of a sheaf. "We 
systematically abstain from using espaces etales ... we never consider a 
sheaf a topological space"  (p. 25).

best, Colin

___________________________________________________________

t 17:34 29/03/2003 -0500, Lawvere wrote:

>Thierry Coquand recently asked me
>
>  > In your "Comments on the Development of Topos Theory" you refer
>  > to a simpler alternative definition of "scheme" due to Grothendieck.
>  > Is this definition available at some place?? Otherwise, it it possible
>  > to describe shortly the main idea of this alternative definition??
>
>Since several people have asked the same question over the years, I
>prepared the following summary which, I hope, will be of general interest:
>
>         The 1973 Buffalo Colloquium talk by Alexander Grothendieck had as
>its main theme that the 1960 definition of scheme (which had required as a
>prerequisite the baggage of prime ideals and the spectral space, sheaves
>of local rings, coverings and patchings, etc.), should be abandoned AS the
>FUNDAMENTAL one and replaced by the simple idea of a good functor from
>rings to sets. The needed restrictions could be more intuitively and more
>geometrically stated directly in terms of the topos of such functors, and
>of course the ingredients from the "baggage" could be extracted when
>needed as auxiliary explanations of already existing objects, rather than
>being carried always as core elements of the very definition..