From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2229 Path: news.gmane.org!not-for-mail From: Colin McLarty Newsgroups: gmane.science.mathematics.categories Subject: Re: Grothendieck's 1973 Buffalo Colloquium Date: Sun, 30 Mar 2003 21:22:43 -0500 Message-ID: <5.2.0.9.0.20030330210746.00b33d90@pop.cwru.edu> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241018511 3141 80.91.229.2 (29 Apr 2009 15:21:51 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:21:51 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Mar 31 09:43:29 2003 -0400 X-Keywords: Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 31 Mar 2003 09:43:29 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 18zzXw-0003vo-00 for categories-list@mta.ca; Mon, 31 Mar 2003 09:41:56 -0400 In-reply-to: X-Sender: cxm7@pop.cwru.edu (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.0.9 Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 54 Xref: news.gmane.org gmane.science.mathematics.categories:2229 Archived-At: I was just looking at Grothendieck's statement about schemes in EGA 3: Pour obtenir un langage qui ``colle" sans effort =E0 l'intuition=20 g\'{e}om\'{e}trique, et \'{e}viter des circonlocutions insupportables \`{a} la longue, nous identifions toujours un=20 pr\'{e}sch\'{e}ma $X$ sur un autre $S$ au foncteur \mbox{\clarrow{(\mathrm{Sch}/S)^\mathrm{o}}{\mathrm{Ens}}} qu'il=20 repr\'{e}sente, "To make the language stick to geometric intuition, and to avoid finally=20 unbearable circumlocutions, we will always identify a scheme X over another= =20 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=20 printed in 1970. I do not yet know if it is printed in the earlier IHES=20 edition. The IHES edition of EGA chapter 0, printed in 1960, does urge the=20 functorial rather than topological space conception of a sheaf. "We=20 systematically abstain from using espaces etales ... we never consider a=20 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..