Grothendieck's 1973 Buffalo Colloquium

Grothendieck's 1973 Buffalo Colloquium

[F W Lawvere]

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.

        Thus his definition is essentially well-known, and indeed is
mentioned in such texts as Demazure-Gabriel, Waterhouse, and Eisenbud;
but it is not carried through to the end, resulting in more
complication, rather than less. I myself had learned the functorial point
of view from Gabriel in 1966 at the Strasbourg-Heidelberg-Oberwolfach
seminar and therefore I was particularly gratified when I heard
Grothendieck so emphatically urging that it should replace the one
previously expounded by Dieudonne' and himself.

        He repeated several times that points are not mere points, but
carry Galois group actions. I regard this as a part of the content of his
opinion (expressed to me in 1989) that the notion of
topos was among his most important contributions. A more general
expression of that content, I believe, is that a generalized "gros" topos
can be a better approximation to geometric intuition than a category
of topological spaces, so that the latter should be relegated to an
auxiliary position rather than being routinely considered as "the" default
notion of cohesive space. (This is independent of the use of localic
toposes, a special kind of petit which represents only a minor
modification of the traditional view and not even any modification in the
algebraic geometry context due to coherence). It is perhaps a reluctance
to accept this overthrow that explains the situation 30 years later, when
Grothendieck's simplification is still not widely considered to be
elementary and "basic".

        To recall some well-known ingredients, let A be the category of
finitely-presented commutative algebras over k (or a larger category
closed under the symmetric algebra functor, for some purposes). Then the
underlying set functor R on A serves as the "line", and any system of
polynomial equations with coefficients in k determines also a functor (sub
space of Rn) in the well-known way; in fact, the idea of spec is simply
identified with the Yoneda embedding of A^op. For example, R has a
subfunctor U of invertible elements and another U' such that
U'(A) = {f|f in A, 1/1-f in A}. The  Grothendieck topology for which U and
U' together cover R yields a subtopos Z known as the gros Zariski topos,
which turns out to be the classifying topos for local k-algebras in any
topos. This Z contains all algebraic schemes over k, but also function
spaces Y^X and distribution spaces Hom(R^X,R) for all X,Y in Z. A basic
open subspace of any space X is determined as the pullback U sub f of U
under any map f: X-->R. One has obviously U sub f intersection U sub g =
U sub fg and the intrinsic notion of epimorphism in Z gives a notion of
covering. Thus for a space (functor) to have a finite open covering, each
piece of which is representable, is a restrictive notion available when
needed.

        A point of X is a map spec(L) --> X where L is a field extension
of k.Thus the "points functor" on spaces goes not to the category of
abstract sets but rather is just the restriction operation to the category
of functors on fields only. This is part of what Grothendieck seems to
have had in mind. A serious discontinuity is introduced by passing to the
single underlying set traditionally considered, which is the inductive
limit of the functor of fields. The fact that the latter process does not
preserve products, and hence for example that an algebraic group "is not a
group", was already for Cartier an indication that the traditional
foundation had an unnatural ingredient, but before topos theory one tried
to live with it (for example, I recall great geometers from the 1950s
struggling to accept the new wisdom that +i and -i is one "point"). The
acceptance of the view that, for non-algebraically-closed k, the
appropriate base topos consists not of pure sets but rather of sheaves on
just the simple objects in A, has in fact many simplifying conceptual and
technical advantages; for example this base (in some sense due to
Galois!) is at least qd in the sense of Johnstone, and even atomic Boolean in
the sense of Barr.

        (Technically, to verify that the above limitation to "algebraic" A
gives the usual results requires the use of a Birkhoff Nullstellensatz
which guarantees that there are "enough" algebras which are
finitely-generated as k-modules. The use of a larger A, insuring for
example that spaces of distributions are often themselves representable,
is quite possible, but the precise description of the kind of double
structure which is then topos-theoretically classified needs to be worked
out. Gaeta's notes of Grothendieck's lecture series at Buffalo point out
that A is more closely suited than most categories to serve as a site for
a geometric category, because it is what is now called "extensive" )

        I believe that Grothendieck's point of view could be applied to
real algebraic geometry as well, in several ways, including the following:
Noting that within any topos the adjoint is available which assigns the
ring R[-1] to any rig R, let us concentrate on the needed nature of
positive quantities R. To include the advantages of differential calculus
based on nilpotent elements, let us allow that the ideal of all elements
having negatives can be non-trivial, and indeed include many
infinitesimals, without disqualifying R from being "nonnegative". The ring
generated by R might appear in a more geometric way as the fiber of R^T,
where T is the representor for the tangent-bundle functor. The
classifying topos for the theory of "real rigs", i.e., those for which
1/1+x is a given global operation, contains the classifying topos for
"really local rigs" in the following sense (where "really" has the double
meaning of (1) a strengthening of localness, but also (2) a concept
appropriate to a real (as opposed to complex) environment): The subspace U
of invertible elements in the generic algebra R has a classifying map
R --> omega which of course as above preserves products; but the
distributive lattice omega is in particular also a rig like R, so we can
require that the classifying map be a rig homomorphism (i.e., also take +
to "union"). (Of course, this elementary condition can be phrased in terms
of subspaces of R and of R^2 without involving omega if desired.) The
preservation of addition is a strengthening, possible for positive
quantities, of the usual notion of localness (which on truth values was
only an inequality).

	Does the right adjoint to ( )^T restrict to this really local rig
classifier?


************************************************************
F. William Lawvere
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
************************************************************

Re: Grothendieck's 1973 Buffalo Colloquium

[Colin McLarty]

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..