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 ************************************************************