Promenade 11

The Magical Spectrum - or Innocence

The two powerful ideas that had the most to contribute to the initiation and development of the new geometry are schemes and toposes . Having made their appearance in a somewhat symbiotic fashion at more or less the same time.

The concept of a locale or of a "Grothendieck topology" ( a preliminary form of the topos) can clearly be discerned in the wake of the scheme. This, in its turn, supplies the needed new language for ideas such as "descent" and "localisation", which are employed at every stage in the development of this theme and of the schematic tools. The more inherently geometric notion of the topos, which one found only implicitely in the work of the following years, really began to define itself clearly from about 1963, with the development of étale cohomology. Bit-by-bit however it took its rightful place as the more fundamental of the two notions. To conclude this guided tour around my opus, I still need to say a few more words about these two principal ideas.

The concept of the scheme is the natural one to start with. As "self-evident" as one could imagine, it comprises in a single concept an infinite series of versions of the idea of an (algebraic) variety, that were previously used( one version for each prime number(*).

(*)It is convenient to include as well the case p = "infinity", corresponding to algebraic varieties of "nul characteristic".
In addition, one and the same "scheme" ( or "variety" in the new sense) can give birth, for each prime number p, to a well-defined "algebraic variety of characteristic p". The collection of these different varieties with different characteristics can thereby be seen as a kind of" (infinite) spectrum of varieties", (one for each characteristic). The "scheme" is in fact this magical spectrum, which connects between them, as so many different "branches", its "avatars", or "incarnations" in all possible characteristics. By virtue of this it furnishes an effective "principle of transition" for tying together these "varieties", arising out of geometries which, up until that point, seemed more or less isolated, cut off from each other. For the present they are all ensconced within a common "geometry" that establishes the connections between them. One might call it Schematic Geometry, the first draft of the "Arithmetic Geometry", which was able to blossom in the coming years.

The very notion of a scheme has a childlike simplicity- so simple, so humble in fact that no one before me had the audacity to take it seriously. So "infantile" in fact, that for many years afterwards, and in spite of all the evidence, for so many of my "learned" colleagues, it was treated as a triviality. In fact I needed several months of lonely investigation to fully convince myself that the idea really "worked" - that this new language,( which, however infantile it might appear, I, in my incurable naivete continued to insist upon as something to be tested) was quite adequate for the understanding of, in a new light, with increased subtlety and in a general setting, some of the most basic geometric intuitions associated with these "geometries of characteristic p". It was a kind of exercise , prejuged by every "well informed" colleague as something idiotic and had the imagination to propose , and even ( nurtured by my private demon...) follow through against all opposition!

Rather than allowing myself to be deterred by the consensus that had laid down the law over what was to be "taken seriously", and what was not, my faith was invested (as it had been in the pas) in the humble voice of phenomena, and that faculty in me which knew how to listen to it. My reward was immediate and above all expectation. In the space of only a few months, without intending to do so, I'd put my finger on several unanticipated yet very powerful tools. They've allowed me, not only to recast ( as if it were play) some old results deemed difficult, in a penetrating light that went far beyond them , but also to approach and solve certain problems in "geometries of characteristic p" that until that moment had appeared inaccessible through all known methods.(*)

(*)The "proceedings" of this "forced inauguration" of the theory of schemes was the topic of my lecture at the International Congress of Mathematicians at Edinborough in 1958. The text of that talk would seem to me to be one of the best introductions to the subject from the aspect of schemes, and such as to perhaps influence a geometrician who reads it to make himself familiar, for better or worse, with the formidable treatise that followed it : Elements of Algebraic Geometry ("Eléments de Géométrie Algébrique" ), which treats in a detailed ( without going into technicalities!), the new foundations and the new techniques of Algebraic Geometry .
In our acquisition of knowledgeof the Universe ( whether mathematical or otherwise) that which renovates the quest is nothing more nor less than complete innocence. It is in ths state of complete innocence that we receive everything from the moment of our birth. Although so often the object of our contempt and of our private fears, it is always in us. It alone can unite humility with boldness so as to allow us to penetrate to the heart of things, or allow things to enter us and taken possession of us.

This unique power is in no way a privilege given to "exceptional talents" - persons of incredible brain power ( for example), who are better able to manipulate, with dexterity and ease, an enormous mass of data, ideas and specialized skills. Such gifts are undeniably valuable, and certainly worthy of envy from those who ( like myself) were not so endowed at birth," far beyond the ordinary".

Yet it is not these gifts, nor the most determined ambition combined with irresistible will-power, that enables one to surmount the "invisible yet formidable boundaries " that encircle our universe. Only innocence can surmount them, which mere knowledge doesn't even take into account, in those moments when we find ourselves able to listen to things, totally and intensely absorbed in child play.


Promenade 12: Topology - or how to survey the fogs

Return to

Récoltes et Semailles

Return to

Home Page