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(*).
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.(*)
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..