Apologies in advance if this question is too elementary for MO. I didn't find an explanation of these ideas in any algebraic geometry books (I don't know French).
The following is an excerpt from this archive:
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".
I'm trying to slowly digest the last paragraph. As a novice in algebraic geometry I'm always looking for geometric and "philosophical" intuition, so I very much want to understand why Grothendieck was insistent on points having Galois group actions.
Why, geometrically (or philosophically?) is it essential and important that points have Galois group actions?