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?
Best Answer
Suppose $k$ is a field, not necessarily algebraically closed. $\text{Spec } k$ fails to behave like a point in many respects. Most basically, its "finite covers" (Specs of finite etale $k$-algebras) can be interesting, and are controlled by its absolute Galois group / etale fundamental group. For example, $\text{Spec } \mathbb{F}_q$, the Spec of a finite field, has the same finite covering theory as $S^1$, which reflects (and is equivalent to) the fact that its absolute Galois group is the profinite integers $\widehat{\mathbb{Z}}$. (So this suggests that one can think of $\text{Spec } \mathbb{F}_q$ itself as behaving like a "profinite circle.")
More generally, suppose you want to classify objects of some kind over $k$ (say, vector spaces, algebras, commutative algebras, Lie algebras, schemes, etc.). A standard way to do this is to instead classify the base changes of those objects to the separable closure $k_s$, then apply Galois descent. The topological picture is that $\text{Spec } k$ behaves like $BG$ where $G$ is the absolute Galois group, $\text{Spec } k_s$ behaves like a point, or if you prefer like $EG$, and the map
$$\text{Spec } k_s \to \text{Spec } k$$
behaves like the map $EG \to BG$. In the topological setting, families of objects over $BG$ are (when descent holds) the same thing as objects equipped with an action of $G$. The analogous fact in algebraic geometry is that objects over $\text{Spec } k$ are (when Galois descent holds) the same thing as objects over $\text{Spec } k_s$ equipped with homotopy fixed point data, which is a generalization of being equipped with a $G$-action which reflects the fact that $k_s$ itself has a $G$-action.
(I need to be a bit careful here about what I mean by "$G$-action" to take into account the fact that $G$ is a profinite group. For simplicity you can pretend that I am instead talking about a finite extension $k \to L$, although I'll continue to write as if I'm talking about the separable closure. Alternatively, pretend I'm talking about $k = \mathbb{R}, k_s = \mathbb{C}$.)
The classification of finite covers is the simplest place to see this: the category of finite covers of $\text{Spec } k_s$ is the category of finite sets, with the trivial $G$-action, so homotopy fixed point data is the data of an action of $G$, and we get that finite covers of $\text{Spec } k$ are classified by finite sets with $G$-action.
But Galois descent holds in much greater generality, and describes a very general sense in which objects over $k$ behave like objects over $k_s$ with a Galois action in a twisted sense.