48
$\begingroup$

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?

Improve this question
$\endgroup$

2 Answers 2

Reset to default
47
$\begingroup$

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.

Improve this answer
$\endgroup$
15
  • 3
    $\begingroup$ The fact that the finite covering theory of $\text{Spec}\,\mathbb{F}_q$ is the same as the finite covering theory of $S^1$ is certainly evidence that they have the same étale fundamental group. But in what sense could this be enough information to determine the étale homotopy type? After all, the finite covering theory of $\text{Spec}\,\mathbb{C}$ is the same as the finite covering theory of $S^{13}$, but that doesn't mean they have the same étale homotopy type. $\endgroup$
    – Tom Church
    Commented Mar 18, 2016 at 15:00
  • 4
    $\begingroup$ @QiaochuYuan: Thanks for the edit. But the claim that "you can think of $\text{Spec}\,\mathbb{F}_q$ as behaving like a 'profinite circle'" still seems to rely on the fact that $\pi^{et}_i(\text{Spec}\,\mathbb{F}_q)=0$ for $i\geq 2$, or at least that $H^i_{et}(\text{Spec}\,\mathbb{F}_q;\,\mathbb{Z}_{\ell})=0$ for $i\geq 2$. These facts are true, as far as I understand, but not trivial, and in any case are not related to the position of Grothendieck the OP asked about. $\endgroup$
    – Tom Church
    Commented Mar 18, 2016 at 15:26
  • 2
    $\begingroup$ @Tom: hmm, I was hoping that the edited "this suggests" would be enough hedging for you. I am certainly not intending to imply that what I wrote is enough to conclude that the etale homotopy type is in fact a profinite circle, only that it suggests this hypothesis. My main motivation for saying "profinite circle" at all is that I wanted to give a concrete example where one can really have a topological picture in mind for what Spec of a field looks like, and it really does not look like a point. $\endgroup$
    – Qiaochu Yuan
    Commented Mar 18, 2016 at 15:31
  • 2
    $\begingroup$ No, as far as I understand, Artin-Verdier duality means that $\text{Spec}\mathbb{Z}$ has cohomological dimension at least 3 (in fact equal to 3, if 2-torsion is ignored). This kind of thing is precisely why I made my original comment. $\endgroup$
    – Tom Church
    Commented Mar 18, 2016 at 16:00
  • 3
    $\begingroup$ A good analogy is that $\operatorname{Gal} ( \mathbb C(x))$ is the inverse limit of the profinite fundamental groups of Zariski open subsets of $\mathbb P^1$, and the complex points of the Zariski open subsets of $\mathbb P^1$ are the classifying spaces of their fundamental groups, so the classifying space of the Galois group is some kind of profinite completion of a inverse limit of complex points of Zariski open subsets of $\mathbb P^1$, which is exactly what we would want the homotopy type to be - take the sphere and remove all the points. $\endgroup$
    – Will Sawin
    Commented Jan 30, 2021 at 14:54
5
$\begingroup$

The points of a Topos have natural transformations between them; restricting to natural isomorphisms you get a groupoid. You can also represent the points of a bounded Topos as principal bundles; I.e. Something with a G -action. Not sure which is being referred to, but you can see this aspect of points without having any rings around.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .