Grothendieck's Galois theory without finiteness hypotheses

Question

This is motivated by the discussion here. The usual definition of the etale fundamental group (as in SGA 1) gives automatic profiniteness: Grothendieck's formulation of Galois theory states that any category with a "fiber functor" to the category of finite sets satisfying appropriate hypotheses is isomorphic to the category of finite, continuous $G$-sets for a well-defined profinite group $G$ (taken as the limit of the automorphism groups of Galois objects, or as the automorphism group of said fiber functor). In SGA1, the strategy is to take finite etale covers of a fixed (connected) scheme with the fiber functor the set of liftings of a geometric point.

One problem with this approach is that $H^1(X_{et}, G)$ for a finite group $G$ (which classifies $G$-torsors in the etale topology) is not isomorphic to $\hom(\pi_1(X, \overline{x}), G)$ unless $G$ is finite. Indeed, this would imply that the cohomology of the constant sheaf $\mathbb{Z}$ would always be trivial, but this is not true (e.g. for a nodal cubic). However, Scott Carnahan asserts on the aforementioned thread that the "right" etale fundamental group of a nodal cubic should be $\mathbb{Z}$, not something profinite.

How exactly does this work? People have suggested that one can define it as a similar inverse limit of automorphism groups, but is there a similar equivalence of categories and an analogous formalism for weaker "Galois categories"? (Perhaps one wants not just all etale morphisms but, say, torsors: the disjoint union of two open immersions might not be the right candidate.) I'm pretty sure that the finiteness is necessary in the usual proofs of Galois theory, but maybe there's something more.

Answer 1

Check out Section 2 of Noohi's paper Fundamental groups of topological stacks with slice property, Algebr. Geom. Topol. 8 (2008) pp 1333–1370, doi:10.2140/agt.2008.8.1333, arXiv:0710.2615.

Answer 2

In "The pro-étale topology for schemes" (http://arxiv.org/abs/1309.1198), Bhatt and Scholze introduce the pro-étale fundamental group which seems to give a good answer to your question (see, e.g., Theorem 1.10). The pro-étale fundamental group also compares well against the usual étale fundamental group and the "SGA3 étale fundamental group".

Answer 3

The basic Grothendieck's assumptions means we are dealing with an connected atomic site $\mathcal{C}$ with a point, whose inverse image is the fiber functor $F: \mathcal{C} \to \mathcal{S}et$:

(i) Every arrow $X \to Y$ in $\mathcal{C}$ is an strict epimorphism.

(ii) For every $X \in \mathcal{C}$ $F(X) \neq \emptyset$.

(iii) $F$ preseves strict epimorphisms.

(iv) The diagram of $F$, $\Gamma_F$ is a cofiltered category.

Let $G = Aut(F)$ be the localic group of automorphisms of $F$.

Let $F: \widetilde{\mathcal{C}} \to \mathcal{S}et$ the pointed atomic topos of sheaves for the canonical topology on $\mathcal{C}$. We can assume that $\mathcal{C}$ are the connected objects of $\widetilde{\mathcal{C}}$.

(i) means that the objects are connected, (ii) means that the topos is connected, (iii) that $F$ is continous, and (iv) that it is flat.

By considering stonger finite limit preserving conditions (iv) on $F$ (corresponding to stronger cofiltering conditions on $\Gamma_F$) we obtain different Grothendieck-Galois situations (for details and full proofs see [1]):

S1) F preserves all inverse limits in $\widetilde{\mathcal{C}}$ of objets in $\mathcal{C}$, that is $F$ is essential. In this case $\Gamma_F$ has an initial object $(a,A)$ (we have a "universal covering"), $F$ is representable, $a: [A, -] \cong F$, and $G = Aut(A)^{op}$ is a discrete group.

S2) F preserves arbritrary products in $\widetilde{\mathcal{C}}$ of a same $X \in \mathcal{C}$ (we introduce the name "proessential for such a point [1]). In this case there exists galois closures (which is a cofiltering-type property of $\Gamma_F)$, and $G$ is a prodiscrete localic group, inverse limit in the category of localic groups of the discrete groups $Aut(A)^{op}$, $A$ running over all the galois objects in $\mathcal{C}$.

S2-finite) F takes values on finite sets. This is the original situation in SGA1. In this case the condition "F preserves finite products in $\widetilde{\mathcal{C}}$ of a same $X \in \mathcal{C}$ holds automatically by condition (iv) ($F$ preserves finite limits), thus there exists galois closures, the groups $Aut(A)^{op}$ are finite, and $G$ is a profinite group, inverse limit in the category of topological groups of the finite groups $Aut(A)^{op}$.

NOTE. The projections of a inverse limit of finite groups are surjective. This is a key property. The projection of a inverse limit of groups are not necessarily surjective, but if the limit is taken in the category of localic groups, they are indeed surjective (proved by Joyal-Tierney). This is the reason we have to take a localic group in 2). Grothendieck follows an equivalent approach in SGA4 by taking the limit in the category of Pro-groups.

S3) No condition on $F$ other than preservation of finite limits (iv). This is the case of a general pointed atomic topos. The development of this case we call "Localic galois theory" see [2], its fundamental theorem first proved by Joyal-Tierney.

[1] ED, On the representation theory of Galois and Atomic topoi, J. Pure Appl. Alg. 186 (2004) pp 233–275, doi:10.1016/S0022-4049(03)00141-5, arXiv:math/0208222

[2] ED, Localic galois theory, Adv. Math. 175 (2003) pp 144–167, doi:10.1016/S0001-8708(02)00046-4, arXiv:math/0012173.

Answer 4

Besides the references already given above, one can also mention §7 of SGA III, Exp.X (Caracterisation Et Classification Des Groupes De Type Multiplicatif, doi:10.1007/BFb0059008), where Grothendieck sketches the theory of an enlarged fundamental group ("groupe fondamental élargi").

This was later on formalized in terms of Galois toposes in Olivier Leroy's Phd "Groupoide Fondamental et Theoreme de van Kampen en Theorie des Topos" which unfortunately does not seem to be widely available. Roughly, it goes at follows. A topos $E$ is locally Galois if it is locally connected and if every object is a sum of locally constant objects: $SLC(E)=E$ (equivalently, if it is generated by its Galois objects [ a Galois object is a locally constant non-empty object that is a pseudo-torsor under its automorphism group]). Such a locally Galois topos can be recovered from the groupoid (in the categorical sense) of its points in the sense that the functor $E\rightarrow (Point(E))^\wedge$ defines an equivalence of toposes (here for a groupoid $C$, $C^\wedge$ denotes the topos of presheaves on $C$). To sum up, $E\mapsto Point(E)$ defines an equivalence between locally Galois toposes and groupoids. You can find more details in Vincent Zoonekynd's paper The fundamental group of an algebraic stack at this address http://zoonek.free.fr/Ecrits/

Then starting from a scheme $X$, you can consider the topos $SLC(\widetilde{X_{et}})$ of locally constant sheaves for the étale topology. This is a Galois topos, the automorphism group of a point is Grothendieck's enlarged fundamental group. If instead the étale topology you stick to the finite étale topology $X_{fet}$, where covers are given by surjective families of finite étale maps, your recover the traditional profinite fundamental group of SGA1.

Answer 5

In Dubuc and de la Vega's very nice write-up On the Galois Theory of Grothendieck (arXiv:math/0009145) they cover the pro-group case when the fibre functor is representable (see section 5.5):

Consider a category $C$ and an object $A \in C$. Axioms on $C$:

R1) $C$ has a terminal object and pullbacks (thus all finite limits).

R2) $C$ has coequalizers.

R3) $C$ has coproducts.

Axioms on $A$ (in terms of the representable functor $[A, −]$):

R4) $[A, −]$ preserves coequalizers.

R5) $[A, −]$ preserves coproducts.

R6) $[A, −]$ reflects isomorphisms.

Note that $[A,-]$, a priori a functor to $\mathbf{Set}$, lifts to $\mathbf{Set}^{\mathrm{Aut}(A)}$, the category of $\mathrm{Aut}(A)$-sets. Given these axioms, we get an adjoint equivalence between $C$ and $\mathbf{Set}^{\mathrm{Aut}(A)}$. Clearly this is a step in the right direction, but not what you are after.

In section 6.2 they mention the more general case—prorepresentable but not finite—and say they will develop it elsewhere. There is also a reference to SGA IV 2.7, where some of this is stated but not proved (as an aside, of course there is the usual warning that topos in SGA means Grothendieck topos, elementary topoi not being invented yet).

I'm not sure where this 'elsewhere' is, but likely to be written down in the guise of representation theorems for Galois topoi (cf arXiv:math/0208222, math/0012173).