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Grothendieck's Galois theory

Context

Topos Theory

Could not include topos theory - contents

Contents

Idea

For a sufficiently nice topological space, the fundamental group at a point can be reconstructed as a group of deck transformations of the universal covering space, which is the same as the automorphisms of the fiber over that point of the projection map. The deck transformations are monodromies induced by loops at the base point;. The functor which assigns to a point the fiber functor over it, generalizes to fiber functors in Tannakian formalism of Grothendieck which defines in more general setups the fundamental groupoid as the group of automorphisms of the appropriate fiber functor. See also fundamental group of a topos.

Grothendieck’s Galois theory was constructed in order to define for schemes an analogue of the familiar correspondence

covering spaces of XX : π 1(X)\pi_1(X)-sets

for a locally path connected, semilocally simply connected topological space XX.

The objects on the left are not difficult to define for schemes (at least naively – one really needs trivialisations over étale covers), but it may not be entirely immediate what the fundamental group defined in terms of loops should be.

The reason Galois’s name is attached to this theory is that in the case of the scheme Spec(k)Spec(k), the objects corresponding to the covering spaces are simply field extensions Spec(k)Spec(k'). The fundamental group of schemes defined in this way is the algebraic fundamental group, and is a profinite group.

Generalizations

The basic idea of Grothendieck’s Galois theory may be extended to objects in an topos – leading to a notion of fundamental group of a topos – and then further to objects in any (∞,1)-topos. For more on this see homotopy group of an ∞-stack.

Details

Preliminaries

Definition

Given an arrow f:xyf:x \to y in a category CC the category of arrows compatible with ff, denoted Comp(f)Comp(f) is the full subcategory of the undercategory xCx \downarrow C on the arrows that coequalize the same pairs g,h:wxg,h:w\rightrightarrows x that ff does.

Definition

An arrow f:xyf:x\to y in a category CC is a strict epimorphism if it is initial in Comp(f)Comp(f).

It is not obvious, but a strict epimorphism is an epimorphism.

Grothendieck’s axioms

In what follows, Let CC be a category and F:CSetF:C \to Set a functor. The axioms presented here are as in

J. P. Murre, Lectures on an introduction to Grothendieck’s theory of the fundamental group, Tata Inst. of Fund. Res. Lectures on Mathematics 40, Bombay, 1967. iv+176+iv pp.

and copied also in

  • Eduardo Dubuc, C. S. de la Vega On the Galois theory on Grothendieck, Bol. Acad. Nac. Cienc. (Cordoba) 65 (2000) 111–136. arXiv

Some terminology: XCX\in C is called finite if F(X)F(X) is a finite set. Let FC\int_F C denote the category of elements of FF, in which an object (X,a)(X,a) is called finite if XX is finite.

  • G0) The full subcategory of FC\int_F C on the finite objects is cofinal.

  • G1) CC has all finite limits

  • G2) CC has an initial object, finite coproducts and quotients by finite groups

  • G3) Given f:xzf:x\to z in CC there is a factorisation xeyizx \stackrel{e}{\to} y \stackrel{i}{\to} z where ee is a strict epimorphism and ii is a mono. Also, yy is assumed to be a direct summand of zz.

  • G4) FF preserves finite limits

  • G5) FF preserves initial object, finite sums, quotients by finite group actions and sends strict epimorphisms to surjections

  • G6) FF reflects isomorphisms


The functor FF is called the fibre functor, and the pair (C,F)(C,F) is sometimes called a Galois category.

It follows from the axioms that FF is a pro-representable functor. The automorphism group of the pro-object PP representing FF is (should be. I’m not familiar enough with pro-objects) a profinite group π\pi. This acts on F(X)=[P,]F(X) = [P,-] by precomposition (talking out of my depth here – it’s getting a bit vague) and so FF lifts to a functor to πSet\pi-Set, and Grothendieck’s result is that this functor is an equivalence of categories.

There are several modifications one can make the above. In the case that CC is the category of covering spaces of a nice enough space, the functor FF is representable by the universal covering space, and so there is a ‘representable’ version of the above, not needing to utilise profinite groups. One can also consider just the connected-objects version, and end up with an equivalence to the category of transitive π\pi-sets.

The classical case of fields

Even for the classical case of the inclusion of fields, Grothendieck’s Galois theorem gives more general statement than the previously known. This is the Grothendieck’s version of the Galois correspondence theorem for fields:

Let KLK \subset L be a finite dimensional Galois extension of fields. Let Gal[L:K]Gal[L : K] denote the group of KK-automorphisms of LL and Gal[L:K]finSetGal [L : K]-finSet the category of finite Gal[L:K]Gal[L : K]-sets. By SplitfinAlg K(L)SplitfinAlg_K(L) denote the finite dimensional KK-algebras which are split over LL; here LL itself is an object. Consider the representable functor h L=Hom K(,L):SplitfinAlg K(L)Seth_L = Hom_K(-,L):SplitfinAlg_K(L)\to Set. It takes values in the subcategory of finite sets and it comes with a canonical Gal[L:K]Gal[L : K]-action. In other words, this functor factors through Gal[L:K]finSetGal [L : K]-finSet. Moreover, the corresponding functor

SplitfinAlg K(L)Gal[L:K]finSet SplitfinAlg_K(L)\to Gal [L : K]-finSet

is an equivalence of categories.

There is an infinitary version as well, generalizing the classical Galois theorem on infinitary Galois extensions.

Thus let KLK\subset L be an arbitrary Galois extension. Now Gal[L:K]Gal[L:K] denotes the profinite Galois group and Gal[L:K]profinSpaceGal[L:K]-profinSpace the category or profinite Gal[L:K]Gal[L:K]-spaces. SplitAlg K(L)SplitAlg_K(L) denotes the category of KK-algebras split over LL (possible infinite-dimensional). Then there is a canonical anti-equivalence of categories

SplitAlg K(L)Gal[L:K]profinSpace SplitAlg_K(L)\to Gal [L : K]-profinSpace

(factorizing a profinite-space version of the representable functor Hom K(,L)Hom_K(-,L)).

A special case of this is the following: the category of étale k-schemes reps. étale group schemes? for a field kk is equivalent to the category of sets equipped with an action of the absolute Galois group reps. to the category of Galois modules of the absolute Galois group.

Galois theorem for locales and topoi

Let EE be a Grothendieck topos. Then there exist an open localic groupoid GG such that EE is equivalent to the category of étale presheaves over GG. One of the classical references is

  • J. P. Murre, Lectures on an introduction to Grothendieck’s theory of the fundamental group, Tata Inst. of Fund. Res. Lectures on Mathematics 40, Bombay, 1967. iv+176+iv pp.

This is a variant of the theorem in the setting of locales from

  • Andre Joyal, M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp.

References

The original development of the theory by Grothendieck is in SGA1.

  • Alexander Grothendieck, (1971), S.G.A.1 - Revetements étales et groupe fondamental, Lecture Notes in Maths. 224, Springer-Verlag.

A more recent treatment can be found in

and more related categorical and topos theoretic aspects in

  • Eduardo Dubuc, Localic Galois theory, Adv. Math. 175:1 (2003), 144–167 doi; On the representation theory of Galois and atomic topoi, JPAA 186:3 (2004) 233–275 doi

A very approachable account is given in

  • Marco Antònio Delgado Robalo, Galois theory towards dessins d’enfants, masters thesis, Lisboa 2009, pdf

(This has the advantage of looking towards Grothendieck’s dessins d’enfants.)

Basic intuition is explained in

  • Pierre Cartier, A mad day’s work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry, Bull. Amer. Math. Soc. 38 (2001), 389-408, pdf

The construction for general toposes is described in section 8.4 of

and, a current state of the art description is in

  • Marta Bunge, Galois groupoids and covering morphisms in topos theory, Galois theory, Hopf algebras, and semiabelian categories, 131–161, Fields Inst. Commun. 43, Amer. Math. Soc., Providence, RI, 2004, links.

A modern approach from classical via Grothendieck up to categorical Galois theory based on precategories and adjunctions is in

The application of a general Tannakian theorem of Saavaedra Rivano, as corrected by Deligne, to the “differential Galois theory” for differential instead of algebraic equations is in the last chapter of Deligne’s Catégories Tannakiennes.

  • George Janelidze, Dietmar Schumacher, Ross Street, Galois theory in variable categories, Applied Categorical Structures 1, Number 1, 103-110, DOI:
  • Federico G. Lastaria, On separable algebras in Grothendieck Galois theory, Le Mathematiche 51:3, 1996, link

category: Galois theory

Revised on March 21, 2014 08:25:04 by Zoran Škoda (193.136.196.12)