Grouppoids of torsors, Galois groups, classifying coverings, noncommutative Kummer's theory

[posic]

Let's start with a bit of abstract nonsense. Grouppoid is a category whose morphisms are all isomorphisms. By connected components of a grouppoid one means isomorphism classes of its objects; the fundamental group of a connected grouppoid is the automorphism group of any object.

The category of all functors from a grouppoid to a grouppoid is again a grouppoid. If S and T are connected grouppoids, then the set of connected components of the grouppoid Funct(S,T) is equal to the set of outer homomorhisms from the fundamental group of S to the fundamental group of T. So the category of connected grouppoids and isomorphism classes of functors between them is equivalent to the category of groups and outer homomorphisms (that is, homomorphisms of groups up to conjugation in the target). Let us denote this category by GrOut.

***

Let X be a connected topological space whose higher homotopy groups are trivial. It's important that no initial point in X is chosen. Then it makes sense to consider the fundamental group of X as an object of the category GrOut, rather than the ordinary category of groups. In fact, the category of such spaces X (with homotopy classes of continuous mappings as morphisms) is equivalent to GrOut.

If one is interested in the 2-category structure, one should assign to X its fundamental grouppoid in one of its numerous incarnations: points of X and gomotopy classes of paths, or universal coverings of X and their isomorphisms, etc. The 2-category of spaces like X with continuous maps as 1-morphisms and homotopy classes of homotopies as 2-morphisms seems to be equivalent to the 2-category of grouppoids.

***

Let F be a field and G be a finite group. I'm interested in the grouppoid S(F,G) which can be defined alternatively (up to an equivalence, in some sense natural) in either of the following ways:
- it is the grouppoid of (right) G-torsors over Spec F in the etale topology;
- it is the grouppoid of functors from the grouppoid of algebraic closures of F to the grouppoid of principal homogeneous (right) G-sets;
- objects are homomorphisms from Gal(F) into G, morphisms are conjugations by elements of G;
- objects are finite-dimensional commutative F-algebras without nilpotent elements (i.e., isomorphic to finite direct products of fields) endowed with an action of G such that the G-invariants coincide with F; morphisms are maps of F-algebras commuting with the action of G.

Notice that the functor assigning the set of connected components of S(F,G) to each finite group G determines the profinite group Gal(F) up to inner isomorphism.

***

Given a finite group G, we would like to "compute" the grouppoid S(F,G) in a uniform way for all fields F (or with relatively mild restrictions on F). More precisely, we want to construct (perhaps in several different ways for a given group G) a grouppoid S_G in the category of schemes (this means a scheme ObS in place of the set of objects and another scheme MoS in place of the set of morphisms) such that the grouppoid of F-points (ObS(F), MoS(F)) is equivalent to S(F,G) (and in a natural way).

Let Y -> X be an etale covering of connected schemes which is a Galois covering with Galois group G. The following construction makes a scheme-grouppoid out of it: set ObS(Y,X)=X and MoS(Y,X)=YxY/G. If I am not mistaken, it seems that for any field F there will be a fully faithful functor from the grouppoid of F-points S(Y,X)(F) into S(F,G).

So we come to the following definition: an etale covering of schemes Y -> X with Galois group G is called a universal G-covering for a field F if any Galois extension of F with the Galois group embedded into G is induced from Y -> X via an F-point of the scheme X.

***

Here is one way to construct universal G-coverings; I hope that there are others. Let k be a field and V be an exact finite-dimensional linear representation of G over k. Let us consider V as a scheme over Spec k. Set Y to be V minus all subspaces with nontrivial pointwise stabilizer in G (in addition, one can throw away any number of affine hyperplanes defined over k) and put X=Y/G. Then Y -> X is a universal G-covering for any field F containing k.

Examples: taking k to be a field containing m-roots of unity and V a one-dimensional exact linear representation of G=Z/m, one obtains Kummer's theory of cyclic extensions. Taking V to be the n-dimensional permutational representation of the symmetric group S_n, one obtains the classification of Galois extensions with Galois groups embedded into S_n via polynomials of degree n with simple roots.

Comments

[(Anonymous)]

Леонид, а есть ли место в Москве, где собираются люди интересующиеся подобными вопросами?
Например какие-нибудь научные семинары и т.п.

Михович Андрей
mikhandr@mail.ru

[posic.livejournal.com]

Это зависит от того, что Вы понимаете под "подобными вопросами". Но вообще есть Независимый Университет и есть семинары по математике в ИТЭФ'е.

[marina_p]

"The category of all functors from a grouppoid to a grouppoid is again a grouppoid."

Я не понимаю: если взять группоид F с двумя объектами и одним изоморфизмом между ними, и группоид G с одним объектом и единственным единичным морфизмом, то ведь у естественного функтора F->G не будет обратного?

[posic.livejournal.com]

Утверждение, что категория таких функторов является группоидом, не имеет отношения к существованию обратного функтора. Это утверждение о том, что все естественные преобразования функторов являются изоморфизмами (для чего достаточно даже, чтобы одна категория -- та, в которую бъют функторы -- была группоидом).

Ваш конкретный функтор F->G является эквивалентностью категорий, но не изоморфизмом категорий. Таким образом, у него нет обратного, но есть квазиобратный функтор. Произвольный функтор между группоидами эквивалентностью категорий не является, конечно.

[marina_p]

Поняла, спасибо. Я думала, что речь о категории, где объекты -- группоиды, а морфизмы -- функторы (она бы называлась категорией группоидов, конечно).

[posic.livejournal.com]

Категория группоидов, конечно, не группоид.