Formation of classes

In mathematics, a formation of classes is a structure used to organize various the groups of Welshman and the modules which appear in the Théorie of the bodies of classes. They were invented by Emil Artin and John Tate.

More precisely, it is the data of a group, acting on a certain module, the whole checking some axiomatic, mainly expressed from a point of view cohomologic. The goal of this concept is to axiomatize the Théorie of the bodies of classes, independently of the various contexts where one wishes to obtain his statements: finished or infinite, total or local body, of null or positive characteristic. The group considered being then the group of absolute Welshman of the body considered, and modulates it being the multiplicative group of the separable fence of this same body.

Definitions

Formation

The data of a group of absolute Welshman of a local body, of its sub-groups of index finished, and action of this group on the multiplicative group of the separable fence of the body considered (the algebraic fence in the case of a Corps of numbers p-adic, since the fact of being in characteristic 0 ensures separability), is the traditional example of formation, that from which the axiomatic one was built. In the paragraph which follows, one indicates between brackets interpretation in this example of the various necessary properties.

A formation is a topological Groupe G with a G - module has . The group G is given provided with a family of sub-groups of finished indices (intended to correspond to the separable extensions finished of the body considered via Welshman correspondence), which one supposes that it is stable by finished intersection (a compositum finished finished extensions is a finished extension), that any sub-group containing an element of the family is in the family (any under-extension of a finished extension is finished), and that it is overall stable by conjugation by elements of G (the image by the action of an element of the group of absolute Welshman of a finished extension, not necessarily galoisienne, is again a finished extension).

One gives oneself then a G - module has , and one stipulates the following condition on the action of G on has : for any point has , the whole of the transformations of G which leave has fixed is an element of the family of sub-group that one set (the extension generated by an element is a finished extension).

The data of G , the family of sub-groups, and the module has , checking these conditions is called formation .

Stage and extension

A stage F / E of a formation is a pair of sub-groups open E , F such as F is a sub-group of E . Always in the spirit of mimer the situation of our fundamental example, one will be authorized with speaking about extension ; in this direction, note that whereas F is included in E as a sub-group, one says that as an extension, a F is on the contrary above E , in the objective to recover the decrease of the Welshman correspondence. An extension is called a normal extension or extension galoisienne if F is a sub-group normal of E , and a cyclic extension so moreover, the quotient group is cyclic. The degree of the extension will be the cardinal of the group quotient E/F . If E is a sub-group of G , then has ^E is defined as being the elements of has fixed by E . We write

H ^N ( F / E )

for the Groupe cohomologic of Touches H ^N ( F / E , has ^F) all times that F / E is a normal extension. In the applications, G is usually the Groupe of absolute Welshman, and in particular is profini, and the sub-groups opened consequently correspond to the finished extensions of the body contained in a certain separable fence.

To speak about formation of classes, one forces the condition following: for any extension galoisienne, $H^1 (Gall (F/E), A_F) =0$ , which is true in our fundamental example by the Théorème 90 of Hilbert, on the one hand; if only this condition is asserted, one will speak sometimes about formation of body . One supposes in addition the existence for each abelian extension F/E of a homomorphism:

inv_E: H^2 (Gall (F/E), A_F) \ rightarrow \ mathbb {Q}/\ mathbb {Z}

injective, with values in the single sub-group of order of the group of arrival. Each group H ² ( F / E ) is then a cyclic group of order. The antecedent of the canonical generator 1 '' of Q/Z is called fundamental class , and is noted U F / E : it is a generator of $H^2 (Gall (F/E), A_F)$ . It is asked finally that be checked the fonctorielle property:

$inv_{E'}\circ Res_{E/E'}=inv_E$
for any finished extension E'/E, where LMBO is the restriction application appearing in Cohomologie galoisienne; this makes it possible to state that the restriction of a fundamental class is still a fundamental class.

In the traditional example, this Homomorphisme is built by regarding the group H ² as a Groupe of Brauer.

Examples of formations of classes

The most important examples of formations of classes (classified coarsely in order of difficulty) are the following:

Theory of the bodies of classes archimédienne : the module has is the group of the complex numbers different from zero, and G is either commonplace, or the cyclic group of a nature 2 generated by the complex conjugation.
finished Bodies: the module has are the entireties (with a G - commonplace action), and G is the group of absolute Welshman of a finished body, which is isomorphous with completion profinie entireties.
local Theory of the bodies of classes of characteristic p > 0: the module has is a separable algebraic fence of the body of the formal series of Laurent on a finished body, and G is the group of Welshman.
Theory of the bodies of classes non-archimédienne of characteristic 0: the module has is the algebraic fence of a Corps of numbers p-adic, and G is a group of Welshman.
total Theory of the bodies of classes of characteristic p > 0: the module has is the union of the groups of the idelic classes of separable extensions finished of some Corps of functions on a finished body, and G is the group of Welshman.
total Theory of the bodies of classes of characteristic 0: the module has is the union of the groups of idelic classes of algebraic bodies of numbers, and G is the group of absolute Welshman of the body of the rational numbers (or a certain body of algebraic numbers) acting on has .

It is easy to check the property of the formation of classes for the case of the finished body and the case of the local body archimédien, but the remaining cases are more difficult. The majority of the difficult work of the theory of the bodies of classes consist in showing that those are indeed formations of classes. This is carried out in several stages, as that is described in the following paragraphs.

The application of reciprocity

That is to say u_F/E the reciprocal image of 1 '' by the application inv_E , i.e. a fundamental class. The cup-product by u_F/E defines then an isomorphism between $H^ {- 2} (Gall (F/E), \ mathbb {Z})$ and $H^ {0} (Gall (F/E), A_F)$ ; this is a consequence of the theorem of cohomologic duality of Touches and Nakayama. However, this first group is identified with the greatest abelian quotient of Gall (F/E) , and groups it of arrival with A_E/N_F/EA_F . This isomorphism is called isomorphism of reciprocity: it is the isomorphism of body of classes.
In the case of our example, one finds well the isomorphism of the local theory of the bodies of classes.

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