Factor system - Alchetron, The Free Social Encyclopedia

In mathematics, a factor system (sometimes called factor set) is a fundamental tool of Otto Schreier’s classical theory for group extension problem. It consists of a set of automorphisms and a binary function on a group satisfing certain condition (so-called cocycle condition). In fact, a factor system constitutes a realisation of the cocycles in the second cohomology group in group cohomology.

Introduction

Suppose G is a group and A is an abelian group. For a group extension

1 → A → X → G → 1 ,

there exists a factor system which consists of a function f : G × G → A and homomorphism σ: G → Aut(A) such that it makes the cartesian product G × A a group X as

( g , a ) ∗ ( h , b ) := ( g h , f ( g , h ) a σ ( h ) b ) .

So f must be a "group 2-cocycle" (symbolically, Ext(G, A) ≅ H²(G, A)). In fact, A does not have to abelian, but more complicated

If f is trivial and σ gives inner automorphisms, then that group extension is split, so X become semi-direct product of G with A.

If a group algebra is given, then a factor system f modifies that algebra to skew-group algebra by group operation xy modifying to f(x, y)xy.

Application: for Abelian field extensions

Let G be a group and L a field on which G acts as automorphisms. A cocycle or factor system is a map c:G × G → L^* satisfying

c ( h , k ) g c ( h k , g ) = c ( h , k g ) c ( k , g ) .

Cocycles are equivalent if there exists some system of elements a : G → L^* with

c ′ ( g , h ) = c ( g , h ) ( a g h a h a g h − 1 ) .

Cocycles of the form

c ( g , h ) = a g h a h a g h − 1

are called split. Cocycles under multiplication modulo split cocycles form a group, the second cohomology group H²(G,L^*).

Crossed product algebras

Let us take the case that G is the Galois group of a field extension L/K. A factor system c in H²(G,L^*) gives rise to a crossed product algebra A, which is a K-algebra containing L as a subfield, generated by the elements λ in L and u_g with multiplication

λ u g = u g λ g , u g u h = u g h c ( g , h ) .

Equivalent factor systems correspond to a change of basis in A over K. We may write

A = ( L , G , c ) .

Every central simple algebra over K that splits over L arises in this way. The tensor product of algebras corresponds to multiplication of the corresponding elements in H². We thus obtain an identification of the Brauer group, where the elements are classes of CSAs over K, with H².

Cyclic algebra

Let us further restrict to the case that L/K is cyclic with Galois group G of order n generated by t. Let A be a crossed product (L,G,c) with factor set c. Let u = u_t be the generator in A corresponding to t. We can define the other generators

u t i = u i

and then we have uⁿ = a in K. This element a specifies a cocycle c by

c ( t i , t j ) = { 1 if i + j < n , a if i + j ≥ n .

It thus makes sense to denote A simply by (L,t,a). However a is not uniquely specified by A since we can multiply u by any element λ of L^* and then a is multiplied by the product of the conjugates of λ. Hence A corresponds to an element of the norm residue group K^*/N_L/KL^*. We obtain the isomorphisms

Br ⁡ ( L / K ) ≡ K ∗ / N L / K L ∗ ≡ H 2 ( G , L ∗ ) .

References

Factor system Wikipedia

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Contents

Introduction

Application: for Abelian field extensions

Crossed product algebras

Cyclic algebra

References