Factor system

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 satisfying 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 is a group and is an abelian group. For a group extension

there exists a factor system which consists of a function and homomorphism such that it makes the cartesian product a group as

So must be a "group 2-cocycle" (and thus define an element in H(G, A), as studied in group cohomology). In fact, does not have to be abelian, but the situation is more complicated for non-abelian groups

If is trivial, then splits over , so that is the semidirect product of with .

If a group algebra is given, then a factor system f modifies that algebra to a skew-group algebra by modifying the group operation to .

Application: for Abelian field extensions

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

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

Cocycles of the form

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

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

The crossed product algebra A is a central simple algebra (CSA) of degree equal to [L : K]. The converse holds: every central simple algebra over K that splits over L and such that deg A = [L : K] 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

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

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

References