Factor system

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In mathematics, a factor system is a function on a group giving the data required to construct an algebra. A factor system constitutes a realisation of the cocycles in the second cohomology group in group cohomology.

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 × GL* satisfying

Cocycles c and c'are equivalent if there exists some system of elements a : GL* 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 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 ug with multiplication

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

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.[1]

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 = ut 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*/NL/KL*. We obtain the isomorphisms

References

  1. Saltman (1999) p.44