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 × G → L^* satisfying

$c(h,k)^{g}c(hk,g)=c(h,kg)c(k,g).$

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

$c'(g,h)=c(g,h)(a_{g}^{h}a_{h}a_{gh}^{-1}).$

Cocycles of the form

$c(g,h)=a_{g}^{h}a_{h}a_{gh}^{-1}$

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 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

$\lambda u_{g}=u_{g}\lambda ^{g},$

$u_{g}u_{h}=u_{gh}c(g,h).$

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

$c(t^{i},t^{j})={\begin{cases}1&{\text{if }}i+j<n,\\a&{\text{if }}i+j\geq n.\end{cases}}$

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

$\mathop {Br} (L/K)\equiv K^{*}/\mathop {N} _{L/K}L^{*}\equiv \mathop {H} ^{2}(G,L^{*}).$