Английская Википедия:Factor system

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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 Шаблон:Math is a group and Шаблон:Math is an abelian group. For a group extension

<math>1 \to A \to X \to G \to 1,</math>

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

<math>(g,a)*(h,b) := (gh, f(g,h)a^{\sigma(h)}b).</math>

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

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

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

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 systemШаблон:Rp is a map c: G × G → L^* satisfying

<math>c(h,k)^g c(hk,g) = c(h,kg) c(k,g) . </math>

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

<math>c'(g,h) = c(g,h) (a_g^h a_h a_{gh}^{-1}) . </math>

Cocycles of the form

<math>c(g,h) = a_g^h a_h a_{gh}^{-1} </math>

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 algebraШаблон:Rp A, which is a K-algebra containing L as a subfield, generated by the elements λ in L and u_g with multiplication

<math>\lambda u_g = u_g \lambda^g ,</math>

<math>u_g u_h = u_{gh} c(g,h) .</math>

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

<math>A = (L,G,c) .</math>

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

<math> u_{t^i} = u^i \, </math>

and then we have uⁿ = a in K. This element a specifies a cocycle c byШаблон:Rp

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

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

<math>\operatorname{Br}(L/K) \equiv K^*/N_{L/K} L^* \equiv H^2(G,L^*) . </math>

References

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book