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.^[1]^[2] 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.^[3]

Introduction

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

1\to A\to X\to G\to 1,

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

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

So $f$ must be a "group 2-cocycle" (and thus define an element in H²(G, A), as studied in group cohomology). In fact, $A$ does not have to be abelian, but the situation is more complicated for non-abelian groups^[4]

If $f$ is trivial, then $X$ splits over $A$ , so that $X$ is the semidirect product of $G$ with $A$ .

If a group algebra is given, then a factor system f modifies that algebra to a skew-group algebra by modifying the group operation $xy$ 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 (Noether) factor system^[5]^: 31 is a map c: G × G → L^* satisfying

c(h,k)^{g}c(hk,g)=c(h,kg)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_{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 c in H²(G,L^*) gives rise to a crossed product algebra^[5]^: 31 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).

The crossed product algebra A is a central simple algebra (CSA) of degree equal to [L : K].^[6] The converse holds: every central simple algebra over K that splits over L and such that deg A = [L : K] arises in this way.^[6] 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.^[7]^[8]

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^[5]^: 33

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

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

References

^ group extension at the nLab
^ Saunders MacLane, Homology, p. 103, at Google Books
^ group cohomology at the nLab
^ nonabelian group cohomology at the nLab
^ ^a ^b ^c Bokhut, L. A.; L’vov, I. V.; Kharchenko, V. K. (1991). "Noncommutative Rings". In Kostrikin, A.I.; Shafarevich, I.R. (eds.). Algebra II. Encyclopaedia of Mathematical Sciences. Vol. 18. Translated by Behr, E. Berlin Heidelberg: Springer-Verlag. doi:10.1007/978-3-642-72899-0. ISBN 9783642728990.
^ ^a ^b Jacobson (1996) p.57
^ Saltman (1999) p.44
^ Jacobson (1996) p.59

Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Universitext. Translated from the German by Silvio Levy. With the collaboration of the translator. Springer-Verlag. ISBN 978-0-387-72487-4. Zbl 1130.12001.
Jacobson, Nathan (1996). Finite-dimensional division algebras over fields. Berlin: Springer-Verlag. ISBN 3-540-57029-2. Zbl 0874.16002.
Reiner, I. (2003). Maximal Orders. London Mathematical Society Monographs. New Series. Vol. 28. Oxford University Press. ISBN 0-19-852673-3. Zbl 1024.16008.
Saltman, David J. (1999). Lectures on division algebras. Regional Conference Series in Mathematics. Vol. 94. Providence, RI: American Mathematical Society. ISBN 0-8218-0979-2. Zbl 0934.16013.

[1] roup extension at the nLab

[2] Saunders MacLane, Homology, p. 103, at Google Books

[3] roup cohomology at the nLab

[4] roup cohomology at the nLab

[Kostrikin-Shafarevich-5] Bokhut, L. A.; L’vov, I. V.; Kharchenko, V. K. (1991). "Noncommutative Rings". In Kostrikin, A.I.; Shafarevich, I.R. (eds.). Algebra II. Encyclopaedia of Mathematical Sciences. Vol. 18. Translated by Behr, E. Berlin Heidelberg: Springer-Verlag. doi:10.1007/978-3-642-72899-0. ISBN 9783642728990.

[Jac57-6] Jacobson (1996) p.57

[Salt44-7] Saltman (1999) p.44

[Jac59-8] Jacobson (1996) p.59

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