Here we shall give a detailed exposition of a general theory of Group representations, pseudo-representations and their deformation. These results will be used later. The reader who knows the theory well can skip this chapter.

Group Representations

A representation of degree n of a group G is a group homomorphism of G into the group of invertible n × n matrices GLn(A) with coefficients in a commutative ring A. When the structure of the group is very complicated or when the group is very large, such as the absolute Galois group over ℚ, it is often easier to study representations rather than the group itself. In this section, we study the basic properties of group representations.

Coefficient rings

Any ring in this section is commutative with the identity 1 = 1A. If we refer to an algebra R, then R may not be commutative. A ring A is called local if there is only one maximal ideal mA in A. An A-module M is artinian (resp. noetherian) if the set of A-submodules of M satisfies the descending (resp. ascending) chain condition. If A itself as an A-module is artinian (resp. noetherian), we just call A artinian (resp. noetherian). For an artinian A-module M, if M ⊃ M1 ⊃ ⊃ Mn = {0} is the maximal descending chain of, A-submodules, the number n is called the length of M and is written as ℓA(M). If A is an artinian ring, A is noetherian (Akizuki's theorem, [CRT] Theorem 1.3.2).