Definitions and First Properties

Let G be a group. A (left) G-module is an abelian group A on which G acts by additive maps on the left; if g ϵ G and a ϵ A, we write ga for the action of g on a. Letting HomG (A, B) denote the G-set maps from A to B, we obtain a category G–mod of left G-modules. The category G-mod may be identified with the category ℤG-mod of left modules over the integral group ring ℤG. It may also be identified with the functor category AbG of functors from the category “G” (one object, G being its endomorphisms) to the category Ab of abelian groups.

A trivial G-module is an abelian group A on which G acts “trivially,” that is, ga = a for all g ϵ G and a ϵ A. Considering an abelian group as a trivial G-module provides an exact functor from Ab to G-mod. Consider the following two functors from G–mod to Ab:

1. The invariant subgroup AG of a G-module A,

2. The coinvariants AG of a G-module A,

Exercise 6.1.1

Show that AG is the maximal trivial submodule of A, and conclude that the invariant subgroup functor −G is right adjoint to the trivial module functor. Conclude that −G is a left exact functor.