Book contents
- Frontmatter
- Acknowledgments
- Contents
- Introduction
- 1 Chain Complexes
- 2 Derived Functors
- 3 Tor and Ext
- 4 Homological Dimension
- 5 Spectral Sequences
- 6 Group Homology and Cohomology
- 7 Lie Algebra Homology and Cohomology
- 8 Simplicial Methods in Homological Algebra
- 9 Hochschild and Cyclic Homology
- 10 The Derived Category
- A Category Theory Language
- References
- Index
6 - Group Homology and Cohomology
Published online by Cambridge University Press: 05 March 2013
- Frontmatter
- Acknowledgments
- Contents
- Introduction
- 1 Chain Complexes
- 2 Derived Functors
- 3 Tor and Ext
- 4 Homological Dimension
- 5 Spectral Sequences
- 6 Group Homology and Cohomology
- 7 Lie Algebra Homology and Cohomology
- 8 Simplicial Methods in Homological Algebra
- 9 Hochschild and Cyclic Homology
- 10 The Derived Category
- A Category Theory Language
- References
- Index
Summary
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:
The invariant subgroup AG of a G-module A,
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.
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- Chapter
- Information
- An Introduction to Homological Algebra , pp. 160 - 215Publisher: Cambridge University PressPrint publication year: 1994