Chain complex of groups

Definition

Over the integers

A chain complex of groups is defined as the following data:

• A collection $G_{n}$ of groups, where $n$ varies over the integers
• A collection $d_{n}:G_{n}\to G_{n-1}$ of group homomorphisms, called the differentials.

such that for any $n$:

$d_{n-1}\circ d_{n}=0$

The chain complex is typically written as:

$\dots \to G_{n}{\stackrel {d_{n}}{\to }}G_{n-1}{\stackrel {d_{n-1}}{\to }}G_{n-2}\to \dots$

Over a contiguous segment of the integers

Suppose $a\leq b$ are integers. A chain complex with index from $b$ to $a$ is a chain complex of groups in the above sense (over all integers) but with $G_{n}$ the trivial group for $n<a$ or $n>b$. This kind of chain complex is usually written in shorthand as:

$1\to G_{b}\to G_{b-1}\to \dots \to G_{a+1}\to G_{a}\to 1$

where 1 denotes the trivial group.

Abstract definition

A chain complex of groups is a group object in the category of chain complexes of pointed sets.

Homomorphism of chain complexes

There are many notions of homomorphism of chain complexes, each discussed below. Each notion of homomorphism defines a corresponding notion of isomorphism.

Homomorphism that preserves the positions

Given two chain complexes $G=(G_{n},d_{n})_{n\in \mathbb {Z} }$ and $H=(H_{n},\partial _{n})_{n\in \mathbb {Z} }$ of groups, a homomorphism $\varphi :G\to H$, also called a chain map or degree zero chain map, is a collection of group homomorphisms:

$\varphi _{n}:G_{n}\to H_{n}$

such that, for all $n$:

$\partial _{n}\circ \varphi _{n}=\varphi _{n-1}\circ d_{n}$

where both sides describe group homomorphisms from $G_{n}$ to $H_{n-1}$.

Homomorphism that allows for a shifting of positions

Let $m$ be an integer. Given two chain complexes $G=(G_{n},d_{n})_{n\in \mathbb {Z} }$ and $H=(H_{n},\partial _{n})_{n\in \mathbb {Z} }$ of groups, $\varphi :G\to H$, a homomorphism of degree $m$, also called a degree $m$ chain map, if it is a collection of group homomorphisms:

$\varphi _{n}:G_{n}\to H_{n+m}$

such that, for all $n$:

$\partial _{n+m}\circ \varphi _{n}=\varphi _{n-1}\circ d_{n}$.

Terminology

• If the image of $d_{n}$ equals the kernel of $d_{n-1}$, then the chain complex is said to be exact at $G_{n-1}$. If this is true for every $n$, the chain complex is termed an exact sequence of groups.
• If the image of $d_{n}$ is a normal subgroup of the kernel of $d_{n-1}$, the quotient group is termed the $(n-1)^{th}$ homology group of the chain complex.
• If the image of $d_{n}$ is a normal subgroup of the whole next group $G_{n-1}$, then the preceding condition holds and we can define the homology. A chain complex of groups where this holds for all $n$ is termed a normal complex of groups.