# Chain complex of groups

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## 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 $a \le b$ 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.