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Chain complex of groups

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Definition

Over the integers

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

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.