Chain complex

Definition

A chain complex over an Abelian category (for instance, the module category over a ring) is defined as follows.

Note that since any Abelian category can be viewed as a subcategory of the category of Abelian groups, we shall view all the objects as Abelian groups with some additional structure.

Data

• A sequence of objects in the Abelian category, aindexed by integers. In other words, for each integer $n$, an object $C_{n}$. These are termed the chain groups.
• A map from each object to its predecessor. In other words, a map $d_{n}:C_{n}\to C_{n-1}$. These are termed the boundary maps.

Conditions

• The boundary maps are all homomorphisms.
• The composite of boundary maps $d_{n-1}\circ d_{n}$ is the zero map. In other words, the kernel of $d_{n-1}$ contains the image of $d_{n}$.

Miscellanea

Note that we use define a chain complex only for positive integer values, or negative integer values. In this case, we can set the remaining groups to be trivial and all the maps to and from them to be trivial.

Constructs

Cycle groups for a chain complex

For a chain complex, the kernel of the map $d_{n}$ is a subgroup of $C_{n}$, and this subgroup is termed the $n^{th}$ cycle group. Its elements are termed cycles. These groups are often denoted by $Z_{n}$.

Boundary groups for a chain complex

For a chain complex, the image of the map $d_{n+1}$ is a subgroup of $C_{n}$, and this subgroup is termed the $n^{th}$ boundary group. Its elements are termed boundaries. These groups are often denoted by $B_{n}$.

Homology groups for a chain complex

For a chain complex, the quotient of the $n^{th}$ cycle group by the $n^{th}$ boundary group is termed the $n^{th}$ homology group. Its elements, viewed as cosets of the boundary group in the cycle group, are termed homology classes. The homology groups are often denoted as $H_{n}$.

Dualizing a chain complex

Given a chain complex of $R$-modules (for some ring $R$), we can associate, to each $R$-module, the module of homomorphisms from it to $R$ (this is the dual module). The dual modules will form another chain complex with the directions of the arrows reversed. This is called a cochain complex. Of course, it can be made into a chain complex by simply negating all the indices.

The $(-n)^{th}$ boundary group of the dual complex is termed the $n^{th}$ coboundary group of the original chain complex, and the $(-n)^{th}$ cycle group of the dual complex is termed the $n^{th}$ cocycle group. The quotient of these is termed the $n^{th}$ cohomology group.