---
tag: 9JA1N3CH80
kind: definition
name: Homology of a chain complex
msc: [18G35]
defines: 2J4JJJYP22
uses: [WQ82C0M3QB, Y9CZ3MTNF1, S417YNEJ6M, H8W38933NC, HTK9WQZJ1G]
sources:
  - url: https://en.wikipedia.org/wiki/Homology_(mathematics)
---

## Definition

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]], for example the category of modules over a ring, and let $(C_\bullet, d)$ be a [[chain complex|WQ82C0M3QB]] in $\mathcal{A}$.
For each integer $n$, let $Z_n = \ker d_n$ be the [[kernel|S417YNEJ6M]] of $d_n$, called the object of *$n$-cycles*.
Since $d_n \circ d_{n+1} = 0$, the morphism $d_{n+1}$ factors through a morphism $\bar d_{n+1} \colon C_{n+1} \to Z_n$.
Its image $B_n = \im \bar d_{n+1}$ is a subobject of $Z_n$, called the object of *$n$-boundaries*.
The *$n$-th homology* of $C_\bullet$ is the [[cokernel|H8W38933NC]] of $\bar d_{n+1}$:
$$H_n(C) = Z_n / B_n = \ker d_n / \im d_{n+1}.$$

## Remarks

The complex is exact at $C_n$ if and only if $H_n(C) = 0$.
A [[chain map|HTK9WQZJ1G]] $f \colon C_\bullet \to D_\bullet$ sends cycles to cycles and boundaries to boundaries, so it induces morphisms $H_n(f) \colon H_n(C) \to H_n(D)$, and $H_n$ is a functor.
For a cochain complex $(C^\bullet, d)$ the same construction gives the *cohomology* $H^n(C) = \ker d^n / \im d^{n-1}$.

Example: let $C_1 = C_0 = \Z$ with $d_1$ multiplication by $2$, and $C_n = 0$ otherwise.
Then $Z_1 = \ker d_1 = 0$, so $H_1(C) = 0$; and $Z_0 = \Z$, $B_0 = 2\Z$, so $H_0(C) = \Z/2\Z$.

Non-example: for the sequence $\Z \xrightarrow{1} \Z \xrightarrow{1} \Z$, the image $\Z$ of the first map is not contained in the kernel $0$ of the second.
So $\ker / \im$ is not defined; the condition $d^2 = 0$ is what makes homology make sense.