# Chain complexes

A fundamental intuitive fact reproduced in this formalism is that the boundary of a boundary is zero. A useful algebraic generalization of this idea is the chain complex, defined to be a sequence of homomorphisms of abelian groups $${\partial_{n}\colon C_{n}\to C_{n-1}}$$ with $${\partial_{n}\partial_{n+1}=0}$$. In our case the abelian groups $${C_{n}}$$ are the $${n}$$-chains, and the chain complex can be illustrated as follows:

A chain complex is an example of an exact sequence, which is any sequence of homomorphisms which sends the image of one object to the kernel of the next. A short exact sequence is of the form

$$\displaystyle 0\longrightarrow N\overset{\phi}{\longrightarrow}E\overset{\pi}{\longrightarrow}G\longrightarrow0,$$

and any longer sequence is called a long exact sequence. $${\phi}$$ is injective and $${\pi}$$ is surjective, so a short exact sequence can be viewed as an embedding of $${N}$$ into $${E}$$ with $${G=E/N}$$. For groups, a short exact sequence is called a group extension, or “$${E}$$ is an extension of $${G}$$ by $${N}$$.” Note that $${N}$$ is normal in $${E}$$ since it is the kernel of $${\pi}$$, and thus $${G\cong E/N}$$. A central extension is one where $${N}$$ also lies in the center of $${E}$$.

 Δ A group extension as above is sometimes described as “$${E}$$ is an extension of $${N}$$ by $${G}$$.” A long exact sequence is sometimes defined as any exact sequence that is not short, or as one which is infinite.