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:

Note that the image of \({\partial_{n+1}}\) is contained in the kernel of \({\partial_{n}}\); if these are in fact equal, the chain complex is an **exact sequence**, defined to be any sequence of homomorphisms for which the image of one object is 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. |