Boundary operator

Definitions

A chain complex${\displaystyle (A_{\bullet },d_{\bullet })}$ is a sequence of abelian groups or modules ${\displaystyle \cdots ,A_{0},A_{1},A_{2},\dots }$ connected by homomorphisms (called boundary operators or differentials) ${\displaystyle d_{n}:A_{n}\to A_{n-1}}$, such that the composition of any two consecutive maps is the zero map. Explicitly, the differentials satisfy ${\displaystyle d_{n}\circ d_{n+1}=0}$ for all ${\displaystyle n}$, or, concisely, ${\displaystyle d^{2}=0}$. The complex may be written out as follows: