Let \(R\) be a ring and let \[ \cdots\to A_{n+1}\mapsvia{d_{n+1}}A_n\mapsvia{d_n}A_{n-1}\to\cdots \] be a sequence of \(R\)-modules and \(R\)-linear maps. The sequence is exact at \(A_n\) if \(\im d_{n+1}=\ker d_n\), and exact if it is exact at every term that has both an incoming and an outgoing map [@Hat02].
The sequence is a chain complex when \(\im d_{n+1}\subseteq\ker d_n\) for all \(n\). A chain complex is exact at \(A_n\) if and only if its homology \(\ker d_n/\im d_{n+1}\) at \(A_n\) is zero.