Let \(R\) be a ring, let \((C_*, \del^C)\) and \((D_*, \del^D)\) be chain complexes of \(R\)-modules, and let \(f, g\colon C_*\to D_*\) be chain maps. A chain homotopy from \(f\) to \(g\) is a family of \(R\)-linear maps \(h_i\colon C_i\to D_{i+1}\), \(i\in\ZZ\), such that \[ f_i - g_i = \del^D_{i+1}\circ h_i + h_{i-1}\circ \del^C_i \quad\text{for all } i\in\ZZ .\] The chain maps \(f\) and \(g\) are chain homotopic if a chain homotopy from \(f\) to \(g\) exists.
[@Hat02].