Chain map

Let \(R\) be a ring and let \((C_*, \del^C)\) and \((D_*, \del^D)\) be chain complexes of \(R\)-modules, with \(\del^C_i\colon C_i\to C_{i-1}\) and \(\del^D_i\colon D_i\to D_{i-1}\). A chain map \(f\colon C_*\to D_*\) is a family of \(R\)-linear maps \(f_i\colon C_i\to D_i\), \(i\in\ZZ\), such that \[ f_{i-1}\circ\del^C_i = \del^D_i \circ f_i \quad\text{for all } i\in\ZZ .\]