Let \(R\) be a ring and \(M\) an \(R\)-module. A projective resolution of \(M\) is an exact sequence of \(R\)-modules \[ \cdots\to P_2\mapsvia{d_2}P_1\mapsvia{d_1}P_0\mapsvia{\varepsilon}M\to0 \] in which every \(P_i\) is a projective module. It is a free resolution if every \(P_i\) is a free module.
Let \(R\) be a ring and \(M\) an \(R\)-module.
\(M\) has a free resolution, and hence a projective resolution.
If \(P_\bullet\to M\) and \(Q_\bullet\to M\) are projective resolutions, there is a chain map \(P_\bullet\to Q_\bullet\) over \(\id_M\), and any such chain map is a chain homotopy equivalence [@Hat02].
Let \(R\) be commutative and \(N\) an \(R\)-module. By (b), the cohomology of the cochain complex \(\Hom_R(P_\bullet,N)\) and the homology of the chain complex \(P_\bullet\otimes_RN\) do not depend on the projective resolution \(P_\bullet\) of \(M\), up to canonical isomorphism; they are \(\Ext^n_R(M,N)\) and \(\Tor_n^R(M,N)\).