Let $(A,d)$ and $(A^{'},d^{'})$ be chain complexes and $f:A \to A^{'}$ $g:A \to A^{'}$ be chain maps. A chain homotopy $D$ between $f$ and $g$ is a sequence of homomorphisms $\{D_{n}:A_{n} \to A_{n+1}^{'}\}$ so that $d_{n+1}^{'} \circ D_{n} + D_{n-1} \circ d_{n}=f_{n}-g_{n}$ for each $n$ Thus, we have the following diagram: $$ \xymatrix{ & A_{n+1} \ar[r]^{d_{n+1}} \ar[d]_{f_{n+1}-g_{n+1}} & A_{n} \ar[dl]^{D_{n}} \ar[r]^{d_{n}} \ar[d] & A_{n-1} \ar[dl]_{D_{n-1}} \ar[d]^{f_{n-1}-g_{n-1}} \\ & A_{n+1}^{'} \ar[r]_{d_{n+1}^{'}} & A_{n}^{'} \ar[r]_{d_{n}^{'}} & A_{n-1}^{'} } $$

If there exists a chain homotopy between $f$ and $g$ then $f$ and $g$ are said to be chain homotopic.