Definition 2

Let $(C, d)$ and $(C', d')$ be two complexes. A morphism $^1$ from $(C, d)$ to $(C', d')$ is a graded A-homomorphism $u$ of degree 0 from $C$ to $C'$ such that

¹ Or morphism of degree 0 (cf. X, p. 81).

For every $n$, we therefore have $d'_n \circ u_n = u_{n-1} \circ d_n$ and ${d''}^n \circ u^n = u^{n+1} \circ d^n$. We have

and we denote by $Z(u) : Z(C) \to Z(C')$, $B(u) : B(C) \to B(C')$, $H(u) : H(C) \to H(C')$ the homomorphisms of A-modules deduced from this; the homogeneous components of these morphisms are denoted $Z_n(u)$, $Z^n(u)$, ...

If $v$ is another morphism from $(C, d)$ to $(C', d')$, then $u + v$ is a morphism from $(C, d)$ to $(C', d')$, and we have

Analogously, if $A$ is an algebra over a commutative ring $k$, and if $\lambda \in k$, then $\lambda u$ is a morphism from $(C, d)$ to $(C', d')$ and we have If $u' : (C', d') \to (C'', d'')$ is another morphism of complexes, then $u' \circ u$ is a morphism from $(C, d)$ to $(C'', d'')$ and we have It is clear that a bijective morphism is an isomorphism.