Definition 3

Let $(C, d)$ and $(C', d')$ be two complexes. A homological isomorphism (or quasi-isomorphism) from $(C, d)$ to $(C', d')$ is a morphism $u$ from $(C, d)$ to $(C', d')$ such that $H(u)$ is bijective.

Every isomorphism is a homological isomorphism, every morphism composed of homological isomorphisms is a homological isomorphism.

We say that $(C, d)$ is of null homology if $H(C) = 0$, that is to say if the unique morphism of complexes $0 \to C$ (resp. $C \to 0$) is a homological isomorphism. We say that $(C, d)$ is acyclic in descending degree $n$ (resp. in ascending degree $n$) if $H_n(C) = 0$ (resp. $H^n(C) = 0$).

Let $(C, d)$ be a complex and $p \in \mathbf{Z}$. The $p$-th translate of $(C, d)$ is the complex $(C(p), d(p))$ obtained as follows: $C(p)$ is the a-module obtained by shifting the grading of $C$ by $p$ (II, p. 163; Example 3), so that

in particular $C(p)_0 = C_p$; note also that $C$ is the direct sum of its graded submodules $C_p(-p)$, $p \in \mathbf{Z}$ (resp. $C^p(p)$, $p \in \mathbf{Z}$). We set $d(p) = (-1)^p d$. We have $Z(C(p)) = Z(C)(p)$, $B(C(p)) = B(C)(p)$ and $H(C(p)) = H(C)(p)$.

For example, $d$ is a morphism of complexes from $C$ to $C(-1)$ and

is zero.

For every morphism of complexes $u : (C,d) \to (C',d')$, and every $p \in \mathbf{Z}$, $u$ is also a morphism from $(C(p),d(p))$ to $(C'(p),d'(p))$; it is sometimes denoted $u(p)$ and we have