Definition 1

A differential complex of $A$-modules is a pair $(C,d)$ formed by a graded $A$-module $C$ and an endomorphism $d:C\to C$ graded of descending degree $-1$ and such that $d\circ d=0$.

It is also called a complex of $A$-modules, or an $A$-complex, or a complex. One often writes C in place of (C, d); the endomorphism d is called the differential of the complex (C, d), or by abuse of language of C.

If C_n (resp. C^n) is the homogeneous component of descending degree (resp. ascending degree) n of C, the datum of d is equivalent to that of the sequence of homomorphisms

(1)

resp.

(1')

such that $d_n \circ d_{n+1} = 0$ for all $n \in \mathbf{Z}$ (resp. $d^n \circ d^{n-1} = 0$ for all $n \in \mathbf{Z}$). By abuse of language, one will also call a complex the datum of such a sequence of $A$-modules and homomorphisms.

One will remark, as a mnemonic device, that when one “follows the direction of the arrows” in the diagrams (1) and (1’), the descending degree decreases and the ascending degree increases.

Every graded $A$-module will tacitly be considered as a complex by endowing it with the zero differential; the complexes thus obtained will be called complexes with zero differential. In particular, every $A$-module M will be endowed with the unique structure of $A$-complex such that $M_0 = M^0 = M$. The complex (C, d) is said to be zero if C is reduced to 0. In the sequel, we denote by 0 a zero complex, chosen once and for all.

Adjoin to the ordered set $\mathbf{Z}$ two elements denoted $-\infty$ and $+\infty$; denote by $\overline{\mathbf{Z}}$ the set obtained, and endow it with the order relation extending that of $\mathbf{Z}$ and such that $-\infty < n < +\infty$ for all $n \in \mathbf{Z}$; every subset of $\overline{\mathbf{Z}}$ possesses a greatest lower bound and a greatest upper bound.

Let C be a complex; one calls the right and left bounds $^1$ of C the elements $b_d(C)$ and $b_g(C)$ of $\overline{\mathbf{Z}}$ defined by

One says that C is zero on the right if $b_d(C) \geqslant 0$, bounded on the right if $b_d(C) \neq -\infty$, zero on the left if $b_g(C) \leqslant 0$, bounded on the left if $b_g(C) \neq +\infty$; one says that C is bounded if One calls the length $^2$ of C and denotes by $l(C)$ the element of $\overline{\mathbf{Z}}$ defined as follows: if C is zero, $l(C) = -\infty$; if C is bounded and nonzero $l(C) = b_g(C) - b_d(C)$; if C

$^1$ The words right and left are relative to the description of C by means of the diagrams (1) and (1’). $^2$ One must not confuse the notion of length of the complex (C, d) with that of length of the module C (II, p. 21).

is not bounded, $l(C) = + \infty$. \* With the conventions of TG, IV, p. 13-17, one always has $l(C) = b_g(C) - b_d(C)$. \*

For example, if $k$ consecutive components of $C$ are non-zero, the others being zero, one has $l(C) = k - 1$ if $k > 0$, $l(C) = - \infty$ if $k = 0$.

One says that the complex $(C, d)$ is free, projective, flat, injective, if each of the modules $C_n$ is so. It will be noted that the complex $(C, d)$ is projective or flat if and only if the module $C$ is so (II, p. 39, prop. 3 and X, p. 8, prop. 4), but that $C$ can be free without the complex $(C, d)$ being so (since a direct factor of a free module is not always free), analogously $(C, d)$ can be injective without $C$ being so (X, p. 170, exercise 21).

Let $(C, d)$ be a complex. Put $Z(C, d) = \mathrm{Ker}\,(d)$, $B(C, d) = \mathrm{Im}\,(d)$; these are graded submodules of $C$, called respectively the module of cycles and the module of boundaries of $(C, d)$; the homogeneous components of $Z(C, d)$ and $B(C, d)$ are denoted $Z_n(C, d) = Z^{-n}(C, d)$, $B_n(C, d) = B^{-n}(C, d)$; one has $Z_n(C, d) = \mathrm{Ker}\,(d_n)$, $B_n(C, d) = \mathrm{Im}\,(d_{n+1})$, $Z^n(C, d) = \mathrm{Ker}\,(d^n)$, $B^n(C, d) = \mathrm{Im}\,(d^{n-1})$.

Since $d \circ d = 0$, one has $B(C) \subset Z(C)$; two cycles are said to be homologous if their difference is a boundary; the graded quotient module $H(C, d) = Z(C, d)/B(C, d)$ is called the homology module of $(C, d)$; its elements are the homology classes; its homogeneous components are denoted $H_n(C, d) = H^{-n}(C, d)$.