Remark

Conversely, let B be a $k$-algebra, L a free B-module with basis $(e_i)_{i \in I}$, and M a B-module. The datum of a linear form $u : L \to B$ is equivalent to that of a family $x = (x_i)_{i \in I}$ of elements of B, by the relation $x_i = u(e_i)$. The underlying complex of $k$-modules of $K_B^*(u, M)$ (resp. $K^B(u, M)$) is then identified, by the isomorphism $\theta^*$ (resp. $\theta_*$) with the Koszul complex $K^*(x, M)$ (resp. $K_*(x, M)$). For example $K^B(u)$ is identified with $K^*(x, B)$.

The notations $H_*(x, M)$, $H_*(x, M)$, etc., are introduced as in No. 1 (X, p. 147), and all the results of Nos. 1 and 2 apply mutatis mutandis, the module $A^I$ being free. We have for example isomorphisms

where $(x)$ denotes the ideal $\sum A x_i$ of A. For example also, if $K_*(x, A)$ is acyclic in degrees $> 0$, we have isomorphisms

Finally, suppose that I is (finite and) totally ordered, for example $I = \{ 1, ..., n \}$; identify $\Lambda^n(A^I)$ with A thanks to the basis element $e_1 \wedge ... \wedge e_n$ and translate the isomorphism $K_p^A(u, M) \to K_A^{n-p}(u, M)$ of X, p. 149. It becomes, by transport by $(\theta_p)$ and $(\theta^{n-p})$, the isomorphism

which associates with $m \in C_1^p(M)$ the element $\tilde{m}$ of $C_1^{n-p}(M)$ such that

if $(\alpha_1, ..., \alpha_p, \beta_1, ..., \beta_{n-p})$ is an even permutation of $\{ 1, ..., n \}$. Let us also remark that when $I = \{ 1, ..., n \}$, we can identify $C_1^p(M)$ with the set of families $m(\alpha_1, ..., \alpha_p)$ of elements of M where $\alpha_1 < \alpha_2 < ... < \alpha_p$; formula (12) remains valid, as does relation (13).