Definition 1

Tag 0E2G, permanent. Cite it as [Bourbaki, Tag 0E2G].

Algebra, chapter X, § 9. Complexes de Koszul

The complex $(\Lambda(L), d_u)$ is denoted by $\mathbf{K}^A(u)$ or $\mathbf{K}(u)$ .

One should note that $\mathbf{K}_n(u) = \Lambda^n(L) = \mathbf{K}^{-n}(u)$ . It is clear that $\mathbf{K}(u)$ is zero on the right and that $H_0(\mathbf{K}(u)) = \mathrm{Coker}\,(u) = A/q$ where $q$ is the ideal $u(L)$ of $A$ .

For every complex of $A$ -modules $C$ , put

\begin{aligned} \mathbf{K}^A(u, C) &= C \otimes_A \mathbf{K}^A(u), \\ H^A(u, C) &= H(C \otimes_A \mathbf{K}^A(u)) \\ H_r^A(u, C) &= H_r(C \otimes_A \mathbf{K}^A(u)), \end{aligned}

\begin{aligned} \mathbf{K}^*_A(u, C) &= \mathrm{Homgr}_A(\mathbf{K}^A(u), C), \\ H^*_A(u, C) &= H(\mathrm{Homgr}_A(\mathbf{K}^A(u), C)), \\ H^r_A(u, C) &= H^r(\mathrm{Homgr}_A(\mathbf{K}^A(u), C)). \end{aligned}

There are therefore canonical homomorphisms of $A$ -modules (X, p. 62 and p. 82)

\begin{aligned} \gamma_0 : H_0(C) \otimes_A A/q &\to H^A_0(u, C), \\ \lambda^0 : H^0_A(u, C) &\to \mathrm{Hom}_A(A/q, H^0(C)). \end{aligned}

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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