Proposition 3

If the A-module L is flat, the sequences $(\mathscr{E}_n)$ above are exact for $n > 0$.

a) Let us first note that, if $p_L$ is the composite homomorphism

where $\alpha$ is the tensor product of the canonical projections and $\beta$ the canonical isomorphism, it remains to prove that $H(p_L)$ is bijective.

b) If $L = 0$ or if $L = A$, the proposition is obvious.

c) Suppose L is free of finite rank; write it as the direct sum $L_1 \oplus ... \oplus L_n$ of free A-modules of rank 1. According to the remark preceding the proposition, the complex $\mathbf{S}(L) \otimes \Lambda(L)$ is isomorphic to the tensor product of the $n$ free complexes $\mathbf{S}(L_i) \otimes \Lambda(L_i)$ whose homology is free according to b). According to X, p. 79, cor. 4, the canonical homomorphism

is bijective. According to b) $H(p_{L_i})$ is bijective for every $i$. Since $\bigotimes_{i=1}^n H(p_{L_i}) = H(p_L) \circ \gamma$,

d) In the general case, L is an inductive limit of a filtered inductive system $(L_i)_{i \in I}$ of free modules of finite rank (X, p. 14, th. 1). Since the canonical bijective homomorphism

is an isomorphism of complexes, the proposition follows from X, p. 28, prop. 1.