Lemma 1

Let C be a complex of right A-modules and E a complex of left A-modules. We assume that $H(C)=0$ and that E is flat and bounded on the right. Then $H(C\otimes_A E)=0$.

For $k\in\mathbf Z$, let $T^{(k)}$ be the subcomplex of $C\otimes_A E$ such that

then $T^{(k-1)}\subset T^{(k)}$ and we have an exact sequence of complexes

where $i_k$ is the canonical injection and where $\pi$ projects the preceding direct sum onto its factor $C_{n-k}\otimes_A E_k=(C\otimes_A E_k(-k))_n$. By Cor. 2 above, we have $H(C\otimes_A E_k(-k))=0$, hence $i_k$ is a homomorphism. We have $T^{(k)}=0$ for $k$ sufficiently small, since $E$ is bounded on the right, hence $H(T^{(k)})=0$ for all $k$ by induction on $k$. Finally, the canonical morphism $\underset{\longrightarrow}{\lim}\,T^{(k)}\to C\otimes_A E$ is obviously an isomorphism, hence $H(C\otimes_A E)=0$ (X, p. 28, Prop. 1).