Exercise 1, § 4

Let B denote the ring $A(\varepsilon)$ (X, p. 27); let C be the complex of $(B,B)$-bimodules such that $C_n=B$ for all $n\in\mathbf Z$, $d(x)=\varepsilon x$ for all $x\in C$. Now prove that the complex C has zero homology, but that $H_n(C\otimes_B C)$ is isomorphic to A for all $n\in\mathbf Z$. Deduce that one cannot remove the hypothesis « E bounded on the right » in lemma 1 and prop. 4, pp. 66-67.