Exercise 3, § 2

Let $C$ be a complex of A-modules; we consider $C$ as an A(\varepsilon)-module (X, p. 27). We denote

the canonical injection. Now prove that the following conditions are equivalent: (i) $C$ is i-projective (X, p. 170, exercise 19). (ii) $C$ is i-injective (loc. cit.). (iii) The complex $C$ is homotopic to zero. (iv) There exists an A-module graded $M$ and an isomorphism of graded A(\varepsilon)-modules $A(\varepsilon) \otimes_A M \to C$. (v) There exists an A-module graded $N$ and an isomorphism of graded A(\varepsilon)-modules $C \to \mathrm{Hom}_A(A(\varepsilon), N)$.