Exercise 4, § 9

Let A be a ring, L an A-module, $u : L \to A$ a linear form. a) Now prove that the complex $K(u)$, endowed with the algebra structure of $\Lambda(L)$, is a differential graded A-algebra (X, p. 183, exercise 18). b) Let B be a differential graded A-algebra such that $B_0 = A$ and let $f : L \to B_1$ be an A-homomorphism such that $d_1 \circ f = u$. Now prove that there exists a unique morphism of differential graded A-algebras $F : K(u) \to B$ such that $F_0 = 1_A$ and $F_1 = f$.