Exercise 18, § 3

We suppose that the ring $A$ is commutative. A differential graded $A$-algebra is an $A$-algebra $S$, graded of type $\mathbf{Z}$, such that $S_n = 0$ for $n < 0$, provided with an antiderivation $d$ of degree $(-1)$, satisfying $d \circ d = 0$. We shall still denote by $S$ the complex of $A$-modules $(S, d)$.

a) Let $C$ be a complex of $A$-modules, zero in degrees $< 0$. Prove that there exists on $S = T(C)$ (resp. $S = S(C)$, resp. $S = \Lambda(C)$) a differential graded $A$-algebra structure, such that the canonical injection of $C$ into $S$ is a morphism of complexes.

b) Prove that the multiplication of $S$ induces a graded $A$-algebra structure on $H(S)$.

c) Let $S, T$ be two differential graded $A$-algebras. We put $D(s \otimes t) = ds \otimes t + (-1)^p s \otimes dt$ for $s \in S_p, t \in T$. Prove that $D$ defines a differential graded $A$-algebra structure on the left tensor product $S \otimes_A T$ (III, p. 49).