Exercise 9, § 7

We suppose that the ring A is commutative. Let B, C be two associative unital A-algebras; we denote by $m_B : B \otimes_A B \to B$ the multiplication of B, and by $m_C$ the multiplication of C.

(a) Prove that the homomorphism

(cf. exercise 7) defines on $\mathrm{Tor}^A(B, C)$ a graded, associative and unital A-algebra structure.

(b) Let S be a differential graded A-algebra and $p : S_0 \to B$ a homomorphism of A-algebras such that $(S, p)$ is a free resolution of B (cf. X, p. 183, exercise 18). Prove that the isomorphism

is an isomorphism of A-algebras if $H(S \otimes_A C)$ is provided with the algebra structure deduced from the differential graded algebra structure of $S \otimes_A C$ (X, p. 183, exercise 18, b) and c)).

(c) If the A-algebras B and C are commutative, prove that the graded algebra $\mathrm{Tor}^A(B, C)$ is alternating (cf. X, p. 183, exercise 19).