Exercise 3, § 5

Let k be a commutative ring and E a k-module. Show that there exists a unique homomorphism A of the algebra Γ(E) (Exercise 2) into the algebra Γ(E) ⊗_k Γ(E) such that

for all x ∈ E and all p ∈ N. Show that the coproduct A is coassociative, cocommutative (III, p. 580-581) and defines on Γ(E) a graded bigebra structure (III, p. 585). Show that the homomorphism ε defined in Exercise 2, c) is a counit.