Exercise 7, § 5

(Extension of scalars) Let k be a commutative ring, L an associative commutative unital k-algebra and E a k-module. a) Let $\Gamma(L \otimes_k E)$ be the gamma algebra of the L-module $L \otimes_k E$ (Exercise 2). Show that there exists a unique homomorphism $\theta$ of $L \otimes_k \Gamma(E)$ into $T(L \otimes_k E)$ compatible with the algebra structure on L and such that

for all $x \in E$ and $p \in \mathbf{N}$. b) Show that $\theta$ is an isomorphism of graded bigebras over L.