Exercise 1, § 4

Let A be a commutative ring, E, F two A-algebras, M a left E-module, N a left F-module and G the algebra $G = E \otimes_A F$. For every ordered pair of endomorphisms $u \in \mathrm{End}_E(M)$, $v \in \mathrm{End}_F(N)$, there exists one and only one G-endomorphism $w$ of $M \otimes_A N$ such that $w(x \otimes y) = u(x) \otimes v(y)$ for $x \in M, y \in N$ and we can write $w = \phi(u \otimes v)$, where $\phi$ is an A-linear mapping (called canonical) of $\mathrm{End}_E(M) \otimes_A \mathrm{End}_F(N)$ into $\mathrm{End}_G(M \otimes_A N)$. Suppose in what follows that A is a field.

(a) Show that the canonical mapping $\phi$ is injective (consider an element $z = \sum_i u_i \otimes v_i$, where the $v_i \in \mathrm{End}_F(N)$ are linearly independent over A and write $(\phi(z))(x_\alpha \otimes y) = 0$ for every element $x_\alpha$ of a basis of M over A and all $y \in N$).

(b) Suppose that M is a finitely generated free E-module. Show that the mapping $\phi$ is bijective in each of the following cases: (1) E is of finite rank over the field A; (2) N is a finitely generated F-module.

(c) Suppose that M is a free E-module and that there exist a $y \in N$ and an infinite sequence $(v_n)$ of endomorphisms of the F-module N such that the vector subspace (over A) of N generated by the $v_n(y)$ is of infinite rank over A. Show that, if the mapping $\phi$ is bijective, every basis of M over E is necessarily finite.

(d) Suppose still that M is a free E-module. Show that, for the G-module $M \otimes_A N$ to be faithful, it is necessary and sufficient that the F-module N be faithful (consider a basis of E over A).