Exercise 14, § 2

Commutative Algebra, I, 45

Bourbaki marks this one as harder than the rest.

Let A, B be two rings, E a left A-module, F an (A, B)-bimodule and G a right B-module; consider the canonical homomorphism (Algebra, Chapter II, § 4, Exercise 5)

\operatorname{is}: \operatorname{Hom}_B(F, G) \otimes_A E \to \operatorname{Hom}_B(\operatorname{Hom}_A(E, F), G)

such that $(\sigma(u \otimes x))(v) = u(v(x))$ for all $x \in E, u \in \operatorname{Hom}_B(F, G), v \in \operatorname{Hom}_A(E, F)$ .

Show that, if G is an injective B-module (Algebra, Chapter II, § 2, Exercise 11) and E finitely presented, is is bijective. (Consider first the case where E is free and finitely generated.)

Solution

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