Bourbaki

Exercise 7, § 4

Algebra, II, 397

Show that, if E1,E2,F1,F2E_1, E_2, F_1, F_2 are four modules over a commutative ring

C, the canonical homomorphism (21) (no. 4) is composed of the homomorphisms

Hom⁡(E1,F1)⊗Hom⁡(E2,F2)→Hom⁡(E1,F2⊗Hom⁡(E2,F2))→Hom⁡(E1,Hom⁡(E2,F1⊗F2))→Hom⁡(E1⊗E2,F1⊗F2) \operatorname{Hom}(E_1, F_1) \otimes \operatorname{Hom}(E_2, F_2) \to \operatorname{Hom}(E_1, F_2 \otimes \operatorname{Hom}(E_2, F_2)) \\ \to \operatorname{Hom}(E_1, \operatorname{Hom}(E_2, F_1 \otimes F_2)) \\ \to \operatorname{Hom}(E_1 \otimes E_2, F_1 \otimes F_2)
where the first two homomorphisms come from the canonical homomorphism (7) (no. 2) and the last is the isomorphism of no. 1, Proposition 1.

Solution

No solution has been written yet.

Transcribed from the printed volume. Not translated.