Corollary 2

Let C be a commutative ring and E, F two finitely generated projective C-modules; then $\operatorname{Hom}_C(E, F)$ is a finitely generated projective C-module.

It may be assumed that there are two finitely generated free C-modules M, N such that $M = E \oplus E', N = F \oplus F'$; it follows from § 1, no. 6, Corollary 1 to Proposition 6 that $\operatorname{Hom}_C(M, N)$ is finitely generated and free and on the other hand that $\operatorname{Hom}_C(M, N)$ is isomorphic to

whence the corollary.