Lemma 10.78.9. Let $R$ be ring. Let $L$, $M$, $N$ be $R$-modules. The canonical map

$\mathop{\mathrm{Hom}}\nolimits _ R(M, N) \otimes _ R L \to \mathop{\mathrm{Hom}}\nolimits _ R(M, N \otimes _ R L)$

is an isomorphism if $M$ is finite projective.

Proof. By Lemma 10.78.2 we see that $M$ is finitely presented as well as finite locally free. By Lemmas 10.10.2 and 10.12.16 formation of the left and right hand side of the arrow commutes with localization. We may check that our map is an isomorphism after localization, see Lemma 10.23.2. Thus we may assume $M$ is finite free. In this case the lemma is immediate. $\square$

Comment #2818 by Jonathan Gruner on

Let $R$ be ring map. -> Let $R$ be a ring.

There are also:

• 2 comment(s) on Section 10.78: Finite projective modules

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).