The Stacks project

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. $\square$

Comments (3)

Comment #2818 by Jonathan Gruner on

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

Comment #9857 by Junyan Xu on

Isn't it easier to use that M is a direct summand of a finite free module?

There are also:

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

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