Lemma 10.78.9 (0DVB)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.78: Finite projective modules
Lemma 10.78.9 (cite)

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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 September 20, 2017 at 22:57

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

Comment #2919 by Johan on October 09, 2017 at 23:54

THanks, Fixed here.

Comment #9857 by Junyan Xu on November 12, 2024 at 06:48

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

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