Let $R$ be a commutative Noetherian ring and $P$ be finitely generated $R$-module.

How to prove the following.

**$P$ is projective if and only if $P\otimes N\cong Hom(Hom(P,R),N)$ for all finitely generated $R$-modules $N$.**

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Let $R$ be a commutative Noetherian ring and $P$ be finitely generated $R$-module.

How to prove the following.

**$P$ is projective if and only if $P\otimes N\cong Hom(Hom(P,R),N)$ for all finitely generated $R$-modules $N$.**

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That the canonical map $P\otimes _{R}N\rightarrow \operatorname{Hom}_{R}(P^*,N)$ is an isomorphism when $P$ is projective (for any $R$-module $N$) is classical, see e.g. Bourbaki *Algebra II*, §4, Proposition 2.

In the other direction, taking $N=R$ shows that $P$ is reflexive. Now take $N=P^*$. We get that the canonical map $P\otimes _{R}P^*\rightarrow \operatorname{Hom}_{R}(P^*,P^*) $ is an isomorphism. By the *Remark* 1 following the above Proposition, this implies that $P^*$ is projective, and therefore that its dual $P$ is projective.

infinitedirect sums? This will give you a feeling for the question, for which $N$ to expect an isomorphism (a finitely generated $N$ is the cokernel of a morphism between finite direct sums of $R$'s, etc.). $\endgroup$ – Sasha Feb 2 '19 at 8:42