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 $N$.**



$P$ is projective if and only if $P\otimes N\cong Hom(Hom(P,R),N)$
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. 


