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Let $R$ be a CRing and let $M,N$ be $R$-modules. Let $M^*:=Hom_R(M,R)$. I have seen the following isomorphism asserted in the case where $R$ is a field and $M$ and $N$ are f.g. vector spaces:

$M^*\otimes_R N\cong Hom_R(M,N)$.

I can give a proof of this using a basis, but in what generality can we say this kind of thing? Does it have a nice arrow-theoretic proof?

For some reason, I can't accept answers, so I apologize for not doing it if I am having trouble. Maybe the mods can force my account to accept the answers? (To answer the gentleman's question, I am having javascript problems, so I fear that it won't help).

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You may not be registered if you can't accept answers. Registering will clear all that up and then I can merge any duplicate accounts that might've been created. – Qiaochu Yuan Feb 3 '11 at 11:39
up vote 5 down vote accepted

Again ol' Bourbaki gives you an answer: when $M$ or $N$ are finitely projective modules, the canonical morphism $M^* \otimes N \longrightarrow \mathrm{Hom}(M,N)$ is an isomorphism (Algebra I, chapter II, 4.2, proposition 2).

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