If $V$ is some finite-dimensional vector space over some field $\mathbb{F}$, then it is well-known that there is an isomorphism
$$(V^{\ast})^{\otimes n}\cong L^{n}_{\mathbb{F}}(V^{n},\mathbb{F})$$
between the tensor product of $n$ copies of $V$ (defined via the universal property) and the set of multiliniear maps of the type $f:V\times\dots\times V\to\mathbb{F}$. Similarely, it is known that there is an isomorphism
$${\bigwedge}^{n}_{\mathbb{F}}V^{\ast}\cong L^{n}_{\mathrm{alt},\mathbb{F}}(V^{n},\mathbb{F})$$
between the $n$th exterior power $V$ (defined via the universal property) and the set of alternating multilinear functions.
Now, I would like to ask the question under which assumptions the above isomorphisms extend to modules. I know that the first isomorphism is still true when we take a module $M$ over some (commutative ring), as long as it is projective and finitely-generated. Is this also the case for the second isomorphism, i.e., is it true that for finitely-generated and projective modules $M$ over some commutative ring $R$ true that
$${\bigwedge}^{n}_{R}M^{\ast}\cong L^{n}_{\mathrm{alt},R}(M^{n},R)?$$
Secondly, are the two isomorphisms true in more general cases? For example, if $R$ is non-commutative?
Best Answer
The question, as stated, is not very natural. This is because $V^{\otimes n}$ is a covariant functor in $V$, whereas $L^n(V,F)$ is a contravariant functor in $V$. So there is no way to construct (or even define the concept of) a natural isomorphism between $V^{\otimes n}$ and $L^n(V,F)$.
What is true, however, is that there is a natural isomorphism $(V^{\otimes n})^* \cong L^n(V,F)$. This comes immediately from the universal property (i.e. the definition) of the tensor product. Namely, the vector space of linear maps $V_1 \otimes \cdots \otimes V_n \to W$ is isomorphic to the vector space of multilinear maps $V_1 \times \cdots \times V_n \to W$. The same holds, for the same reasons, for modules over a commutative ring.
For a finite-dimensional vector space $V$ there is some isomorphism between $V$ and $V^*$, but this cannot be made canonical. For finitely generated projective modules $V$ over a commutative ring, we don't even have an isomorphism between $V$ and $V^*$ in general. This also shows that the answer to both of your (current) questions is No, already for $n=1$.
In the non-commutative case, notice that you need to take a bimodule in order to make sense of the tensor products.