I´m looking for homomorphisms between exterior powers of a **free** module M of rank m
  



ΛmR M → Λm-1R M


Exactly, I´m looking for an **explicit** isomorphism
  



M → Hom R (ΛmR M , Λm-1R M)


I compare the ranks and the things go, but I can not imagine a concrete expression.


Suggestions are welcome



Homomorphism between exterior powers of a free module of finite rank
If $R$ is commutative, let $e\_1, \ldots, e\_m$ be an $R$-basis of $M$. Then $\Lambda^m M$ is a free $R$-module of rank one generated by the vector $e\_1 \wedge e\_2 \wedge \ldots \wedge e\_m$ and $\Lambda^{m-1}M$ is free of rank $m$ with basis $$\lbrace e\_1 \wedge \ldots \wedge \hat{e\_i} \wedge \ldots \wedge e\_m \rbrace\_{i=1}^{m}$$ where the hat denotes an omitted basis vector. Here is an explicit isomorphism: $$e\_i \in M\mapsto \Big(e\_1 \wedge e\_2 \wedge \ldots \wedge e\_m \mapsto e\_1 \wedge \ldots \wedge \hat{e\_i} \wedge \ldots \wedge e\_m \Big) \in \mathrm{Hom}\_R(\Lambda^m M, \Lambda^{m-1} M)$$


