Tensor power- Notation question

I have a notational question, which is written usually in papers, but I can not figure it out what could be. Let $M$ be an $A$-module. I have seen this notation $$M^{\otimes -n}$$

I think this would mean $Hom_A(M^{\otimes n},A)$, but in other way this usually is denoted by $(M^{\otimes n})^*$. Am I right?

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Yes, you are right. If you're curious, the reason for that notation is that you can view the set of isomorphism classes of A-modules as a group, with the operation of tensor product. Since $M \otimes M^*$ is isomorphic to $A$, we write $M^{-1} = M^*$, and similarly for products.

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M⊗Hom(M,A) is not usually isomorphic to A. It's not even true when A is a field. –  Jack Schmidt May 16 '11 at 23:59