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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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Perhaps the other answer was describing the Picard group of a commutative ring A. If M is a finitely generated projective A-module of rank 1, then the tensor product MA HomA(M, A) is isomorphic to A. In particular, the set of isomorphism classes of finitely generated projective modules of rank 1 forms a group under tensor product.

If A is an integral domain, then the M we consider are just the isomorphism classes of projective modules that are isomorphic to ideals of A. This forms the ideal class group.

In particular for principal ideal domains or fields, this is not a useful concept.

Also, none of this works for general modules, not even finitely generated projective modules of rank larger than 1, so I would be very cautious of assuming this is true for a negative tensor power of some random module.

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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

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