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Let $M$ and $N$ be a finitely generated projective module over a ring $R$. Suppose we have a non degenerate bilinear pairing $\langle \ \cdot \ ,\ \cdot\ \rangle: M \times N \to R$.

I want to show $M$ is isomorphic to the dual $N^*$ of $N$.

The injectivity of $M$ into $N^*$ follows from the non degeneracy of the pairing by definiting a map $x \mapsto \langle x, \cdot\rangle$. What I cannot prove is surjectivity.

If I impose a condition that $R$ is an injective module then I think surjectivity also follows. But I want to prove it without any condition on $R$

Any help is appriciated.

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    You need (at least) the bilinear pairing to be surjective. Otherwise you get examples like: $R$ the ring of algebraic integers of a number field $K$, [K:\mathbb{Q}]<\infty. $M,N$ any non-zero ideals of $R$ (they are all projective), $\langle\ ,\ \rangle =$ the product.2012-07-06

1 Answers 1

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If by non-degenerate you mean that $\langle x , - \rangle = 0 \Leftrightarrow x = 0$ (or $\langle - , y \rangle \Leftrightarrow y = 0$), your question is equivalent to: Is every injective homomorphism between finitely generated projective $R$-modules an isomorphism? The answer is no, even for maps $R \to R$.

If by non-degenerate you mean that $\langle x , - \rangle = 0 \Leftrightarrow x = 0$ and $\langle - , y \rangle \Leftrightarrow y = 0$, then your question is equivalent to: If $M \to N$ is an injective homomorphism of f.g. proj. modules such that $N^* \to M^*$ is also injective, then $M \to N$ is an isomorphism. But again this is false for maps $R \to R$.

The "correct" definition of a perfect pairing $M \times N \to R$ is the following: Both maps $M \to N^*$ and $N \to M^*$ are isomorphisms.

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    I've edited my answer.2012-07-06