The Stacks project

Lemma 15.23.3. Let $R$ be a discrete valuation ring and let $M$ be a finite $R$-module. Then the map $j : M \to \mathop{\mathrm{Hom}}\nolimits _ R(\mathop{\mathrm{Hom}}\nolimits _ R(M, R), R)$ is surjective.

Proof. Let $M_{tors} \subset M$ be the torsion submodule. Then we have $\mathop{\mathrm{Hom}}\nolimits _ R(M, R) = \mathop{\mathrm{Hom}}\nolimits _ R(M/M_{tors}, R)$ (holds over any domain). Hence we may assume that $M$ is torsion free. Then $M$ is free by Lemma 15.22.11 and the lemma is clear. $\square$


Comments (2)

Comment #3601 by Remy on

I think this is probably true over any ring . Indeed, if is a surjection, then naturality of the evaluation map shows that the diagram commutes. Since the top map is an isomorphism and the right vertical map is surjective, we conclude that the bottom map is surjective.

Comment #3602 by Remy on

Oh I see, I don't know why the right map is surjective.


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