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I was reading JS Milne's book on Arithmetic duality theorems and he states on page 105 that for a finitely generated torsion-free G-module (G is actually a galois group) M we have $$Hom_{cts}(M,\mathbb{C}^{\times})=Hom(M,\mathbb{C}^{\times})$$ where the cts denotes continuous homomorphisms and the second are just homomorphims.

I can quite see why this is true, so I was wondering if I could get some soft of explanation of why this is true since I dont have much of an intuition when it comes to continuous homomorphisms.

Thank you

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What topology could a finitely generated (torsion free) module could carry? –  Asaf Karagila Nov 19 '12 at 21:29
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