Consider the $\mathbb{Z}$-module $M=\bigoplus{\mathbb{Z}/p\mathbb{Z}}$, where the direct sum is taken over the set of all prime numbers. How do I show that the localizations $M_\mathfrak{p}$ are finitely generated $\mathbb{Z}_\mathfrak{p}$-modules for any prime ideal $\mathfrak{p}$ of $\mathbb{Z}$?
localizations of a direct sum module
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$\begingroup$
commutative-algebra
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1Hint: If $q\neq p$ are primes, then $q$is a $p$-adic unit. What can you say about a module that is annihilated by a unit? – 2011-09-11
1 Answers
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Recall the following facts:
(1) Direct limits, direct sums and tensor products commute up to isomorphism
(2) The localization of a $R$-module $M$ at a prime $\mathfrak{p}$ is isomorphic to the module obtained by extending $M$ to scalars in $R_{\mathfrak{p}}$
(3) $(\mathbb{Z}/p\mathbb{Z}) \otimes_{\mathbb{Z}}\mathbb{Z}_q$ is isomoprhic to $\mathbb{Z}/p\mathbb{Z}$ if $p=q$ and $0$ otherwise.
From these three facts, it follows that
$M_{\mathfrak{p}} = M \otimes_{\mathbb{Z}} \mathbb{Z}_{\mathfrak{p}} = (\displaystyle\lim_{\rightarrow} \text{ } \displaystyle \bigoplus_{p
and your claim follows.
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0I like this answer a lot (you should add the remarks on $\prod$, which are excellent, to the main answer if you get the chance), but it seems like the solution of the problem is mixed up with what is more or less a proof of "arbitrary direct products commute with tensor products", which could be confusing. Let me know if I'm wrong! – 2011-09-12