While looking over the universal coefficient theorem, I was wondering whether we can assume that groups of the form $\mathbb Z^{\mathbb N}$ are free. If we replace $\mathbb Z$ by a field, the answer is yes, since we can take maximal linearly independent sets by Zorn’s lemma, and by taking denominators to show that this set spans the whole space. This technique seems to fail in an essential way over $\mathbb Z$, but is there any nice invariant we can use to properly show that $\mathbb Z^{\mathbb N}$ is not free?
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4The group $\mathbb Z^{\mathbb N}$ is not free. – Christoph Jun 01 '19 at 12:25
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@DietrichBurde: it sounds weird to mark a question as duplicate of a closed question – YCor Jun 01 '19 at 15:29
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@YCor Yes, but the question is formulated there like here: "From this post I see that $\mathbb{Z}^{\mathbb{N}}$ is not free. Someone can explain me why?". The original post is a bit different from the wording and does not use the symbol $\Bbb Z^{\Bbb N}$ (although it is mentioned by Martin in the comments). – Dietrich Burde Jun 01 '19 at 15:34
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@DietrichBurde I see. So, I voted to reopen the given post (https://math.stackexchange.com/questions/877645/a-not-free-mathbbz-module) and hope it will be. – YCor Jun 01 '19 at 15:42
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@YCor I did, too. – Dietrich Burde Jun 01 '19 at 15:47
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@YCor I can't see any issue with marking a question as a duplicate of a closed question as long as the last one has a good answer. A question can be closed for many reasons: lack of context, no effort, etc, and reopening it for being chosen as a dupe is a wrong reason. – user26857 Jun 01 '19 at 16:52
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@user26857 I see a clear issue. To close a question prevents from adding answers. If you close a (good) question as duplicate of a closed question (closed on the ground of lacking context), you prevent possible answers to be posted to either question. – YCor Jun 01 '19 at 16:55
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@YCor This "good question" is in fact a well known problem which is solved on this site and on MO many times and in various ways. But if you are right then I'd consider the reopening of the present question which is more actual and people can be tempted to answer it. – user26857 Jun 01 '19 at 16:58
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2@user26857 I have no definite opinion on this particular case. I'm aware this specific problem appears at many places. I just claim that I consider as a bug that a question can be closed as duplicate of a closed question. Yet I'm lazy to open a discussion on meta – YCor Jun 01 '19 at 17:03
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Finally, it's marked as duplicate of https://math.stackexchange.com/questions/320444/why-isnt-an-infinite-direct-product-of-copies-of-bbb-z-a-free-module which is not closed – YCor Jun 02 '19 at 20:11