Timeline for Divisible torsion $\mathbb{Z}$-modules
Current License: CC BY-SA 4.0
31 events
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| Jul 22, 2019 at 7:26 | history | edited | YCor | CC BY-SA 4.0 |
made question more readable, added tag
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| Jul 22, 2019 at 6:20 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
Removed the deprecated (abstract-algebra) tag - see the tag info: https://mathoverflow.net/tags/abstract-algebra/info (if there are some other suitable tags, choose them instead.)
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| Apr 7, 2014 at 12:31 | comment | added | user76758 | @KevinVentullo: Whoops, I had not noticed the last sentence of the question. Since that endomorphism ring is $\widehat{Z}$, as noted by Strickland, for the purpose of tensoring against a torsion module it could just as well be $\mathbf{Z}$ (as I had been thinking). | |
| Apr 7, 2014 at 8:28 | vote | accept | Fat | ||
| Apr 7, 2014 at 8:28 | vote | accept | Fat | ||
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| Apr 7, 2014 at 8:27 | vote | accept | Fat | ||
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| Apr 7, 2014 at 7:04 | vote | accept | Fat | ||
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| Apr 7, 2014 at 7:03 | vote | accept | Fat | ||
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| Apr 7, 2014 at 7:03 | vote | accept | Fat | ||
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| Apr 7, 2014 at 7:02 | vote | accept | Fat | ||
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| Apr 7, 2014 at 7:02 | vote | accept | Fat | ||
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| Apr 7, 2014 at 7:02 | vote | accept | Fat | ||
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| Apr 7, 2014 at 7:02 | vote | accept | Fat | ||
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| Apr 7, 2014 at 7:02 | vote | accept | Fat | ||
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| Apr 7, 2014 at 5:18 | comment | added | Kevin Ventullo | @user76758 I'm not sure I understand your first comment. The author says the tensor is taken over $End(\mathbb{Q}/\mathbb{Z})$. For your second comment, you're right, I should remark that the image of $\mathbb{Q}_p/\mathbb{Z}_p$ under any homomorphism lands inside a cofinitely generated sub. | |
| Apr 7, 2014 at 4:48 | answer | added | user76758 | timeline score: 5 | |
| Apr 7, 2014 at 4:39 | comment | added | user76758 | @KevinVentullo: The step where you say direct limits "commute with everything" also has a small gap insofar as moving a direct limit out of the second variable of a Hom is not a generally valid thing, so its validity in the present setting (which is true) does require an argument (e.g., via Pontryagin duality or other means). | |
| Apr 7, 2014 at 4:35 | comment | added | user76758 | @KevinVentullo: The end of your argument has a slight "cheat": the tensor product begins as one over $\mathbf{Z}$, and one has to directly compute (by various elementary means) the endomorphism ring to see that it collapses away in the tensor product (so tensoring over the endomorphism ring is a red herring). | |
| Apr 7, 2014 at 1:27 | answer | added | Neil Strickland | timeline score: 4 | |
| Apr 6, 2014 at 23:22 | comment | added | Kevin Ventullo | Outline of proof: Everything breaks into a direct sum of the $p$-torsion parts over all primes $p$, so it's enough to prove it after localizing at $p$. Now $V$ is a direct limit of its ``cofinitely generated'' subs, i.e. submodules which look like a sum of finitely many $\mathbb{Q}_p/\mathbb{Z}_p$'s. Direct limits commute with everything, so it's basically enough to prove it for $V\cong \mathbb{Q}_p/\mathbb{Z}_p$. But here it's just the isomorphism $\mathbb{Q}_p/\mathbb{Z}_p \otimes_{End(\mathbb{Q}_p/\mathbb{Z}_p)} End(\mathbb{Q}_p/\mathbb{Z}_p) \cong \mathbb{Q}_p/\mathbb{Z}_p$. | |
| Apr 6, 2014 at 23:14 | comment | added | Denis Nardin | In case it is interesting I have a proof of the surjectivity that I'm unable to upgrade to a full proof. Every element $v$ such that $nv=0$ is the image of $(\frac{1}{n}+\mathbb{Z})\otimes \varphi$ where $\varphi$ is any extension to $\mathbb{Q}/\mathbb{Z}$ of the map $\mathbb{Z}/n\to V$ sending $1+n\mathbb{Z}$ to $v$. The extension exists since $V$ is divisible and hence injective. | |
| Apr 6, 2014 at 22:21 | comment | added | Dag Oskar Madsen | I think you just added an important detail to the question. | |
| Apr 6, 2014 at 22:17 | comment | added | Fat | There are 32 references in this paper | |
| Apr 6, 2014 at 22:07 | history | edited | Fat | CC BY-SA 3.0 |
added 41 characters in body
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| Apr 6, 2014 at 22:04 | comment | added | Fat | No without proof | |
| Apr 6, 2014 at 22:03 | comment | added | Fernando Muro | Stated without proof? Have you tried to find the proof in other references? | |
| Apr 6, 2014 at 22:00 | comment | added | Fat | It is stated in Robert wisbaur paper(static modules and equivlances) in page 8 that this map is an isomorphism | |
| Apr 6, 2014 at 20:02 | comment | added | Fernando Muro | How do you know it is bijective? | |
| Apr 6, 2014 at 15:12 | review | First posts | |||
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| Apr 6, 2014 at 15:01 | history | edited | Joe Silverman | CC BY-SA 3.0 |
Changed math to LaTeX
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| Apr 6, 2014 at 14:52 | history | asked | Fat | CC BY-SA 3.0 |