Timeline for Different definitions of the rank of a module
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| S Sep 11, 2019 at 22:19 | history | suggested | Mike Pierce | CC BY-SA 4.0 |
formatting
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| Sep 11, 2019 at 20:06 | review | Suggested edits | |||
| S Sep 11, 2019 at 22:19 | |||||
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Mar 12, 2016 at 18:55 | history | edited | brunoh | CC BY-SA 3.0 |
Another definition
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| Jan 6, 2014 at 19:51 | comment | added | Pete L. Clark | @brunoh: Thanks, the result is in there! So I must have flipped past it before, but the preamble is a bit technical so I don't think I ever looked at it in detail. (Also it comes right after Hilbert's Nullstellensatz. I have always found Matsumura's ordering of the topics to be strange.) At this point though I wish that others who knew more about these topics than I do would be willing to speak a bit more loudly. I think everyone who is listening would benefit... | |
| Jan 6, 2014 at 19:49 | vote | accept | brunoh | ||
| Jan 6, 2014 at 19:39 | comment | added | brunoh | @PeteL.Clark in his famous book on Commutative Ring Theory Matsumura proved this result of Forster and Swan (theorem 5.7) and the paper of M. Mohan. | |
| Jan 6, 2014 at 19:26 | comment | added | Pete L. Clark | @Mohan: Most of all, I think your comment is actually an answer to the question, and the site works more smoothly if answers are left as answers. (They are easier to see; they can be edited by other users, more easily searched for...) Also you did provide more information in your last comment, so there seems to be more to say. Finally, it took me a little while (okay, two minutes) to track down precise references to the papers of Forster and Swan. I suspect that more people would look at these papers if you included a precise reference and/or a link. None of this is required, but why not? | |
| Jan 6, 2014 at 16:45 | comment | added | Mohan | @PeteL.Clark: I have already mentioned the result and you can easily find their papers on the web. What else would you like to know? There are also generalizations of these for polynomial rings, called Eisenbud-Evans conjectures, proved by Sathaye and myself (this was my Ph. d. thesis). | |
| Jan 6, 2014 at 0:15 | comment | added | Pete L. Clark | @Mohan: I didn't know that result and find it very striking. Could you please leave an answer that describes it? | |
| Jan 5, 2014 at 21:36 | comment | added | Mohan | Though one uses the definitions 2 or 3 for rank in general, here is a nice result and usefulness of the first definition due to O. Forster and R. G. Swan. If $M$ is a finitely generated module over a Noetherian ring $R$, define $\mu(M)$ to be the maximum of $r_p(M)+\dim R/p$ as $p$ varies over all prime ideals. Then $M$ can be generated by $\mu(M)$ elements. | |
| Jan 5, 2014 at 20:32 | answer | added | Pete L. Clark | timeline score: 13 | |
| Jan 5, 2014 at 19:49 | history | edited | brunoh | CC BY-SA 3.0 |
Small mistake corrected
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| Jan 5, 2014 at 19:37 | comment | added | brunoh | Thank you for your information. Usually I don't like when under the same name different definitions give different results. But if it is not a problem in everyday work ... In any case, I was thinking that when M is locally free of finite type only, 1 & 2 are identical. When R is a domain in addition, then they all are identical. So my only problem is when R is a domain, M finite type not projective, 2. makes sense if we define the rank of Mp over Rp using 3., but then it is different from 1. IMO, 1. is not really about the rank because it cancels too much information, and more about generators | |
| Jan 5, 2014 at 17:27 | comment | added | abz | All definitions are widely used. For example, when speaking of the rank of the Mordell-Weil group of an elliptic curve, one means 3). It's usually clear in context what is meant. | |
| Jan 5, 2014 at 16:11 | history | asked | brunoh | CC BY-SA 3.0 |