Timeline for How to introduce notions of flat, projective and free modules?
Current License: CC BY-SA 2.5
22 events
| when toggle format | what | by | license | comment | |
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| Jun 22, 2011 at 14:43 | comment | added | David White | I'm not 100% sure there is really a correct answer to this question. Should it therefore be community wiki instead? I've seen several great answers below and I imagine different professors of algebra reading this page would consider different answers to be "the best" based on what they wanted to impart to their students | |
| Nov 27, 2010 at 20:28 | vote | accept | Pete L. Clark | ||
| Nov 20, 2010 at 13:46 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
added 231 characters in body
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| Nov 19, 2010 at 23:47 | answer | added | Carl Weisman | timeline score: 1 | |
| Nov 19, 2010 at 19:56 | answer | added | Greg Marks | timeline score: 6 | |
| Nov 19, 2010 at 13:35 | answer | added | Sebastian Petersen | timeline score: 2 | |
| Nov 19, 2010 at 3:42 | answer | added | roy smith | timeline score: 21 | |
| Nov 19, 2010 at 2:05 | answer | added | Hailong Dao | timeline score: 21 | |
| Nov 19, 2010 at 0:39 | comment | added | Mariano Suárez-Álvarez | IMHO, the motivation of flatness based on the single case in which one considers the problem of when extension of scalars along a map of rings $A\to B$ preserves exactness is quite good already. In fact, I think that extension of scalars is a great way to motivate tensor products, because it is very very concrete and natural. | |
| Nov 19, 2010 at 0:27 | comment | added | Manny Reyes | @Mariano: I just thought it might be relevant since the OP explicitly says "this is not a homological algebra course." Incidentally, see the post of Andreas Blass for the characterization of flatness to which I was referring. | |
| Nov 19, 2010 at 0:13 | comment | added | Mariano Suárez-Álvarez | Why would you avoid mentioning "homological algebra"(whereby one means, in this context "preservation of exactness properties") when introducing a property which is essentially of homological nature? | |
| Nov 18, 2010 at 23:56 | answer | added | Anton Fonarev | timeline score: 7 | |
| Nov 18, 2010 at 23:10 | comment | added | Manny Reyes | This may or may not be helpful to you, but there are many different characterizations of flatness in Lam's book Lectures on Modules and Rings, Ch. 4. Flatness can be viewed purely in terms of linear equations as in Thm. 4.24, the "equational criteria for flatness." This may not add much intuition, but it at least avoids any mention of homological algebra. | |
| Nov 18, 2010 at 22:59 | answer | added | Andreas Blass | timeline score: 26 | |
| Nov 18, 2010 at 22:57 | answer | added | Martin Brandenburg | timeline score: 7 | |
| Nov 18, 2010 at 22:51 | comment | added | Kevin Ventullo | For projective modules, you might start with the observation that many of the nice properties of free modules follow only from the lifting property. | |
| Nov 18, 2010 at 22:44 | answer | added | roy smith | timeline score: 13 | |
| Nov 18, 2010 at 22:08 | answer | added | Timothy Wagner | timeline score: 7 | |
| Nov 18, 2010 at 21:57 | comment | added | Pete L. Clark | @Mikhail: I myself am a number theorist, so I am partial to Dedekind domains. I might introduce them early on the course, yes... | |
| Nov 18, 2010 at 21:57 | answer | added | Sean Tilson | timeline score: 29 | |
| Nov 18, 2010 at 21:55 | comment | added | Mikhail Bondarko | In my opinion, the very basic projective non-free modules are (fractional) ideals in number rings (and Dedekind domains); this particular case of Pic is very intriguing for those who are interested in algebraic number theory. By the way, do you want to introduce the defintion of a Dedekind domain (early in your course)? | |
| Nov 18, 2010 at 21:37 | history | asked | Pete L. Clark | CC BY-SA 2.5 |