Timeline for What is the right definition of the Picard group of a commutative ring?
Current License: CC BY-SA 3.0
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| Mar 27, 2018 at 20:03 | history | edited | Georges Elencwajg | CC BY-SA 3.0 |
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| Mar 27, 2018 at 19:27 | comment | added | Georges Elencwajg | Finally (2)c) only describes some of the modules in (2)a). So (2)c) is not equivalent to (2)a) in general, but it is equivalent under some mild conditions, for example if $R$ is a domain. | |
| Mar 27, 2018 at 19:27 | comment | added | Georges Elencwajg | @Gro-Tsen. I have added an edit which points to one source of the messiness. I think that nobody doubts that 2) and 3) in my answer are the correct definitions. The trouble begins with alternative definitions which are thought to be equivalent but aren't. In Pete's question (1) and (2)a) are equivalent (and equivalent to what I have described as the correct definition) iff , as in Bourbaki, "rank one" includes the condition that $M$ be finitely generated. As to (2) b), it is not equivalent to (1) nor 2)a) nor 2)c) because it doesn't even imply projective. (to be continued) | |
| Mar 27, 2018 at 18:58 | history | edited | Georges Elencwajg | CC BY-SA 3.0 |
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| Mar 27, 2018 at 12:30 | comment | added | Gro-Tsen | So if we call (1), (2a), (2b), (2c) the four conditions in Peter L. Clark's original question, we have (1⇔2a) by the reference to Bourbaki cited in Clark Barwick's answer, but (2c) is not equivalent to (1/2a) nor to (2b) by the counterexamples you cite. Did I get this right? But are (1/2a) and (2b) equivalent or not? Furthermore, since you propose yet another definition of the Picard group, we now have potentially FOUR different Picard groups (or at least, Picard sets — maybe they aren't all groups) and I am tempted to ask about all possible maps between them. This is getting really messy. | |
| Mar 27, 2018 at 6:57 | history | edited | Olivier | CC BY-SA 3.0 |
Changed $r_i$ to $f_i$
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| Mar 26, 2018 at 21:51 | history | edited | Georges Elencwajg | CC BY-SA 3.0 |
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| Mar 26, 2018 at 21:35 | history | edited | Georges Elencwajg | CC BY-SA 3.0 |
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| Mar 26, 2018 at 18:36 | history | answered | Georges Elencwajg | CC BY-SA 3.0 |