Timeline for Using Dunwoody's results on cohomological dimension to learn about a von Neumann regular group ring
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Apr 5, 2018 at 18:49 | comment | added | Benjamin Steinberg | Thanks. I saw in Goodearl that countably generated left ideals in vnr rings are projective. | |
| Apr 5, 2018 at 18:47 | comment | added | rschwieb | I think you've deleted the comment already but Kaplansky's proof appears in the paper On the Dimension of Modules and Algebras, X: A Right Hereditary Ring which is not left Hereditary (1958) It's for arbitrary VNR rings, no relation to group rings afaict | |
| Apr 5, 2018 at 16:47 | comment | added | rschwieb | My first comment is that I think part of the challenge of #1 is proving $G$ must be locally finite, but you've used that as an assumption. However, again, I still appreciate the version you've given. Also, thanks for these pointers to the Almost Stability and Stallings Ends Theorems | |
| Apr 5, 2018 at 16:37 | comment | added | rschwieb | The goal of question 1 is to recover Connell's theorem on VNR group rings from Dunwoody's theorem (hopefully.) The goal of question 2 is to recover Renault's theorem restricted to right self-injective VNR rings using Dunwoody. I consider the fact the augmentation ideal is projective to be already established by Dunwoody's theorem. So the big question is "why does injectivity of $R[G]$ tip the scales and make $G$ finite?" I'm not sure if this solution has those aims yet, but I appreciate the explanation all the same, so please don't think of omitting anything already written :) | |
| Apr 5, 2018 at 16:27 | history | edited | Benjamin Steinberg | CC BY-SA 3.0 |
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| Apr 5, 2018 at 15:49 | history | edited | Benjamin Steinberg | CC BY-SA 3.0 |
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| Apr 5, 2018 at 15:44 | history | answered | Benjamin Steinberg | CC BY-SA 3.0 |