Timeline for Cohomological dimension
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| May 31, 2010 at 6:18 | comment | added | Torsten Ekedahl | Indeed,there is a wonderful proof that is short enough to fit into one of these comments... Consider a finite projective resolution of a module. Shifted it is also a finite resolution of the left hand side which is injective. As all but the last modules in that resolution are injective the last on must also be. Hence any module with a finite projective resolution must be injective and hence projective. | |
| May 30, 2010 at 22:17 | comment | added | Leonid Positselski | A simple explanation for all nontrivial finite groups having infinite homological dimension is that the homological dimension of any Frobenius algebra is either zero or infinity. | |
| May 30, 2010 at 20:28 | comment | added | Torsten Ekedahl | Then you will no doubt enjoy learning about groups with periodic cohomology. They are completely classified and Swan proved that a finite group has periodic cohomology iff it admits a free action on a CW-complex homotopic to a sphere. | |
| May 30, 2010 at 19:55 | history | edited | Greg Kuperberg | CC BY-SA 2.5 |
added 18 characters in body
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| May 30, 2010 at 19:53 | comment | added | Greg Kuperberg | Hmmm....I don't seem to know as much about this as I thought. | |
| May 30, 2010 at 19:20 | comment | added | Torsten Ekedahl | Very few finite groups have periodic cohomology. The conclusion that all non-trivial finite $G$ have cohomology in arbitrarily high dimension is of course still true. | |
| May 30, 2010 at 19:08 | history | answered | Greg Kuperberg | CC BY-SA 2.5 |