Timeline for Splitting of the Universal Coefficients sequence
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 16, 2010 at 0:21 | comment | added | Jeff Strom | Thanks to both of you, these have given me some good stuff to think about. | |
| Jun 14, 2010 at 17:30 | comment | added | Torsten Ekedahl | (cont'd) The end game is then the same; the derived categories of $\mathbb Z$-modules and of $\mathbb Z$-modules are (I hope) equivalent. | |
| Jun 14, 2010 at 17:29 | comment | added | Torsten Ekedahl | I think Chris' view point is sufficiently different from the one I presented to merit mention. I however also agree that it is conceptually very close: In my case a map (of simplicial sets says) $X \to K(A,n)$ which by adjunction is the same as $\mathbb Z[X] \to K(A,n)$ and we have a map of abelian simplicial groups, i.e., of chain complexes. In Chris case we have a (stable) map $X \to K(A,n)$ which by adjunction is the same as $X\wedge H\mathbb Z \to K(A,n)$, a map of $H\mathbb Z$-module spectra. (cont'd) | |
| Jun 14, 2010 at 14:12 | comment | added | Chris Schommer-Pries | Since the splitting is not natural, you won't be able to see it by looking at the Eilenberg-Maclane spaces alone. If you could find the splitting as a map between them it would induce a natural transformation. I agree with Ekedahl, that the only way I see to get the spliting is to realize that a map from X into K(A,n)s factors through $X \wedge H \mathbb{Z} $, and that this is quasi-isomorphic to a sum of K(A,n)s in the category of HZ-module spectra. But this is just a fancy way of saying you need to use the chain complex to see the splitting. | |
| Jun 13, 2010 at 17:26 | comment | added | Torsten Ekedahl | I at least don't know of any way to do it without it. | |
| Jun 13, 2010 at 17:04 | comment | added | Jeff Strom | You seem to be suggesting that we you don't use chain complexes, we can't prove the splitting. | |
| Jun 13, 2010 at 7:08 | history | edited | Robin Chapman | CC BY-SA 2.5 |
cured tex problems
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| Jun 13, 2010 at 5:24 | history | answered | Torsten Ekedahl | CC BY-SA 2.5 |