Timeline for Why does one invert $G_m$ in the construction of the motivic stable homotopy category?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Mar 8, 2010 at 13:49 | vote | accept | Peter Arndt | ||
| Feb 25, 2010 at 16:46 | comment | added | Dustin Clausen | Remember how duality works for suspension spectra of compact manifolds in the ordinary stable homotopy category: X is dual to X^(-T_X), i.e. you need Thom spectra of virtual bundles. Here the duality comes from the purity map X x X --> X^(T_x). Now, in the motivic case, there a notion of Thom space for which purity works, namely take your vector bundle mod the complement of its zero section. These are the things you need to invert, and they look like smash products of P^1's. (p.s. I learned this stuff form a lecture of Jacob Lurie at Harvard...) | |
| Feb 25, 2010 at 12:23 | comment | added | Peter Arndt | The phrase "we want suspension spectra of varieties to be dualizable under smash product" sounds like something I am after. To get a notion of suspension spectrum, I already have to choose what I want to invert. So are you saying then that any choice of T for which schemes embedded into the category of T-spectra have duals must have a G_m-factor? Why? | |
| Feb 24, 2010 at 23:51 | comment | added | Dustin Clausen | toly: true!! just in the stable instead of abelian context, but the motivation is the same. | |
| Feb 24, 2010 at 23:45 | comment | added | Anatoly Preygel | .. and these two motivations are exactly the same as those mentioned (for the Lefschetz motive) in the last two paragraphs of the question! | |
| Feb 24, 2010 at 23:15 | comment | added | Dustin Clausen | wait was i right about K-theory? is it monoidal for smash product? i dunno. | |
| Feb 24, 2010 at 23:07 | comment | added | Dustin Clausen | BTW the phenomenon of wanting to invert extra spheres in nonstandard stable settings is not special to motivic theory. For example in equivariant stable homotopy theory, you gotta invert those representation spheres, and if you want to do stable homotopy theory over a space it's probably handy to invert sphere bundles... | |
| Feb 24, 2010 at 23:03 | history | answered | Dustin Clausen | CC BY-SA 2.5 |