Timeline for Some questions about the map $K_0(\text{Var})\to K_0(\text{Mot})$
Current License: CC BY-SA 3.0
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| Feb 17, 2017 at 9:55 | comment | added | Mikhail Bondarko | You probably don't get an example with abelian varieties for rational coefficients. Yet you can try to think whether you can have $X^2\cong Y^2$ with $X\not\cong Y$ in a Tannakian category. | |
| Feb 16, 2017 at 2:22 | comment | added | peterx | Thanks a lot! So your result defines the morphism to the motivic Grothendieck group in very high generality, then. I'm not very familiar with motivic homotopy, but I'll give it a read. I tried thinking about what the polarized Tannakian structure buys us, but struggled; it's hard to see how anything can come easily from the formalism, since certainly twisting by a general simple motive will not be full, and trying to detect whether there is an isomorphism seems to require real geometry. I'm not sure if an abelian varieties counterexample can work; as isogenies become isomorphisms. | |
| Feb 15, 2017 at 18:01 | history | edited | Mikhail Bondarko | CC BY-SA 3.0 |
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| Feb 15, 2017 at 15:44 | history | edited | Mikhail Bondarko | CC BY-SA 3.0 |
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| Feb 15, 2017 at 7:53 | history | answered | Mikhail Bondarko | CC BY-SA 3.0 |