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Are there examples of:

  1. two smooth projective schemes over a field having homotopy equivalent algebraic K-theory spectra and having different rational Voevodsky motives;
  2. two smooth projective schemes over a field having the same integral Voevodsky motives and having non-homotopy equivalent algebraic K-theory spectra?

Also, if we take the field to be algebraically closed, is there a Weil cohomology theory whose values (as graded vector spaces) are determined by algebraic K-theory?

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    $\begingroup$ One idea is that dual abelian varieties have equivalent derived categories by Fourier Mukai hence equivalent algebraic $K$ theories, but should have dual motives. Unfortunately I'm not knowledgeable enough to confirm that this works. $\endgroup$
    – Phil Tosteson
    Commented May 20, 2019 at 22:47
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    $\begingroup$ @PhilTosteson I am also not knowledgeable enough; I think Orlov says they should have rationally equivalent motives (so looks like it does not fly, but maybe I am just being an idiot). $\endgroup$
    – user138661
    Commented May 21, 2019 at 13:00

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