Seminar

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A theorem of Borel says, in a suitable sense, that the rational K-theory of a number field lives in just one cohomological degree. A classical calculation of Bass and Tate suggests that adjoining a transcendental variable creates exactly one more.

This raises a deceptively simple question: for a finitely generated field, how many motivic cohomological degrees can there be?

I will explain how generic motives make the pattern particularly transparent, why the nonzero groups can nevertheless be enormous, and how the resulting open problem is related to the conjectural t-structure on mixed motives. At the end, curves and their Jacobians will suggest a broader version of the question.