Tate Motives and the Vanishing Conjectures for Algebraic K-Theory
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We give axioms for a triangulated ℚ-tensor category T, generated by “Tate objects” ℚ(a), which ensure the existence of a canonical weight filtration on T, and additional axioms which give rise to an abelian subcategory A generated by the ℚ(a). We show in addition that A is a Tannakian category, with fiber functor to graded ℚ-vectorspaces given by taking the associated graded with respect to the weight filtration. We then apply this to our construction of a triangulated motivic category over a field k, to show that, assuming the vanishing conjectures of Soulé and Beilinson are true for k, there is a Tannakian category TM k which has many of the properties of the conjectural category of mixed Tate motives. In particular, the category TM k exists for k a number field.
KeywordsExact Sequence Short Exact Sequence Full Subcategory Tensor Category Weight Filtration
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- [BBD]A.A. Beilinson, J.N. Bernstein, P. Deligne, “Faisceaux pervers”, in ‘Asterisque 100’, Soc. Math. France 1982Google Scholar
- [BGS]A.A. Beilinson, V.A. Ginzberg, V.V. Schechtman, “Koszul Duality”, J. Geom. Phys. `5’(1988) no. 3, 317–350.Google Scholar
- [Bo]A. Borel, “Stable real cohomology of arithmetic groups”, Ann. Sci. Éc. Norm. Sup. Ser. 4 `7’(1974) 235–272.Google Scholar
- [D]P. Deligne, “Tannakian Categories”, in `Hodge Cycles, Motives and Shimura Varieties’, LNM 900, Springer 1982.Google Scholar
- [G]L] M. Levine, “The derived motivic category”, preprint (1991).Google Scholar
- [L2]A. Goncharov, “Bloch’s higher Chow groups revisited”, submitted (1992) to the proceedings of the 1992 Strasbourg K-theory conference.Google Scholar
- [S]N. Saavedra Rivano, “Catégories Tannakiennes”, LNM 265, Springer 1972. [V] J.L. Verdier, “Catégories triangulées, état 0”, in ‘SGA 4 1/2’ LNM O 262–308.Google Scholar