Techniques, computations, and conjectures for semi-topological K-theory
Techniques, computations, and conjectures for semi-topological K-theory
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DOI：
10.1007/s00208-004-0569-3
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发表时间：
2004-08
影响因子：
1.4
通讯作者：
E. Friedlander;Christian Haesemeyer;M. Walker
中科院分区：
文献类型：
--
作者：
E. Friedlander;Christian Haesemeyer;M. Walker
We establish the existence of an “Atiyah-Hirzebruch-like” spectral sequence relating the morphic cohomology groups of a smooth, quasi-projective complex variety to its semi-topologicalK-groups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that relates the motivic cohomology and algebraicK-theory of varieties, and it is also compatible with the classical Atiyah-Hirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the Borel-Moore (singular) homology of complex varieties introduced by H. Gillet and C. Soulé – to compute the semi-topologicalK-theory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational three-folds, and related varieties, the semi-topologicalK-groups and topologicalK-groups are isomorphic in all degrees permitted by cohomological considerations. We also formulateintegralconjectures relating semi-topologicalK-theory to topologicalK-theory analogous to more familiar conjectures (namely, the Quillen-Lichtenbaum and Beilinson-Lichtenbaum Conjectures) concerning mod-nalgebraicK-theory and motivic cohomology. In particular, we prove a local vanishing result for morphic cohomology which enables us to formulate precisely a conjectural identification of morphic cohomology by A. Suslin. Our computations verify that these conjectures hold for the list of varieties above.
