Rational isomorphisms between K-theories and cohomology theories
Rational isomorphisms between K-theories and cohomology theories
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K-理论与上同调理论之间的有理同构
DOI：
10.1007/s00222-003-0300-0
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发表时间：
2003
影响因子：
3.1
通讯作者：
M. Walker
中科院分区：
文献类型：
--
作者：
E. Friedlander;M. Walker
在特征为0的域上，通过无限环空间的映射（Segre映射）证明了有理代数k理论群与光滑变化的有理动机上同构。此外，还证明了在有理同伦群上的相关chen字符映射是一个环同构。给出了拟射影复变上的函子自然变换的一个有用的一般判据，从而推导出半拓扑奇异复的同伦等价。由于半拓扑k -论和形态上同构可以表示为与代数k -论和动机上同构相关的半拓扑奇异复，这个判据提供了光滑复变种的半拓扑k -论群和形态上同构群之间的一个理性同构。结果包括半拓扑k -论上Chern特征的Riemann-Roch定理，以及在k -论术语上对奇异上同群的“拓扑过滤”的解释。
The well known isomorphism relating the rational algebraic K-theory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the “Segre map”) of infinite loop spaces. Moreover, the associated Chern character map on rational homotopy groups is shown to be a ring isomorphism. A technique is introduced that establishes a useful general criterion for a natural transformation of functors on quasi-projective complex varieties to induce a homotopy equivalence of semi-topological singular complexes. Since semi-topological K-theory and morphic cohomology can be formulated as the semi-topological singular complexes associated to algebraic K-theory and motivic cohomology, this criterion provides a rational isomorphism between the semi-topological K-theory groups and the morphic cohomology groups of a smooth complex variety. Consequences include a Riemann-Roch theorem for the Chern character on semi-topological K-theory and an interpretation of the “topological filtration” on singular cohomology groups in K-theoretic terms.