Rational isomorphisms between K-theories and cohomology theories

• 成果类型: Article
• 署名作者: Friedlander, EM; Walker, ME
• 署名单位: Northwestern University; University of Nebraska System; University of Nebraska Lincoln
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-003-0300-0
• 发表日期: 2003
• 页码: 1-61
• 关键词: algebraic cycles; HOMOLOGY; cocycles; SPACES

摘要

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 Chem 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 Chem character on semi-topological K-theory and an interpretation of the topological filtration on singular cohomology groups in K-theoretic terms.