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Mixed motives and algebraic \(K\)-theory. (English) Zbl 0661.14001
Regensburg (FRG): Univ. Regensburg, Fachbereich Mathematik, 272 p. (1988).
The concept of motives was introduced by A. Grothendieck to explain phenomena in different cohomology theories of algebraic varieties in a coherent way, and developed by P. Deligne and others. Work of A. Beilinson suggests that mixed motives are related to higher algebraic \(K\)-theory, like cycles are related to \(K_ 0.\)
The paper under review consists of three parts.
In part I a category of mixed motives in the setting of absolute Hodge cycles is defined.
In part II the author investigates relations between algebraic cycles, algebraic \(K\)-theory, and mixed structures in cohomology of arbitrary varieties.
In part III the author presents some plausible conjectures on Chern characters from \(K\)-theory into \(\ell\)-adic cohomology for varieties over finite fields or global fields, and proves these in some very specific cases.
Reviewer: Li Fu-an

MSC:
14A20 Generalizations (algebraic spaces, stacks)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
19D99 Higher algebraic \(K\)-theory