# zbMATH — the first resource for mathematics

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
##### Keywords:
mixed motives; Chern characters; $$K$$-theory