Mixed Motives, Motivic Cohomology, and Ext-Groups

Uwe Jannsen
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Uwe Jannsen: Universität zu Köln, Mathematisches Institut

A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 667-679 from Springer

Abstract: Abstract The standard cohomology theories for algebraic varieties show some fascinatingly parallel behavior. A well-known instance is the phenomenon of “independence of ℓ” for ℓ-adic cohomology. But there areeven more mysterious relations between the latter and Hodge theory. The theory of motives is the suitable setting for describing and studying these issues. Whereas Grothendieck’s theory of pure motives is designed as a universal cohomology theory for smooth projective varieties and is linked to algebraic cycles, the theory of mixed motives concerns the cohomology of arbitrary varieties and was related to algebraic K-theory by Beilinson and Deligne. A new phenomenon of the mixed case is the existence of nontrivial extensions. The aim of these notes is to discuss some of these ideas and to report on the problems and a few results.

Date: 1995
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DOI: 10.1007/978-3-0348-9078-6_60

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