Mixed Motives and their Realization in Derived Categories

The conjectural theory of mixed motives would be a universal cohomology theory in arithmetic algebraic geometry. The monograph describes the approach to motives via their well-defined realizations. This includes a review of several known cohomology theories. A new absolute cohomology is introduced and studied.
The book assumes knowledge of the standard cohomological techniques in algebraic geometry as well as K-theory. So the monograph is primarily intended for researchers. Advanced graduate students can use it as a guide to the literature.

Contenu
Basic notions.- Derived categories of exact categories.- Filtered derived categories.- Gluing of categories.- Godement resolutions.- Singular cohomology.- De Rham cohomology.- Hodge realization.- 1-adic cohomology.- Comparison functors: 1-adic versus singular realization.- The mixed realization.- The tate twist.- ?-product and internal hom on D MR .- The Künneth morphism.- The Bloch-Ogus axioms.- The Chern class of a line bundle.- Classifying spaces.- Higher Chern classes.- Operations of correspondences.- Grothendieck motives.- Polarizability.- Mixed motives.