Birational motives, I: pure birational motives

Abstract

This is a considerably expanded version of the "pure" part of our 2002 preprint. We define a category of pure birational motives over a field, depending on the choice of an adequate equivalence relation on algebraic cycles. It is obtained by "killing" the Lefschetz motive in the corresponding category of effective motives. For rational equivalence, it encompasses Bloch's decomposition of the diagonal. We study the induced Chow-Künneth decompositions in this category, and establish relationships with Rost's cycle modules and the Albanese functor for smooth projective varieties.

Correction in proof of Th. 2.4.1; to appear in Annals of K-theory

References in corpus (3)

• Unirational threefolds with no universal codimension 2 cycle
• Varieties over a finite field with trivial Chow group of 0-cycles have a rational point
• Duality of Albanese and Picard 1-motives

Cited by in corpus (3)

• Birational motives, II: Triangulated birational motives
• Albanese kernels and Griffiths groups
• Torsion birational motives of surfaces and unramified cohomology