We develop a theory of modulus sheaves with transfers, which generalizes
Voevodsky's theory of sheaves with transfers. This paper and its sequel are
foundational for the theory of motives with modulus, which is developed in
[KMSY20].
Comment: 64 pages
Using the 'slice filtration', defined by effectivity conditions on Voevodsky's triangulated motives, we define spectral sequences converging to their motivic cohomology andétale motivic cohomology. These spectral sequences are particularly interesting in the case of mixed Tate motives as their E 2 -terms then have a simple description. In particular this yields spectral sequences converging to the motivic cohomology of a split connected reductive group. We also describe in detail the multiplicative structure of the motive of a split torus.
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-K\"unneth
decompositions in this category, and establish relationships with Rost's cycle
modules and the Albanese functor for smooth projective varieties.Comment: Correction in proof of Th. 2.4.1; to appear in Annals of K-theor
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