SLICE FILTRATION AND TORSION THEORY IN MOTIVIC COHOMOLOGY

Abstract

OF THE DISSERTATION Slice Filtration and Torsion Theory in Motivic Cohomology by KNIGHT FU Dissertation Director: Charles Weibel We show that the category HI of homotopy invariant Nisnevich sheaves with transfers and the category CycMod are each equipped with a strong filtrations and a strong cofiltration. To do so, we first define pre-coradicals and coradicals on well-powered abelian categories, and show that every isomorphism class of coradical is associated to a canonical torsion theory. We then summarize the theory of motivic cohomology needed to define HI, its symmetric monoidal structure ⊗H and its partial internal hom HomHI. Along the way, we recall the construction of the slice filtration on DM eff,−, and extend the filtration structure on DMeff,− to DM. We then define and construct the torsion filtration on HI by constructing a sequence of coradicals. We explain how the torsion filtration is related to the slice filtration on DMeff,−. We extend the torsion filtration to the category HI∗ of homotopic modules. Appealing to the categorical equivalence between HI∗ and CycMod, we obtain the torsion filtration onCycMod. Finally, we generalize the conditions under which torsion filtrations exist for the heart of a tensor triangulated category.