Motivic Complexes and Relative Cycles

Abstract

This part is based on Suslin and Voevodsky’s theory of relative cycles that we develop in categorical terms, in the style of EGA. The climax of the theory is obtained in the study of a pullback operation for suitable relative cycles which is the incarnation of intersection theory in this language. Properties of this pullback operation, and on the conditions necessary to its definition, are made again inspired by intersection theory. We study the compatibility of this pullback operation with projective limits of schemes. In Section 9, the theory of relative cycles is exploited to introduce Voevodsky’s category of finite type schemes over an arbitrary base with morphisms finite correspondences. It is showed that this defines a premotivic category. In Section 10, we define and study sheaves with transfers in an abelian context for various topologies. The main result is that in good cases, they form a premotivic abelian category as defined in Section 1. A theorem comparing rational Nisnevich sheaves with transfers with rational qfh-sheaves for geometrically unibranch schemes is established. In the last section of Part 3, we introduce motivic complexes based on the preceding section. We study the corresponding motivic cohomology in low degrees and prove it is an orientable theory in the sense of motivic homotopy theory. We construct the six functors for motivic complexes and show several of their expected properties.