Rost Motives and H90

Abstract

This chapter introduces the notion of a Rost motive, which is a summand of the motive of a Rost variety 𝑋. It highlights the theorem that, assuming that Rost motives exist and H90(n − 1) holds, then 𝐻𝑛+1 ét(𝑘, ℤ(𝑛)) injects into 𝐻𝑛+1 ét(𝑘(𝑋), ℤ(𝑛)). While there may be many Rost varieties associated to a given symbol, there is essentially only one Rost motive. The Rost motive captures the part of the cohomology of a Rost variety 𝑋. Since a Rost motive is a special kind of symmetric Chow motive, the chapter begins by recalling what this means. It then introduces the notion of 𝔛-duality. This duality plays an important role in the axioms defining Rost motives, as well as a role in the construction of the Rost motive in the next chapter. Finally, this chapter assumes that Rost motives exist and proves a key theorem.