Existence of Rost Motives

Abstract

This chapter fixes a Rost variety 𝑋 for a sequence. It constructs a Rost motive 𝑀 = (𝑋, 𝑒) with coefficients ℤ(𝓁) under the inductive assumption that BL(n − 1) holds and discusses three important axioms. It introduces a candidate for the Rost motive and demonstrates how a motive satisfies two axioms. To further aid in the proof, the chapter argues that End(𝑀) is a local ring and then verifies an axiom proving that 𝑀 is a Rost motive whenever 𝑋 is a Rost variety. Finally, the chapter considers the historical background behind these equations. It reveals the eponymous Rost motive and considers Voevodsky's own construction of the Rost motive.