An Overview of the Proof

This chapter provides the main steps in the proof of Theorems A and B regarding the norm residue homomorphism. It also proves several equivalent (but more technical) assertions in order to prove the theorems in question. This chapter also supplements its approach by defining the Beilinson–Lichtenbaum condition. It thus begins with the first reductions, the first of which is a special case of the transfer argument. From there, the chapter presents the proof that the norm residue is an isomorphism. The definition of norm varieties and Rost varieties are also given some attention. The chapter also constructs a simplicial scheme and introduces some features of its cohomology. To conclude, the chapter discusses another fundamental tool—motivic cohomology operations—as well as some historical notes.

Cohomology Operations

This chapter concerns cohomology operations. Although motivic cohomology was originally defined for smooth varieties over the perfect field 𝑘, it is more useful to view it as a functor defined on the pointed 𝔸1-homotopy category Ho ·, discussed previously in chapter 12. After defining cohomology operations and giving a few examples, the chapter turns to an axiomatic treatment of the motivic Steenrod operations. The motivic Milnor operations are then presented. Thereafter, this chapter demonstrates that the sequence of Milnor operations 𝑄𝑖 is exact on the reduced cohomology of the suspension Σ‎𝔛 attached to a Rost variety 𝑋, using the degree map 𝑡𝒩. It concludes with Voevodsky's motivic degree theorem.

The Norm Residue Theorem in Motivic Cohomology

This book presents the complete proof of the Bloch–Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups. Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The book draws on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduces the key figures behind its development. It proceeds to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. It then addresses symmetric powers of motives and motivic cohomology operations. The book unites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.

Motivic Classifying Spaces

This chapter focuses on motivic classifying spaces. It first connects the motives 𝑆∞ tr(𝕃𝑛) to cohomology operations on 𝐻2𝑛, 𝑛, at least when char(𝑘)=0. This parallels the Dold–Thom theorem in topology, which identifies the reduced homology ̃𝐻*(𝑋, ℤ) of a connected space 𝑋 with the homotopy groups of the infinite symmetric product 𝑆∞𝑋. A similar analysis shows that 𝔾𝑚 represents 𝐻1,1(−, ℤ), which allows us to describe operations on 𝐻1,1. The chapter then introduces the notion of scalar weight operations on 𝐻2𝑛, 𝑛. Afterward, it develops formulas for 𝑆𝓁tr(𝕃𝑛). These formulas imply that 𝑆∞ tr(𝕃𝑛) is a proper Tate motive, so there is a Künneth formula for them. The chapter culminates in a theorem demonstrating that β‎𝑃𝑏 is the unique cohomology operation of scalar weight 0 in its bidegree.