Rost’s Chain Lemma

This chapter states and proves Rost's Chain Lemma. The proof (due to Markus Rost) does not use the inductive assumption that BL(n − 1) holds. Throughout this chapter, 𝓁 is a fixed prime, and 𝑘 is a field containing 1/𝓁 and all 𝓁th roots of unity. It fixes an integer 𝑛 ≥ 2 and an 𝑛-tuple (𝑎1, ..., 𝑎𝑛) of units in 𝑘, such that the symbol ª = {𝑎1, ..., 𝑎𝑛} is nontrivial in the Milnor 𝐾-group 𝐾𝑀 𝑛(𝑘)/𝓁. The chapter produces the statement of the Chain Lemma by first proving the special case 𝑛 = 2. The notion of an 𝓁-form on a locally free sheaf over 𝑆 is then introduced, before the chapter shows how 𝓁-forms may be used to define elements of 𝐾𝑀 𝑛(𝑘(𝑆))/𝓁.

Existence of Rost Motives

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.

Existence of Norm Varieties

This chapter constructs norm varieties for symbols ª = {𝑎1, ...,𝑎𝑛} over a field 𝑘 of characteristic 0, and starts the proof that norm varieties are Rost varieties. It first recalls the definition of a norm variety for a symbol ª in 𝐾𝑀 𝑛(𝑘)/𝓁; if 𝑛 ≥ 2 and 𝑘 is 𝓁-special, norm varieties are geometrically irreducible. Next, the chapter uses the Chain Lemma to produce a specific ν‎ n−1-variety ℙ(𝒜), and a pencil Q of splitting varieties over 𝔸1—{0} whose fibers 𝑄𝑊 are fixed point equivalent to ℙ (𝒜). Using a bordism result, this chapter shows that any equivariant resolution 𝑄(ª) of 𝑄𝑊 is a ν‎ n−1-variety. Next, one of Rost's degree formulas is used to show that any norm variety for ª is ν‎ n−1 because 𝑄(ª) is. Finally, a norm variety for ª is constructed by induction on 𝑛, making use of the global inductive assumption that BL(n − 1) holds.

The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I

This chapter takes up the proof of Lemma 4.24, which has two cases, A and B. The strategy goes as follows: It first repeats the ideas from Chapter 5 and derives a new local equation from the assumption of Proposition 4.13. However, it finds that this new equation is very far from proving the claim of Lemma 4.24. It then has to return to the hypothesis of Proposition 4.13 and extract an entirely new equation from its conformal variation. It proceeds with a detailed study of this new equation (again using the inductive assumption of Proposition 4.13); the result is a second new local equation which again is very far from proving the claim of our lemma. Next, it formally manipulates this second new local equation and adds it to the first one, and observes certain miraculous cancellations, which yield new local equations that can be collectively called the grand conclusion. Lemma 4.24 in case A then immediately follows from the grand conclusion.

The Inductive Step of the Fundamental Proposition: The Simpler Cases

This chapter takes up the proof of Lemmas 4.16 and 4.19, which are easier than Lemma 4.24. The proof of these two lemmas relies on the study of the first conformal variation of the assumption of Proposition 4.13. This study involves many complicated calculations and also the appropriate use of the inductive assumption of Proposition 4.13.

On the finiteness and uniqueness of certain 2-tame N-groups

Unlike ring modules certain faithful N-groups are unique. The main theorem is that if N is a 2-tame ring-free near-ring where N/J(N) has DCCR, then all faithful 2-tame N-groups are finite and N-isomorphic. The finiteness of such an N-group follows easily from the fact it has a composition series. It is then shown that the length of a composition series depends only on N. This fact is used at key points in the proof. The situations where the N-group has or has not a minimal submodule require different analysis. The first case makes use of other interesting results and the second makes strong use of the inductive assumption.