NORM VARIETIES AND THE CHAIN LEMMA

MARKUS ROST

Notes taken by Christian Haesemeyer. Preliminary version.

Recall from the fall term lectures ([4]) that we want to prove the following theorem. Here p is a fixed prime, k a field of characteristic 0, containing the p-th roots of unity.

Theorem 0.1. Let n ≥ 2 and 0 ≠ \{\underline{a}\} = \{a_{1}, …, a_{n}\} ∈ K_{n}^{M}(k)/p. Then

1) There exists an absolutely irreducible projective p-generic splitting variety for \{\underline{a}\} of dimension p^{n−1} − 1.

Assume further that k is p-special. If X is a projective p-generic splitting variety for \{\underline{a}\} of dimension p^{n−1} − 1, then

2) X is a ν_{n−1}-variety;

3) each element of \overline{A}_{0}(X, K_{1}) is of the form [x, λ], where x ∈ X is a closed point of degree p and λ ∈ k(x)^{×}.

In the fall term, the splitting varieties in the theorem were constructed (see [4, ??]) and the necessary properties proved, modulo two results. For the notation and terminology used, see the notes from the fall term lectures. First, the chain lemma.

Theorem 0.2 (Chain Lemma). Assume k is p-special. Let {a_{1}, …, a_{n}} ∈ K_{n}^{M}(k)/p be a nontrivial symbol. Then there exists a smooth projective cellular variety S/k and a collection of invertible sheaves J = J_{1}, J′_{1}, …, J_{n−1}, J′_{n−1} equipped with nonzero p-forms γ = γ_{1}, …, γ′_{n−1} satisfying the following conditions.

(1) dimS = p(p^{n−1} − 1) = p^{n} − p;

(2) {a_{1}, …, a_{n}} = {a_{1}, …, a_{n−2}, γ_{n−1}, γ′_{n−1}} ∈ K_{n}^{M}(k(S))/p, and for each 2 ≤ i ≤ n − 1, we have {a_{1}, …, a_{i−1}, γ_{i}} = {a_{1}, …, a_{i−2}, γ_{i−1}, γ′_{i−1}} ∈ K_{i}^{M}(k(S))/p. In particular, {a_{1}, …, a_{n}} = {γ, γ′_{1}, …, γ′_{n−1}} ∈ K_{n}^{M}(k(S))/p;

(3) γ ∉ Γ(S, J)^{⊗(−p)};

(4) for any s ∈ V (γ_{i}) ∪ V (γ′_{i}), the field k(s) splits {a_{1}, …, a_{n}};

(5) I(V (γ_{i})) + I(V (γ′_{i})) ⊂ pℤ for all i;

(6) deg(c_{1}(J)^{dimS}) is relatively prime to p.

And, the norm principle.

Proposition 0.3 (Norm Principle). Again, suppose k is p-special and that X is a p-generic slitting variety of dimension p^{n−1} − 1 for some symbol \{\underline{a}\}. Let [x, λ] ∈ \overline{A}_{0}(X, K_{1}) be such that [k(x) : k] = p^{m} for m > 1. Then there exist x_{i} ∈ X, λ_{i} ∈ k(x_{i})^{×} with [k(x_{i}) : k] < [k(x) : k] for all i such that [x, λ] = ∑_{i}[x_{i}, λ_{i}].

The goal of these lectures is to furnish proofs for these two results.

Date: February 14, 2005.

1. Lecture 1. January 19, 2005.

This lecture makes a start on proving the chain lemma in weight 2, that is, in the case of central simple algebras, as the constructions can be made very explicit in that case. The general case will be tackled in subsequent lectures.

As a matter of notation, we fix a prime p, sometimes also called l; a field k of characteristic different from p, containing the p-th roots of unity (in fact, we will assume that k is p-special); and a primitive p-th root of unity ζ.

Suppose a, b ∈ k^{×}. Then we define a central simple algebra

(1.1)  B = A_{ζ}(a, b) = k{X, Y |X^{p} = a, Y^{p} = b, Y X = ζXY }.

Observe that A_{ζ}(1, 1) ≅ M_{p}(k), with generators X_{0} a permutation matrix of order p and Y_{0} = diag(1, ζ, ζ^{2}, …, ζ^{p−1}).

There is a reduced norm map Nrd : B → k (in the split case B = M_{p}(k), this is just the determinant) with the property that for any maximal commutative subalgebra E ⊂ B, Nrd|_{E} = N_{E/k} the reduced norm is just the ordinary norm. We will mainly be interested in the case E = k[X] ≅ k[t]/(t^{p} − 1).

Using the reduced norm, we can define a characteristic polynomial Nrd(t·1+x) = t^{p} + T_{1}(x)t^{p−1} + ⋯ + T_{p}(x) for x ∈ B, with T_{p}(x) = Nrd(x).

Definition 1.2. Let B be a central simple algebra of degree p. A Kummer element x ∈ B is an element such that T_{i}(x) = 0 for i = 1, …, p − 1 and Nrd(x) ≠ 0. In particular, x^{p} = (−1)^{p+1}Nrd(x).

Remark 1.3. In the split case, when k = \overline{k} and thus B = M_{p}(k), any Kummer element is conjugate to some scalar multiple of X_{0} or Y_{0}.

Definition 1.4. A Kummer pair relative ζ in the central simple algebra B of degree p is a pair (x, y) of elements of B^{×} such that xy = ζyx (equivalently, yxy^{−1} = ζx).

We list some elementary properties of Kummer pairs.

Lemma 1.5. If (x, y) is a Kummer pair, then x and y are Kummer elements.

Proof. As yxy^{−1} = ζx, the eigenvalues of x (as element of B\overline{k}) are of the form λ, ζλ, …, ζ^{p−1}λ for some λ ≠ 0, and hence the characteristic polynomial of x is of the form t^{p} − λ^{p}. □

Lemma 1.6 (parametrization lemma). Let (x, y) be a Kummer pair. Then (x, z) is a Kummer pair if and only if z ∈ y · (k[x])^{×}.

Proof. Because x is semisimple, the centralizer C_{B}(x) of x in B is exactly k[x]. It follows from the definition of a Kummer pair that (x, z) is a Kummer pair if and only if z is invertible and yz^{−1} is in C_{B}(x). □

Lemma 1.7. If k = \overline{k}, then any Kummer pair (x, y) in M_{p}(k) is (simultaneously) conjugate to a pair (cX_{0}, dY_{0}) for some c, d ∈ k^{×}

Proof. By Lemma 1.5, we can assume that x = c′X_{0} for some c′ ∈ k^{×}. Now an application of Lemma 1.6 and Hilbert’s Theorem 90 allows to complete the proof. □

The choice of a Kummer pair in a central simple algebra of degree p allows us to realize it in the form of 1.1.

Lemma 1.8. Let (x, y) be a Kummer pair in B. The assignment X ↦ x, Y ↦ y defines an isomorphism of algebras

A_{ζ}(x^{p}, y^{p}) → B

Next, we study varieties of Kummer elements and Kummer chains.

Given a central simple algebra B of degree p, we define a variety of Kummer elements 𝔎 = 𝔎(B) ⊂ ℙ(B) ≅ ℙ^{p^{2}−1} by

𝔎(A) = {[x] ∈ ℙ_{A}(B ⊗ A)|x ∈ (B ⊗ A)^{×} and T_{i}(x) = 0, i = 1, …, p − 1}

for a (commutative) k-algebra A. Note that 𝔎 is defined by p − 1 equations inside ℙ(B) and hence dim(𝔎) = p^{2} − p.

For a Kummer pair (x, y) in B we define a Kummer chain of length n starting from (x, y) to be a sequence (x_{1}, …, x_{n}) of elements in B^{×} such that x_{i}x_{i−1} = ζ^{(−1)^{i}}x_{i−1}x_{i} for i = 0, …, n, with x_{0} = y. That is, (x_{i}, x_{i−1}) is a Kummer pair relative ζ^{(−1)^{i}}.

To connect all this to K_{2}, we make the following observations. Let (x_{1}, …, x_{n}) be a Kummer chain starting at (x, y), let x_{−1} = x and x_{0} = y and define a_{i} = x_{i}^{p} ∈ k^{×} for all i. By Lemma 1.8, we have isomorphisms

B ≅ A_{ζ^{(−1)i}}(a_{i−1}, a_{i}), i = 0, … n.

In K_{2}(k)/p, we have the relations {a_{−1}, a_{0}} = −{a_{0}, a_{1}} = ⋯ = ±{a_{n−1}, a_{n}} and a_{i} ∈ a_{i−2}N(k(\sqrt[p]{a_{i−1}})). That is, the symbols \{a_{−1}, a_{0}\} and \{a_{n−1}, a_{n}\} are connected by a chain of elementary changes.

The variety of Kummer chains of length n starting at (x, y) is the locally closed subvariety 𝔎_{n} = 𝔎_{n}(B, y) ⊂ ℙ(B)^{×n} with points

𝔎_{n}(A) = {([x_{1}], …, [x_{n}])|x_{i} ∈ (B ⊗ A)^{×} and x_{i}x_{i−1} = ζ^{(−1)^{i}}x_{i−1}x_{i}, i = 1, …, n}

for a (commutative) k-algebra A (here, x_{0} = y ⊗ 1). We observe that 𝔎_{n}(B, y) depends on B, ζ and y, but not on x.

Lemma 1.9. A sequence (x_{1}, …, x_{n}) is a Kummer chain starting at (x, y) if and only if x_{1} ∈ k[y]^{×} · x, and x_{i} ∈ k[x_{i−1}]^{×} · x_{i−2} for i = 2, …, n and x_{0} = y.

Proof. This follows from Lemma 1.6 and the observation that a pair (z, w) is a Kummer pair relative ζ if and only if (w, z) is a Kummer pair relative ζ^{−1}. □

This lemma implies that the morphism 𝔎_{n} → 𝔎_{n−1} given by dropping the last component is a torus bundle of rank p − 1; in particular, we have that dim(𝔎_{n}) = n(p − 1).

Let h_{n} : 𝔎_{n} → 𝔎 be the projection onto the last component. Then a point [z] ∈ 𝔎(k) is in the image of h_{n}(k) if and only if z can be obtained from (x, y) via a Kummer chain of length n. The morphism h_{p} : 𝔎_{p} → 𝔎 is a morphism of varieties of the same dimension. We will prove the following result, which will in turn be used to prove the chain lemma (Theorem 0.2) in weight 2 (the cellular variety S in that theorem will be the iterated projective space bundle associated to the iterated torus bundle 𝔎_{n}).

Theorem 1.10. The degree of h_{p} : 𝔎_{p} → 𝔎 is prime to p. More precisely, deg(h_{p}) ≡ −1 mod p.

Proof. Will be given in a future lecture. □

We remark that in special cases, we can be more precise. In general, we do not know the exact value of deg(h_{p}), even though the construction of the morphism is very explicit. We do know that for p = 2, we have deg(h_{2}) = 1 and for p = 3, we have deg(h_{3}) = 2 (see [1]). Moreover, if p = 3, then h_{4} has a rational section (see [3], [2]).

2. Lecture 2. January 26, 2005.

In this lecture, we turn to an approach of constructing the cellular variety of the chain lemma 0.2 that works in all weights. At the end of the lecture, we will relate this to the constructions using Kummer chains explained in the first lecture.

We start by recalling a number of well-known facts about vector bundles and Chern classes; in the process, we introduce some notation.

The Chow ring of a projective bundle. Let X/k be a quasiprojective variety, and π_{V} : V → X a (geometric) vector bundle. We write π_{ℙ(V)} : ℙ(V) → X for the associated projective space bundle (that is, if V = Spec(S^{∗}(M)) for some locally free O_{X}-module M, then ℙ(V) = Proj(S^{∗}(M))).

The tautological line bundle of ℙ(V) → X is the subbundle 𝕃(V) ⊂ π_{ℙ(V)}^{∗}V = V ×_{X} ℙ(V) whose fiber over a point (x, L) ∈ ℙ(V) (here x ∈ X and L ⊂ V_{x} is a line in the fiber V_{x} of V over x) is L. This is the geometric vector bundle with sheaf of sections O_{V} (−1). Observe that the natural map (induced by projection) ϕ_{V} : 𝕃(V) → V is the blow-up of the zero section of V in V.

Now assume that X is smooth, r = rk(V), and z = c_{1}(𝕃(V)) ∈ CH^{1}(ℙ(V)) is the first Chern class of the tautological line bundle. A standard computation shows that

CH^{∗}(ℙ(V)) = CH^{∗}(X)[z]/(c_{V} (z))

where c_{V} (z) = z^{r} − c_{1}(V)z^{r−1} + ⋯ + (−1)^{r}c_{r}(V). The classes c_{i}(V) ∈ CH^{i}(X) are the Chern classes of V. In particular, CH^{∗}(ℙ(V)) is a free module over CH^{∗}(X) with basis 1, z, …, z^{r−1}. We write c(V) = 1 + c_{1}(V) + ⋯ + c_{r}(V) for the total Chern class of V.

Example 2.1. Let H → X be a line bundle, x = c_{1}(H) ∈ CH^{1}(X) and V = A(H) := ⨁_{i=0}^{p−1} H^{⊗i}. Then the total Chern class of V is

c(V) = ∏_{i=0}^{p−1}(1 + ix) ≡ (1 − x^{p−1}) mod p.

Hence CH^{∗}(ℙ(V))/p ≅ (CH^{∗}(X)/p)[z]/(z^{p} − zx^{p−1}).

Euler class of a vector bundle. Again, let V → X be a vector bundle of rank r, with X equidimensional (but not necessarily smooth) of dimension d over k. Let i : X → V be the zero section.

Definition 2.2. The Euler class of V is the class e(V) = i^{∗}i_{∗}([X]) ∈ CH_{d−r}(X). Note that i^{∗} makes sense because the zero section is a regular embedding.

Remark 2.3. When the Chern classes of V make sense (e.g., if X is smooth), then e(X) = c_{r}(X).

Now assume d = r and X is proper. In that case e(V) ∈ CH_{0}(X) and we can use the degree map deg : CH_{0}(X) → ℤ to obtain an integer deg(e(V)), called the degree of V.

Example 2.4. For a line bundle L → X we call δ(L) = deg(L^{⊕d}) ∈ ℤ the degree of L; observe that δ(L) = deg(c_{1}(L)^{d}).

Example 2.5. If V/k is a vector space of dimension n, then the degree of the tautological line bundle over ℙ(V) is δ(𝕃(V)) = (−1)^{n−1}.

Construction of Varieties in the Chain Lemma. The goal is to construct certain iterated projective bundles together with line bundles and differential forms on them as needed in the chain lemma 0.2. Recall from Example 2.1 that for a line bundle H, A(H) denotes the vector bundle ⨁_{i=0}^{p−1} H^{⊗i}.

We start with a variety X_{0} and line bundles H_{−1}, H_{0} on X_{0} (for example, we could choose X_{0} = Spec(k)).

From this initial data, we construct inductively varieties X_{r} and line bundles H_{r} on X_{r}, for r ≥ 1, together with morphisms f_{r} : X_{r} → X_{r−1}; all f_{r} will be projective space bundles of relative dimension p − 1.

Namely, we define

f_{r} = π_{ℙ(A(H_{r−1}))} : X_{r} := ℙ(A(H_{r−1})) → X_{r−1}.

Setting L_{r} := 𝕃(A(H_{r−1})) we define

H_{r} := (f_{r−1} ∘ f_{r})^{∗}(H_{r−2}) ⊗_{O_{Xr}} L_{r}.

(For r = 1, let f_{0} = id_{X_{0}}.)

Let us consider the easiest example of this construction, when X_{0} = Spec(k).

In this case, H_{−1}, H_{0} are obviously trivial, and dimX_{r} = r(p−1). The next result computes δH_{r} mod p; it is the basis of all our future computations of degrees.

Proposition 2.6. Suppose 1 ≤ r ≤ p. Then δ(H_{r}) ≡ 1 mod p.

Proof. The proof has two parts: First, we compute an explicit presentation of the Chow ring CH^{∗}(X_{r})/p; then we use this explicit computation and some elementary algebra to calculate δ(H_{r}).

So, let x_{i} = c_{1}(H_{i}) ∈ CH^{1}(X_{i})/p and z_{i} = c_{1}(L_{i}) ∈ CH^{1}(X_{i})/p. From Example 2.1 recall that

CH^{∗}(X_{r})/p = (CH^{∗}(X_{r−1})/p)[z_{r}]/(z_{r}^{p} − z_{r}x_{r−1}^{p−1})

whence

CH^{∗}(X_{r})/p ≅ 𝔽_{p}[x_{1}, …, x_{r}, z_{1}, …, z_{r}]/I

where

I = (z_{i}^{p} − z_{i}x_{i−1}^{p−1}, x_{i} − x_{i−2} − z_{i}|i = 0, 1, … r)

(here we set x_{−1} = x_{0} = 0 to simplify notation).

So, as 𝔽_{p} vector space, CH^{∗}(X_{r})/p has a basis consisting of the monomials z_{1}^{n_{1}} ⋯ z_{r}^{n_{r}} for 0 ≤ n_{i} ≤ p − 1. In particular, CH_{0}(X_{r})/p is generated by one element u = z_{1}^{p−1} ⋯ z_{r}^{p−1} and x_{r}^{r(p−1)} ≡ du mod p for some d ∈ 𝔽_{p}.

We can compute deg(u) by induction. For r = 1, Example 2.5 shows that deg((z_{1})^{p−1}) = (−1)^{p−1}. The projection formula implies that

deg(u) = deg(z_{r}^{p−1} · (z_{1}^{p−1} ⋯ z_{r−1}^{p−1})) = deg((f_{r})_{∗}(z_{r}^{p−1}) · (z_{1}^{p−1} ⋯ z_{r−1}^{p−1}))

(note we abuse notation here, writing z_{i} for the classes z_{i} ∈ CH^{1}(X_{i}), f_{i+1}^{∗}(z_{i}) ∈ CH^{1}(X_{i+1}) etc.). Applying Example 2.5 one more time, we conclude that (f_{r})_{∗}(z_{r}^{p−1}) = (−1)^{p−1}[X_{r−1}]; by induction, deg(u) = (−1)^{(p−1)r} ≡ 1 mod p.

This means we can calculate δ(H_{r}) = deg(x_{r}^{r(p−1)}) ≡ d mod p. We will prove by induction on r that d = 1. This is obvious if r = 1. Suppose 1 < r ≤ p. Recall that x_{r} = z_{r} + x_{r−2}. So

x_{r}^{r(p−1)} = (z_{r} + x_{r−2})^{p(r−1)+(p−r)} ≡ (z_{r}^{p} + x_{r−2}^{p})^{r−1} · (z_{r} + x_{r−2})^{p−r} mod p

(because p − r ≥ 0). Employing the relation z_{r}^{p} ≡ z_{r}x_{r−1}^{p−1} mod p, we get that

x_{r}^{r(p−1)} ≡ (z_{r}x_{r−1}^{p−1} + x_{r−2}^{p})^{r−1}(z_{r} + x_{r−2})^{p−r} ≡ z_{r}^{p−1}x_{r−1}^{(r−1)(p−1)} + T mod p

where T is not divisible by z_{r}^{p−1} and hence T ≡ 0 mod p for dimension reasons. By inductive hypothesis,

x_{r−1}^{r(p−1)} = z_{r}^{p−1}x_{r−1}^{(r−1)(p−1)} = z_{r}^{p−1} · (z_{1}^{p−1} ⋯ z_{r−1}^{p−1}) = u

in CH^{∗}(X_{r})/p, as claimed. □

Connection with Kummer Chains. Recall from the first lecture that given a central simple algebra B over k of degree p and a Kummer pair (x, y) in B we constructed quasiprojective varieties 𝔎_{r}, for 0 ≤ r, together with maps g_{r+1} : 𝔎_{r+1} → 𝔎_{r} that are torus bundles of rank p−1; we also constructed another variety 𝔎 and a morphism h_{p} : 𝔎_{p} → 𝔎 of varieties of the same dimensions, and formulated a theorem asserting that the degree of this map was invertible mod p (cf. Theorem 1.10). To prove this assertion, we will apply the degree formula [4, ??] to a rational map X_{p} → \overline{𝔎} ⊂ ℙ(B) and the line bundles H_{p} and 𝕃(B)|\overline{𝔎} and show that 𝔎_{p} has an open embedding into X_{p}.

Proposition 2.7. Let B be a central simple algebra over k of degree p and (x, y) a Kummer pair in B. For all r ≥ 0, there is natural open embedding i_{r} : 𝔎_{r} = 𝔎_{r}(B, y) ⊂ X_{r} such that the diagram

𝔎_{r+1} ⟶^{g_{r+1}} 𝔎_{r}
i_{r+1}↓  ^{i_{r}} ↓
X_{r+1} ⟶^{f_{r+1}} X_{r}

commutes.

Proof. Instead of a rigorous proof, we give more of an explanation. To make any sense of what follows, view all appearing k-vector spaces, k-alegbras and Zariski open subsets of them as k-vector space schemes, k-algebra schemes and open subschemes. For example, if E/k was a quadratic field extension, then E^{×} and E −{0} would have the same k-rational points, but they are not isomorphic as schemes: the E-valued points of E^{×} (viewed as scheme) are (E ⊗_{k} E)^{×} = E^{×} × E^{×}, while the E-valued points of E − {0} are ((E ⊗_{k} E) − {0}) = (E × E) − {(0, 0)}.

First, we define i_{1}. Since X_{0} = Spec(k), X_{1} is just a projective space, which we identify with ℙ(k[y] · x). The embedding i_{1} : 𝔎_{1} → X_{1} is now induced by the embedding k[y]^{×} · x → (k[y] − {0}) · x. With this identification of X_{1}, the fiber of H_{1} = L_{1} over a point [x_{1}] ∈ ℙ(k[y] · x) is exactly the line k · x_{1}. Therefore the fiber of A(H_{1}) over [x_{1}] can be naturally identified with k[x_{1}]·y. In this way, we obtain a natural open embedding i_{2} : 𝔎_{2} → X_{2}, over X_{1}; fiberwise, it is induced by the open embedding k[x_{1}]^{×} ·y → (k[x_{1}]−{0})·y. We proceed inductively in the same way to obtain the embeddings i_{3}, i_{4}, . . .. On A-valued points, for a commutative k-algebra A, we obtain an identification of X_{r}(A) with sequences (([x_{1}], [x_{2}], …, [x_{r}]) such that x_{i} ∈ (A[x_{i−1}] − {0}) · x_{i−2} ⊂ B ⊗_{k} A for 1 ≤ i ≤ r, where we write x_{−1} = x and x_{0} = y; with this identification, i_{r}(A) becomes the obvious embedding of the subset 𝔎_{r}(A) of X_{r}(A).

Note however that X_{r} does not depend on the choice of (B, y) (even though the description of X_{r}(A) given above does), while the open subscheme 𝔎_{r} does. □

Recall that for a proper k-scheme Y we define I(Y) as the ideal in ℤ generated by the image of the degree map deg : CH_{0}(Y) → ℤ.

Proposition 2.8. Let 0 ≤ r ≤ p and suppose the central simple algebra B is a skew field over k. Then I(X_{r} − i_{r}(𝔎_{r})) ⊂ pℤ.

Proof. Let F/k be a finite field extension. Given the identifications indicated in the proof of Proposition 2.7, a point in X_{r}(F) can be given as a sequence ([x_{1}], [x_{2}], …, [x_{r}]) such that x_{i} ∈ (F [x_{i−1}] − {0}) · x_{i−2} ⊂ B ⊗_{k} F for 1 ≤ i ≤ r, where we write x_{−1} = x and x_{0} = y.

On the other hand, such a point will be in 𝔎_{r}(F) provided x_{i} ∈ (F [x_{i−1}]^{×}) · x_{i−2} for 1 ≤ i ≤ r. Observe that x_{j} is a Kummer element, for −1 ≤ j ≤ r, by Lemma 1.5; because B is a skew field and of degree p over k, this implies that F [x_{j}] is a field if gcd([F : k], p) = 1. In that case, F [x_{j}]^{×} = F [x_{j}] − {0}. Hence, 𝔎_{r}(F) = X_{r}(F) or p|[F : k] (note, however, that 𝔎_{r}(E) ≠ X_{r}(E) over any field E splitting B). This finishes the proof. □

Recall that 𝔎 = 𝔎(B) was the variety of Kummer elements of B. We are interested in the morphism h_{p} : 𝔎_{p} → 𝔎 given by projection onto the last component. To apply some version of the degree formula, we extend this to a morphism of projective varieties X_{p} → X, defining X(A) = {[x] ∈ ℙ_{A}(B ⊗ A)|T_{i}(x) = 0, i = 1, …, p − 1} for a commutative k-algebra A (recall that T_{i}(x) denotes coefficients of the characteristic polynomial of x). Clearly, we have an open embedding 𝔎 → X, and h_{p} extends to a morphism h̃_{p} : X_{p} → X. Note also that by an argument as in Proposition 2.8 we see that I(X − 𝔎) ⊂ pℤ if B is a skew field. Let L be the restriction of the tautological line bundle 𝕃(B) of ℙ(B) to X; the fiber of L over [x] is k · x ⊂ B.

Lemma 2.9. There is an isomorphism of line bundles on 𝔎_{p}, h_{p}^{∗}(L|𝔎) ≅ H_{p}|𝔎_{p}.

Now we can use the degree formula for the morphism h_{p} with completion h̃_{p} and the line bundles L and H_{p}, applying Proposition 2.8 to conclude:

Proposition 2.10. Assume that B is a skew field. Then δ(H_{p}) ≡ deg(h_{p})δ(L) mod p.

In turn, this implies Theorem 1.10 in case B is a skew field, in light of the computations of δ(L) in Example 2.5 and of δ(H_{p}) in Proposition 2.6. To obtain Theorem 1.10 in the general case, it suffices to observe that 𝔎 and 𝔎_{p} are absolutely irreducible so that the degree of h_{p} does not change when scalars are extended; hence the case where B is a skew field implies the case where k is algebraically closed (so B is split), and that in turn implies the assertion for arbitrary B.

3. Lecture 3. February 2, 2005.

This lecture consists of two parts. First, we complete the proof of the chain lemma in weight 2 by introducing the necessary forms into the picture. In the second part, we give some heuristic ideas concerning the chain lemma in higher weights.

Notation. As before, V → X denotes a (geometric) vector bundle. We write Θ for the trivial line bundle over Spec(k). A p-form or form of degree p γ : V → Θ is a fiberwise linear map Sym_{p}(V) → T heta along f : X → Spec(k). (Equivalently, this is a section of the bundle \underline{Hom}(Sym_{p}(V), f^{∗}Θ) = Sym_{p}(V)^{∗}.) Here Sym_{p}(V) = S_{p}(V) = Γ^{p}(V) = (V^{⊗p})^{Σ_{p}} is the bundle of p-th divided powers of V.

Given a line bunlde L → X and a p-form γ on L, we define a bundle of O_{X}- algebras A_{γ} = A_{γ}(L) together with a p-form N_{γ} as follows.

A_{γ} = T_{O_{X}} (L)/ < x^{⊗p} − γ(x) >

(here T_{O_{X}} (L) is the tensor algebra bundle of L) and N_{γ} is the fiberwise norm on A_{γ}. Note that the natural morphism Spec(A_{γ}) → X is flat of degree p, as A_{γ} is a bundle of degree p algebras.

The Chain Lemma in Weight 2. Recall that we constructed in the previous lecture, starting with a variety X_{0} and line bundles H_{−1} and H_{0} on X_{0} a sequence of projective bundles

X_{p} ⟶^{f_{p}} X_{p−1} ⟶^{f_{p−1}} ⋯ ⟶^{f_{1}} X_{0}

together with line bundles L_{r} and H_{r} on X_{r}. We will inductively define p-forms ϵ_{r} on L_{r} and γ_{r} on H_{r}, starting from given forms γ_{−1} and γ_{0}. Assume γ_{−1}, …, γ_{r−1} are already defined. Recall that L_{r} = 𝕃(A(H_{r−1})) and H_{r} = (f_{r−1} ∘ f_{r})^{∗}(H_{r−2}) ⊗ L_{r}. Observe that A(H_{r−1}) ≅ A_{γ_{r−1}}(H_{r−1}) as vector bundle, and on the latter we have the p-form N_{γ_{r−1}}. Now we define ϵ_{r} to be the composition

Sym_{p}(𝕃(A(H_{r−1}))) → Sym_{p}(A(H_{r−1})) ⟶^{N_{γ_{r−1}}} Θ

where the first map is induced by the canonical inclusion 𝕃(A(H_{r−1})) → f_{r}^{∗}(A(H_{r−1})).

In turn we define γ_{r} = (f_{r−1}∘f_{r})^{∗}(γ_{r−2})⊗ϵ_{r}) (note there is a natural isomorphism Sym_{p}(L ⊗ H) ≅ Sym_{p}(L) ⊗ Sym_{p}(H) for line bundles L and H, so that the tensor product of p-forms ϵ and γ on L and H, respectively, defines qa p-form ϵ ⊗ γ on L ⊗ H).

Now we go back to the example of this construction where X_{0} = Spec(k) and H_{−1} = H_{0} = k are just trivial line bundles. A p-form on k is determined, up to isomorphism, by an element a ∈ k^{×}/(k^{×})^{p}. Choose a, b ∈ k^{×} to define the forms γ_{−1}(t) = at^{p} and γ_{0}(t) = bt^{p} and write u = {a, b} ∈ K_{2}(k)/p for the resulting symbol.

As explained in above, we obtain from this initial data a sequence (X_{r}, L_{r}, H_{r}, ϵ_{r}, γ_{r}) of (smooth projective cellular) varieties, line bundles, and p-forms. The next proposition will finish the proof of the chain lemma for weight 2. For a scheme Z, a point Q on some Z-scheme f : S → Z and a vector bundle V on Z we write V |Q for the k(Q) vector space f^{∗}(V)_{Q} ⊗_{O_{S,Q}} k(Q).

Proposition 3.1. Let P ∈ X_{r} be a point with residue field E = k(P); choose generators x_{i} ∈ H_{i}|P and y_{i} ∈ L_{i}|P for the one-dimensional E vector spaces H_{i}|P and L_{i}|P, for −1 ≤ i ≤ r in such a way that x_{i} = x_{i−2} ⊗ y_{i}.

(1) If for some 1 ≤ i ≤ r we have γ_{i}|P = 0 (than is, γ_{i}(x_{i}) = 0), then

{a, b}_{E} = 0 ∈ K_{2}(E)/p.
(2) If γ_{i}|P ≠ 0 for −1 ≤ i ≤ r, then
{a, b}_{E} = (−1)^{r}{γ_{r−1}(x_{r−1}), γ_{r}(x_{r})} ∈ K_{2}(E)/p.

Proof. By induction on r. Both parts are obvious if r = 0. To prove the first part, we may assume that γ_{i}|P ≠ 0 for 1 ≤ i ≤ r − 1, but γ_{r}|P = 0. We have x_{r} = x_{r−2} ⊗ y_{r} and by the definition of γ_{r}, we conclude

0 = γ_{r}(x_{r}) = γ_{r−2}(x_{r−2})N_{γ_{r−1}}(y_{r})

whence N_{γ_{r−1}}(y_{r}) = 0. Now N_{γ_{r−1}} is, by definition, simply the norm of the Kummer extension E[\sqrt[p]{γ_{r−1}(x_{r−1})}]/E and y_{r} ≠ 0; we conclude that this Kummer extension is split, that is, γ_{r−1}(x_{r−1}) is a p-th power in E. Consequently, {γ_{r−2}(x_{r−2}), γ_{r−1}(x_{r−1})}_{E} = 0 ∈ K_{2}(E)/p, but this symbol is, by induction, equal to ±{a, b}_{E} in K_{2}(E)/p. That finishes the proof of the first assertion.

For the second claim, we can assume by induction that

{a, b}_{E} = ±{γ_{r−2}(x_{r−2}), γ_{r−1}(x_{r−1})}_{E}.

Now γ_{r}(x_{r}) = γ_{r−2}(x_{r−2})N_{γ_{r−1}}(y_{r−1}). But {γ_{r−1}(x_{r−1}), N_{γ_{r−1}}(y_{r−1})} = 0; indeed, for any field k containing the p-th roots of unity, any element a ∈ k^{×} and any y ∈ K_{a} = k[\sqrt[p]{a}], the symbol \{a, N_{K_{a}/k}(y)\} = 0 ∈ K_{2}(k)/p. We conclude that

{γ_{r−2}(x_{r−2}), γ_{r−1}(x_{r−1})}_{E} ≡ −{γ_{r−1}(x_{r−1}), γ_{r}(x_{r})}_{E} mod p;

this concludes the proof of the second assertion. □

We now proved the chain lemma Theorem 0.2 in weight 2. Indeed, writing {a, b} for the symbol {a_{1}, a_{2}} in 0.2, we have S = X_{p}, J = J_{1} = H_{p}, J′_{1} = f_{p}^{∗}(H_{p−1}) and γ_{1} (in 0.2) is the form γ_{p} of 3.1, while γ′_{1} in 0.2 is the γ_{p−1} of 3.1. Now Theorem 0.2, (1) is immediate from the construction of S = X_{p}; 0.2, (2) is Proposition 3.1, (2); 0.2, (3), is just the fact that γ = γ_{p} is not constant; 0.2, (4) is 3.1, (1); 0.2, (5) is a consequence of Proposition 2.8; and 0.2, (6) is Proposition 2.6.

Preview: Higher Weights. Warning:  The rest of this lecture is only
poorly understood by the person writing these notes.

In weight two, splitting varieties have an explicit description (as Severi-Brauer varieties); also for the prime p = 2, the splitting varieties in all weights can be described using Pfister quadrics. We indicate an explicit construction of (rational) forms that may play the role of Pfister forms for odd primes, and the role of the reduced norm form in wieghts bigger than 2.

Let a_{1}, …, a_{n} ∈ k^{×} be a sequence of units in k; wirte u_{r} = {a_{1}, …, a_{r}} ∈ K_{r}^{M}(k)/p.. Inductively, we define rational functions Φ_{r} = Φ_{a_{1},...,a_{r}} in p^{r} variables, as follows.

We set Φ_{0}(t) = t^{p}. Assuming Φ_{0}, …, Φ_{r−1} have been defined, let

Ψ_{r}(X_{ij}|1 ≤ i ≤ p − 1 and 1 ≤ j ≤ p^{r−1}) = ∏_{i=1}^{p−1}(1 − a_{r}Φ_{r−1}(X_{ij}))

and

Φ_{r}(X_{ij}|0 ≤ i ≤ p − 1 and 1 ≤ j ≤ p^{r−1}) = Φ_{r−1}(X_{0j})Ψ_{r}(X_{ij}|1 ≤ i ≤ p − 1).

The next lemma is a generalization of the fact, mentioned before, that the symbol {a, b} vanishes mod p if b is a norm from the Kummer extension defined by a - this was the basic fact used to prove the relations between symbols in Proposition 3.1.

Lemma 3.2. Let b be a nonzero value of Φ_{n}. Then u_{n} · {b} = 0 ∈ K_{n+1}^{M}(k)/p.

Proof. Use induction and the Steinberg relations. □

For the following remarks, let P̃ hi_{0} = Φ_{0},

Ψ̃_{r}(X_{ij}|1 ≤ i ≤ p − 1 and 1 ≤ j ≤ p^{r−1}) = ∏_{i=1}^{p−1}(1 − a_{r}Φ̃_{r−1}(X_{ij}))^{(−1)^{i}}

and

Φ̃_{r}(X_{ij}|0 ≤ i ≤ p − 1 and 1 ≤ j ≤ p^{r−1}) = Φ̃_{r−1}(X_{0j})Ψ̃_{r}(X_{ij}|1 ≤ i ≤ p − 1).

Note that Lemma 3.2 holds for Φ̃_{n} as well as for Φ_{n}.

Remark 3.3. If p = 2, then the form Φ̃_{n} is birationally equivalent to the Pfister form ≪ a_{1}, …, a_{n} ≫.

If p = 3, then Φ̃_{1} is birationally equivalent to the norm N_{k[ \sqrt[3]{a}]/k}, Φ̃_{2} is birationally equivalent to the reduced norm Nrd of the algbra A_{ζ}(a_{1}, a_{2}) and Φ̃_{3} is equivalent to the norm form of the exceptional Jordan algebra J(a_{1}, a_{2}, a_{3}). Here two rational functions are birationally equivalent if they can be transformed into one another by an automorphism (over the base field k) of the field of rational functions.

Moreover, we have the following result.

Lemma 3.4. If p = 3, then the set of nonzero values of Φ̃_{4} is a subgroup of k^{×}.

We also have a result that is analogous to Proposition 3.1, with the norm forms replaced by Φ_{n}.

Proposition 3.5. There is a smooth proper cellular variety X of dimension p^{n} −1, a line bundle L → X and a p-form γ on L such that:

(1) deg(L) ≢ 0 mod p.

(2) x ∈ V (γ) ⇒ (u_{n})_{k(x)} ≡ 0 mod p.

(3) γ is birationally equivalent to Φ_{n} (in some sense yet to be understood precisely by the note-taker - presumably, γ and Φ_{n} both define elements of the function field of L (note this is a rational function field as X is cellular) and one can be transformed to the other by an automorphism of that field.)

References

[1] Wieland Fischer. On twisted cyclic algebras and the chain equivalence of kummer elements. Dissertation, Universität Regensburg, 1999.

[2] Holger P. Petersson and Michel L. Racine. An elementary approach to the Serre-Rost invariant of Albert algebras. Indag. Math. (N.S.), 7(3):343–365, 1996.

[3] Markus Rost. The chain lemma for Kummer elements of degree 3. C. R. Acad. Sci. Paris Sér. I Math., 328(3):185–190, 1999.

[4] A. Suslin and V. Joukhovistky. Norm Varieties. Notes on Lectures given at IAS in the Fall Term 2004.