On Galois Cohomology, Norm Functions and Cycles

Markus Rost Bielefeld, September 2006

Galois Cohomology

p a prime F a field, char F ≠ p F̄ a separable closure of F G_{F} = Gal(F̄ /F) the absolute Galois group

H^{n}(F, Z /p) = H^{n}(G_{F} , Z /p)

H^{n}(F, Z /p) = \underrightarrow{lim}_{L/F} H^{n}(Gal(L/F), Z /p)

(Limit over the finite Galois field extensions L of F)

F^{×} = F \ {0} the multiplicative group of F F contains a primitive p-th root ζ_{p} of unity μ_{p} ⊂ F^{×} the subgroup generated by ζ_{p}

H^{n}(F, μ_{p}^{⊗m}) = H^{n}(F, Z /p) ⊗ μ_{p}^{⊗m}

Computation of H^{1}(F, μ_{p}):

H^{1}(F, μ_{p}) = Hom (G_{F} , μ_{p})

Hilbert Satz 90, Kummer theory:

F^{×}/(F^{×})^{p} ⟶^{≃} H^{1}(F, μ_{p})

a → (a) = [F (\sqrt[p]{a})/F]

Bloch-Kato conjecture: For any field F with char F ≠ p, the Galois cohomology ring

_{n≥0}^{⨁} H^{n}(F, μ_{p}^{⊗n})

is generated by H^{1}(F, μ_{p})

The basic relation in H^{2}(F, μ_{p}^{⊗2}):

(a) ∪ (1 − a) = 0 (a ∈ F \ {0, 1}) Proof: E = F (α), α^{p} = a

(a) ∪ (1 − a) = (a) ∪ (N_{E/F} (1 − α))

= N_{E/F} ((a)_{E} ∪ (1 − α)) = N_{E/F} ((α^{p}) ∪ (1 − α)) = 0

Milnor’s K-ring of a field F:

K_{∗}^{M}F = K_{0}F ⊕ K_{1}F ⊕ K_{2}F ⊕ ⋯

= T_{Z} (F^{×})/⟨a ⊗ (1 − a), a ∈ F \ {0, 1}⟩

K_{0}F = Z  (integers)
K_{1}F = F^{×} (multiplicative group)

Bloch-Kato conjecture: The ring homomorphism

K_{∗}^{M}F/p ⟶ _{n≥0}^{⨁} H^{n}(F, μ_{p}^{⊗n})

a_{1} ⊗ ⋯ ⊗ a_{n} → (a_{1}) ∪ ⋯ ∪ (a_{n})

is bijective The elements

(a_{1}) ∪ ⋯ ∪ (a_{n}) ∈ H^{n}(F, μ_{p}^{⊗n})

with a_{1}, …, a_{n} ∈ F^{×} are called symbols Bloch-Kato conjecture (mod p, weight n): H^{n}(F, μ_{p}^{⊗n}) is additively generated by symbols Proofs:

n = 1  classical, Hilbert’s Satz 90
p = 2, n = 2  Merkurjev (1982)
n = 2  Merkurjev/Suslin (1982)
p = 2, n = 3  Merkurjev/Suslin, Rost (1986)
p = 2  Voevodsky (1996–2002)
∀ p, n???  Voevodsky/Rost (1997–2007 ?)

H^{2}(F, μ_{p}) and the Brauer group Br(F) = group of similarity classes of central simple algebras over F Br(F) = set of isomorphism classes of skew fields with center F (finite F -dimension) Cyclic algebras: ζ_{p} ∈ F, a, b ∈ F^{×} A(a, b) = ⟨ X, Y | X^{p} = a, Y^{p} = b, Y X = ζ_{p}XY ⟩ There is a natural isomorphism

H^{2}(F, μ_{p}) ⟶^{≃} _{p} Br(F)

p Br(F) = p-torsion subgroup of Br(F) If μ_{p} ⊂ F, symbols correspond to cyclic algebras:

H^{2}(F, μ_{p}^{⊗2}) ⟶^{≃} _{p} Br(F)

(a) ∪ (b) → [A(a, b)] The H^{3}-invariant for semisimple algebraic groups G (Rost, Serre; 1993) H^{1}(F, G) = isomorphism classes of principal homogeneous G-spaces over F The H^{3}-invariant is a collection of maps

Θ: H^{1}(F, G) ⟶ H^{3}(F, Q_{G} ⊗ μ_{N(G)}^{⊗2}) functorial in F and G Q_{G} = Weyl invariant quadratic forms on the root lattice Q_{G} = Z for simple G Example: G = G_{2} (char F ≠ 2):

H^{1}(F, G_{2}) = isomorphism classes of octonion algebras over F The nontoral subgroup

(Z /2)^{3} ⟶^{j} G_{2}

yields

H^{1}(F, Z /2)^{3} ⟶^{j} H^{1}(F, G_{2}) ⟶^{Θ} H^{3}(F, Z /2)

((a), (b), (c)) → [O(a, b, c)] → (a) ∪ (b) ∪ (c)

Example: G = F_{4} (char F ≠ 3, μ_{3} ⊂ F):

H^{1}(F, F_{4}) = isomorphism classes of exceptional Jordan algebras over F The nontoral subgroup

(Z /3)^{3} ⟶^{j} F_{4}

yields

H^{1}(F, μ_{3})^{3} ⟶^{j} H^{1}(F, F_{4}) ⟶^{Θ} H^{3}(F, Z /3)

((a), (b), (c)) → [J(a, b, c)] → (a) ∪ (b) ∪ (c)

Multiplicative Norm Functions

Given a symbol

u = (a_{1}) ∪ ⋯ ∪ (a_{n}) ∈ H^{n}(F, μ_{p}^{⊗n}) Need some sort of multiplicative function Φ in p^{n} variables generalizing the classical examples: Example: n = 2: Φ is the reduced norm form

Φ = Nrd : A(a_{1}, a_{2}) → F

of the cyclic algebra corresponding to u Example: n = 3, p = 2: Φ is the norm form of the octonion algebra O(a_{1}, a_{2}, a_{3}) Example: n = 3, p = 3: Φ is the norm form of the exceptional Jordan algebra J(a_{1}, a_{2}, a_{3}) Example: p = 2: Φ is the Pfister quadratic form Φ = ⟨⟨a_{1}, …, a_{n}⟩⟩ Recall from (complex) cobordism:

s_{d}(X) ∈ Z is Milnor’s characteristic number

If d = dim X = p^{m} − 1, then s_{d}(X) ∈ pZ

Using algebraic cobordism and degree formulas one shows:

Theorem: If u ≠ 0, there exists a rational function

Φ: A ⟶ A^{1}

on some variety A such that:

• (u)_{F (A)} ∪ (Φ) = 0 in H^{n+1} (F (A), μ_{p}^{⊗(n+1)} )

• dim A = p^{n}

• For any smooth compactification X of the generic fiber of Φ one has

\frac{s_{d}(X)}{p} ≠ 0 mod p

(A, Φ) is unique “up to extensions of degree prime to p” (at least for n = 2 or p = 2)

Multiplicativity of Φ: Ideally this means

Φ(μ(x, y)) = Φ(x)Φ(y)

for some (bilinear, rational?) map

μ: A × A ⟶ A

Look for a correspondence

μ: A × A ⟵^{f} W ⟶^{g} A

with (deg f, p) = 1

This involves:

• Existence of generic splitting varieties of symbols (Voevodsky, see next pages)

• Algebraic cobordism (Morel/Levine)

• Parameterization of the “subfields” of the “algebra A with norm Φ”—motivated by chain lemma for exceptional Jordan algebras (Serre, Petersson/Racine 1995)

Construction of certain Cycles

u = (a_{1}) ∪ ⋯ ∪ (a_{n}) ∈ H^{n}(F, μ_{p}^{⊗n}) a symbol X a splitting variety of u: u_{F (X)} = 0 Using Bloch-Kato conjecture in weight n − 1, get an element

η_{u} ∈ CH^{b}(X^{2}) b = \frac{p^{n−1} − 1}{p − 1}

in the Chow group of b-codimensional cycles Example: n = 2: X = Severi-Brauer variety of cyclic algebra A(a_{1}, a_{2}) (η_{u})^{p−1} = Diagonal (X) + decomp. elements Example: p = 2: X = Quadric with quadratic form ⟨⟨a_{1}, …, a_{n−1}⟩⟩ ⊥ ⟨−a_{n}⟩

η_{u} = “Rost projector” + decomp. elements H_{M}^{r,s}: motivic cohomology (Suslin, Voevodsky) X = simplicial scheme : X \overleftarrow{←} X^{2} ← ; ← ; ← X^{3} ⋯ β = Bockstein Q_{i} Steenrod/Milnor operations (Voevodsky) The map j is an isomorphism assuming the Bloch-Kato conjecture in weight n − 1 Construction of η_{u}:

u ∈ ker [H^{n}(F, μ_{p}^{⊗(n−1)} ) ⟶ H^{n}(F (X), μ_{p}^{⊗(n−1)} )]

≃↑j
H_{M}^{n,n−1}(X , Z /p)
↓β ∘ Q_{1} ∘ ⋯ ∘ Q_{n−2}
H_{M}^{2b+1,b} (X , Z )
↓proj

Homology of [CH^{b}(X) → CH^{b}(X^{2}) → CH^{b}(X^{3})] Problem: Find some variety X such that:

(1) (u)_{F (X)} = 0 in H^{n}(F (X), μ_{p}^{⊗n})

(2) d = dim X = p^{n−1} − 1

(3) The integer

c(X) = (π_{1})_{∗}(η_{u}^{p−1}) ∈ CH^{0}(X) = Z is nonzero mod p

Then X would be a generic splitting variety (up to extensions of degree prime to p)

Theorem: There exists X with (1), (2) and

\frac{s_{d}(X)}{p} ≠ 0 mod p

Voevodsky announced essentially that

c(X) = \frac{s_{d}(X)}{p} mod p