On Galois Cohomology, Norm Functions and Cycles Markus Rost - PDF document

On Galois Cohomology, Norm Functions and Cycles Markus Rost Bielefeld, September 2006

Galois Cohomology p a prime F a field, char F � = p ¯ F a separable closure of F G F = Gal( ¯ F/F ) the absolute Galois group H n ( F, Z /p ) = H n ( G F , Z /p ) H n ( F, Z /p ) = lim H n (Gal( L/F ) , Z /p ) − → L/F (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 ⊗ m ) = H n ( F, Z /p ) ⊗ µ ⊗ m p p 1

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 ) /F ] a → ( a ) = [ F ( p Bloch-Kato conjecture: For any field F with char F � = p , the Galois cohomology ring H n ( F, µ ⊗ n � ) p n ≥ 0 is generated by H 1 ( F, µ p ) 2

The basic relation in H 2 ( F, µ ⊗ 2 p ) : ( 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 H n ( F, µ ⊗ n � ∗ F/p − → ) p n ≥ 0 a 1 ⊗ · · · ⊗ a n → ( a 1 ) ∪ · · · ∪ ( a n ) is bijective 3

The elements ( a 1 ) ∪ · · · ∪ ( a n ) ∈ H n ( F, µ ⊗ n ) p with a 1 , . . . , a n ∈ F × are called symbols Bloch-Kato conjecture (mod p , weight n ): H n ( F, µ ⊗ n ) is additively generated by symbols p 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 ?) 4

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 alge- bras: ≃ H 2 ( F, µ ⊗ 2 p ) − → p Br( F ) ( a ) ∪ ( b ) → [ A ( a, b )] 5

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 ⊗ 2 N ( G ) ) functorial in F and G Q G = Weyl invariant quadratic forms on the root lattice Q G = Z for simple G 6

Example: G = G 2 (char F � = 2): H 1 ( F, G 2 ) = isomorphism classes of octonion algebras over F The nontoral subgroup j ( Z / 2) 3 − → G 2 yields j Θ H 1 ( F, Z / 2) 3 → H 1 ( F, G 2 ) → 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 excep- tional Jordan algebras over F The nontoral subgroup j ( Z / 3) 3 − → F 4 yields j Θ H 1 ( F, µ 3 ) 3 → H 1 ( F, F 4 ) → H 3 ( F, Z / 3) − − (( a ) , ( b ) , ( c )) → [ J ( a, b, c )] → ( a ) ∪ ( b ) ∪ ( c ) 7

Multiplicative Norm Functions Given a symbol u = ( a 1 ) ∪ · · · ∪ ( a n ) ∈ H n ( F, µ ⊗ n ) p 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 � � 8

Recall from (complex) cobordism: s d ( X ) ∈ Z is Milnor’s characteristic number If d = dim X = p m − 1, then s d ( X ) ∈ p Z Using algebraic cobordism and degree formulas one shows: Theorem: If u � = 0, there exists a rational function → A 1 Φ: A − on some variety A such that: • ( u ) F ( A ) ∪ (Φ) = 0 in H n +1 ( F ( A ) , µ ⊗ ( n +1) ) p • dim A = p n • For any smooth compactification X of the generic fiber of Φ one has s d ( X ) � = 0 mod p p ( A, Φ) is unique “up to extensions of degree prime to p ” (at least for n = 2 or p = 2) 9

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 f g µ : A × A ← − W − → 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 alge- bras (Serre, Petersson/Racine 1995) 10

Construction of certain Cycles u = ( a 1 ) ∪ · · · ∪ ( a n ) ∈ H n ( F, µ ⊗ n ) a symbol p X a splitting variety of u : u F ( X ) = 0 Using Bloch-Kato conjecture in weight n − 1, get an element b = p n − 1 − 1 η u ∈ CH b ( X 2 ) 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 11

H r,s M : motivic cohomology (Suslin, Voevodsky) ← X 2 ← ← X 3 · · · X = simplicial scheme : X ← ← 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 ⊗ ( n − 1) → H n ( F ( X ) , µ ⊗ ( n − 1) ) − )] p p �  ≃  j  H n,n − 1 ( X , Z /p ) M  � β ◦ Q 1 ◦ · · · ◦ Q n − 2   H 2 b +1 ,b ( X , Z ) M   � proj  Homology of [CH b ( X ) → CH b ( X 2 ) → CH b ( X 3 )] 12

Problem: Find some variety X such that: (1) ( u ) F ( X ) = 0 in H n ( F ( X ) , µ ⊗ n ) p (2) d = dim X = p n − 1 − 1 (3) The integer c ( X ) = ( π 1 ) ∗ ( η p − 1 ) ∈ CH 0 ( X ) = Z u 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 s d ( X ) � = 0 mod p p Voevodsky announced essentially that c ( X ) = s d ( X ) mod p p 13