Milnor Conjectures and Quadratic Forms R. Parimala Department of Mathematics Emory University Second Abel Conference: A Mathematical Celebration of John Milnor February 1, 2012 IMA, University of Minnesota Introduction “Milnor had had a deep and affectionate interest in quadratic form theory. His paper Algebraic K-theory and quadratic forms continues to guide much of the current research in the algebraic theory of quadratic forms.” — H. Bass (1993) bq : V × V ! k associated bilinear form 1 b (v; w) = q(v + w) − q(v) − q(w) q 2 T (q) = bq(ei ; ej ) Gram matrix of q, fe1; ··· ; eng a basis of V (V ; q) is nondegenerate if T (q) is invertible. Witt rings and filtration F field, char(F) 6= 2 (V ; q) quadratic form over F (V ; q) is nondegenerate if T (q) is invertible. Witt rings and filtration F field, char(F) 6= 2 (V ; q) quadratic form over F bq : V × V ! k associated bilinear form 1 b (v; w) = q(v + w) − q(v) − q(w) q 2 T (q) = bq(ei ; ej ) Gram matrix of q, fe1; ··· ; eng a basis of V Witt rings and filtration F field, char(F) 6= 2 (V ; q) quadratic form over F bq : V × V ! k associated bilinear form 1 b (v; w) = q(v + w) − q(v) − q(w) q 2 T (q) = bq(ei ; ej ) Gram matrix of q, fe1; ··· ; eng a basis of V (V ; q) is nondegenerate if T (q) is invertible. 0 m 0 n 0 Witt equivalence. q ∼ q () q ? h ' q ? h () q0 ' q0 W (F) = Witt equivalence classes of quadratic forms under ? and ⊗. I(F) ideal of W (F) of even-dimensional forms In(F) = I(F)n Witt rings and filtration h = (F 2; xy) hyperbolic plane r Theorem. (Witt decomposition) (V ; q) = (V0; q0) ? h q0 anisotropic, uniquely determined by q r = Witt index of q. Witt rings and filtration h = (F 2; xy) hyperbolic plane r Theorem. (Witt decomposition) (V ; q) = (V0; q0) ? h q0 anisotropic, uniquely determined by q r = Witt index of q. 0 m 0 n 0 Witt equivalence. q ∼ q () q ? h ' q ? h () q0 ' q0 W (F) = Witt equivalence classes of quadratic forms under ? and ⊗. I(F) ideal of W (F) of even-dimensional forms In(F) = I(F)n In(F) is additively generated by n-fold Pfister forms Witt rings and filtration n-fold Pfister form: hha1; a2;:::; anii = h1; −a1i ⊗ h1; −a2i ⊗ · · · ⊗ h1; −ani Witt rings and filtration n-fold Pfister form: hha1; a2;:::; anii = h1; −a1i ⊗ h1; −a2i ⊗ · · · ⊗ h1; −ani In(F) is additively generated by n-fold Pfister forms • H0(F; Z=2Z) = Z=2Z × × • H1(F; Z=2Z) = F =F 2 (Kummer isomorphism) × (a) 2 H1(F; Z=2Z) denotes the square class of a 2 F 2 • H (F; Z=2Z) = 2Br(F) The cup product (a):(b) represents the quaternion algebra with generators i; j and relations i2 = a, j2 = b, ij = −ji. Galois cohomology ΓF = Gal(Fs=F) n n H (F; Z=2Z) = H (ΓF ; Z=2Z) = n( ( = ); = ) −!lim H Gal L F Z 2Z L=F finite Galois × × • H1(F; Z=2Z) = F =F 2 (Kummer isomorphism) × (a) 2 H1(F; Z=2Z) denotes the square class of a 2 F 2 • H (F; Z=2Z) = 2Br(F) The cup product (a):(b) represents the quaternion algebra with generators i; j and relations i2 = a, j2 = b, ij = −ji. Galois cohomology ΓF = Gal(Fs=F) n n H (F; Z=2Z) = H (ΓF ; Z=2Z) = n( ( = ); = ) −!lim H Gal L F Z 2Z L=F finite Galois • H0(F; Z=2Z) = Z=2Z 2 • H (F; Z=2Z) = 2Br(F) The cup product (a):(b) represents the quaternion algebra with generators i; j and relations i2 = a, j2 = b, ij = −ji. Galois cohomology ΓF = Gal(Fs=F) n n H (F; Z=2Z) = H (ΓF ; Z=2Z) = n( ( = ); = ) −!lim H Gal L F Z 2Z L=F finite Galois • H0(F; Z=2Z) = Z=2Z × × • H1(F; Z=2Z) = F =F 2 (Kummer isomorphism) × (a) 2 H1(F; Z=2Z) denotes the square class of a 2 F Galois cohomology ΓF = Gal(Fs=F) n n H (F; Z=2Z) = H (ΓF ; Z=2Z) = n( ( = ); = ) −!lim H Gal L F Z 2Z L=F finite Galois • H0(F; Z=2Z) = Z=2Z × × • H1(F; Z=2Z) = F =F 2 (Kummer isomorphism) × (a) 2 H1(F; Z=2Z) denotes the square class of a 2 F 2 • H (F; Z=2Z) = 2Br(F) The cup product (a):(b) represents the quaternion algebra with generators i; j and relations i2 = a, j2 = b, ij = −ji. The discriminant × ×2 1 e1 : I(F) ! F =F = H (F; Z=2Z) n e1(V ; q) = (−1) det T (q) , 2n = dim V 2 ker(e1) = I (F) Classical invariants for quadratic forms The dimension mod 2 0 e0 : W (F) ! Z=2Z = H (F; Z=2Z) e0(V ; q) = dim V (mod 2) ker(e0) = I(F) Classical invariants for quadratic forms The dimension mod 2 0 e0 : W (F) ! Z=2Z = H (F; Z=2Z) e0(V ; q) = dim V (mod 2) ker(e0) = I(F) The discriminant × ×2 1 e1 : I(F) ! F =F = H (F; Z=2Z) n e1(V ; q) = (−1) det T (q) , 2n = dim V 2 ker(e1) = I (F) 2 Example: e2(h1; −ai ⊗ h1; −bi) = (a) · (b) 2 H (F; Z=2Z) Classical invariants for quadratic forms The Clifford invariant 2 2 e2 : I (F) ! 2Br(F) = H (F; Z=2Z) e2(V ; q) = [C(q)] 2 2Br(F) C(q) = Clifford algebra of (V ; q) = T (V )=hv ⊗ v − q(v)i. Classical invariants for quadratic forms The Clifford invariant 2 2 e2 : I (F) ! 2Br(F) = H (F; Z=2Z) e2(V ; q) = [C(q)] 2 2Br(F) C(q) = Clifford algebra of (V ; q) = T (V )=hv ⊗ v − q(v)i. 2 Example: e2(h1; −ai ⊗ h1; −bi) = (a) · (b) 2 H (F; Z=2Z) 3 • Is ker(e2) = I (F)? Classical invariants for quadratic forms Classical questions. • Is e2 onto? Is 2Br(F) generated by quaternion algebras? Classical invariants for quadratic forms Classical questions. • Is e2 onto? Is 2Br(F) generated by quaternion algebras? 3 • Is ker(e2) = I (F)? Milnor conjecture (1970). The map en extends to a homomorphism n n en : I (F) ! H (F; Z=2Z) which is onto with kernel In+1(F). Milnor Conjecture (MC) Theorem. (Arason 1975) The assignment n en(hha1;:::; anii) = (a1) · (a2) ····· (an) 2 H (F; Z=2Z) is well-defined on the isomorphism classes of n-fold Pfister forms. Milnor Conjecture (MC) Theorem. (Arason 1975) The assignment n en(hha1;:::; anii) = (a1) · (a2) ····· (an) 2 H (F; Z=2Z) is well-defined on the isomorphism classes of n-fold Pfister forms. Milnor conjecture (1970). The map en extends to a homomorphism n n en : I (F) ! H (F; Z=2Z) which is onto with kernel In+1(F). Milnor Conjecture (MC) Equivalently, there is an isomorphism M n n+1 M n (en): I (F)=I (F) −! H (F; Z=2Z) n≥0 n≥0 of the graded Witt ring and the graded cohomology ring. • MC is a theorem due to Voevodsky (2003) and Orlov–Vishik–Voevodsky (2007). • MC as stated above is a consequence of two conjectures of Milnor relating Milnor ring K∗F with mod 2 Galois cohomology ring and the graded Witt ring. Milnor Conjecture (MC) • MC for n = 2 is a theorem of Merkurjev (1981)—the first major breakthrough for a general field. • MC as stated above is a consequence of two conjectures of Milnor relating Milnor ring K∗F with mod 2 Galois cohomology ring and the graded Witt ring. Milnor Conjecture (MC) • MC for n = 2 is a theorem of Merkurjev (1981)—the first major breakthrough for a general field. • MC is a theorem due to Voevodsky (2003) and Orlov–Vishik–Voevodsky (2007). Milnor Conjecture (MC) • MC for n = 2 is a theorem of Merkurjev (1981)—the first major breakthrough for a general field. • MC is a theorem due to Voevodsky (2003) and Orlov–Vishik–Voevodsky (2007). • MC as stated above is a consequence of two conjectures of Milnor relating Milnor ring K∗F with mod 2 Galois cohomology ring and the graded Witt ring. Milnor’s definition of K2 is inspired by Steinberg, Moore, and Matsumoto. His definition of Kn for n > 2 is "purely ad hoc"! Milnor K -theory K0(F) = Z × K1(F) = F × × K∗(F) = T (F )=ha ⊗ (1 − a); a 2 F ; a 6= 1i fa1;:::; ang denotes the image of a1 ⊗ · · · ⊗ an in Kn(F) Milnor K -theory K0(F) = Z × K1(F) = F × × K∗(F) = T (F )=ha ⊗ (1 − a); a 2 F ; a 6= 1i fa1;:::; ang denotes the image of a1 ⊗ · · · ⊗ an in Kn(F) Milnor’s definition of K2 is inspired by Steinberg, Moore, and Matsumoto. His definition of Kn for n > 2 is "purely ad hoc"! Norm residue map and MC1 p prime, p 6= char F Definition. (Bass–Tate) The Kummer isomorphism × ×p ∼ 1 F =F −! H (F; µp) extends to a homomorphism called the norm residue map n ⊗n hn;F : Kn(F)=p Kn(F) −! H (F; µp ) sending fa1; a2;:::; ang to (a1) · (a2) ····· (an). Theorem. (Voevodsky 2003) MC1 holds. Norm residue map and MC1 Milnor Conjecture (MC1). For p = 2, hn;F is an isomorphism for all n. Theorem. (Merkurjev 1981) For p = 2, h2;F is an isomorphism. Norm residue map and MC1 Milnor Conjecture (MC1).