Notes to Forms of Characteristic 2 [version 16 Jul 2018 ed.]

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Table of contents :
1. Basics......Page 1
2. Structure theory......Page 3
3. Pfister forms......Page 5
4. Invariants......Page 8
5. Bijectivity of d`3́9`42`"̇613A``45`47`"603Alog......Page 11
6. A theorem of Kato......Page 21
References......Page 33

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NOTES TO FORMS OF CHARACTERISTIC 2 ZHENGYAO WU (吴正尧)

Abstract. These are notes for my talk at Southern University of Science and Technology in China (SUSTC) on 14 April 2018. (Revised 8 July 2018). Sections 1-4 are about quadratic forms [EKM08]; sections 5-6 are about Milnor Conjecture for symmetric bilinear forms in characteristic 2 [Kat82b].

Contents 1.

Basics

1

2.

Structure theory

3

3.

Pfister forms

5

4.

Invariants

8

5.

Bijectivity of d log

11

6.

A theorem of Kato

21

References

33

Let F be a field. In this talk, we ignore proofs about char F 6= 2. For a ∈ F ∗ , let hhaii = h1, −ai. For a, b ∈ F , let [a, b] denote ax2 +xy +by 2 , x, y ∈ F . When char F = 2, the Artin-Schreier map is ℘ : F → F , ℘(x) = x2 + x. Let V be a finite dimensional vector space over F and V ∗ = HomF (V, F ) its dual. Let V ∗ ⊗F V ∗ Quad(V ) be the set of quadratic forms on V . Let Sym2 (V ∗ ) = (f ⊗ g − g ⊗ f ) be the set of symmetric bilinear forms on V . For b ∈ Sym2 (V ∗ ), Rad(b) = {v ∈ V | b(v, w) = 0 ∀w ∈ V.}. For q ∈ Quad(V ), Rad(q) = {v ∈ Rad(bq ) | q(v) = 0}. 1. Basics 1.1 Lemma (1) Suppose char F 6= 2. Then hhaii is isotropic iff a ∈ (F ∗ )2 . (2) Suppose char F = 2. Then [1, a] is isotropic iff a ∈ ℘(F ). Proof. (2) Suppose xy 6= 0. Then x2 + xy + ay 2 = 0 iff a = ℘ (xy −1 ). 1



2

ZHENGYAO WU (吴正尧)

1.2 Lemma Consider the F -linear map ϕ : Quad(V ) → Sym2 (V ∗ ) q 7→ bq (x, y) = q(x + y) − q(x) − q(y) (0) bq (x, x) = 2q(x). (1) Suppose char F 6= 2. Then ϕ induces a linear isomorphism. (2) Suppose char F = 2. Then ϕ does not induce a linear isomorphism. Proof. (2) For q(x) = x2 , we have bq (x, y) = (x + y)2 − x2 − y 2 = 2xy = 0. Thus ϕ is not injective.



1.3 Lemma (1) Suppose char F 6= 2. Then Rad(q) = Rad(bq ). (2) Suppose char F = 2. Then Rad(q) ⊂ Rad(bq ) and there exists a quadratic form q such that Rad(q) 6= Rad(bq ). Proof. (2) For q(x) = x2 , bq = 0. We have Rad(q) = 0 and Rad(bq ) = V .



1.4 Definition A quadratic form q is totally singular if bq = 0. (1) Suppose char F 6= 2. Then q is totally singular iff q = 0. (2) Suppose char F = 2. There exists q is totally singular but q 6= 0 (e.g. q(x) = x2 ). 1.5 Lemma Suppose char F = 2. Then q is totally singular iff it is diagonalizable. In this case, every basis of q is orthogonal. ! x Proof. Suppose q( ) = xy. Then y bq (

x1

!

y1

,

x2 y2

! ) = (x1 + x2 )(y1 + y2 ) − x1 x2 − y1 y2 = x1 y2 + x2 y1 6= 0.

So if bq = 0, then q(X1 , X2 , . . . , Xn ) = a1 X12 + a2 X22 + · · · + an Xn2 . 1.6 Definition A quadratic form q is regular if Rad(q) = 0.



NOTES TO FORMS OF CHARACTERISTIC 2

3

A quadratic form q is nonsingular if Rad(bq ) = 0. A quadratic form q is nondegenerate if Rad(q) = 0 and dimF (Rad(bq )) ≤ 1. 1.7 Lemma (1) Suppose char F 6= 2. Then [a, b] is nondegenerate iff ∆ = 1 − 4ab 6= 0. (2) Suppose char F = 2. Then [a, b] is always nondegenerate. ! x Proof. (2) The quadratic form [a, b]( ) = ax2 +xy +by 2 defines a symmetric y bilinear form ! !! x1 x2 , 7→ x1 y2 + x2 y1 . y1 y2 ! ! x1 x2 Suppose ∈ Rad(b[a,b] ). Then x1 y2 + x2 y1 = 0 for all ∈ F 2 . Take y1 y2 ! ! ! ! x2 1 x2 0 = , we get y1 = 0; Take = , we get x1 = 0. Hence y2 0 y2 1 Rad(b[a,b] ) = 0 and Rad([a, b]) = 0.  1.8 Lemma (1) Suppose char F 6= 2. Then hhaii ' hhbii iff ba−1 ∈ (F ∗ )2 . (2) Suppose char F = 2. Then hhaii ' hhbii iff b = x2 + ay 2 for some x ∈ F , y ∈ F ∗. Proof. (2) Let {e1 , e2 } be a basis of q = hhaii such that q(e1 ) = 1, q(e2 ) = a and bq (e1 , e2 ) = 0. First, we show that hhaii ' hhx2 + ay 2 ii. Base change to {e1 , xe1 + ye2 }, we have q(e1 ) = 1, q(xe1 + ye2 ) = x2 + ay 2 and bq (e1 , xe1 + ye2 ) = 2xq(e1 ) + yb(e1 , e2 ) = 0. Conversely, since hhaii ' hhbii, b is represented by hhaii and hence b = x2 + ay 2 .

 2. Structure theory

2.1 Lemma Suppose char F = 2. Let q be a quadratic form over a field F . If W = Rad(bq ) 6= 0, then q|W ⊥ ' [a1 , b1 ] ⊥ · · · ⊥ [an , bn ] where ai , bi ∈ F .

4

ZHENGYAO WU (吴正尧)

Proof. Take 0 6= v ∈ W ⊥ . Then there exists v 0 such that bq (v, v 0 ) = c 6= 0. Let w = c−1 v 0 . Then bq (v, w) = 1. Let an = q(v), bn = q(w). Since [an , bn ] is nondegenerate, we have q|W ⊥ ' q ∗ ⊥ [an , bn ] such that dim(q ∗ ) = dim(q|W ⊥ ) − 2. By induction q|W ⊥ ' q 0 ⊥ [a1 , b1 ] ⊥ · · · ⊥ [an , bn ] where dim(q 0 ) ≤ 1 and ai , bi ∈ F . If dim(q 0 ) = 1, then bq0 = 0 since char F = 2, a contradiction to Rad(bq |W ⊥ ) = 0. Thus q 0 = 0.



2.2 Lemma (1) Suppose char F 6= 2. Then q is nondegenerate iff it is regular. (2) Suppose char F = 2. If dim(q) is even, then q is nondegenerate iff Rad(bq ) = 0. If dim(q) is odd, then q is nondegenerate iff dimF (Rad(bq )) = 1 and q| Rad(bq ) 6= 0. Proof. (2) Let W = Rad(bq ). Then dim(q|W ⊥ ) is even and hence dim(q) ≡ dim W mod 2.



2.3 Corollary (1) Suppose char F 6= 2. Then every nondegenerate quadratic form q over F is diagonalizable q ' ha1 , . . . , an i. i.e. q is of the form a1 X12 + · · · + an Xn2 , ai ∈ F ∗ . (2) Suppose char F = 2. Then for every nondegenerate quadratic form q over F: If dim(q) = 2n is even, then q ' [a1 , b1 ] ⊥ · · · ⊥ [an , bn ], ai , bi ∈ F . If dim(q) = 2n + 1 is odd, then q ' hci ⊥ [a1 , b1 ] ⊥ · · · ⊥ [an , bn ], where c ∈ F ∗ , bhci = Rad(bq ) and ai , bi ∈ F . 2.4 Corollary Let F be a quadratically closed field. (1) Suppose char F 6= 2. Then every anisotropic form is isometric to h1i. (2) Suppose char F = 2. Then every anisotropic form is isometric to h1i or [1, a], where a ∈ F − ℘(F ). Proof. (2) Every nondegenerate odd dimensional quadratic form q is isotropic since dimF (Rad(bq )) = 1, q|Rad(bq ) 6= 0. Suppose dim(q) = 2. Then q ' [a1 , b1 ]. Since q is anisotropic, a1 6= 0. Since F is quadratically closed, a1 = c2 for some c ∈ F . Then [a1 , b1 ] ' [1, a1 b1 ] by base change (e, f ) 7→ (c−1 e, cf ). Here a1 b1 ∈ F − ℘(F ) since q is anisotropic.

NOTES TO FORMS OF CHARACTERISTIC 2

5

Suppose dim(q) = 2n and n > 1. Suppose [a1 , b1 ] ' [1, c] and [a2 , b2 ] ' [1, d] for some c, d ∈ F . We have [1, c] ⊥ [1, d] ' [1, c + d] ⊥ [0, c]. by base change (e, f, e0 , f 0 ) 7→ (e, f + f 0 , e + e0 , f 0 ). Here [0, c] is hyperbolic, q must be isotropic.



2.5 Lemma Let H = xy be the hyperbolic plane over F . (1) If char F 6= 2, then H ' hh1ii = h1, −1i. (2) If char F = 2, then H = [0, 0] ' [0, a] for all a. Proof. (2) The isomorphism [0, 0] → [0, a] is defined by base change (e, f ) 7→ (e, ae + f ).  2.6 Lemma (1) If char F 6= 2, then Witt cancellation holds. (2) If char F = 2, then Witt cancellation does not hold in general. Proof. (2) Consider [1, a], where a ∈ F − ℘(F ). So [1, a] is anisotropic and hence [1, a] 6' H. However, suppose [1, a] ⊥ h1i has basis (e, f, g). Then e + g is a nontrivial zero. Hence [1, a] ⊥ h1i ' [0, a] ⊥ h1i ' H ⊥ h1i by base change (e, f, g) 7→ (e + g, f, g).



2.7 Lemma Let q be a quadratic form over F . Let qan be the anisotropic part of q. (1) If char F 6= 2, then qan is nondegenerate. (2) If char F = 2, then qan may be degenerate. Proof. (2) (Thanks to Fei Xu from Capital Normal University) q(x, y) = x2 + ay 2 , (a 6∈ F 2 ) is anisotropic but degenerate since bq = 0.



3. Pfister forms 3.1 Definition A quadratic F -algebra A is an F -algebra of dimension 2. Suppose a ∈ A − F

6

ZHENGYAO WU (吴正尧)

has minimal polynomial t2 + bt + c. Then define a = −b − a, TrA (a) = −b, NA (a) = c. A quadratic algebra is ´ etale if it is isomorphic to either F × F , or a quadratic separable field extension of F . 3.2 Proposition Let A, B be quadratic ´etale F -algebras. Then X

A?B ={

xi ⊗ yi ∈ A ⊗F B |

X

xi ⊗ yi =

X

xi ⊗ yi }.

´ 2 (F ) of isomorphism classes of quais a quadratic ´etale F -algebra. The set Et dratic ´etale F -algebras with [A][B] = [A?B] form an abelian group of exponent 2. 3.3 Lemma ´ 2 (F ). In particular, H 1 (F, Z/2Z) ' Et ´ 2 (F ) is given by a 7→ [Fa ] (1) Kummer. If char F 6= 2, then F ∗ /(F ∗ )2 ' Et where a ∈ F ∗ , Fa = F [t]/(t2 − a). Let j be the class of t in Fa . Then x + yj = x − yj, Tr(x + yj) = 2x, N(x + yj) = x2 − ay 2 , for all x, y ∈ F. ´ 2 (F ) is given by a 7→ [Fa ] (2) Artin-Schreier. If char F = 2, then F/℘(F ) ' Et where a ∈ F , Fa = F [t]/(t2 + t + a). Let j be the class of t in Fa . Then x + yj = x + y + yj, Tr(x + yj) = y, N(x + yj) = x2 + xy + ay 2 , for all x, y ∈ F. Proof. (2) Homomorphism. [F0 ] = [F × F ] = 0, [Fa ][Fb ] = [Fa+b ]. In fact, let ja be the class of t in Fa and let jb be the class of t in Fb . In Fa ? Fb , ja ⊗ 1 = ja ⊗ 1 = (ja + 1) ⊗ 1 = ja ⊗ 1 + 1 ⊗ 1 =⇒ 1 ⊗ 1 = 0. ja ⊗ jb = ja ⊗ jb = (ja + 1) ⊗ (jb + 1) = ja ⊗ jb + 1 ⊗ jb + ja ⊗ 1 + 1 ⊗ 1 =⇒ 1 ⊗ jb = ja ⊗ 1 is the identity element of Fa ? Fb . (ja ⊗ jb )2 = ja2 ⊗ jb2 = (ja + a) ⊗ (jb + b) = ja ⊗ jb + a ⊗ jb + ja ⊗ b + a ⊗ b = ja ⊗ jb + a(1 ⊗ jb ) + b(ja ⊗ 1) + ab(1 ⊗ 1) = ja ⊗ jb + (a + b)(1 ⊗ jb ) Injectivity. Suppose Fa ' F × F . Then j ∈ F and hence a = ℘(j) ∈ ℘(F ). Surjectivity. Let K = F (d) be a quadratic separable extension of F . Suppose the minimal polynomial of d is t2 + bt + c. We have b 6= 0 otherwise c = e2 for

NOTES TO FORMS OF CHARACTERISTIC 2

7

some e ∈ F and t2 +e2 = (t+e)2 . Then K = F ( db ) and the minimal polynomial of

d b

is t2 + t +

c . b2

Hence [K] = [Fc/b2 ].

Finally, let u = x + yj in Fa . Then j = (

u+x . Since j 2 + j + a = 0, y

u+x 2 u+x ) + + a = 0, u2 + yu + (x2 + xy + ay 2 ) = 0. y y 

3.4 Proposition (1) If char F 6= 2, then Fa ' F × F iff hhaii is isotropic iff a ∈ (F ∗ )2 . (2) If char F = 2, then Fa ' F × F iff hha]] is isotropic iff a ∈ ℘(F ). 3.5 Definition The norm of a quadratic ´etale algebra is called a 1-fold Pfister form. (1) If char F 6= 2, then it is hhaii = h1, −ai, i.e. x2 − ay 2 . (2) If char F = 2, then it is hha]] = [1, a], i.e. x2 + xy + ay 2 . 3.6 Definition Let L/F be a Galois field extension of degree 2 with Gal(L/F ) = {1, σ}. Suppose b ∈ F ∗ . Let (L/F, b) = L ⊕ Lj such that j 2 = b and jl = σ(l)j for all l ∈ L. Then (L/F, b) is called a quarternion algebra. Let j = −j and l = σ(l) for all l ∈ L. Then it gives an involution on (L/F, b), called the canonical involution. Let Q be a quaternion algebra with canonical involution. For x ∈ Q, define TrdQ (x) = x + x and NrdQ (x) = xx. 3.7 Lemma (1) If char F 6= 2, then write Q = (L/F, b) =

a, b

!

. Here L = F (i) such F that i2 = a ∈ F ∗ and ij = −ji. For x = x0 + x1 i + x2 j + x3 ij, we have x = x0 − x1 i − x2 j − x3 ij, TrdQ (x) = 2x0 , NrdQ (x) = x20 − ax21 − bx22 + abx23 . " # a, b . Here L = F (s) such (2) If char F = 2, then write Q = (L/F, b) = F sj that s2 + s + ab = 0. Let i = . Then i2 = a and ij + ji = 1. For b x = x0 + x1 i + x2 j + x3 ij,

x = x0 +x3 +x1 i+x2 j+x3 ij, TrdQ (x) = x3 , NrdQ (x) = (x20 +x0 x3 +abx23 )+(ax21 +x1 x2 +bx22 ).

8

ZHENGYAO WU (吴正尧)

Proof. (2) We have s + σ(s) = 1, sσ(s) = ab and ij =

sj 2 = s. b

sσ(s)j 2 abb sjsj = = 2 = a. b2 b b 2 2 sjj + jsj sj + σ(s)j (s + σ(s))j 2 1b ij + ji = = = = = 1. b b b b jσ(s) jσ(s) sj Since j = j, i = = = = i, we have ij = j i = ji = ij + 1. b b b Finally, NrdQ (x) follows from the multiplication table “1st column times 1st i2 =

row” 1

i

j

ij

1

1

i

j

ij

i

i

a

ij

aj

j

j

ij + 1

b

bi + j

ij

ij

i + aj

bi

ij + ab 

3.8 Definition The reduced norm of a quaternion algebra is called a 2-fold Pfister form. (1) If char F 6= 2, then it is h1, −a, −b, abi = hhaii ⊗ hhbii = hha, bii. (2) If char F = 2, then it is [1, ab] ⊥ [a, b] ' hhaii ⊗ hhab]] = hha, ab]]. 3.9 Proposition An algebra is quaternion iff it is a central simple algebra of dimension 4. A quaternion algebra is either!isomorphic to M2 (F ) or a division algebra. a, b (1) If char F 6= 2, then ' M2 (F ) iff hha, bii is isotropic, i.e. hyperbolic. F " # a, b ' M2 (F ) iff hha, ab]] is isotropic, i.e. hyperbolic. (2) If char F = 2, then F 3.10 Definition We define n-fold Pfister forms inductively. (1) If char F 6= 2, then hha1 , . . . , an ii = hha1 , . . . , an−1 ii ⊗ hhan ii. (2) If char F = 2, then hha1 , . . . , an ]] = hha1 , . . . , an−1 ii ⊗ hhan ]]. 4. Invariants 4.1 Definition Let q : V → F be a quadratic form. Its Clifford algebra is C(q) = T (V )/I

NOTES TO FORMS OF CHARACTERISTIC 2

where T (V ) =

∞ `

9

V ⊗n and I is the ideal of T (V ) generated by x⊗x−q(x), x ∈

n=0

V. The algebra C(q) is Z/2Z-graded and C(q) ' C0 (q) ⊕ C1 (q). We call C0 (q) the even Clifford algebra of q. 4.2 Lemma (0) C(0) =

V

(V ). a, b

(1) If char F 6= 2, then C(ha, bi) =

and C0 (ha, bi) = F−ab .

F "

a, b

(2) If char F = 2, then C([a, b]) =

F !

Proof. (2) Let V = F 2 , e0 =

!

1

# and C0 ([a, b]) = Fab . !

0

, e1 = . Then C([a, b]) has basis 0 1 {1, i, j, ij} where i is the class of e0 , j is the class of e1 and ij is the class of e0 ⊗ e1 . We have i2 = a since e20 = [a, b](e0 ) = a; j 2 = b since e21 = [a, b](e1 ) = b; ij + ji = 1 since e0 e1 + e1 e0 = (e0 + e1 )2 − e20 − e21 = b[a,b] (e0 , e1 ) = 1.

Here C0 ([a, b]) = F ⊕ F ij ' Fab since (ij)2 = ijij = i(ij + 1)j = i2 j 2 + ij = ab + ij.  4.3 Definition Let q be a nondegenerate even dimensional quadratic form over F . Then the ´ 2 (F ) ' center Z(C0 (q)) is a quadratic ´etale algebra over F , its class in Et H 1 (F, Z/2Z) is called the discriminant of q, we write disc(q). 4.4 Lemma (1) If char F 6= 2, then for q = ha1 , . . . , a2n i, ai ∈ F ∗ , disc(q) = (−1)n a1 · · · a2n ∈ F ∗ /(F ∗ )2 is also called the signed determinant. (2) If char F = 2, then for q = [a1 , b1 ] ⊥ · · · ⊥ [an , bn ], ai , bi ∈ F , disc(q) = a1 b1 + · · · + an bn ∈ F/℘(F ) is also called the Arf invariant. Proof. We grant that Z(C0 (q1 ⊥ q2 )) = Z(C0 (q1 )) ? Z(C0 (q2 )). Then disc(q1 ⊥ q2 ) = disc(q1 ) disc(q2 ).

10

ZHENGYAO WU (吴正尧)

Since Z(C0 [ai , bi ]) = Fai bi , disc(q) corresponds ´ 2 (F ). [Fa1 b1 ] · · · [Fan bn ] = [Fa1 b1 +···+an bn ] ∈ Et  4.5 Definition Two central simple F -algebras A, B are Brauer equivalent if Mm (A) ' Mn (B) for some integers m, n > 0. Let Br(F ) be the quotient of the set of isomorphism classes of central simple algbras modulo Brauer equivalence. 4.6 Proposition We have an abelian group Br(F ) with [A][B] = [A ⊗ B],

0 = [Mn (F )],

−[A] = [Aop ].

4.7 Definition Let q be a nondegenerate even dimensional quadratic form over F . The class [C(q)] in the 2-torsion part Br2 (F ) ' H 2 (F, Z/2Z(1)) is called the Clifford invariant of q, we write clif(q). 4.8 Theorem Let W (F ) be the Witt group of nondegenerate symmetric bilinear forms over F . Let I the ideal of W (F ) generated by even dimensional forms. Let GW∗ (F ) = ` n n+1 ` I /I the graded Witt ring. Let K∗ (F ) = Kn (F ) be the ring of n≥0 n≥0 ` Milnor K-groups of F . Let k∗ (F ) = kn (F ) = K∗ (F )/2K∗ (F ). n≥0

(1) [OVV07, Th.4.1] If char F 6= 2, then s∗ : k∗ (F ) → GW∗ (F ) {a1 , . . . , an } 7→ hha1 , . . . , an ii mod I n+1 is an isomorphism. (2) [Kat82b] If char F = 2, then s∗ : k∗ (F ) → GW∗ (F ) {a1 , . . . , an } 7→ h1, a1 i ⊗ · · · ⊗ h1, an i mod I n+1 is an isomorphism. We will prove Kato’s theorem in the next two sections.

NOTES TO FORMS OF CHARACTERISTIC 2

11

4.9 Remark For Milnor’s conjecture, see [Mil70, Th. 4.1] and the last paragraph of [Mil71]. For Voevodsky’s proof see [Voe03a, Voe03b]. If char F 6= 2, then s0 is the dimension modulo 2; s1 is the signed discriminant; s2 is the Clifford invariant; for s3 see [Ros86] and [MS90a, MS90b]. If char F = 2, the graded structure of the Witt group W q(F ) of quadratic forms over F (In general GW q(F ) 6' GW (F )) is achieved in [EKM08, 13.7] for n = 1, [Sah72] for n = 2 and [Kat82b] for general n. 5. Bijectivity of d log Let F be a field. Suppose char F = 2. Let S = {x2 | x ∈ F }. Then S is a subfield of F . Let (bλ )λ∈Λ be a family of elements of F . Let uλ : S(bλ ) → F be canonical injections. 5.1 Definition The family (bλ )λ∈Λ is a 2-basis of F over S if O

uλ : S((bλ )λ∈Λ ) =

λ∈Λ

O

S(bλ ) → F

λ∈Λ

is an isomorphism as S-vector spaces. 5.2 Proposition Q

The set (bλ )λ∈Λ is a 2-basis of F over S iff {

λ∈Λ

s(λ)



| s : Λ → {0, 1},

P

s(λ)
i, note that λ ≤ β(k) does not impliy λ ≤ α(k) in Λ). 5.4 Definition Let b = (bλ )λ∈Λ be a 2-basis of F over S. (1) For λ ∈ Λ,

12

ZHENGYAO WU (吴正尧)

• Let Fλ = S((bµ )µ≤λ ). • Let F