Some Algebraic Aspects

of Quadratic Forms over Fields

of Characteristic Two

Ricardo Baeza^{1}

Received: May 10, 2001

Communicated by Ulf Rehmann

Abstract. This paper is intended to give a survey in the algebraic theory of quadratic forms over fields of characteristic two. The relationship between differential forms and quadratics and bilinear forms over such fields discovered by Kato is used to reduced some problems on quadratics forms to concrete questions about differential forms, which in general are easier to handle.

1991 Mathematics Subject Classification: 11 E04, 11 E81, 12 E05, 12 F20 Keywords and Phrases: Keywords and Phrases: Bilinear forms, Quadratic forms, Differential forms, Witt-groups, Function fields.

1 Introduction.

In his historical account on the algebraic theory of quadratic forms (s [Sch ]), Scharlau remarks that fields of characteristic two have remained the pariahs of the theory. Nevertheless, as he also mentions right before the above remark (s. loc. cit.), some aspects of the theory over these fields are more interesting and richer, because of the interplay of symmetric bilinear and quadratic forms, as well as both separable and purely inseparable quadratic extensions have to be considered. The purpose of this brief survey article is to show how these aspects work, and how some questions related to Milnor’s conjecture for fields with 2 ≠ 0, can be answered in a more elementary way in the case of characteristic two.

1Partially supported by Fondecyt 1000392 and Programa Formas Cuadraticas, Universidad de Talca.

Documenta Mathematica · Quadratic Forms LSU 2001 · 49–63

We will focus our attention on the W (F)-module structure of W_{q}(F), where W (F) is the Witt-ring of a field F with 2 = 0 and W_{q}(F) is the Witt-group of quadratic forms over F (s. [Mi]_{2}, [Sa] and section 2). If I ⊂ W (F) is the maximal ideal of W (F), then we have the graded Witt-ring

gr_{I}W (F) = ⨁_{n=0}^{∞} I^{n}/I^{n+1}

and the graded gr_{I}W (F)- module

gr_{I}W_{q}(F) = ⨁_{n=0}^{∞} I^{n}W_{q}(F)/I^{n+1}W_{q}(F).

The structure of this module is explained in sections 3 and 4. Section 3 deals with the relationship established by Kato between differential forms over F and symmetric bilinear and quadratic forms. If k_{∗}(F) denotes Milnor’s graded k-ring of F, we introduce in section 4 a graded k_{∗}(F)-module, defined by generators and relations, which describes the graded gr_{I}W (F)-module gr_{I}W_{q}(F). In section 5 we examine the behaviour of this module under certain field extensions, particularly function field extensions of quadrics defined by Pfisterforms. As an application of these results we mention, how Knebusch’s degree conjecture for fields with 2 = 0 follows from them. The results of section 5, (c.f. (5.10), (5.11), (5.14), (5.16)), cited from [Ar-Ba]_{3} and [Ar-Ba]_{4} have not been published yet, but these manuscripts can be found at the server ”Linear Algebraic Groups and Related Structures” http://www.mathematik.unibielefeld.de/LAG/.

2 Basic definitions.

Let F be a field of characteristic two. A symmetric bilinear form b : V ×V ⟶ F defined on an n-dimensional F -vector space V is non-singular if b(x, y) = 0 for all x ∈ V implies y = 0. (V, b) is anisotropic if b(x, x) ≠ 0 for all x ≠ 0, and in this case it is easy to see that (V, b) admits an orthogonal basis (s. [Mi]_{2} for example). If a ∈ F^{∗} = F \ {0} we will denote by < a > the one dimensional form axy, and by < a_{1}, ⋯, a_{n} > (a_{i} ∈ F^{∗}) the orthogonal sum < a_{1} >⊥ ⋯ ⊥< a_{n} > A non singular quadratic form on V is a map q : V ⟶ F such that q(λx) = λ^{2}q(x) and b_{q}(x, y) = q(x + y) − q(x) − q(y) is a symmetric non singular bilinear form on V. Since b_{q}(x, x) = 0, n must be even. The most simple non singular quadratic forms over F are the forms ax^{2} + xy + by^{2} with a, b ∈ F (i.e. q : F e ⊕ F f ⟶ F, q(e) = a, q(f) = b, b_{q}(e, f) = b_{q}(f, e) = 1 ), which we will denote by [a, b]. Any non singular quadratic form over F is of the form [a_{1}, b_{1}] ⊥ ⋯ ⊥ [a_{m}, b_{m}]. Scaling a quadratic form q by a ∈ F^{∗} means (aq)(x) = aq(x). This extends to an operation of bilinear forms on quadratic forms by < a_{1}, ⋯ a_{n} > ·q = a_{1}q ⊥ ⋯ ⊥ a_{n}q. Besides the dimension, the most simple invariant of a symmetric bilinear form b =< a_{1}, ⋯ a_{n} > is its discriminant d(b) = a_{1} ⋯ a_{n} ∈ F^{∗}/F^{∗^{2}} If q = [a_{1}, b_{1}] ⊥ ⋯ ⊥ [a_{n}, b_{n}] is a quadratic form the analogue of the discriminant is its Arf-invariant A(q) = a_{1}b_{1} + ⋯ + a_{n}b_{n} ∈ F/℘F, where ℘F = {a^{2} − a\a ∈ F }. One can write [a, b] =< a > [1, ab] if a ≠ 0, so that in general one usually writes a quadratic form q as q =< a_{1} > [1, b_{1}] ⊥ ⋯ ⊥< a_{n} > [1, b_{n}], and hence its Arf-invariant is A(q) = b_{1} + ⋯ + b_{n} ∈ F/℘F (s. [A], [Ba]_{1}, [Sa]). For quadratic forms (V, q) we have also the Clifford - algebra C(q), which defines

an element w(q) ∈ Br(F) = Brauer group of F. If q = ⊥^{m}_{1} < a_{i} > [1, b_{i}],

then w(q) = ⊗^{m}_{1} (a_{i}, b_{i}] ∈ Br(F), where (a, b] denotes the quaternion algebra

F ⊕ F e ⊕ F f ⊕ F ef with e^{2} = a, f^{2} + f = b, ef + fe = e. A symmetric bilinear form (V, b) is called metabolic if V contains a subspace W ⊆ V with W = W^{⊥} (dim W = \frac{1}{2} dim V). Two bilinear forms b_{1}, b_{2} are Witt-equivalent if b_{1} ⊥ m_{1} ≅ b_{2} ⊥ m_{2}, where m_{1}, m_{2} are metabolic. The set of classes W (F) of symmetric non singular bilinear forms is a ring, additively generated by the classes < a >, a ∈ F^{∗} with relations < a > + < b > = < a + b > + < ab(a + b) > if a + b ≠ 0, < a > + < a > = 0 and < a > · < b > = < ab >. We denote by I_{F} ⊂ W (F) the maximal ideal of even dimensional forms (s. [Mi]_{2}, [Sa] for basic facts on W (F)). A quadratic form (V, q) is hyperbolic if V contains a totally isotropic subspace W ⊂ V with dim W = \frac{1}{2} dim V. The form [0, 0] = ℍ is the hyperbolic plane and every hyperbolic space is of the form ℍ ⊥ … ⊥ ℍ. The forms q_{1}, q_{2} are Wittequivalent if q_{1} ⊥ r × ℍ ≅ q_{2} ⊥ s × ℍ (r, s ≥ 0) and we denote by W_{q}(F) the Witt-group of such classes. The action defined above of bilinear forms on quadratic forms induces a W (F)-module structure on W_{q}(F). I_{F} is additively generated by the 1-fold Pfister forms < 1, a >, a ∈ F^{∗}, so that for all n ≥ 1, I_{F}^{n} is generated by the n-fold bilinear Pfister forms ≪ a_{1}, ⋯, a_{n} ≫ = < 1, a_{1} > ⊗ ⋯ ⊗ < 1, a_{n} >. These ideals define submodules I_{F}^{n} · W_{q}(F) of W_{q}(F), which are additively generated by the n-fold quadratic Pfister forms ≪ a_{1}, ⋯ a_{n}, a |] = ≪ a_{1}, ⋯ a_{n} ≫ ⊗[1, a], a_{i} ∈ F^{∗}, a ∈ F (s. [Ba]_{1}, [Sa] for details on these forms). Thus we have now two filtrations

W (F) ⊇ I_{F} ⊃ I_{F}^{2} ⊃ ⋯ ⊃ I_{F}^{n} ⋯

W_{q}(F) ⊇ IW_{q}(F) ⊃ I^{2}W_{q}(F) ⊃ ⋯ ⊃ I^{n}W_{q}(F) ⊃ ⋯

and we will be mainly concerned with the quotients I_{F}^{n}/I_{F}^{n+1} and I^{n}W_{q}(F)/I^{n+1}W_{q}(F) which we denote by \overline{I}_{F}^{n} and \overline{I^{n}W_{q}}(F) respectively.

One easily checks that dim : \overline{I}_{F}^{0} ⟶^{∼} ℤ/2ℤ, d : \overline{I}_{F} ⟶^{∼} F^{∗}/F^{∗^{2}}

and A : \overline{I^{0}W_{q}}(F) ⟶^{∼} F/℘F. The main result of [Sa] states that w : \overline{IW_{q}}(F) ⟶^{∼} Br(F)_{2} = 2-torsion part of Br(F). The surjectivity of w is a consequence of well-known results on p-algebras for p = 2 (s. [Al]), and the injectivity is shown in [Sa] by an elementary induction argument (notice that the isomorphism \overline{IW_{q}}(F) →^{∼} Br(F)_{2} is the analogue of Merkurjev’s result I_{F}^{2} /I_{F}^{3} →^{∼} Br(F)_{2} for fields with 2 ≠ 0). The higher groups \overline{I}_{F}^{n} and \overline{I^{n}W_{q}}(F) will be studied in the next section.

3 Differential forms and its relationship to quadratic and bilinear forms

The basic reference for what follows is Kato’s fundamental paper [Ka]_{1}. Let Ω_{F}^{1} be the F -vector space generated (over F) by the symbols da, a ∈ F, with the relations d(ab) = bda + adb. In particular d(F^{2}) = 0, and hence the map d : F ⟶ Ω_{F}^{1} is F^{2}-linear. Let Ω_{F}^{n} = ⋀^{n} Ω_{F}^{1} be the F -space of n-differential forms over F. The map d : F ⟶ Ω_{F}^{1} extends to d : Ω_{F}^{n} ⟶ Ω_{F}^{n+1} for all n ≥ 1 by d(xdx_{1} ∧ ⋯ ∧ dx_{n}) = dx ∧ dx_{1} ∧ ⋯ ∧ dx_{n}. Recall that a 2-basis of F

is a set {a_{i} , i ∈ I} ⊂ F such that the elements {a^{ε} = ∏_{i ∈ I} a_{i}^{ε_{i}}, ε = (ε_{i}, i ∈

I), ε_{i} ∈ {0, 1} and almost all ε_{i} = 0} form a F^{2}-basis of F. If {a_{1}, a_{2}, … } is

a 2-basis of F, then the forms \frac{da_{i_{1}}}{a_{i_{1}}} ∧ … ∧ \frac{da_{i_{n}}}{a_{i_{n}}} 1 ≤ i_{1} < ⋯ < i_{n} form

a F -basis of Ω_{F}^{n}. Fixing such a 2-basis, we define

[Ω_{F}^{n}]^{2} = \{ ∑_{i_{1}<···<i_{n}} c_{i_{1}···i_{n}}^{2} \frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{i_{n}}}{a_{i_{n}}} , c_{i_{1}···i_{n}} ∈ F \}

which depends on the choice of the 2-basis. Then in [Ca] it is shown that the space Z_{F}^{n} = ker(d : Ω_{F}^{n} ⟶ Ω_{F}^{n+1}) has a direct-sum decomposition Z_{F}^{n} = [Ω_{F}^{n}]^{2} ⊕ dΩ_{F}^{n−1}.

One now defines a homomorphism (3.1) C : Z_{F}^{n} ⟶ Ω_{F}^{n}

by

C( ∑_{i_{1}<···<i_{n}} c_{i_{1}···i_{n}}^{2} \frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{i_{n}}}{a_{i_{n}}} + dη) = ∑_{i_{1}<···<i_{n}} c_{i_{1}···i_{n}} \frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{i_{n}}}{a_{i_{n}}}

C obviously does not depend on the choice of the 2-basis and induces an isomorphism \overline{C} : Z_{F}^{n}/dΩ_{F}^{n−1} ⟶^{∼} Ω_{F}^{n} of abelian groups. We will call C the Cartier-operator. Let us define now the homomorphism ℘ = \overline{C} ^{−1} − 1 : Ω_{F}^{n} ⟶ Ω_{F}^{n} /dΩ_{F}^{n−1}, which is given on generators by ℘(x\frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}}) = (x^{2} − x)\frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}} mod dΩ_{F}^{n−1}. One can define a 2-basis dependent homomorphism ℘ : Ω_{F}^{n} → Ω_{F}^{n} as follows. Fix a 2-basis B = {a_{1}, a_{2}, ⋯ } of F. Then we set

℘ ( ∑_{i_{1}<···<i_{n}} c_{i_{1}···i_{n}} \frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{i_{n}}}{a_{i_{n}}} )

= ∑_{i_{1}<···<i_{n}}(c_{i_{1}···i_{n}}^{2} − c_{i_{1}···i_{n}})\frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{i_{n}}}{a_{i_{n}}} .

If for ω = ∑_{i_{1}<···<i_{n}} c_{i_{1}···i_{n}} \frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{i_{n}}}{a_{i_{n}}} we set

ω^{[2]} = ∑_{i_{1}<···<i_{n}}c_{i_{1}···i_{n}}^{2} \frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{i_{n}}}{a_{i_{n}}} ,

then ℘ω = ω^{[2]} − ω. Obviously if we change the 2-basis, the image of ω ∈ Ω_{F}^{n} under the new ℘- operator differs from ℘ω by an exact form. We will use this type of operator in section 5.

Let ν_{F} (n) = Ker(℘) and H^{n+1}(F) = Coker(℘), so that 0 → ν_{F} (n) → Ω_{F}^{n} →^{℘}

Ω_{F}^{n} /dΩ_{F}^{n−1} → H^{n+1}(F) → 0 is exact. An obvious characterization of ν_{F} (n) is the following

(3.2) Lemma. ν_{F} (n) = {ω ∈ Ω_{F}^{n} \dω = 0, C(ω) = ω}

In [Ka]_{1} it is shown that ν_{F} (n) is additively generated by the pure logarithmic differentials \frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}}, which is a direct consequence of lemma 2 in [Ka]_{2}. Since we will refer frequently to this lemma, we will state it explicitly. Let B = {a_{i}, i ∈ I} be a 2-basis of F and endow I with a totally ordering. For any j ∈ I set F_{i} resp. F_{≤j} for the subfield of F generated over F^{2} by the elements a_{i} with i < j resp. i ≤ j. Endow with the lexicographic ordering the set ∑_{n} of functions α : {1, ⋯ n} → I with α(i) < α(j) whenever i < j. Then {da_{α(1)} ∧ ⋯ ∧ da_{α(n)}, α ∈ ∑_{n}} is a F -basis of Ω_{F}^{n} and for any α ∈ ∑_{n} set Ω_{F,α}^{n} resp. Ω_{F,<α}^{n} for the subspace of Ω_{F}^{n} generated by the elements da_{β(1)} ∧ ⋯ ∧ da_{β(n)} with β ≤ α resp. β < α. Then Kato’s lemma 2 in [Ka]_{2} asserts

(3.3) Lemma. Let y ∈ F, α ∈ ∑_{n} and ω_{α} = \frac{da_{α(1)}}{a_{α(1)}} ∧ ⋯ ∧ \frac{da_{α(n)}}{a_{α(n)}} ∈ Ω_{F}^{n}, be such that

(y^{2} − y)ω_{α} ∈ Ω_{F,<α}^{n} + dΩ_{F}^{n−1}.

Then there exist v ∈ Ω_{F,<α}^{n} and a_{i} ∈ F_{α(i)}^{∗}, 1 ≤ i ≤ n, with

yω_{α} = v + \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} .

It is clear that the last remark above follows immediately from this result, which we will quote as Kato’s lemma in what follows.

One of the main results of [Ka]_{1} is the fact that there exist two natural isomorphisms

(3.4) α_{F} : ν_{F} (n) ⟶ \overline{I}_{F}^{n}

(3.5) β_{F} : H^{n+1}(F) ⟶ \overline{I^{n}W_{q}}(F)

given on generators by

α_{F} (\frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}} ) = \overline{≪ x_{1}, ⋯ x_{n} ≫}

β_{F} (\overline{x\frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}}} ) = \overline{≪ x_{1}, ⋯ x_{n}, x |]}

Thus α and β translate many questions on bilinear and quadratic forms to corresponding problems in differential forms, which some times are easier to handle, in particular if one is able to choose a suitable 2-basis of the field F. Nevertheless the use of the isomorphism α can be some times difficult, since in order to compute α(ω) one must first write ω ∈ ν_{F} (n) as a sum of pure logarithmic differential forms.

4 Milnor’s K-theory.

For any field F Milnor defined in [Mi]_{1} its K-groups K_{n}(F) in a purely algebraic manner as follows (s. also Pfister’s survey [Pf] for more details). Let K_{1}(F) be the multiplicative group of F written additively, i.e. l :

F^{∗} →^{∼} K_{1}(F), l(ab) = l(a) + l(b) for a, b ∈ F^{∗}. Set K_{0}(F) = ℤ and

K_{n}(F) = K_{1}(F)^{⊗n}/ℑ_{n} (n ≥ 2), where ℑ_{n} is the subgroup of K_{1}(F)^{⊗n} generated by elements of the form l(a_{1}) ⊗ ⋯ ⊗ l(a_{n}) with a_{i} + a_{j} = 1 for some i ≠ j. Denote by l(x_{1}) ⋯ l(x_{n}) the image of l(x_{1}) ⊗ ⋯ ⊗ l(x_{n}). Thus the main defining relation of these groups is l(a)l(1 − a) = 0 in K_{2}(F) for a ≠ 0, 1.

Let k_{n}(F) = K_{n}(F)/2K_{n}(F) and form the commutative ring k_{∗}(F) = k_{0}(F) ⊕ k_{1}(F) ⊕ ⋯ with k_{0}(F) = ℤ/2ℤ, k_{1}(F) →^{∼} F^{∗}/F^{∗^{2}}. Milnor defines epimorphisms s_{n} : k_{n}(F) → \overline{I}_{F}^{n} by

s_{n}(l(a_{1}) ⋯ l(a_{n})) = \overline{≪ a_{1}, ⋯ a_{n} ≫}

and conjectures that they are isomorphisms for all n. If 2=0 in F, then there are also natural homomorphisms (s. [Ka]_{1})

d log : k_{n}(F) ⟶ ν_{F} (n)

given by d log(l(a_{1}) ⋯ l(a_{n})) = \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}}.

A consequence of Kato’s lemma is that d log is an epimorphism. In [Ka]_{1} it is shown that d log is an isomorphism, which combined with the isomorphism (3.3) gives us the following main result of [Ka]_{1} (4.1) Theorem (Kato) For any field F with 2=0 there is a commutative diagram of isomorphisms

k_{n}(F)  \underrightarrow{d log}  ν_{F} (n)
\begin{matrix}⧹ \\ s_{n} ↘ \end{matrix}  \begin{matrix}⧸ \\ ↙ α_{F} \end{matrix}
\overline{I}_{F}^{n}

The defining relation l(a)l(a − 1) = 0 (a ≠ 0, 1) of the groups k_{n}(F) corresponds in the case 2 ≠ 0 to the basic fact that the quaternion algebra (a, 1 − a) splits. Here (x, y) denotes the quaternion algebra F ⊕ F e ⊕ F f ⊕ F ef, e^{2} = x, f^{2} = y, ef = −fe.

But if 2=0 we do not have such interpretation and the groups k_{n}(F) are suitable only to describe symmetric bilinear forms and for quadratic forms, we need another universal object, which we introduce now. Thus in order to obtain groups which are appropriate to describe the quotients \overline{I^{n}W_{q}}(F) by generators and relations one is led to alter Milnor’s definition of k_{n} taking into account the basic relations of quaternion algebras over a field with 2=0. This has been done in [Ar-Ba]_{1}. Let a ∈ F^{∗}, b ∈ F. The quaternion algebra (a, b] is the algebra F ⊕ F e ⊕ F f ⊕ F ef with e^{2} = a, f^{2} + f = b and ef + fe = e. It holds (ax^{2}, b + y + y^{2}] ≅ (a, b], and (a, b] splits if and only if a ∈ D_{F} ([1, b]) = {x^{2} + xy + by^{2}/ x, y ∈ F }, and a ≠ 0. Thus the bilinear map

ϕ : F^{∗}/F^{∗^{2}} × F/℘F ⟶ Br(F)_{2}, ϕ(ā, b̄) = (a, b]

satisfies ϕ(ā, b̄) = 0 iff a ∈ D_{F} ([1, b]). The universal symbol for ϕ can be constructed as follows. Let k_{1}(F) = F^{∗}/F^{∗^{2}}, h_{1}(F) = F/℘F and set

h_{2}(F) = \frac{k_{1}(F) ⊗ h_{1}(F)}{< l(a) ⊗ t(b) a ∈ D_{F} [1, b], a ≠ 0 >}

(here t(b) is the image of b in h_{1}(F) = F/℘F). Thus one obtains a natural homomorphism

ϕ_{F} : h_{2}(F) ⟶ Br(F)_{2}

which is in fact an isomorphism (s. [Ar-Ba]_{1},[Sa]). On the other hand we also have a bilinear map

k_{1}(F) × h_{1}(F) ⟶ H^{2}(F)

given by (l(a), t(b)) ⟶ b\frac{da}{a}, which induce a natural homomorphism

d log : h_{2}(F) ⟶ H^{2}(F).

This homomorphism is also an isomorphism (s. loc. cit), so that the group h_{2}(F), H^{2}(F), Br(F)_{2}, \overline{IW}_{q}(F) are all isomorphic and we have a commutative diagram of isomorphisms

(4.2) \begin{matrix}h_{2}(F) & \underrightarrow{ϕ_{F}} & Br(F)_{2} \\ d log ↓ & & ↑ω \\ H^{2}(F) & \overrightarrow{β_{F}} & \overline{IW}_{q}(F) \end{matrix}

Let now

h_{n}(F) = k_{1}(F)^{⊗(n−1)} ⊗ h_{1}(F)/R_{n}

where R_{n} is the subgroup generated by the elements l(a_{1}) ⊗ ⋯ ⊗ l(a_{n−1}) ⊗ t(b) such that either a_{i} + a_{i+1} = 1 for some i or a_{i} ∈ D_{F} [1, b]. We denote by l(a_{1}) ⋯ l(a_{n−1})t(b) in h_{n}(F) the image of l(a_{1}) ⊗ ⋯ ⊗ l(a_{n−1}) ⊗ t(b). The natural product k_{r}(F) × h_{s}(F) → h_{r+s}(F) induces a k_{∗}(F)-module structure on h_{∗}(F) = h_{1}(F) ⊕ h_{2}(F) ⊕ · · ·. There are natural epimorphisms

s_{n} : h_{n}(F) ⟶ \overline{I^{n−1}W}_{q}(F)

d log : h_{n}(F) ⟶ H^{n}(F)

given by

s_{n}(l(a_{1}) ⋯ l(a_{n−1})t(b)) = \overline{≪ a_{1}, ⋯ a_{n−1}, b |]}

d log(l(a_{1}) ⋯ l(a_{n−1})t(b)) = \overline{b\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n−1}}{a_{n−1}}}

In [Ar-Ba]_{1} it is shown that d log is an isomorphism, and combining it with Kato’s isomorphism β_{F}, we conclude also that s_{n} is an isomorphism. Thus we have (s. [Ar-Ba]_{1} and [Ka]_{1})

(4.3) Theorem. For all n there is a commutative diagram of isomorphisms

h_{n}(F)  \underrightarrow{s_{n}}  \overline{I^{n−1}W}_{q}F (n)

\begin{matrix}⧹ \\ d log ↘ \end{matrix}  \begin{matrix}⧸ \\ ↙ β_{F} \end{matrix}
H^{n}(F)

Remark. The groups k_{n}(F) and h_{n}(F) are related through Galois cohomology. If F_{s} is a separable closure of F and G_{F} = Gal(F_{s}/F) then k_{n}(F_{s}) is a G_{F} - module and it holds (s. [Ar-Ba]_{1})

H^{0}(G_{F} , k_{n}(F_{s})) ≅ k_{n}(F)

H^{1}(G_{F} , k_{n}(F_{s})) ≅ h_{n+1}(F)

(s. [Ar]).

5 Behaviour of quadratic and bilinear forms under field extensions.

A natural question is the behaviour of the groups Ω_{F}^{n} , ν_{F} (n), H^{n+1}(F) resp \overline{I}_{F}^{n}, \overline{I^{n}W_{q}}(F) under field extensions. Since the isomorphisms α_{F} , β_{F} (s. (3.4) and (3.5)) are functorial, we only need to study the behaviour of the groups ν_{F} (n), H^{n+1}(F), to get information about \overline{I}_{F}^{n} and \overline{I^{n}W_{q}}(F) (but, as mentioned before, care must be taken with the use of α_{F}). If L/F is a field extension, we denote by Ω_{L/F}^{n} the kernel Ker(Ω_{F}^{n} → Ω_{L}^{n}), and similarly we define ν_{L/F} (n), H^{n+1}(L/F), \overline{I}_{L/F}^{n} and \overline{I^{n}W_{q}}(L/F). By the remark above

α_{F} : ν_{L/F} (n) →^{∼} \overline{I}_{L/F}^{n} and β_{F} : H^{n+1}(L/F) →^{∼} \overline{I^{n}W_{q}}(L/F). The easiest group

to handle is Ω_{L/F}^{n} because a suitable choice (if possible!) of a 2-basis of F and L gives quickly the answer. Since

(5.1) ν_{L/F} (n) = ν_{F} (n) ∩ Ω_{L/F}^{n}

one also gets information about ν_{L/F} (n) knowing Ω_{L/F}^{n}. Let us now review what we know about these kernels for some field extensions.

(i) Purely Transcendental extensions. If L = F (X), X any set of variables over F, and B is a 2-basis of F, then B ∪ {X} is a 2-basis of F (X). In particular Ω_{F}^{n} → Ω_{F (X)}^{n} is injective and Ω_{F (X)/F}^{n} = 0. Hence ν_{F (X)/F} (n) = 0. Using Kato’s lemma (3.3) one can also show H^{n+1}(F (X)/F) = 0 (s. [Ar-Ba]_{3})

(ii) Quadratic extensions. Let L = F (\sqrt{b}), b ∈ F \F^{2} be a purely inseparable quadratic extension of F. Choose a 2-basis B = {b_{i}, i ∈ I} with b = b_{i_{0}}, some i_{0} ∈ I. Then \{b_{i}, i ∈ I − \{i_{0}\}, \sqrt{b} \} is a 2-basis of F (\sqrt{b}) and it is easy to check that

(5.2) Ω_{F (\sqrt{b})/F}^{n} = Ω_{F}^{n−1} ∧ \frac{db}{b}

Hence ν_{F (\sqrt{b})/F} (n) = \{ω ∧ \frac{db}{b} / ω ∈ Ω_{F}^{n−1}, ω ∧ \frac{db}{b} ∈ ν_{F} (n)\}. It follows from (5.11) below that

(5.3) ν_{F (\sqrt{b})/F} (n) = \{ω ∧ \frac{db}{b} / ω ∈ Ω_{F}^{n−1} and ℘ω ∈ a[Ω_{F}^{n−1}]^{2}+

dΩ_{F}^{n−2} + Ω_{F}^{n−2} ∧ da}

(s. section 3 for the definition of ℘ω).

The corresponding result for \overline{I} ^{n} is now (s. (5.12) below for a more general statement)

(5.4) \overline{I}_{F (\sqrt{b})/F}^{n} = ∑_{x∈F^{2}(b)^{∗}} \overline{I}_{F}^{n−1} < 1, x >

Let us now examine the kernel H^{n+1}(F (\sqrt{b})/F). We have (s.[Ar-Ba]_{3})

(5.5) H^{n+1}(F (\sqrt{b})/F) = \overline{Ω_{F}^{n−1} ∧ \frac{db}{b}}

The proof of this fact is again based on Kato’s lemma and runs briefly as follows. Take B = {b_{1} = b, b_{2}, ⋯ } a 2-basis of F (one can assume w.l.o.g. that B is enumerable or even finite), so that B′ = \{\sqrt{b_{1}}, b_{2}, ⋯ \} is a 2-basis of F (\sqrt{b}). \overline{ω} ∈ H^{n+1}(F (\sqrt{b})/F) means ω ∈ Ω_{F}^{n} and ω = ℘u + dv with u ∈ Ω_{F (\sqrt{b})}^{n}, v ∈ Ω_{F (\sqrt{b})}^{n−1}. Order B′ such that \sqrt{b} > b_{i}, i = 2, 3 · · · . Since

\overline{Ω_{F}^{n−1} ∧ \frac{db}{b}} ⊆ H^{n+1}(F (\sqrt{b})/F) we may assume that db does not appear in the 2-basis expansion of ω and let α ∈ ∑_{n} be the leading index of ω (notice α(i) > 1 for all i = 1, ⋯ n), and let β ∈ ∑_{n} be the leading index of u. Using Kato’s lemma one may assume β ≤ α, and we obtain

(℘u_{α} + ω_{α})\frac{db_{α}}{b_{α}} ≡ dv mod Ω_{F (\sqrt{b}),<α}^{n}

(here \frac{db_{α}}{b_{α}} means \frac{db_{α(1)}}{b_{α(1)}} ∧ ⋯ ∧ \frac{db_{α(n)}}{b_{α(n)}} )

with v ∈ Ω_{F (\sqrt{b})}^{n−1}. Since b_{α(i)} < \sqrt{b} for all i, we conclude comparing coefficients that the leading coefficient of dv is in F, so that u_{α} is defined over F. Thus v may be taken also in Ω_{F}^{n−1}. Since Ω_{F (\sqrt{b})/F}^{n} = Ω_{F}^{n−1} ∧ \frac{db}{b}, we conclude in Ω_{F}^{n}

ω_{α} \frac{db_{α}}{b_{α}} ≡ ℘(u_{α})\frac{db_{α}}{b_{α}} + dv mod Ω_{F, <α}^{n} + Ω_{F}^{n−1} ∧ \frac{db}{b}

Inserting this relation in ω, we can lower the highest index in ω. This concludes the proof of the claim. The corresponding kernel for I^{n}W_{q} is now

(5.6) \overline{I^{n}W_{q}}(F (\sqrt{b})/F) = \overline{≪ b ≫ I^{n−1}W_{q}(F)}

For quadratic separable extensions of F the corresponding kernels are much easier to compute. Let L = F (z), z^{2} + z = b (b ∉ ℘F) be a quadratic separable extension of F. Since we can alter b by elements of ℘F, we can assume b ∈ F^{2}. Thus z ∈ L^{2} and we see that any 2-basis of F remains a 2basis of L. In particular Ω_{L}^{n} = Ω_{F}^{n} ⊕ z · Ω_{F}^{n}. Thus Ω_{L/F}^{n} = 0 and also ν_{L/F} = 0. The computation of H^{n+1}(L/F) is in this case also very easy. We claim

(5.7) H^{n+1}(L/F) = \overline{bν_{F} (n)}

For the proof, take \overline{ω} ∈ H^{n+1}(F) with ω = ℘u + dv, u ∈ Ω_{L}^{n}, v ∈ Ω_{L}^{n−1} and set u = u_{1} + zu_{2}, v = v_{1} + zv_{2} with u_{i} ∈ Ω_{F}^{n} , v_{i} ∈ Ω_{F}^{n−1}. Inserting in the above equation it follows ℘u_{2} = dv_{2} ∈ dΩ_{F}^{n−1}, and this means u_{2} ∈ ν_{F} (n). Moreover ω = bu_{2}^{[2]} + ℘u_{1} + dv_{1} in Ω_{F}^{n}. But u_{2} ∈ ν_{F} (n) implies u_{2}^{[2]} ≡ u_{2} ( mod dΩ_{F}^{n−1}) and since b ∈ F^{2}, it follows ω ≡ bu_{2} mod (℘Ω_{F}^{n} + dΩ_{F}^{n−1}), ie \overline{ω} = \overline{bu_{2}}. This proves (5.7). The corresponding result for quadratic forms is

(5.8) \overline{I^{n}W_{q}}(L/F) = \overline{I_{F}^{n} · [1, b]}

(iii) Function fields of Pfister forms. Let us fix an anisotropic bilinear n-fold Pfister-form ϕ =≪ a_{1}, ⋯, a_{n} ≫. This means that {a_{1}, ⋯, a_{n}} are part of 2-basis of F. Let L = F (ϕ) be the function field of the quadric \{ϕ(x, x) = 0\}. Thus L = F (X)(\sqrt{T}), where X = \{X_{μ}, μ ∈ S_{n}\} and T = ∑_{μ}

a^{μ}X_{μ}^{2}, a^{μ} = Π^{n}_{i = 1} a_{i}^{μ(i)}, for all μ ∈ S_{n} where S_{n} denotes the set of maps

μ : {1, ⋯, n} → {0, 1} whith some μ(i) = 1 In [Ar-Ba]_{3} it is shown that

(5.9)  Ω_{L/F}^{m} = 0  if m < n

(5.10)  Ω_{L/F}^{m} = Ω_{F}^{m−n} ∧ \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}}  if m ≥ n

In particular ν_{L/F} (m) = 0 if m < n. The case m ≥ n has been considered in [Ar-Ba]_{4} and the result is:

(5.11) ν_{L/F} (m) = \{ω ∧ \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} / ω ∈ Ω_{F}^{m−n}, ℘ω ∈ ∑_{ε ≠ 0} a^{ε}[Ω_{F}^{m−n}]^{2}+

dΩ_{F}^{m−n−1}+ ∑^{n}_{i = 1} Ω_{F}^{m−n−1} ∧ da_{i}}

If m = n, this result looks nicer, namely

ν_{L/F} (n) = \{a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} / a^{2} − a ∈ F^{2}(a_{1}, ⋯ a_{n})′ \}

where F^{2}(a_{1}, ⋯, a_{n})′ ⊂ F^{2}(a_{1}, ⋯ a_{n}) is the subgroup consisting in the ele-

ments ∑_{ε ≠ 0} c_{ε}^{2}a_{1}^{ε_{1}} ⋯ a_{n}^{ε_{n}}, ε = (ε_{1}, ⋯ ε_{n}) ∈ {0, 1}^{n}.

The corresponding result for bilinear forms is

(5.12) I_{L/F}^{m} = ⟨\overline{ψ ≪ x_{1}, ⋯ x_{n} ≫}/ ψ ∈ \overline{I}_{F}^{m−n}, x_{1}, ⋯, x_{n} ∈ F^{2}(a_{1}, ⋯ a_{n})^{∗}⟩

The case m = n is particularly interesting, because

\overline{I}_{L/F}^{n} = \{\overline{≪ x_{1}, ⋯ x_{n} ≫}/ x_{i} ∈ F^{2}(a_{1}, ⋯ a_{n})^{∗}\}

implies the following corollary

(5.13) Corollary. Given x_{1}, ⋯ x_{n}, y_{1}, ⋯ y_{n} ∈ F^{2}(a_{1}, ⋯ a_{n})^{∗}, then there exist z_{1}, ⋯ z_{n} ∈ F^{2}(a_{1}, ⋯ a_{n})^{∗} such that

≪ x_{1}, ⋯, x_{n} ≫ + ≪ y_{1}, ⋯, y_{n} ≫ ≡ ≪ z_{1}, ⋯ z_{n} ≫ mod I_{F}^{n+1}

This is a kind of relative n-linkage property of the subfields F^{2}(a_{1}, ⋯, a_{n}) of F.

Let us now turn our attention to H^{n+1}. The main result of [Ar-Ba]_{3} is

(5.14) Theorem. If ϕ =≪ a_{1}, ⋯ a_{n} ≫ is anisotropic over F, then

H^{n+1}(F (ϕ)/F) = \overline{F \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}}}

The proof of this fact, although elementary, is rather long. For \overline{ω} ∈ H^{n+1}(F (ϕ)/F) we get an equation ω = ℘u+dv with u ∈ Ω_{F (ϕ)}^{n} and v ∈ Ω_{F (ϕ)}^{n−1}.

Writing F (ϕ) = L(y), L = F (X_{μ}, μ ∈ S_{n}), y^{2} = T = ∑_{μ∈S_{n}} a^{μ}X_{μ}^{2}, a^{μ} =

a_{1}^{μ(1)} ⋯ a_{n}^{μ(n)}, we choose a 2-basis B = {a_{i}, i ∈ I} of F containing a_{1}, ⋯ a_{n}, so that B ∪ {X_{μ}, μ ∈ S_{n}} is a 2-basis of L and then we fix a 2-basis B′ = B \ {a_{1}} ∪ {X_{μ}, μ ∈ S_{n}} ∪ {y} of F (ϕ). We order the elements of this basis such that all X_{μ} > B \ {a_{1}} and y > X_{μ} for all μ (i.e. y is maximal). Using these choices, and Kato’s lemma, one sees that u and v can be chosen free of differentials of the form dX_{μ} or dy, and moreover that the scalar coefficients of u and v do not contain y in the 2-basis expansion. Thus u and v are defined over L = F (X_{μ}). But since H^{n+1}(F (ϕ)/L) = \overline{Ω_{L}^{n−1} ∧ dT} by (5.5),we have

(5.15) ω = ℘u + dv + λ ∧ dT

in Ω_{L}^{n}, with some λ ∈ Ω_{L}^{n−1}. Expanding with respect to the 2-basis B∪{X_{μ}, μ ∈ S_{n}} and comparing coefficients, one can show that u, v, λ can be taken in Ω_{F}^{n} ⊗ M and Ω_{F}^{n−1} ⊗ M respectively, where M = F (X_{μ}^{2}, μ ∈ S_{n}). This is the start for long descent argument which leads to an equation ω = ℘u_{0} + dv_{0} + bda_{1} ∧ ⋯ ∧ da_{n} whith b ∈ F and u_{0}, v_{0} defined over F The corresponding result for quadratic forms is

(5.16) Theorem

\overline{I^{n}W_{q}}(F (ϕ)/F) = \{\overline{≪ a_{1}, ⋯, a_{n}, a |]} / a ∈ F \}

As it is shown in [Ar-Ba]_{2}, this result implies the following one. Let p =≪ a_{1}, ⋯, a_{n}, a |] be now an anisotropic quadratic n-fold Pfister form and let F (p) be the function field of the quadric {p(x) = 0}. Then

(5.17) Theorem

H^{n+1}(F (p)/F) = {0, p̄}

Remark. One may expect that (5.14) generalizes to the following assertion

H^{m+1}(F (ϕ)/F) = \overline{Ω_{F}^{m−n} ∧ \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}}} , m ≥ n.

Documenta Mathematica · Quadratic Forms LSU 2001 · 49–63

6 An application:

generic splitting of quadratic forms. One can develop a generic splitting theory for non singular quadratic forms over a field with 2 = 0 in the same way as it has been done for the case 2 ≠ 0 in [Kn]_{1,2}, because in the case 2 = 0 one has:

(i) the analogue of Pfister’s subform theorem (s. [Am], [Ba]_{3} and [Le])

(ii) The analogue of Knebusch’s norm theorem (s. [Ba]_{2}).

With these tools one defines a generic splitting tower of a non singular quadratic form q over F and obtains a leading form, which is similar to a Pfister form. The degree of this form is called the degree of q. Now define ℑ(n) = {q̄ ∈ W_{q}(F) /deg q ≥ n}. Then ℑ(n) is a W (F)-submodule of W_{q}(F) and one easily sees that I^{n}W_{q}(F) ⊆ ℑ(n). In [Ar-Ba]_{3} it is shown that the equality ℑ(n) = I^{n}W_{q}(F) for all n (over a field of any characteristic) is equivalent with the statement of theorem (5.17) above for any n. Thus we have

(6.1) Theorem For any field F with 2 = 0, it holds

ℑ(n) = I^{n}W_{q}(F)

Remark. The corresponding result for (5.17) over fields with 2 ≠ 0 has been announced by Orlov-Vishik-Voevodsky (s. [Pf]).

References

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Ricardo Baeza

Instituto de Matematica Universidad de Talca Casilla 721 Talca, Chile rbaeza@inst-mat.utalca.cl 64

Documenta Mathematica · Quadratic Forms LSU 2001