LINKAGE OF FIELDS IN CHARACTERISTIC 2

ROBERTO ARAVIRE AND RICARDO BAEZA

Abstract

Let F be a field with 2 = 0 and φ =≪ a_{1}, …, a_{n} ≫ an n-fold anisotropic bilinear Pfister form over F with function field F (φ). In this paper we compute ker[I_{F}^{n}/I_{F}^{n+1} → I_{F (φ)}^{n}/I_{F (φ)}^{n+1}] where I_{F} ⊂ W (F) is the maximal ideal in the Witt ring W (F) of F. We use this computation to prove a n-linkage property of the subfields F^{2}(a_{1}, …, a_{n}).

0. Introduction

In this paper F will denote throughout a field with 2 = 0. Let Ω_{F}^{∗} = ⨁_{n=0}^{∞} Ω_{F}^{n} be the F -algebra of differential forms over F and let d : Ω_{F}^{n} → Ω_{F}^{n+1} (n ≥ 0) be the differential operator (see [Ca], [Ka], [A-Ba 1]). In [Ka] (see also [Mi]) a homomorphism ℘ : Ω_{F}^{n} → Ω_{F}^{n} /dΩ_{F}^{n−1} is defined on generators as follows

℘(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}.

(if n = 0 we have the usual Artin-Schreier operator ℘(x) = x^{2} − x). Let ν_{F} (n) = ker(℘) and H^{n+1}(F) = coker(℘). In [Ka] 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}} (x_{i} ∈ F^{∗} = F \\{0\}) and that there exists a natural isomorphism

α : ν_{F} (n) ≃ \overline{I}_{F}^{n} := I_{F}^{n}/I_{F}^{n+1} (1)

given on generators by α(\frac{dx_{1}}{x_{1}} ∧⋯∧ \frac{dx_{n}}{x_{n}}) :=≪ x_{1}, …, x_{n} ≫ mod I_{F}^{n+1}. Here I_{F} denotes the maximal ideal of even dimensional forms in the Witt ring W (F) of non-singular symmetric bilinear forms over F and ≪ x_{1}, …, x_{n} ≫ is the n-fold Pfister form < 1, x_{1} > ⊗ ⋯ ⊗ < 1, x_{n} > (see [Sa], [A-Ba 1] for basic definitions). If L/F is a field extension, let ν_{L/F} (n) denote ker(ν_{F} (n) → ν_{L}(n)) and \overline{I}_{L/F}^{n} = ker(\overline{I}_{F}^{n} → \overline{I}_{L}^{n}). Thus α induces an isomorphism α : ν_{L/F} (n) ≃ \overline{I}_{L/F}^{n}.

2000 Mathematics Subject Classification: 11E04, 11E81, 12E05, 12F20, 15A63.

Keywords and phrases: Bilinear forms, differential forms, Witt-groups, function fields.

First author was partially supported by Fondecyt 100 0391 and Programa Formas Cuadráticas, U. de Talca.

Second author was partially supported by Fondecyt 100 0392 and Programa Formas Cuadráticas, U. de Talca.

The purpose of this note is to compute ν_{L/F} (n) when L is the function field of an anisotropic bilinear Pfister form and to relate this computation to some linking property of subfields of F. The computation of H^{n+1}(L/F) = ker(H^{n+1}(F) → H^{n+1}(L)) is much more involved and has been done in [A-Ba 2]. For any bilinear form φ over F we will denote by F (φ) the function field of the quadric {φ(x, x) = 0}.

In Section 1 and 2 we compute ν_{F (φ)/F} (n) and \overline{I}_{F (φ)/F}^{n} where φ =≪ a_{1}, …, a_{n} ≫. In section 3 we extend these computations to ν_{F (φ)/F} (m), \overline{I}_{F (φ)/F}^{m}, for arbitrary m ≥ 1. We will use the following notions and notations taken from [Ca]. A 2-basis of F is a subset {a_{1}, a_{2}, … } ⊂ F such that the elements {a^{ε} = ∏_{i} a_{i}^{ε_{i}} | ε = (ε_{i})} (here ε = (ε_{i}), runs over all sequences with ε_{i} = 0 or 1, and with 0 almost everywhere), form a basis of F over F^{2} as a vector space. This is equivalent with the fact that the forms \frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{in}}{a_{in}}, i_{1} < ⋯ < i_{n} are a F -basis of Ω_{F}^{n} for all n ≥ 0. Fixing such a 2-basis, let [Ω_{F}^{n}]^{[2]} = ⨁_{i_{1}<···<i_{n}} F^{2} \frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{in}}{a_{in}}. Then the space Z_{F}^{n} = ker(d : Ω_{F}^{n} → Ω_{F}^{n+1}) has the direct sum decomposition Z_{F}^{n} = [Ω_{F}^{n}]^{[2]} ⊕ dΩ_{F}^{n−1} and we get a homomorphism C : Z_{F}^{n} → Ω_{F}^{n} given 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}}}

which obviously induces an isomorphism C : Z_{F}^{n}/dΩ_{F}^{n−1} ≃ Ω_{F}^{n}. Although the decomposition of Z_{F}^{n} depends on the choice of the 2-basis, the map C does not. We will call C the Cartier operator. It is easy to see that ν_{F} (n) is characterized by: w ∈ ν_{F} (n) if and only if dw = 0 and C(w) = w.

1. ν_{L/F} (n) for L = F (≪ a_{1}, …, a_{n} ≫)

Let φ =≪ a_{1}, …, a_{n} ≫ be an anisotropic n-fold bilinear Pfister form over F. This means that {a_{1}, a_{2}, …, a_{n}} are part of a 2-basis {a_{1}, a_{2}, … } of F. Let F^{2}(a_{1}, …, a_{n}) be the subfield of F generated by a_{1}, …, a_{n} over F^{2} and let F^{2}(a_{1}, …, a_{n})′ the additive subgroup of pure elements ⨁_{ε≠0} F^{2}a^{ε}, ε = (ε_{1}, …, ε_{n}) ∈ {0, 1}^{n}. Thus F^{2}(a_{1}, …, a_{n}) = D_{F} (φ) is the set of elements of F represented by φ (including 0) and F^{2}(a_{1}, …, a_{n})′ = D_{F} (φ)′ are the elements represented by the pure part φ′ of φ. Let L = F (φ) and set Ω_{L/F}^{n} = ker(Ω_{F}^{n} → Ω_{L}^{n}). Then we have

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

In [A-Ba 2] we have computed Ω_{L/F}^{m} for any m ≥ 0. The result is

1.1. Proposition.

Ω_{L/F}^{m} = \{ \begin{matrix}0 & , & if m < n \\ Ω_{F}^{m−n} ∧ \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} & , & if m ≥ n. \end{matrix}

1.2. Corollary.

ν_{L/F} (m) = \{ \begin{matrix}0 & , & if m < n \\ Ω_{F}^{m−n} ∧ \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} ⋂ ν_{F} (m) & , & if m ≥ n. \end{matrix}

For the sake of completeness we will give a sketch of the proof of 1.1 at the end of this section. The rest of this section is devoted to compute ν_{L/F} (n) = F \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} ∩ ν_{F} (n).

1.3. Lemma. For any a ∈ F the following assertions are equivalent

1. a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} ∈ ν_{F} (n).

2. ℘(a) = a^{2} − a ∈ F^{2}(a_{1}, …, a_{n})′.

Proof. Assume (1). Choose a 2-basis of F containing a_{1}, …, a_{n}, say {a_{1}, …, a_{n}, …, a_{N}} (we can assume without restriction that this basis is finite). Then a = ∑_{ε} c_{ε}^{2}a^{ε}, where ε = (ε_{1}, …, ε_{N}) runs over {0, 1}^{N} and a^{ε} = a_{1}^{ε_{1}} ⋯ a_{N}^{ε_{N}}. It follows

da = ∑_{i=1}^{N} D_{i}(a)da_{i}

with D_{i}(a) = ∑_{ε} c_{ε}^{2}a_{1}^{ε_{1}} ⋯ â_{i} ⋯ a_{N}^{ε_{N}} where ε runs over all ε’s with ε_{i} = 1. Since a\frac{da_{1}}{a_{1}} ∧⋯∧ \frac{da_{n}}{a_{n}} ∈ ν_{F} (n) ⊂ ker(d), we obtain d(a)∧da_{1}∧⋯∧da_{n} = 0, and this implies d(a) = c_{1}da_{1} + ⋯ + c_{n}da_{n} with some c_{i} ∈ F. Hence D_{i}(a) = 0 for all i > n and we conclude c_{ε} = 0 whenever ε_{i} = 1 for some i > n. Thus we have

a = ∑_{ε=(ε_{1},... ,ε_{n})} c_{ε}^{2}a^{ε} ∈ F^{2}(a_{1}, …, a_{n})

and

a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} = ∑_{ε} c_{ε}^{2} \frac{da_{1}}{a_{1}^{1−ε_{1}}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}^{1−ε_{n}}} .

From our assumption (1) and the definition of the Cartier operator mentioned in the introduction, we obtain

a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} = C (a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} )
= ∑_{ε} C (c_{ε}^{2} \frac{da_{1}}{a_{1}^{1−ε_{1}}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}^{1−ε_{n}}} ) .

For any ε ≠ 0, the form c_{ε}^{2}_{\frac{da_{1}}{a_{1}^{1−ε_{1}}}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}^{1−εn}} is exact and hence

C(c_{ε}^{2}_{\frac{da_{1}}{a_{1}^{1−ε_{1}}}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}^{1−εn}}) = 0. Thus we obtain

a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} = C (c_{0}^{2} \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} )
= c_{0} \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}}

i.e. a = c_{0}. Therefore a = a^{2} + ∑_{ε≠0} c_{ε}^{2}a^{ε}, and this is precisely (2).

Conversely, assuming a = a^{2} + ∑_{ε≠0} c_{ε}^{2}a^{ε}, ε = (ε_{1}, …, ε_{n}) ∈ {0, 1}^{n}, we obtain

a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} = a^{2} \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} + ∑_{ε≠0} c_{ε}^{2} \frac{da_{1}}{a_{1}^{1−ε_{1}}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}^{1−ε_{n}}}

and hence C(a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}}) = a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} and this implies (1).

Putting 1.2 and 1.3 together we obtain

1.4. Theorem. Let L = F (≪ a_{1}, …, a_{n} ≫). Then

ν_{L/F} (n) = \{a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} |a ∈ F with ℘(a) ∈ F^{2}(a_{1}, …, a_{n})′\} .

Proof of Proposition (1.1). The function field L = F (φ), where φ =≪ a_{1}, …, a_{n} ≫, can be described as follows: for any ε ∈ {0, 1}^{n}, different from (0, …, 0), let X_{ε} be a variable and set K = F (X_{ε}) for the field generated by all this variables over F. Then L = K(\sqrt{T}), where T = ∑_{ε≠0} a^{ε}X_{ε}^{2} (T is the pure part of ≪ a_{1}, …, a_{n} ≫). We proceed now in three steps:

1. Ω_{K/F}^{m} = 0. This is clear choosing a 2-basis B of F and enlarging it to a 2-basis B ∪ {X_{ε}, ε ≠ 0} of K.

2. If E = F (\sqrt{a}), a ∈ F \F^{2}, then Ω_{E/F}^{m} = Ω_{F}^{m−1} ∧ da. Since a ∉ F^{2}, we can choose a 2-basis of F containing a, say B = {a, c_{i} i ∈ I}. Then \{\sqrt{a}, c_{i} i ∈ I\} is a 2-basis of E. If w ∈ Ω_{E/F}^{m}, set w = (∑_{i_{1}<···<i_{m−1}} a_{i_{1}···i_{m−1}}dc_{i_{1}}∧⋯∧dc_{i_{m−1}})∧da+∑_{i_{1}<···<i_{m}} a_{i_{1}···i_{m}}dc_{i_{1}}∧ ⋯∧dc_{i_{m}}. Then in Ω_{E}^{m} we get ∑_{i_{1}<···<i_{m}} a_{i_{1}···i_{m}}dc_{i_{1}} ∧⋯∧dc_{i_{m}} = 0 and hence a_{i_{1}···i_{m}} = 0 for all i_{1} < ⋯ < i_{m}. This proves the claim.

3. Ω_{L/F}^{m} = Ω_{F}^{m−n} ∧ \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}}, and if m < n, Ω_{L/F}^{m} = 0.

We just consider the case m ≥ n. We choose a 2-basis B = {a_{1}, …, a_{n}, … } of F and take w ∈ Ω_{L/F}^{m}. From (2) we get w = u∧dT, u ∈ Ω_{K}^{m−1}, where dT = k_{1}da_{1}+⋯+k_{n}da_{n}. k_{1}, …, k_{n} being certain quadratic polynomials over F. Inserting da_{1} = k_{1}^{−1}(k_{2}da_{2}+⋯+k_{n}da_{n})+k_{1}^{−1}dT into u, we see that we may assume u free from terms containing da_{1} in its basis expansion with respect to the 2-basis B∪{X_{ε}, ε ≠ 0} of K. Write w = w_{0}+w_{1}∧da_{1} with forms w_{0}, w_{1} free from forms containing da_{1}. Then in Ω_{K}^{m} we have w_{0} + w_{1} ∧ da_{1} = u ∧ k_{1}da_{1} + u ∧ (k_{2}da_{2} + ⋯ + k_{n}da_{n}), i.e. (w_{1} + k_{1}u) ∧ da_{1} = w_{0} + u ∧ (k_{2}da_{2} + ⋯ + k_{n}da_{n}). Since the form on the right of this equation does not contain da_{1} in the 2-basis expansion, we obtain w_{1} = k_{1}u and therefore k_{1}w = w_{1} ∧ dT. Everything in this equation is defined over F [X_{ε}, ε ≠ 0], so that a simple specialization of the variables proves that da_{1} divides w in Ω_{F}^{m}. Thus w is divisible by da_{1}, …, da_{n} and hence by da_{1} ∧ … ∧ da_{n}. This proves the claim.

2. I_{L/F}^{n} for L = F (≪ a_{1}, …, a_{n} ≫)

Using the isomorphism α : ν_{L/F} (n) ≅ \overline{I}_{L/F}^{n} and 1.4 we can now describe \overline{I}_{L/F}^{n}. But in order to apply α to a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} ∈ ν_{L/F} (n) we must express this form in terms of pure logarithmic differentials.

2.1. Examples. 1. If n = 1, i.e. a = a^{2} + c^{2}a_{1} (see (1.4)), then

a\frac{da_{1}}{a_{1}} = \frac{a^{2}a_{1}}{aa_{1}} \frac{da_{1}}{a_{1}} = \frac{a^{2}a_{1}}{c^{2}a_{1}^{2} + a^{2}a_{1}} \frac{da_{1}}{a_{1}} = \frac{d(c^{2}a_{1}^{2} + a^{2}a_{1})}{c^{2}a_{1}^{2} + a^{2}a_{1}}

and hence α(a\frac{da_{1}}{a_{1}} ) =≪ c^{2}a_{1}^{2} + a^{2}a_{1} ≫ mod I_{F}.

2. If n = 2, i.e. a = a^{2} + c_{1}^{2}a_{1} + c_{2}^{2}a_{2} + c_{3}^{2}a_{1}a_{2} = a′ + a_{1}a′′, with a′ = a^{2} + c_{2}^{2}a_{2}, a′′ = c_{1}^{2} + c_{3}^{2}a_{2} ∈ F^{2}(a_{2}), a simple computation shows that d(aa_{1}) ∧ da_{2} = a′da_{1} ∧ da_{2}, and hence

a\frac{da_{1}}{a_{1}} ∧ \frac{da_{2}}{a_{2}} = \frac{a^{2}}{a′} \frac{d(aa_{1})}{aa_{1}} ∧ \frac{da_{2}}{a_{2}} = \frac{a^{2}}{a′} \frac{da_{2}}{a_{2}} ∧ \frac{d(aa_{1})}{aa_{1}}
= \frac{d(a_{2}a′)}{a_{2}a′} ∧ \frac{d(aa_{1})}{aa_{1}}.

Thus α(a\frac{da_{1}}{a_{1}} ∧ \frac{da_{2}}{a_{2}} ) =≪ a_{2}a′, aa_{1} ≫ mod I_{F}^{2} .

Let us now consider the general situation. According to (1.3) we have a = a^{2} + ∑_{ε≠0} c_{ε}^{2}a^{ϵ} = a′ + a_{1}a′′ with a′, a′′ ∈ F^{2}(a_{2}, …, a_{n}). We claim that there exist elements x_{i} ∈ F^{2}(a_{i}, …, a_{n})^{∗} with a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} = \frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}}. The cases n = 1, 2 are clear because of (2.1). We have

a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} = \frac{a^{2}}{a′} \frac{da_{2}}{a_{2}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} ∧ \frac{d(aa_{1})}{aa_{1}}.

Notice that the form \frac{a^{2}}{a′} \frac{da_{2}}{a_{2}} ∧⋯∧ \frac{da_{n}}{a_{n}} is defined over F^{2}(a_{2}, …, a_{n}). Set \overline{a} = a^{2}/a′. We show that ℘(\overline{a}) ∈ F^{2}(a_{2}, …, a_{n})′, so the claim follows by induction. In fact, we have ℘(\overline{a}) = \overline{a}^{2} + \overline{a} = a^{2}(a^{2} + a′)/a^{2}, and since

a′ = a^{2} + ∑_{η≠0} c_{η}^{2}a_{2}^{η_{2}} ⋯ a_{n}^{η_{n}},

η ∈ \{0, 1\}^{n−1}, it follows ℘(\overline{a}) = ∑_{η≠0} (\frac{ac_{η}}{a′} )^{2} a_{2}^{η_{2}} ⋯ a_{n}^{η_{n}} ∈ F^{2}(a_{2}, …, a_{n})′. From a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} = \frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}} we obtain α(a\frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}}) = ≪ x_{1}, …, x_{n} ≫ mod I_{F}^{n+1}, and this shows \overline{I}_{L/F}^{n} ⊂ \{≪ x_{1}, …, x_{n} ≫ mod I_{F}^{n+1} | x_{1}, …, x_{n} ∈ F^{2}(a_{i}, …, a_{n})^{∗}}. It is easy to check that the other inclusion also holds. Thus we have shown.

2.2. Theorem. If L = F (≪ a_{1}, …, a_{n} ≫), then

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

Using the fact that \overline{I}_{L/F}^{n} is a group, we obtain

2.3. Corollary. For any x_{1}, …, x_{n}, y_{1}, …, y_{n} ∈ F^{2}(a_{1}, …, a_{n})^{∗}, 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}.

One can interpret this result as a sort of n-linkage property of the subfields F^{2}(a_{1}, …, a_{n}) relative to the field F in the sense of the following definition. Let E ⊂ F be a subfield. E is called n-linked relative to F if for all x_{1}, …, x_{n}, y_{1}, …, y_{n} ∈ E^{∗}, there exist z_{1}, …, z_{n} ∈ E^{∗} with ≪ x_{1}, …, x_{n} ≫ + ≪ y_{1}, …, y_{n} ≫=≪ z_{1}, …, z_{n} ≫ mod I_{F}^{n+1} in W (F).

3. ν_{L/F} (m) for L = F (≪ a_{1}, …, a_{n} ≫)

The aim of this section is to extend the results of section 2 to arbitrary m, i.e. we will compute ν_{L/F} (m) for m ≥ 1. Since ν_{L/F} (m) = Ω_{L/F}^{m} ∩ ν_{F} (m), we conclude ν_{L/F} (m) = 0 if m < n (see (1.1)). Thus we assume m ≥ n, and hence ν_{L/F} (m) = Ω_{F}^{m−n} ∧ \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} ∩ ν_{F} (m). In order to characterize the forms w ∈ Ω_{F}^{m−n} with w ∧ \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} ∈ ν_{F} (m) we generalize the operator ℘(a) = a^{2} − a as follows (see [Ka], [A-Ba 2]). Let {a_{1}, …, a_{n}, …, a_{N}} be a 2-basis of F which we assume without restriction to be finite. Then any form η ∈ Ω_{F}^{q}, (q ≥ 1) can be written in a unique way as

η = ∑_{i_{1}<···<i_{q}} c_{i_{1}···i_{q}} \frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{i_{q}}}{a_{i_{q}}} .

We define ℘ : Ω_{F}^{q} → Ω_{F}^{q} by

℘(η) = ∑_{i_{1}<···<i_{q}} ℘(c_{i_{1}···i_{q}})\frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{i_{q}}}{a_{i_{q}}}
= η^{[2]} − η

where η^{[2]} = ∑_{i_{1}<···<i_{q}} c_{i_{1}···i_{q}}^{2} \frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{iq}}{a_{iq}}.

This definition obviously depends on the choice of the 2-basis {a_{i}}, but if one changes the 2-basis, the new ℘ operator values differs from the former values by exact forms, i.e. ℘(η) mod dΩ_{F}^{q−1} is independent of the 2-basis. With this notation we have

3.1. Lemma. Let m ≥ n and w ∈ Ω_{F}^{m−n}. Fix a 2-basis {a_{1}, …, a_{n}, …, a_{N}} of F. Then the following assertions are equivalent

1. w ∧ \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} ∈ ν_{F} (m).

2. ℘(w) ∈ ∑_{ε≠0} a^{ε} [Ω_{F}^{m−n}]^{[2]} + dΩ_{F}^{m−n−1} + ∑_{i=1}^{n} Ω_{F}^{m−n−1} ∧ da_{i} (where a^{ε} = a_{1}^{ε_{1}} ⋯ a_{n}^{ε_{n}} and [Ω_{F}^{q}]^{[2]} denotes the group of all η^{[2]}, η ∈ Ω_{F}^{q}).

If m = n, we recover (1.3). Since the proof of (3.1) follows the same pattern as the proof of (1.3), we will skip some details.

Proof of (3.1) Assume (2), i.e. we have

w = w^{[2]} + ∑_{ε≠0} a^{ε}A_{ε}^{[2]} + dB + ∑_{i=1}^{n} E_{i} ∧ da_{i}

with A_{i} ∈ Ω_{F}^{m−n}, B, E_{1}, …, E_{n} ∈ Ω_{F}^{m−n−1}. It follows d(w ∧ η) = 0, where η = \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}}. Applying the Cartier Operator to w ∧ η we get

C(w ∧ η) = C(w^{[2]} ∧ η) = w ∧ η,

because a^{ε}A_{ε}^{[2]} ∧η, dB ∧η are exact forms. This implies w ∧η ∈ ν_{F} (m). Let us now assume (1). Set w = ∑_{i_{1}<···<i_{m−n}} c_{i_{1}...i_{m−n}} \frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{i_{m−n}}}{a_{m−n}}. In what follows all computations will be modulo < da_{1}, …, da_{n} >. Let k > n be the maximal index with w = R_{0} + a_{k}R_{1}, R_{0}, R_{1} differential forms generated by \frac{da_{n+1}}{a_{n+1}} , ⋯, \frac{da_{N}}{a_{N}} over F^{2}(a_{1}, …, a_{k−1}). Let us write R_{0} = M_{0} + M_{1} ∧ \frac{da_{k}}{a_{k}}, R_{1} = M_{2} + M_{3} ∧ \frac{da_{k}}{a_{k}} with forms M_{0}, …, M_{3} not containing the differential da_{k} and with coefficients in F^{2}(a_{1}, …, a_{k−1}). It follows

dw = dM_{0} + a_{k}dM_{2} + dM_{1} ∧ \frac{da_{k}}{a_{k}} + a_{k}(dM_{3} + M_{2}) ∧ \frac{da_{k}}{a_{k}}

and since d(w ∧ η) = 0, we get d(M_{0} ∧ η) = 0, d(M_{2} ∧ η) = 0, d(M_{1} ∧ \frac{da_{k}}{a_{k}} ∧ η) = 0 and (dM_{3} + M_{2}) ∧ \frac{da_{k}}{a_{k}} ∧ η = 0. Because of the choice made above, we obtain d(M_{0}) = 0, dM_{2} = 0, dM_{1} = 0, dM_{3} + M_{2} = 0. Thus w = M_{0} + M_{1} ∧ \frac{da_{k}}{a_{k}} + d(a_{k}M_{3}). Let w′ = M_{0} + M_{1} ∧ \frac{da_{k}}{a_{k}}. It follows d(w′ ∧ η) = dw′ ∧ η = 0. Moreover w′ is generated by \frac{da_{n+1}}{a_{n+1}} , …, \frac{da_{N}}{a_{N}} over F^{2}(a_{1}, …, a_{k−1}).

Repeating the above procedure with w′ we finally arrive at a decomposition

w = w_{0} + dM

with w_{0} generated by \frac{da_{n+1}}{a_{n+1}} , …, \frac{da_{N}}{a_{N}} over F^{2}(a_{1}, …, a_{n}), i.e. w = ∑_{μ} c_{μ} \frac{da_{μ}}{a_{μ}} +dM, with c_{μ} ∈ F^{2}(a_{1}, …, a_{n}), and μ running over all m−ntuples μ = (i_{1}, …, i_{m−n}) of integers with n + 1 ≤ i_{1} < ⋯ < i_{m−n} ≤ N and where a_{μ} = a_{i_{1}} ⋯ a_{i_{m−n}}, da_{μ} = da_{i_{1}} ∧ ⋯ ∧ da_{i_{m−n}}.

Let c_{μ} = ∑_{ε} c_{μ,ε}^{2}a^{ε}, where ε = (ε_{1}, …, ε_{n}) ∈ {0, 1}^{n}, a^{ε} = a_{1}^{ε_{1}} ⋯ a_{n}^{ε_{n}}. Then

w = ∑_{ε} a^{ε} (∑_{μ} c_{μ,ε}^{2} \frac{da_{μ}}{a_{μ}} ) + dM.

LINKAGE OF FIELDS IN CHARACTERISTIC 2 9

Set A_{ε} = ∑_{μ} c_{μ,ε} \frac{da_{μ}}{a_{μ}}. Thus we have

w = A_{0}^{[2]} + ∑_{ε≠0} a^{ε}A_{ε}^{[2]} + dM.

Using the fact that a^{ε}A_{ε}^{[2]} ∧ η, dM ∧ η ∈ dΩ_{F}^{m−n−1}, we conclude

C(w ∧ η) = C (A_{0}^{[2]} ∧ η) = A_{0} ∧ η,

i.e. w ∧ η = A_{0} ∧ η. It follows w = A_{0} + H with H ∈< \frac{da_{1}}{a_{1}} , …, \frac{da_{n}}{a_{n}} >.

Therefore

w = w^{[2]} + ∑_{ε≠0} a^{ε}A_{ε}^{[2]} + dM + H′

with H′ ∈ ∑_{i=1}^{n} Ω_{F}^{m−n−1} ∧ \frac{da_{i}}{a_{i}}. This proves the lemma.

3.2. Corollary. Let L = F (≪ a_{1}, …, a_{n} ≫). Then 1. If m < n, ν_{L/F} (m) = 0.

2. If m ≥ n, then

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

Using this decomposition of ν_{L/F} (m) and lemma (2.5) in [Ka], we can show 3.3. Corollary. Let L = F (≪ a_{1}, …, a_{n} ≫). Then

1. If m < n, \overline{I}_{L/F}^{m} = 0

2. If m ≥ n, \overline{I}_{L/F}^{m} = \{\overline{ψ ≪ x_{1}, …, x_{n} ≫} | ψ ∈ I_{F}^{m−n}, x_{1}, …, x_{n} ∈ F^{2}(a_{1}, …, a_{n})^{∗}\}

We will omit the details of the proof.

References

[A-Ba 1] Aravire, R., Baeza, R.: Milnor’s K-theory and quadratic forms over fields of characteristic two. Comm. Algebra, 20 (4), 1087-1107 (1992).

[A-Ba 2] Aravire, R., Baeza, R.: The behavior of quadratic and differential forms under function field extensions in characteristic two. Preprint 2000.

[Ca] Cartier, P.: Questions de rationalité des diviseurs en géometrie algebrique. Bull. Soc. Math. France, 86, 177-251 (1958).

[Ka] Kato, K.: Symmetric bilinear forms, quadratic forms and Milnor K-theory in characteristic two. Inv. Math. 66, 493-510 (1982).

[Mi] Milne, J.S.: Duality in flat cohomology of a surface . Ann. Sc. Ec. N. Sup. 9, 171-202 (1976).

[Sa] Sah, C.H.: Symmetric bilinear forms and quadratic forms. J. of Algebra 20, 144-160 (1972).

Roberto Aravire Departamento de Ciencias Físicas y Matemáticas Universidad Arturo Prat, Casilla 121, Iquique, Chile e-mail: raravire@cec.unap.cl.

Ricardo Baeza Instituto de Matemática y Física Universidad de Talca, Casilla 747, Talca, Chile. Departamento de Matemáticas, Facultad de Ciencias Universidad de Chile, Casilla 653, Santiago, Chile. e-mail: rbaeza@inst-mat.utalca.cl.