The behavior of quadratic and differential forms under function field extensions in

characteristic two

R. Aravire  R. Baeza

Contents

Abstract i Introduction 1

1 The algebra of differential forms 4 2 The behavior of differential forms under some field exten-

sions 14 3 Some technical results 27 4 The kernel H^{n+1}(F (ϕ)/F) 39 5 Quadratic forms and differential forms 61 6 Generic splitting of quadratic forms and the degree conjec-

ture 66 References 70

Abstract

Let F be a field of characteristic 2. Let Ω_{F}^{n} be the F -space of absolute differential forms over F. There is a homomorphism ℘ : Ω_{F}^{n} → Ω_{F}^{n} /dΩ_{F}^{n−1} given 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}

Let H^{n+1}(F) = Coker(℘). We study the behavior of H^{n+1}(F) under the function field F (ϕ)/F, where ϕ =≪ b_{1}, …, b_{n} ≫ is a n-fold Pfister form and F (ϕ) is the function field of the quadric ϕ = 0 over F. We show that

ker(H^{n+1}(F) → H^{n+1}(F (ϕ)) = \overline{F · \frac{db_{1}}{b_{1}} ∧ ⋯ ∧ \frac{db_{n}}{b_{n}}}

Using Kato’s isomorphism of H^{n+1}(F) with the quotient I^{n}W_{q}(F)/ I^{n+1}W_{q}(F), where W_{q}(F) is the Witt group of quadratic forms over F and I ⊂ W (F) the maximal ideal of even dimensional bilinear forms over F, we deduce from the above result the analogue in characteristic 2 of Knebusch’s degree conjecture, i.e. I^{n}W_{q}(F) is the set of all classes \overline{q} with deg(q) ≥ n.

Acknowledgments. This work has been supported by Proyecto Fondecyt 197 0214 and Programa Formas Cuadráticas, Universidad de Talca (second author) and Proyecto Fondecyt 196 0880, 198 0681 (first author).

Keywords and phrases: quadratic forms, differential forms, bilinear forms, Witt-groups, function fields, generic splitting fields of quadratic forms, degree of quadratic forms.

1991 Mathematics Subject Classification. 11E04, 11E81, 12E05, 12F20, 15A63, 15A75.

Introduction

Since Knebusch’s seminal papers on generic splitting of quadratic forms appeared (see [ Kn 1], [ Kn 2]), few work until recently has been done on his degree conjecture, which asserts that the n-th power I^{n} of the ideal I of even dimensional forms in the Witt-ring W (F) of symmetric non singular bilinear forms of a field F with 2 ≠ 0, coincides with the ideal of forms of degree ≥ n. Compare [ Kn 1], [ Ar-Kn], the 1998-preprint [ OVV] and [ A-Ba 2]. A similar theory of generic splitting of quadratic forms over a field of characteristic 2 can be developed (see section 6 of this work) and the corresponding degree conjecture can be stated. The aim of this work is to prove this analogue of Knebusch’s conjecture for fields with 2=0.

The advantage of working with fields of characteristic 2 is the fact discovered by K. Kato (see [ Ka 1]), that there is a strong relationship between quadratic forms and differential forms defined over such fields (see section 5 of this work). Thus many problems concerning quadratic forms in characteristic 2 can be translated into the language of differential forms, which are sometimes easier to handle, in particular choosing a suitable 2-basis of the ground field. Let us briefly recall Kato’s above mentioned correspondence. Let F be a field of characteristic 2 and let Ω_{F}^{n} be the space of n-differential forms over F (see [ Ca], [ Groth]). Let d : Ω_{F}^{n−1} → Ω_{F}^{n} be the usual differential operator which extends d : F → Ω_{F}^{1} , a ↦ da. Then there is a well defined homomorphism ℘ : Ω_{F}^{n} → Ω_{F}^{n} /dΩ_{F}^{n−1} given 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}

(see [ Ka 1], [ Mi] or section 1 of this paper).

Choosing a 2-basis of F one can easily lift ℘ to a homomorphism ℘ : Ω_{F}^{n} → Ω_{F}^{n} which of course depends on the 2-basis ( see section 1). Let ν_{F} (n) = ker(℘) and H^{n+1}(F) = Coker(℘), so that there is a exact sequence

0 → ν_{F} (n) → Ω_{F}^{n} → Ω_{F}^{n} /dΩ_{F}^{n−1} → H^{n+1}(F) → 0.

In [ Ka 1] it is shown that there exists a natural isomorphism of groups

H^{n+1}(F) →^{∼} I^{n}W_{q}(F)/I^{n+1}W_{q}(F)

given by

\overline{b\frac{ba_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}}} ↦ \overline{≪ a_{1}, …, a_{n}; b]]}.

Here W_{q}(F) denotes the W (F)-module of non singular quadratic forms over F and ≪ a_{1}, …, a_{n}; b]] denotes the quadratic n-fold Pfister form defined by a_{1}, …, a_{n} ∈ F^{∗}, b ∈ F (see [ Ka 1], [ A-Ba 1] or section 5 of this work).

The generic splitting theory developed in section 6 enables us to define the degree deg(q) of a non singular quadratic form over a field F with 2=0 along the same lines as Knebusch does for fields with 2 ≠ 0. We show also that I^{n}W_{q}(F) ⊆ \{\overline{q} ∈ W_{q}(F)| deg(q) ≥ n\}. In [ A-Ba 2] we have shown that equality holds (i.e. the degree conjecture holds) if one has the following equality:

ker [\overline{I^{n}W_{q}(F)} → \overline{I^{n}W_{q}(F (ϕ))}] = \overline{ϕ · W_{q}(F)}

for any anisotropic n-fold bilinear Pfister form ϕ over F. Here \overline{I^{n}W_{q}(F)} means I^{n}W_{q}(F)/I^{n+1}W_{q}(F) and F (ϕ) is the function field of ϕ over F.

This last equality is according to Kato’s isomorphism equivalent with

ker [H^{n+1}(F) → H^{n+1}(F (ϕ)] = \overline{F · \frac{db_{1}}{b_{1}} ∧ ⋯ ∧ \frac{db_{n}}{b_{n}}} (*)

in H^{n+1}(F), where ϕ =≪ b_{1}, …, b_{n} ≫. The proof of (*) will be finished in section 4 of this work. The corresponding result for fields with 2 ≠ 0 has been announced in [ OVV].

In section 1 we review some well known definitions and results concerning differential forms over a field with 2 = 0. We introduce the Cartier operator C and the ℘-homomorphism and we prove some technical results about divisibility of forms by forms in the F -algebra Ω_{F} = ⨁_{n=0}^{∞} Ω_{F}^{n} (see (1.16)). In section 2 we start to study the behavior of Ω^{n} and H^{n+1} under field extensions. The computation of ker(Ω_{F}^{n} → Ω_{E}^{n}) for some field extension E/F is not so difficult if one can choose a suitable 2-basis of E. We prove for example ker(Ω_{F}^{m} → Ω_{F (ϕ)}^{m}) = Ω_{F}^{m−n} ∧ db_{1} ∧ ⋯ ∧ db_{n} if m ≥ n ( and = 0 otherwise). This is the first evidence for the equality (*). The computation of ker(H^{n+1}(F) → H^{n+1}(E)) is much harder, even for very simple extensions E/F. We use frequently Kato’s fundamental lemma (see [ Ka 2] ), which is stated here without proof as lemma (2.15). As an important consequence of these computations we obtain that a form w ∈ Ω_{F}^{n} belongs to ker(H^{n+1}(F) → H^{n+1}(F (ϕ))) if and only if w satisfies a certain “differential equation” in the space Ω_{L}^{n} where L/F is a purely transcendental extension.

In fact, let S_{n} be the set of all maps μ : {1, …, n} → {0, 1} with μ(i) = 1 for at least one i. Let L = F (X_{μ}, μ ∈ S_{n}) where X_{μ} are independent variables. Let M = F (X_{μ}^{2}, μ ∈ S_{n}) ⊂ L and set Ω_{F} [M] for the sub-space of Ω_{L} (over M) generated by the forms db, b ∈ F, over M. For example if T = ∑_{μ∈S_{n}} b^{μ}X_{μ}^{2} with b^{μ} = ∏_{i=1}^{n} b_{i}^{μ(i)}, then T ∈ M and dT ∈ Ω_{F} [M]. Then we show that w ∈ Ω_{F}^{n} is contained in ker(H^{n+1}(F) → H^{n+1}(F (ϕ))) if and only if w satisfies an equation in Ω_{L} of the form

w = ℘(u) + dv + λ ∧ dT

with u, v, λ ∈ Ω_{F} [M] (see (2.25)). Section 3 is of technical nature and prepares the way for the proof of our main result in section 4. Section 4 is the heart of this work. We start with a relation w = ℘(u)+dv +λ∧dT where u, v, λ ∈ Ω_{F} [M] and we develop a descent procedure to finally end with a relation w = ℘(u_{0}) + dv_{0} + λ_{0} ∧ db_{1} ∧ ⋯ ∧ db_{n} where u_{0}, v_{0}, λ_{0} ∈ Ω_{F}. This is exactly the content of (*). In section 5 we explain briefly the basic relations between quadratic and differential forms. It is of expository character and details can be found in [ Ka 1], [ A-Ba 2]. Finally in section 6, as mentioned above, we extend Knebusch’s generic splitting theory to fields with 2 = 0 and prove the analogue of his degree conjecture.

1 The algebra of differential forms

We will consider in this paper only fields of characteristic 2. Let F be such a field. Let Ω_{F}^{1} be the F -vector space of absolute differential 1-forms, i.e. Ω_{F}^{1} is the F -vector space generated by the symbols da, a ∈ F, with the relations d(a + b) = da + db, d(ab) = adb + bda for a, b ∈ F. In particular d(F^{2}) = 0, where F^{2} = {a^{2}|a ∈ F } and d : F → Ω_{F}^{1} is a F^{2}-derivation.

Let us denote by Ω_{F}^{n} the n-exterior power ⋀^{n}(Ω_{F}^{1} ). Thus Ω_{F}^{n} is a F - vector space generated by the forms da_{1} ∧ ⋯ ∧ da_{n}. The operator d can be extended to a F^{2}-linear map d : Ω_{F}^{n} → Ω_{F}^{n+1} by d(ada_{1} ∧ ⋯ ∧ da_{n}) = da ∧ da_{1} ∧ ⋯ ∧ da_{n}. We will write Ω_{F} or Ω_{F}^{∗} for the F -algebra ⊕_{n=0}^{∞}Ω_{F}^{n} . We denote by Z_{F} the F^{2}-sub algebra {w ∈ Ω_{F}^{∗} |dw = 0}. Since d^{2} = 0, we obtain the ideal B_{F} = dΩ_{F}^{∗} in Z_{F} of exact forms. Now let us fix a 2-basis B = {b_{i}, i ∈ I} of F over F^{2}, i.e. if we take an ordering of I, then the monomials b_{i_{1}} ⋯ b_{i_{r}}, i_{1} < ⋯ < i_{r}, form a F^{2} -basis of F (see [ Ca] or [ Groth] for details about p-basis). Then it is well known that the (logarithmic) differential forms \{\frac{db_{i_{1}}}{b_{in}} ∧ ⋯ ∧ \frac{db_{in}}{b_{in}} , i_{1} < ⋯ < i_{n}\} form a F −basis of Ω_{F}^{n}. The following fact is obvious.

(1.1) Lemma. Let η ∈ Ω_{F} be a form which does not contain db for some b ∈ B, in its expansion with respect to the above basis. Then η ∧ db = 0 implies η = 0.

Let us denote by Ω_{F}^{[2]} the F^{2}-sub algebra of Ω_{F}^{∗} generated by the logarithmic differential db/b, b ∈ B. Ω_{F}^{[2]} depends on the choice of the 2-basis B. Then a well known result of Cartier (see [ Ca]) asserts that as an F^{2}- algebra we have

Z_{F} = B_{F} ⊕ Ω_{F}^{[2]} (1.2)

Moreover this decomposition is compatible with the graduation of Ω_{F}^{∗}. Any w ∈ Z_{F} can be written uniquely as

w = dη + ∑_{i_{1}<···<i_{n}} a_{i_{1}···i_{n}}^{2} \frac{db_{i_{1}}}{b_{i_{1}}} ∧ ⋯ ∧ \frac{db_{i_{n}}}{b_{i_{n}}}

so that we can define a homomorphism

C : Z_{F} → Ω_{F}^{∗} (1.3) by

C(w) = ∑_{i_{1}<···<i_{n}} a_{i_{1}···i_{n}} \frac{db_{i_{1}}}{b_{i_{1}}} ∧ ⋯ ∧ \frac{db_{i_{n}}}{b_{i_{n}}}

C is the well known Cartier-operator (see [ Ca]) and it is uniquely determined by the following properties

C(a^{2} \frac{db}{b}) = a\frac{db}{b} (1.4)

C(dη) = 0 for η ∈ Ω_{F}^{·} (1.5)

C(a^{2}w) = aC(w) , a ∈ F , w ∈ Z_{F} (1.6)

C(w ∧ λ) = C(w) ∧ C(λ) (1.7)

In particular C does not depend on the choice of the 2−basis and ker(C) = B_{F}, Im(C) = Ω_{F}^{∗}. Thus we get a ring isomorphism (compatible with the graduation)

C : Z_{F} /B_{F} →^{∼} Ω_{F}^{∗} (1.8)

For a fixed 2-basis B = {b_{i}, i ∈ I} of F we define the square operator

s : Ω_{F}^{∗} → Ω_{F}^{∗} (1.9)

by

s (∑_{σ} a_{σ} \frac{db_{σ}}{b_{σ}} ) = ∑_{σ} a_{σ}^{2} \frac{db_{σ}}{b_{σ}}

where σ runs over tuples of indices i_{1} < ⋯ < i_{p} and b_{σ} = b_{i_{1}} ⋯ b_{i_{p}}, db_{σ} = db_{i_{1}} ∧ ⋯ ∧ db_{i_{p}}. Of course s depends on the choice of B. We will write w^{[2]} instead of s(w) when we have a fixed 2-basis of F. Using (1.9) we also define the following operator

\begin{aligned}℘ : & Ω_{F}^{∗} → Ω_{F}^{∗} \\ & ℘(w) = w^{[2]} + w \end{aligned} (1.10)

Since s is additive, ℘ is additive too and depends on the choice of the 2-basis B. But any other choice of a 2-basis changes ℘(w) by an exact form, i.e. we get a well defined group-homomorphism

℘ : Ω_{F}^{∗} → Ω_{F}^{∗} /dΩ_{F}^{∗}

which for every n defines a homomorphism

℘ : Ω_{F}^{n} → Ω_{F}^{n} /dΩ_{F}^{n−1} (1.11)

Using (1.8) one easily checks that

℘ = C^{−1} − id

We now derive some useful properties of the Cartier operators, which will be used frequently in the next sections.

(1.12) Proposition. Let {b_{1}, …, b_{n}} be elements of a 2-basis B of F. Let λ ∈ Ω_{F}^{∗} be such that d(λ ∧ db_{1} ∧ ⋯ ∧ db_{n}) = 0. Then there is δ ∈ Ω_{F}^{∗} with

C(λ ∧ db_{1} ∧ ⋯ ∧ db_{n}) = δ ∧ db_{1} ∧ ⋯ ∧ db_{n}.

Proof. We apply induction with respect to n. Assume first n = 1, i.e. let u = λ ∧ db be a closed form and b ∈ B. From du = 0 we conclude from (1.2) that u = dn + m^{[2]} with some forms m, n. Write m = m_{0} + m_{1} ∧ \frac{db}{b} where m_{0}, m_{1} are forms which do not contain db in the basis representation with respect to B. We can also write n = n_{0} + bn_{1} + n_{2} ∧ db + bn_{3} ∧ db where n_{0}, n_{1}, n_{2}, n_{3} do not contain odd powers of b in their coefficients nor db in their 2-basis expansion. Thus

u = m_{0}^{[2]} + m_{1}^{[2]} ∧ \frac{db}{b} + dn_{0} + bdn_{1} + n_{1} ∧ db
+ dn_{2} ∧ db + bdn_{3} ∧ db
= λ ∧ db

Comparing terms with db we obtain

\begin{matrix}m_{0}^{[2]} + dn_{0} + bdn_{1} = 0 \\ u = λ ∧ db = m_{1}^{[2]} ∧ \frac{db}{b} + n_{1} ∧ db + dn_{2} ∧ db + bdn_{3} ∧ db \end{matrix}

The second equation and the assumption du = 0 imply d(n_{1} ∧ db) = dn_{1} ∧ db = 0.

But since n_{1} does not contain db as well b as odd power in its coefficients, we see that dn_{1} does not contain db in its basis expansion. Thus (1.1) implies dn_{1} = 0. Thus n_{1} ∧ db = d(bn_{1}), and since dn_{2} ∧ db = d(n_{2} ∧ db), bdn_{3} ∧ db = d(bn_{3} ∧ db) are exact, we conclude

u = m_{1}^{[2]} ∧ \frac{db}{b} + dv

with some form v. Applying the Cartier-operator to u we obtain

C(u) = m_{1} ∧ \frac{db}{b}

and δ = b^{−1}m_{1} does the job. Let us now assume the proposition for all integers less than n. Let u = λ∧db_{1}∧⋯∧db_{n} with du = 0 and b_{1}, …, b_{n} ∈ B. By induction

C(u) = δ ∧ db_{2} ∧ ⋯ ∧ db_{n} = μ ∧ db_{1} ∧ db_{3} ∧ ⋯ ∧ db_{n}

with some forms δ, μ. We can assume that δ is free from terms containing db_{2}, …, db_{n}, respectively μ is free from terms containing db_{1}, db_{3}, …, db_{n}, in their basis expansion. Set δ = δ_{0} + δ_{1} ∧ db_{1} where δ_{0}, δ_{1} are free from terms containing db_{1}. Then

\begin{aligned}δ_{0} ∧ db_{2} ∧ ⋯ ∧ db_{n} & + & δ_{1} ∧ db_{1} ∧ db_{2} ∧ ⋯ ∧ db_{n} \\ & = & μ ∧ db_{1} ∧ db_{3} ∧ ⋯ ∧ db_{n}. \end{aligned}

Since δ_{0} ∧ db_{2} ∧ ⋯ ∧ db_{n} is free from terms containing db_{1} we conclude δ_{0} ∧ db_{2} ∧ ⋯ ∧ db_{n} = 0 and C(u) = δ_{1} ∧ db_{1} ∧ db_{2} ∧ ⋯ ∧ db_{n}. This proves the claim. □

(1.13) Proposition. Let {b_{1}, …, b_{n}} be elements belonging to a 2-basis B of F. Let v ∈ Ω_{F} be such that

dv ∈ Ω_{F} ∧ db_{1} ∧ ⋯ ∧ db_{n}.

Then there exist forms z_{μ} ∈ Z_{F} and u ∈ Ω_{F} with

dv = (∑_{μ≠1} b^{μ}z_{μ} + b_{1} ⋯ b_{n}du) ∧ db_{1} ∧ ⋯ ∧ db_{n}

where μ runs over all functions {1, …, n} → {0, 1} distinct from 1 given by 1(i) = i for all 1 ≤ i ≤ n. Here we have set b^{μ} = ∏_{i=1}^{n} b_{μ(i)}. Moreover the forms z_{μ} and u can be chosen free from terms containing db_{1}, …, db_{n} as well as coefficients containing odd powers of b_{1}, …, b_{n} in their 2-basis expansion.

Proof. We show the claim by induction on n. Let us first assume n = 1, i.e. dv ∈ Ω_{F} ∧ db with b = b_{1} ∈ B. We write

v = v_{0} + bv_{1} + v_{2} ∧ db + bv_{3} ∧ db

where v_{0}, v_{1}, v_{2}, v_{3} are free from terms containing db and also do not contain odd powers of b in the expansion of their coefficients with respect to the 2-basis B. Then we have

\begin{aligned}dv & = & dv_{0} + bdv_{1} + v_{1} ∧ db + dv_{2} ∧ db + bdv_{3} ∧ db \\ & = & dv_{0} + bdv_{1} + (v_{1} + dv_{2} + bdv_{3}) ∧ db \end{aligned}

By the choice of v_{0}, v_{1} we see that dv_{0}, dv_{1} do not contain terms involving db in their basis expansion with respect to B. But by hypothesis dv ∈ Ω_{F} ∧db, so that we get

dv_{0} = 0, dv_{1} = 0

since dv_{0} and dv_{1} do not have coefficients containing odd powers of b in their 2-basis expansion. Hence

dv = (v_{1} + dv_{2} + bdv_{3}) ∧ db

is of the desired form.

Let us assume the assertion for any integer less than n. Then from dv ∈ Ω_{F} ∧ db_{1} ∧ ⋯ ∧ db_{n} we conclude

dv = ( ∑_{v∈S_{n−1},ν≠1} b^{ν}z_{ν} + b_{1} ⋯ b_{n−1}du) ∧ db_{1} ∧ ⋯ ∧ db_{n−1}

where S_{r} = {ν : {1, …, r} → {0, 1}} is the set of all functions {1, …, r} → {0, 1}. Here all z_{ν} are closed and the z_{ν}, u are free from b_{1}, …, b_{n−1}. We write

\begin{aligned}z_{ν} & = & c_{ν,0} + b_{n}c_{ν,1} + c_{ν,2} ∧ db_{n} + b_{n}c_{ν,3} ∧ db_{n} \\ u & = & u_{0} + b_{n}u_{1} + u_{2} ∧ db_{n} + b_{n}u_{3} ∧ db_{n} \end{aligned}

with c_{ν,i}, u_{i} free from b_{n} (i.e. do not contain db_{n} as well as odd powers of b_{n} in the 2-basis expansion of their coefficients). Since dz_{ν} = 0, we have

0 = dc_{ν,0} + b_{n}dc_{ν,1} + c_{ν,1} ∧ db_{n} + dc_{ν,2} ∧ db_{n} + b_{n}dc_{ν,2} ∧ db_{n}

and the choice of the c_{ν,i} is imply

\begin{matrix}dc_{ν,0} = 0, dc_{ν,1} = 0 \\ c_{ν,1} + dc_{ν,2} + b_{n}dc_{ν,3} = 0. \end{matrix}

This last equation implies

c_{ν,1} = dc_{ν,2}, dc_{ν,3} = 0.

Hence

dv = ( ∑_{ν∈S_{n−1},≠1} b^{ν}(c_{ν,0} + b_{n}dc_{ν,2}) + b1(du_{0} + b_{n}du_{1})) ∧ db_{1} ∧ ⋯ ∧ db_{n−1} +

+ ( ∑_{ν∈S_{n−1},≠1} b^{ν}(c_{ν,2} + b_{n}c_{ν,3}) + b1(u_{1} + du_{2} + b_{n}du_{3})) ∧ db_{1} ∧ ⋯ ∧ db_{n}

The expression in the first parenthesis does not contain db_{1}, …, db_{n} and since dv ∈ ⟨db_{1} ∧ ⋯ ∧ db_{n}⟩, we conclude (see (1.1))

∑_{v∈S_{n−1},≠1} b^{ν}(c_{ν,0} + b_{n}dc_{ν,2}) + b_{1} ⋯ b_{n−1}du_{0} + b_{1} ⋯ b_{n}du_{1} = 0

Since by hypothesis c_{ν,0}, c_{ν,2}, u_{0}, u_{1} do not contain coefficients with odd powers of b_{n} in their 2-basis expansion, we conclude

∑_{ν∈S_{n−1},≠1} b^{ν}dc_{ν,2} + b_{1} ⋯ b_{n−1}du_{1} = 0

By induction all c_{ν,2}, u_{1} are free from b_{1}, …, b_{n−1}. Then we obtain dc_{ν,2} = 0, du_{1} = 0. Since

\begin{aligned}dv & = & (∑_{ν∈S_{n−1},≠1} b^{ν}c_{ν,2} + ∑_{ν∈S_{n−1},≠1} b_{n}b^{ν}c_{ν,3} \\ & & + b1(u_{1} + du_{2}) + b1b_{n}du_{3}) ∧ db_{1} ∧ ⋯ ∧ db_{n} \end{aligned}

is of desired form, this concludes the proof of the lemma. □

(1.14) Remark. Under the hypothesis of the lemma above, one can give a more precise description of the form v. By the lemma we have

dv = ( ∑_{μ∈S_{n},≠1} b^{μ}z_{μ} + b1du) ∧ db_{1} ∧ ⋯ ∧ db_{n}

where b1 = b_{1} ⋯ b_{n}, dz_{μ} = 0 for all μ ≠ 1. Let μ be such a function and let 1 ≤ i ≤ n with μ(i) = 0. Then

\begin{matrix}d(b_{i}b^{μ}z_{μ} ∧ db_{1} ∧ ⋯ ∧ db_{i−1} ∧ db_{n}) \\ = b^{μ}z_{μ} ∧ db_{1} ∧ ⋯ ∧ db_{n} + b_{i}db^{μ} ∧ z_{μ} ∧ db_{1} ∧ ⋯ ∧ db_{i−1} ∧ db_{i+1} ∧ ⋯ ∧ db_{n} \end{matrix}

Since b^{μ} contains only b_{j} with j ≠ i, we see that the last term is 0 and hence

d(b_{i}b^{μ}z_{μ} ∧ db_{1} ∧ ⋯ ∧ db_{i} ∧ ⋯ ∧ db_{n}) = b^{μ}z_{μ} ∧ db_{1} ∧ ⋯ ∧ db_{n}.

Also b1du ∧ db_{1} ∧ ⋯ ∧ db_{n} = d(b1u ∧ db_{1} ∧ ⋯ ∧ db_{n}) and we conclude

dv = d (∑ \begin{matrix}μ∈S_{n},≠1 \\ μ(i)=0 \end{matrix}b_{i}b^{μ}z_{μ} ∧ db_{1} ∧ ⋯ ∧ d̂b_{i} ∧ ⋯ ∧ db_{n} + b1u ∧ db_{1} ∧ ⋯ ∧ db_{n})

i.e.

v = ∑ \begin{matrix}μ∈S_{n},≠1 \\ μ(i)=0 \end{matrix}b_{i}b^{μ}z_{μ} ∧ db_{1} ∧ ⋯ ∧ d̂b_{i} ∧ ⋯ ∧ db_{n} + b1u ∧ db_{1} ∧ ⋯ ∧ db_{n} + z (1.15)

with z ∈ Z_{F} a closed form.

The next result characterizes divisibility by pure forms and will be useful in the next sections. We say that a form ω ∈ Ω_{F} divides the form λ if λ = η ∧ ω with some form η ∈ Ω_{F}. Then we have

(1.16) Proposition. Let b_{1}, …, b_{n} be elements of F contained in a 2-basis B of F. Let w ∈ Ω_{F}^{m} be a m-form. If db_{1}, …, db_{n} divide w, then db_{1} ∧ ⋯ ∧ db_{n} divides w.

Proof. Let B = {b_{1}, …, b_{n}, … } be the given 2-basis. We show the claim by induction on n. For n = 1 the claim is obvious. Assume the proposition for n − 1. Then we have w = η ∧ db_{1} ∧ ⋯ ∧ db_{n−1}. Write η = η_{0} + η_{1} ∧ db_{n} with forms η_{0}, η_{1} which do not contain db_{n} in their basis expansion. Then w = η_{0} ∧ db_{1} ∧ ⋯ ∧ db_{n−1} + η_{1} ∧ db_{1} ∧ ⋯ ∧ db_{n} is divisible by db_{n} by hypothesis. Since η_{0} ∧ db_{1} ∧ ⋯ ∧ db_{n−1} does not contains db_{n}, we conclude η_{0} ∧ db_{1} ∧ ⋯ ∧ db_{n−1} = 0 and w = η_{1} ∧ db_{1} ∧ ⋯ ∧ db_{n}. □

(1.17) Remark. Let us consider a rational function field L = F (X_{μ}; μ ∈ A)) with A finite. Let B = {b_{i}|i ∈ I} be a 2-basis of F. Then B ∪ {X_{μ}, μ ∈ A} is a 2-basis of L. Let B ⊆ A be a subset and N = F (X_{μ}, μ ∈ B), respectively M = F (X_{μ}^{2}, μ ∈ B) ⊂ N. Let X be any variable X_{μ} with μ ∉ B. We will be later interested in forms contained in Ω_{L} which are generated over M by the differentials db, b ∈ F. Thus we will define Ω_{F} M = Ω_{F} ⊗ M ⊂ Ω_{L}, respectively Ω_{F} M[X^{2}] = Ω_{F} ⊗ M[X^{2}] ⊂ Ω_{L}. This last set is the M[X^{2}]- submodule

⨁_{i_{1}<···<i_{m}} M[X^{2}]db_{i_{1}} ∧ ⋯ ∧ db_{i_{m}}

(for some ordering < in I) of Ω_{L}. For every p(X) ∈ N[X] irreducible and monic we set for every n ≥ 0

(1.18) p^{−∞}Ω_{F}^{n} M[X^{2}] = \{\frac{w}{p^{s}} |w ∈ Ω_{F}^{n} M[X^{2}], s ≥ 1, deg_{X} w ≤ s deg_{X} p\}

if p ∈ M[X^{2}], and

(1.19) p^{−∞}Ω_{F}^{n} M[X^{2}] = \{\frac{w}{p^{2s}} |w ∈ Ω_{F}^{n} M[X^{2}], s ≥ 1, deg_{X} w ≤ 2s deg_{X} p\}

if p ∉ M[X^{2}].

Here we have set deg_{X} w = 2t whenever we have w = w_{0} + w_{1}X^{2} + ⋯ + w_{t}X^{2t} with w_{0}, …, w_{t} ∈ Ω_{F}^{n}, w_{t} ≠ 0 for some w ∈ Ω_{F}^{n} M[X^{2}].

Now we have

(1.20) Lemma. The sum

Ω_{F}^{n} M[X^{2}] + ∑_{p} p^{−∞}Ω_{F}^{n} M[X^{2}] ⊆ Ω_{L}^{n}

is direct. Here p runs over all irreducible polynomials contained in N[X].

Proof. Let us assume

u_{0} + ∑_{p} \frac{u_{p}}{p^{s_{p}}} = 0

in Ω_{L}, with u_{0}, u_{p} ∈ Ω_{F} M[X^{2}], deg_{X} u_{p} < s_{p} deg_{X} p for all p. Thus

∏_{p} p^{s_{p}}u_{o} + ∑_{p} (∏_{q≠p} q^{s_{q}}) u_{p} = 0

holds in Ω_{F} M[X^{2}]. Recall that s_{p} is even if p ∉ M[X^{2}], and hence p^{s_{p}} is contained always in M[X^{2}].

We fix now some p_{0} and we get in Ω_{F} M[X^{2}]

(∏_{p} p^{s_{p}}) u_{0} = ( ∏_{q≠p_{0}} q^{s_{q}}) u_{p_{0}} + ∑_{p≠p_{0}} (∏_{q≠p} q^{s_{q}}) u_{p}.

This implies that p_{0}^{s_{p_{0}}} divides the term (∏_{q≠p_{0}} q^{s_{q}}) u_{p_{0}}, and since Ω_{F} M[X^{2}]

is a free module, p_{0}^{s_{p_{0}}} divides u_{p_{0}} in Ω_{F} M[X^{2}]. Since deg_{X} u_{p_{0}} < s_{p_{0}} deg_{X} p_{0}, it follows u_{p_{0}} = 0. This proves the lemma. □

The relevant point in the above decomposition is that the operators ℘ and d respect this decompositions, whenever ℘ is defined with respect to a 2-basis containing a 2-basis of F. Thus we have:

dΩ_{F} M[X^{2}] ⊆ Ω_{F} M[X^{2}]
℘Ω_{F} M[X^{2}] ⊆ Ω_{F} M[X^{2}]
d(p^{−∞}Ω_{F} M[X^{2}]) ⊆ p^{−∞}Ω_{F} M[X^{2}]
℘(p^{−∞}Ω_{F} M[X^{2}]) ⊆ p^{−∞}Ω_{F} M[X^{2}]

(1.21)

If we do not specify a particular 2-basis, the relations above must be understood as follows. For any field F the maps Ω_{F} → Ω_{F} given by w ↦ w^{[2]} depends on the choice of a 2-basis. If we choose another 2-basis of F and we denote by w^{(2)} the same operation with respect to this new 2-basis, we have

w^{[2]} = w^{(2)} + dv

with some v ∈ Ω_{F}. Therefore expressions of the form w^{[2]} + dv, which will often occur in the sequel, make sense if the particular choice of the form v does not matter. Thus, we will not make sometimes an explicit choice of a 2-basis when dealing with such expressions. Of course the same remark applies to expressions of the form ℘w + dv. For example if u ∈ Ω_{F} and f ∈ F^{∗}, then for any choice of a 2-basis, the forms ℘(u ∧ \frac{df}{f}) and ℘(u) ∧ \frac{df}{f} differ by an exact form.

In the concrete situation above, if we choose u ∈ Ω_{F} M[X^{2}] and f ∈ M[X^{2}], then there is some v ∈ Ω_{F} M[X^{2}] with

℘ (u ∧ \frac{df}{f} ) = ℘(u) ∧ \frac{df}{f} + \frac{dv}{f^{2}}

2 The behavior of differential forms under some field extensions

We continue in this section the algebraic study of differential forms by considering their behavior under field extension. Any field extension F ↪ L induces a natural homomorphism Ω_{F}^{n} → Ω_{L}^{n} for all n ≥ 0. We will denote by Ω^{n}(L/F) the kernel of this homomorphism. It is clear that d(Ω^{n}(L/F)) is contained in Ω^{n+1}(L/F). Fixing any 2-basis of F we see that the same holds true for the operator ℘ and s(= square). We are particularly interested in the following field extension of F. Let ϕ =≪ b_{1}, …, b_{n} ≫= ⟨1, b_{1}⟩ ⊗ ⋯ ⊗ ⟨1, b_{n}⟩ be an anisotropic bilinear Pfister form (see [ Ba 1], [ A-Ba 1]). The fact that ϕ is anisotropic means that {b_{1}, …, b_{n}} are algebraically independent over F^{2} and hence can be chosen as part of a 2-basis of F. This will be always assumed in the sequel. The function field F (ϕ) of the quadric ϕ = 0 is constructed as follows. Let S_{n} be the set of all maps μ : {1, …, n} → {0, 1} with μ(i) = 1 for at least one index i and choose a variable X_{μ} for each μ ∈ S_{n}. Let L = F (X_{μ}, μ ∈ S_{n}) the rational function field over F in the variables X_{μ} and set T = ∑_{μ∈S_{n}} b^{μ}X_{μ}^{2}, where b^{μ} = ∏_{i=1}^{n} b_{i}^{μ(i)} (T is the so called pure part of ϕ). Then the field

F (ϕ) = L(\sqrt{T}) (2.1)

is the function field of ϕ. Obviously ϕ ⊗ F (ϕ) is isotropic, although it is not necessarily hyperbolic, but metabolic. We are interested in the behavior of Ω^{m} under the field extension F ↪ F (ϕ). To this end we will compute Ω^{m}(L/F), Ω^{m}(F (ϕ)/L) and finally Ω^{m}(F (ϕ)/F). We will denote by K the field F (ϕ) in what follows.

(2.2) Lemma. Let F (X)/F be a pure transcendental extension of F. Then Ω^{m}(F (X)/F) = 0.

Proof. We may assume that F (X)/F has transcendence degree one. Let B = {b_{i}, i ∈ I} be a 2-basis of F. Then B ∪ {X} is a 2-basis of F (X). If

w = ∑ c_{i_{1},... ,i_{m}} \frac{db_{i_{1}}}{b_{i_{m}}} ∧ ⋯ ∧ \frac{db_{i_{m}}}{b_{i_{m}}}

is in Ω^{m}(F (X)/F), we have w = 0 in Ω_{F (X)}^{m}. But since

\{\frac{db_{i_{1}}}{b_{i_{1}}} ∧ ⋯ ∧ \frac{db_{i_{m}}}{b_{i_{m}}} , i_{1} < ⋯ < i_{m}\} .

is part of a F (X)-basis of Ω_{F (X)}^{m}, we conclude c_{i_{1}···i_{m}} = 0 for all i_{1} < ⋯ < i_{m}, i.e. w = 0 in Ω_{L}^{m}. □

(2.3) Remark. If B is a 2-basis of F, then B ∪{X_{μ}, μ ∈ S_{n}} is a 2-basis of L.

(2.4) Lemma. Let k be any field of characteristic 2 and t ∈ k \ k^{2}. Then for m ≥ 1

Ω^{m}(k(\sqrt{t})/k) = Ω_{k}^{m−1} ∧ dt.

Proof. Since t = (\sqrt{t})^{2} we have dt = 0 in Ω_{k(\sqrt{t})}, and therefore Ω_{k}^{m−1} ∧ dt ⊆ Ω^{m}(k(\sqrt{t})/k). Since t ∉ k^{2}, we can choose a 2-basis of k, say B, such that t ∈ B, i.e. B = \{t, c_{j}, j ∈ J\}. Then \{\sqrt{t}, c_{j}, j ∈ J\} is a 2-basis of k(\sqrt{t}). Let w ∈ Ω^{m}(k(\sqrt{t})/k). Then we have

w = ∑_{σ} a_{σ}dc_{σ} + (∑ b_{τ}dc_{τ}) ∧ dt

where σ runs over all maps {1, …, m} → J and τ over all maps {1, …, m− 1} → J which are monotone (i.e. i < j implies σ(i) < σ(j) in some ordering < of J). Moreover dc_{σ} means dc_{σ(1)} ∧ ⋯ ∧ dc_{σ(m)}, etc. and a_{σ}, b_{τ} ∈ k. Then in Ω_{k(\sqrt{t})}^{m} we get ∑_{σ} a_{σ}dc_{σ} = 0. Since all c_{i}, i ∈ J are part of a 2-basis of k(\sqrt{t}) we conclude a_{σ} = 0 for all σ, i.e. w = (∑ b_{τ}dc_{τ}) ∧ dt. This proves the claim. □

In particular we obtain

(2.5) Corollary. Ω^{m}(K/L) = Ω_{L}^{m−1} ∧ dT.

(2.6) Remark. In the case of a quadratic separable extension of k, say E = k +kz with z^{2} +z = a, we may assume that a is a square in k and hence dz = 0. In this case z is a square in E and a 2-basis of k remains a 2-basis of E. Therefore the argument in the proof of (2.4) shows Ω^{m}(E/k) = {0}.

(2.7) Lemma. For all m ≥ n

Ω^{m}(F (ϕ)/F) = Ω_{F}^{m−n} ∧ db_{1} ∧ ⋯ ∧ db_{n}

Otherwise Ω^{m}(F (ϕ)/F) = 0.

Proof. We choose a 2-basis of F containing {b_{1}, …, b_{n}}. Let us take w in Ω^{m}(F (ϕ)/F), i.e. w ∈ Ω_{F}^{m} with w = 0 in Ω_{F (ϕ)}^{m}. From (2.4) we infer that w = u ∧ dT with some u ∈ Ω_{L}^{m−1}. Now dT = k_{1}db_{1} + ⋯ + k_{n}db_{m} with some polynomials k_{1}, …, k_{n} (see (2.9) below). Thus replacing db_{1} in u we see that one can assume that u does not contain db_{1} in its basis expansion with respect to the 2-basis B ∪ {X_{μ}, μ ∈ S_{n}}. Let us write in Ω_{F}^{m}

w = w_{0} + w_{1} ∧ db_{1}

with forms w_{0}, w_{1} not containing db_{1}. Then in Ω_{L}^{m}

\begin{aligned}w_{0} + w_{1} ∧ db_{1} & = & u ∧ k_{1}db_{1} + u ∧ (k_{2}db_{2} + ⋯ + k_{n}db_{n}) \\ (w_{1} + k_{1}u) ∧ db_{1} & = & w_{0} + u ∧ (k_{2}db_{2} + ⋯ + k_{n}db_{n}) \end{aligned}

Since b_{1} is part of the 2-basis of L and the right hand side of this equation does not contain db_{1}, we obtain (w_{1} + k_{1}u) ∧ db_{1} = 0 in Ω_{L}^{m}. But w_{1} + k_{1}u also does not contain db_{1}, so that w_{1} + k_{1}u = 0. Thus u = k_{1}^{−1}w_{1}, and w = k_{1}^{−1}w_{1} ∧ dT. We get

k_{1}w = w_{1} ∧ dT (2.8)

which is a relation between forms all of whose coefficients are polynomials. Now on such differential forms we can specialize the values of the variables X_{μ}, obtaining forms defined over F. Letting X_{μ} → 0 for all μ ≠ 1 and X_{1} → 1 and using k_{1} → 1, k_{i} → 0 for i ≥ 2, we get from (2.8) in Ω_{F}^{m}

w = w̄_{1} ∧ db_{1}

with some form w̄_{1}. Thus db_{1}, …, db_{n} divide the form w . The lemma follows from (1.16). □

(2.9) Remark. The differential of the polynomial T is

dT = k_{1}db_{1} + ⋯ + k_{n}db_{n}

where k_{1}, …, k_{n} are polynomials in F [X^{2}] given by

k_{i} = ∑_{μ∈S_{n},μ(i)=1} b_{i}^{−1}b^{μ}X_{μ}^{2}, 1 ≤ i ≤ n.

Let us recall that the Cartier-operator defines an isomorphism C : Z_{F}^{n}/B_{F}^{n} → Ω_{F}^{n} where Z_{F}^{n} are the closed forms in Ω_{F}^{n} and B_{F}^{n} = dΩ_{F}^{n−1} are the exact forms. Following Kato (see [ Ka 1] and [ Mi]) we introduce the operator

℘ : Ω_{F}^{n} → Ω_{F}^{n} /dΩ_{F}^{n−1} (2.10)

given by

℘(w) = C^{−1}(w) − \overline{w}

i.e. ℘ = C^{−1} − \overline{id}.

For example

℘ (a\frac{db_{1}}{b_{1}} ∧ ⋯ ∧ \frac{db_{n}}{b_{n}} ) = \overline{(a^{2} − a)\frac{db_{1}}{b_{1}} ∧ ⋯ ∧ \frac{db_{n}}{b_{n}}} (2.11)

In order to complete (2.10) to an exact sequence we introduce the groups (see [ Ka 1])

ν_{F} (n) = ker(℘) (2.12)

H^{n+1}(F) = Coker (℘) (2.13)

and we have the exact sequence

0 → ν_{F} (n) → Ω_{F}^{n} → Ω_{F}^{n} /dΩ_{F}^{n−1} → H^{n+1}(F) → 0. (2.14)

Taking a 2-basis B = {b_{i}, i ∈ I} of F we get the square-operator s : Ω_{F}^{n} → Ω_{F}^{n} given by s : (∑_{σ} a_{σ} \frac{db_{σ}}{b_{σ}} ) = ∑_{σ} a_{σ}^{2} \frac{db_{σ}}{b_{σ}} (see section 1). We define the basis-dependent operator ℘ : Ω_{F}^{n} → Ω_{F}^{n} by ℘(w) = s(w) − w and easily check that

H^{n+1}(F) = Ω_{F}^{n} /(dΩ_{F}^{n−1} + ℘Ω_{F}^{n}).

Another choice of a 2-basis would change ℘(w) by an exact form. Thus dΩ_{F}^{n−1} + ℘Ω_{F}^{n} is basis independent.

The next two propositions are due to Kato (see [ Ka 1]) and will be used continuously in what follows.

Let {b_{i}}_{i∈I} be a 2-basis of F and endow I with the structure of a totally ordered set. For j ∈ I let F_{j} (resp. F_{<j} ) be the subfield of F generated over F^{2} by the elements b_{i} with i ≤ j (resp. i < j). For fixed n let ∑_{n} be the set of all functions α : {1, …, n} → I with α(i) < α(j) whenever 1 ≤ i < j ≤ n. We endow ∑_{n} with the lexicographic ordering, namely α < β ( α, β ∈ ∑_{n}) if and only if there exists some i such that α(i) < β(i) and α(j) ≤ β(j) for all j ≤ i. The F -vector space Ω_{F}^{n} has the basis {db_{α(1)} ∧ ⋯ ∧ db_{α(n)}, α ∈ ∑_{n}} and we can introduce in Ω_{F}^{n} the following filtration: for α ∈ ∑_{n} let Ω_{F,α}^{n}, (resp.Ω_{F,<α}^{n}), be the subspace of Ω_{F}^{n} generated by the elements db_{β(1)} ∧ ⋯ ∧ db_{β(n)} with β ≤ α, (resp. β < α). Using this notation we formulate the following basic result due to Kato (see also [ Ka 2]).

(2.15) Lemma. Let y ∈ F, α ∈ ∑_{n} and

w_{α} = \frac{db_{α(1)}}{b_{α(1)}} ∧ ⋯ ∧ \frac{db_{α(n)}}{b_{α(n)}} ∈ Ω_{F}^{n}

be such that

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

Then

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

for some v ∈ Ω_{F,<α}^{n} and some a_{i} ∈ F_{α(i)}^{∗}, 1 ≤ i ≤ n.

We will refer to this result in the sequel as Kato’s lemma. An immediate consequence of (2.15) is

(2.16) Corollary.

ν_{F} (n) = \{∑_{i} \frac{da_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{da_{i_{n}}}{a_{i_{n}}} |a_{i_{k}} ∈ F^{∗}\} .

We will now study the behavior of H^{n+1} under field extensions. If F ↪ L is a field extension we get the obvious maps ν_{F} (n) → ν_{L}(n) and H^{n+1}(F) → H^{n+1}(L). We write H^{n+1}(L/F) for the kernel of H^{n+1}(F) → H^{n+1}(L). The main goal of this paper is the computation of H^{n+1}(F (ϕ)/F) for an anisotropic bilinear n-fold Pfister form ϕ defined over F. This will be done in section 4. We will now consider only some easier field extensions.

(2.17) Lemma. Let F (X)/F be a pure transcendental extension. Then

H^{n+1}(F (X)/F) = 0

Proof. Let B = {b_{i}, i ∈ I} be a fixed 2-basis of F. We may assume that F (X)/F has transcendence degree one. Then B ∪ {X} is a 2-basis of F (X). We fix an ordering in I (and hence in B) and we choose the ordering of B ∪ {X} with X > b_{i} for all i ∈ I. Let w̄ ∈ H^{n+1}(F) with w̄ = 0 in H^{n+1}(F (X)). Thus in Ω_{F (X)}^{n} we have

w = ℘u + dv

for some u ∈ Ω_{F (X)}^{n} and v ∈ Ω_{F (X)}^{n−1}. Here ℘ is defined with respect to the 2-basis B ∪ {X}. Hence ℘u = w + dv with w ∈ Ω_{F}^{n}. Let α ∈ Σ_{n,F} ⊂ Σ_{n,F (X)} be the leading index of w, that is w = ∑_{γ∈Σ_{n,F}} w_{γ} \frac{db_{γ}}{b_{γ}} with w_{α} ≠ 0 and w_{γ} = 0 for all γ > α, and β ∈ Σ_{n,F (X)} the leading index of u. If β > α, since ℘(u_{β})\frac{db_{γ}}{b_{γ}} ∈ dΩ_{F (X)}^{n−1} + Ω_{F (X),<β}^{n}, we apply Kato’s lemma and conclude that u_{β} \frac{db_{γ}}{b_{γ}} = u′ + \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}}, with a_{i} ∈ F_{β(i)} and u′ ∈ Ω_{F (X),<β}^{n}. By this way we can replace u by a differential form with lower leading index. This means that we may assume β ≤ α. Then we have

(℘(u_{α}) + w_{α}) \frac{db_{α}}{b_{α}} = d(v) mod Ω_{F (X),<α}^{n}

with v ∈ Ω_{F (X)}^{n−1}. Since b_{α(i)} < X for all i ∈ I, we conclude that we may assume that the coefficients of d(v) are in F (X^{2}). Because if we write v = v_{0} + Xv_{1} + v_{2} ∧ dX + Xv_{3} ∧ dX where v_{0}, v_{1}, v_{2}, v_{3} are generated over F (X^{2}) by the differentials db_{i}, i ∈ I, we get

dv = dv_{0} + Xdv_{1} + (v_{1} + dv_{2} + Xdv_{3}) ∧ dX

and hence v_{1} + dv_{2} + Xdv_{3} = 0. By the choice of the forms v_{1}, v_{2}, v_{3} we infer from the last relation that v_{1} + dv_{2} = 0 and dv_{3} = 0. Therefore inserting v_{1} = dv_{2} into v we obtain

v = v_{0} + d(Xv_{2}) + Xd(Xv_{3})

and since d(Xd(Xv_{3})) = dX ∧ d(Xv_{3}) = 0, it follows dv = dv_{0}. Thus we may replace v by v_{0} and we can assume that v is generated over F (X^{2}) by the differentials db_{i}, i ∈ I. Then u is also generated over F (X^{2}) by the differentials db_{i}, i ∈ I.

Therefore the relation ℘(u) + w = d(v) holds in the subspace

Ω_{F}^{n} [X^{2}] ⊕ ⨁_{p} p^{−∞}Ω_{F}^{n} [X^{2}] of Ω_{F (X)}^{n},

where p runs over all irreducible polynomials in F [X^{2}]. Let us write

u_{α} = u_{α,0} + ∑_{p} u_{α,p} , v = v_{0} + ∑_{p} v_{p} ,

with u_{α,0} ∈ F [X^{2}], v_{0} ∈ Ω_{F}^{n−1}[X^{2}], resp. u_{p} ∈ p^{−∞}F [X^{2}], v_{p} ∈ p^{∞}Ω_{F}^{n−1}[X^{2}]. Then

w_{α} \frac{db_{α}}{b_{α}} = ℘u_{α,0} \frac{db_{α}}{b_{α}} + dv_{0} + ∑_{p} (℘u_{α,p} \frac{db_{α}}{b_{α}} + dv_{p}) mod Ω_{F (X),<α}^{n}

and since ℘, d are compatible with the above direct sum, we see that

w_{α} \frac{db_{α}}{b_{α}} = ℘u_{α,0} \frac{db_{α}}{b_{α}} + dv_{0} mod Ω_{F,<α}^{n}[X^{2}]

holds in Ω_{F}^{n} [X^{2}]. We write now u_{α,0} = \overline{u_{α,0}} + X^{2s}u_{s}, v_{0} = \overline{v_{0}} + X^{2r}v_{r} with u_{s} ∈ F, v_{r} ∈ Ω_{F}^{n−1} and 2s = deg(u_{α,0}), 2r = deg(v_{0}), deg(\overline{u_{α,0}}) < 2s, deg(\overline{v_{0}}) < 2r. Thus

w_{α} \frac{db_{α}}{b_{α}} = ℘\overline{u_{α,0}} \frac{db_{α}}{b_{α}} + d\overline{v_{0}} + [X^{4s}u_{s}^{[2]} + X^{2s}u_{s}]\frac{db_{α}}{b_{α}} + X^{2r}dv_{r} mod Ω_{F,<α}^{n}[X^{2}].

Let us first assume 4s > 2r. Then comparing coefficients we get u_{s}^{[2]} \frac{db_{α}}{b_{α}} ∈ Ω_{F,<α}^{n} and applying the Cartier-operator it follows u_{s} = 0, which is a contradiction. Similarly if 2r > 4s, then dv_{r} ∈ Ω_{F,<α}^{n}, including X^{2r}dv_{r} in Ω_{F,<α}^{n}, we conclude that we can lower the degree of v_{0}. Thus we are lead to the case 4s = 2r. If 4s = 2r > 0, then we get u_{s}^{[2]} \frac{db_{α}}{b_{α}} = dv_{r} mod Ω_{F,<α}^{n} and applying the Cartier operator we obtain u_{s} \frac{db_{α}}{b_{α}} ∈ Ω_{F,<α}^{n}, that is u_{s} \frac{db_{α}}{b_{α}} = 0, which is a contradiction. Thus we have s = r = 0, i.e. w_{α} \frac{db_{α}}{b_{α}} = ℘u_{0} \frac{db_{α}}{b_{α}} + dv_{0} mod Ω_{F,<α}^{n} with u_{0} ∈ F and v_{0} ∈ Ω_{F}. Replacing in w, this see \overline{w} can be represented by a differential form with lower leading index. This shows \overline{w} = 0 in H^{n+1}(F), and concludes the proof of the lemma. □

In particular we have H^{m+1}(L/F) = (0) for all m, where L is the field extension F (X_{μ}, μ ∈ S_{n}) introduced at the beginning of this section.

We want now to compute H^{m+1}(F (ϕ)/L), where F (ϕ) = L(\sqrt{T}) (see (2.1)). To this end we prove the following general fact.

(2.18) Lemma. Let k be field of characteristic 2, and b ∈ k \ k^{2}. Then

H^{m+1}(k(\sqrt{b})/k) = \overline{Ω_{k}^{m−1} ∧ db}

Proof. Since b ∈ k \ k^{2} we can take b part of a 2-basis of k. Let B = \{b_{1} = b, b_{2}, … \} be a 2-basis of k. Then B′ = \{\sqrt{b}, b_{2}, … \} is a 2-basis of k(\sqrt{b}). Take now w̄ ∈ H^{m+1}(k(\sqrt{b})/k), i.e. w̄ ∈ H^{m+1}(k) with w̄ = 0 in H^{m+1}(k(\sqrt{b})). This means

w = ℘u + dv

with u ∈ Ω_{k(\sqrt{b})}^{m}, v ∈ Ω_{k(\sqrt{b})}^{m−1}. We order the 2-basis B′ with \sqrt{b} > b_{i} for all i (≠ 1). Let α ∈ Σ_{m,k}, α(i) > 1 for all i = 1, …, m, be the leading index of w and β ∈ Σ_{m,k(\sqrt{b})} the leading index of u. Thus we can assume, as in the proof of (2.17), β ≤ α. In this case we have

(℘(u_{α}) + w_{α}) \frac{db_{α}}{b_{α}} = d(v) mod Ω_{k(\sqrt{b}),<α}^{n}

with v ∈ Ω_{k(\sqrt{b})}^{n−1}. Since b_{α(i)} < \sqrt{b} for all i, we conclude that the leading coefficient of d(v) is in k, then u_{α} is also in k and we may assume that v ∈ Ω_{k}^{m−1}.

Since ker[Ω_{k}^{m} → Ω_{k(\sqrt{b})}^{m}] = Ω_{k}^{m−1} ∧ d(b), we conclude that

w_{α} \frac{db_{α}}{b_{α}} = ℘(u_{α})\frac{db_{α}}{b_{α}} + d(v) mod Ω_{k,<α}^{n} + Ω_{k}^{m−1} ∧ d(b)

in Ω_{k}^{m}. Replacing in w, this shows \overline{w} can be represented by a differential form with lower leading index. This shows \overline{w} ∈ \overline{Ω_{k}^{m−1} ∧ db} in H^{n+1}(k), and concludes the proof of the lemma. □

(2.19) Corollary.

H^{m+1}(F (ϕ)/L) = \overline{Ω_{L}^{m−1} ∧ dT}

Let us close this section with some remarks concerning the computation of H^{n+1}(F (ϕ)/F). We write y = \sqrt{T} , so that K = F (ϕ) = L[y]. Fix a 2-basis B = {b_{i}, i ∈ I} of F, so that B ∪ {X_{μ}, μ ∈ S_{n}} is a 2-basis of L. We order the elements of B according to an order of I and the elements of {X_{μ}, μ ∈ S_{n}} for example using the lexicographic ordering and we set B < {X_{μ}, μ ∈ S_{n}} for an ordering in B ∪ {X_{μ}, μ ∈ S_{n}}. Since y^{2} = T = ∑_{μ} b^{μ}X_{μ}^{2} in K, we see that the elements of B are not 2-independent over K^{2}. Let us fix some b = b_{1} ∈ B. Then B \ {b_{1}} ∪ {X_{μ}, μ ∈ S_{n}} ∪ {y} is a 2-basis of K. We order this basis such that y is the maximal element. In particular we have the operator ℘ on Ω_{K} defined with respect to this 2-basis. Take now w̄ ∈ H^{n+1}(K/F). Then

w = ℘u + dv (2.20)

with u ∈ Ω_{K}^{n} and v ∈ Ω_{K}^{n−1}. From ℘u = w + dv, and using Kato’s lemma with a filtration defined by the above ordering, we see that one can assume in (2.20) that u and v are differential forms generated over K by the differential db_{i}, i ∈ I (i ≠ 1), i.e. they do not contain differentials of the type dX_{μ} or dy. Looking at the 2-basis expansion of both sides of (2.20) we easily conclude that u and v do not contain y in their coefficients, i.e. they are contained in L. Therefore the forms u, v (and w) are defined over L and are generated over L by the differentials db_{i}, i ∈ I. From (2.4) we conclude that w + ℘u + dv ∈ Ω^{n}(K/L) ⊂ Ω_{L}^{n} as a form in Ω_{L}^{n}, and using (2.4) we obtain in Ω_{L}^{n}

w = ℘u + dv + λ ∧ dT (2.21)

with λ ∈ Ω_{L}^{n−1}. We will show below that the coefficients of u, v, λ in the 2-basis expansion of L do not contain odd powers of the variables X_{μ} and that λ also is generated by the differentials db_{i}, i ∈ I. Therefore if we define the subfield of L

M = F (X_{μ}^{2}, μ ∈ S_{n}) ⊂ L

we see that all forms u, v, λ, dT and w are generated over M by the differentials db_{i}, i ∈ I, i.e. they are contained in the subspace Ω_{F} ⊗ M of Ω_{L}. We show now this assertion. The fact that λ does not contain differentials dX_{μ}, μ ∈ S_{n} follows from (2.21) and the fact that all other forms, including dT, do not contain such forms. Let us write u = u_{0} + Xu_{1} , v = v_{0} + Xv_{1}, λ = λ_{0} + Xλ_{1}, for some fixed X = X_{μ}, and where u_{i}, v_{i}, λ_{i} (i = 0, 1) do not contain odd powers of X in their coefficients, with respect to their 2-basis representation. Then in Ω_{L}^{n}

w = ℘u_{0} + X^{2}u_{1}^{[2]} + Xu_{1} + dv_{0} + Xdv_{1} + v_{1} ∧ dX + λ_{0} ∧ dT + Xλ_{1} ∧ dT

and this implies

w = ℘u_{0} + X^{2}u_{1}^{[2]} + dv_{0} + λ_{0} ∧ dT (2.22)

u_{1} + dv_{1} + λ_{1} ∧ dT = 0 (2.23)

v_{1} ∧ dX = 0. (2.24)

Since v_{1} does not contain dX, we obtain from (2.24) v_{1} = 0 and therefore u_{1} = λ_{1} ∧ dT. Inserting this in (2.22) it follows

w = ℘u_{0} + dv_{0} + (λ_{0} + X^{2}T λ_{1}^{[2]}) ∧ dT,

since from u_{1} = T λ_{1} ∧ \frac{dT}{T} we get u_{1}^{[2]} = T^{2}λ_{1}^{2} ∧ \frac{dT}{T} = T λ_{1}^{[2]} ∧ dT. The form λ_{0} + X^{2}T λ_{1}^{[2]} does not contain odd powers of X , and the above equation therefore shows that we can eliminate all odd powers of X from u, v and λ. Since X = X_{μ} was arbitrary, this proves the claim. Therefore we have shown

(2.25) Proposition. Let w̄ ∈ H^{n+1}(K/F) and fix a 2-basis B = {b_{i}, i ∈ I} of F. Then there exist forms u, v, λ in Ω_{F} ⊗ M ⊂ Ω_{L}, i.e. generated over M = F (X_{μ}^{2}, μ ∈ S_{n}) by the differentials db_{i}, i ∈ I with

w = ℘u + dv + λ ∧ dT

where ℘ is defined with respect to any 2-basis of L containing the basis B

The next two sections will be devoted to the study of this equation.

(2.26) Remark. The behavior of ν_{F} (m) under field extensions is easier to handle than that of H^{m+1}(F). For example (2.7) implies Ω^{m}(F (ϕ)/F) = Ω_{F}^{m−n}∧db_{1}∧⋯∧db_{n} if ϕ =≪ b_{1}, …, b_{n} ≫. Hence ker(ν_{F} (m) → ν_{F (ϕ)}(m)) = Ω_{F}^{m−n}∧db_{1}∧⋯∧db_{n}∩ν_{F} (m). Thus this kernel is determined by the following

(2.27) Lemma. For any a ∈ Ω_{F}^{m−n} the following statements are equivalent:

(i) a ∧ \frac{db_{1}}{b_{1}} ∧ ⋯ ∧ \frac{db_{n}}{b_{n}} ∈ ν_{F} (m)

(ii) ℘(a) ∈ ∑_{ϵ∈S_{n},ϵ≠0} b^{ϵ}[Ω_{F}^{m−n}]^{[2]} + dΩ_{F}^{m−n−1} + ∑_{i=1}^{n} Ω_{F}^{m−n−1} ∧ db_{i}

where ϵ = (ϵ_{1}, …, ϵ_{n}) runs over all sequences with ϵ_{i} = 0 or 1.

Proof. Choose a 2-basis B of F with b_{1}, …, b_{n} ∈ B. Without restriction we can assume B finite, i.e. B = {b_{1}, …, b_{n}, …, b_{N}} and let us denote by η the differential form \frac{db_{1}}{b_{1}} ∧ ⋯ ∧ \frac{db_{n}}{b_{n}}.

1. First we prove that (ii) implies (i). Let a be an element in Ω_{F}^{m−n} such that ℘(a) ∈ ∑_{ϵ∈S_{n},ϵ≠0} b^{ϵ}[Ω_{F}^{m−n}]^{[2]} + dΩ_{F}^{m−n−1} + ∑_{i=1}^{n} Ω_{F}^{m−n−1} ∧ db_{i}, i.e. a can be written as

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

with A_{ϵ} ∈ Ω_{F}^{m−n}, B ∈ Ω_{F}^{m−n−1}, E_{i} ∈ Ω_{F}^{m−n−1}. Since (∑_{ϵ≠0} b^{ϵ}A_{ϵ}^{[2]}) ∧ η and dB ∧ η are in d(Ω_{F}^{m−1}), and E_{i} ∧ db_{i} ∧ η = 0, we obtain that

d(a ∧ η) = 0

and

C(a ∧ η) = C((a^{[2]} + ∑_{ϵ≠0} b^{ϵ}A_{ϵ}^{[2]} + dB + ∑_{i=1}^{n} E_{i} ∧ db_{i}) ∧ η)
= C(a^{[2]} ∧ η) + ∑_{ϵ≠0} C(b^{ϵ}A_{ϵ}^{[2]} ∧ η) + C(dB ∧ η) = a ∧ η

This implies that a ∧ η ∈ ν(m)_{F} and proves (i).

2. Now let a be in Ω_{F}^{m−n} such that a ∧ η ∈ ν_{F} (m). We write a as

a = ∑_{μ} c_{μ} \frac{db_{μ}}{b_{μ}}

where \frac{db_{μ}}{b_{μ}} = \frac{db_{μ(1)}}{b_{μ(1)}} ∧ ⋯ ∧ \frac{db_{μ(m−n)}}{b_{μ(m−n)}} , and c_{μ} ∈ F. Note that a ∧ η ∈ ν_{F} (m) implies that d(a ∧ η) = 0 and C(a ∧ η) = a ∧ η.

Be k the maximal index k > n such that

a = R_{0} + b_{k}R_{1}

where R_{0}, R_{1} are differential forms generated by \frac{db_{1}}{b_{1}} , …, \frac{db_{N}}{b_{N}}, with coefficients in F^{2}(b_{1}, …, b_{k−1}). Also we decompose R_{0} and R_{1} as R_{0} = M_{0} + M_{1} ∧ \frac{db_{k}}{b_{k}} and R_{1} = M_{2} + M_{3} ∧ \frac{db_{k}}{b_{k}}. Then d(a ∧ η) = 0 implies that

[d(M_{0}) + b_{k}d(M_{1}) + b_{k}M_{1} ∧ \frac{db_{k}}{b_{k}} + (d(M_{2}) + b_{k}d(M_{3})) ∧ \frac{db_{k}}{b_{k}} ] ∧ η = 0

which means that

d(M_{0}) ∧ η = 0
d(M_{1}) ∧ η = 0
d(M_{2}) ∧ η = 0
M_{1} ∧ η = dM_{3} ∧ η.

From the last relation we obtain M_{1} = d(M_{3}) + E where E is in Ω_{F}^{m−n−1} ∧ \frac{db_{1}}{b_{1}} + ⋯ + Ω_{F}^{m−n−1} ∧ \frac{db_{n}}{b_{n}}. Replacing in the above decomposition of a ∧ η we get

a = M_{0} + M_{2} ∧ \frac{db_{k}}{b_{k}} + d(b_{k}M_{3}) + E.

Now we work with a′ = M_{0} + M_{2} ∧ \frac{db_{k}}{b_{k}} which is also generated by \frac{db_{1}}{b_{1}} , …, \frac{db_{N}}{b_{N}}, with coefficients in F^{2}(b_{1}, …, b_{k−1}). Since d(M_{0}) ∧ η = 0 and d(M_{2}) ∧ η = 0 we have that a′ ∧ η = 0. By repeating of the above procedure on a′ we conclude that a can be written as

a = M + d(G) + H

where M is generated by \frac{db_{1}}{b_{1}} , …, \frac{db_{N}}{b_{N}} with coefficients in F^{2}(b_{1}, …, b_{n}), G ∈ Ω_{F}^{m−n−1} and H in Ω_{F}^{m−n−1} ∧ \frac{db_{1}}{b_{1}} + ⋯ + Ω_{F}^{m−n−1} ∧ \frac{db_{n}}{b_{n}}. This means that a can written as

a = ∑_{μ∈Σ_{m−n}} c_{μ} \frac{db_{μ}}{b_{μ}} + d(G) + H

where each c_{μ} is in F^{2}(b_{1}, …, b_{n}), i.e. c_{μ} = ∑_{ϵ} c_{μ,ϵ}^{2}b^{ϵ} with ϵ = (ϵ_{1}, …, ϵ_{n}) running over all sequences with ϵ_{i} = 0 or 1. Reordering the above relation we obtain

a = ∑_{ϵ} b^{ϵ} (∑_{μ} c_{μ,ϵ}^{2} \frac{db_{μ}}{b_{μ}} ) + d(G) + H.

Note that each (∑_{μ} c_{μ,ϵ}^{2} \frac{db_{μ}}{b_{μ}} ) is in Ω_{F}^{[2]} so we will denote it by A_{ϵ}^{[2]}. By this way we write a as

a = ∑_{ϵ} b^{ϵ}A_{ϵ}^{[2]} + d(G) + H = A_{0}^{[2]} + ∑_{ϵ≠0} b^{ϵ}A_{ϵ}^{[2]} + d(G) + H.

Finally we compute C(a ∧ η). Since b^{ϵ}A_{ϵ}^{[2]} ∧ η, d(G) ∧ η ∈ d(Ω_{F}^{m−1}) and H ∧ η = 0, we have

C(a ∧ η) = C(A_{0}^{[2]} ∧ η) = A_{0} ∧ η.

Using that C(a ∧ η) = a ∧ η, we obtain a = A_{0} + H′ with H′ in Ω_{F}^{m−n−1} ∧ \frac{db_{1}}{b_{1}} + ⋯ + Ω_{F}^{m−n−1} ∧ \frac{db_{n}}{b_{n}}. This means that

a = a^{[2]} + ∑_{ϵ≠0} b^{ϵ}A_{ϵ}^{[2]} + d(G) + H′′

where H′′ = H + H′, completing the proof. □

Considering the case m = n we obtain

(2.28) Corollary.

ker(ν_{F} (n) → ν_{F (ϕ)}(n)) = \{a\frac{db_{1}}{b_{1}} ∧ ⋯ ∧ \frac{db_{n}}{b_{n}} |℘(a) ∈ D_{F} (ϕ′)\}

3 Some technical results

Let us first recall some notation. Let S_{n} be the set of maps μ : {1, …, n} → {0, 1} with μ(i) = 1 for at least some index i. If μ is defined by μ(i) = 1, μ(j) = 0 for j ≠ i we write i instead of μ. Let L = F (X_{μ}, μ ∈ S_{n}) and for a fixed 0 ≤ s ≤ n we set N = F (X_{μ}|μ ≠ 0, …, s) , M = F (X_{μ}^{2}|μ ≠ 0, …, s), (if s = 0 we get N = L and M is the field introduced in the last section).

(3.1) Lemma. Let X = X_{s}. If f ∈ M[X^{2}] is decomposed in monic irreducible polynomials in M[X], say

f = a ∏_{p} p^{n_{p}} (a ∈ M)

then for each p holds p ∈ M[X^{2}] or n_{p} ≡ 0 (mod 2).

Proof. Let p ∈ M[X] be an irreducible factor of f and assume p ∉ M[X^{2}]. This means D_{X}(p) ≠ 0, where D_{X}(p) is the derivative of p with respect to X. Let us assume n_{p} = 2t + 1 for some t ≥ 0. Since f ∈ M[X^{2}] implies D_{X}(f) = 0, we obtain

D_{X} (ap^{2t+1} ∏_{q≠p} q^{n_{q}}) = 0

ap^{2t+1}D_{X} (∏_{q≠p} q^{n_{q}}) + a (∏_{q≠p} q^{n_{q}}) p^{2t}D_{X}(p) = 0

(∏_{q≠p} q^{n_{q}}) D_{X}(p) = pD_{X} (∏_{q≠p} q^{n_{q}})

But deg D_{X}(p) < deg_{X}(p) implies that p divides ∏_{q≠p} q^{n_{q}} in M[X], which is impossible. This proves the claim. □

For any irreducible monic polynomial p ∈ N[X](X = X_{s}) we will write N(p) for the quotient field N[X]/(p). Let us fix a 2-basis B = {b_{i}, i ∈ I} of F, so that B ∪ {X_{μ} \ μ ≠ 1, …, s} is a 2-basis of N. If p ∈ M[X^{2}], there is some b_{i} ∈ B such that B \ {b_{i}} ∪ {X̄_{μ}, μ ≠ 1, …, s − 1} is a 2-basis of N(p), where X̄_{μ} denotes the image of X_{μ} in N(p). If p ∉ M[X^{2}] then B is part of a 2-basis of N(p) and in fact there is some index i_{0} ≠ 1, …, s − 1 such that B ∪ {X̄_{μ}, μ ≠ i_{0}, 1, …, s − 1} is a 2-basis of N(p). The natural map N[X] → N(p) induces a homomorphism Ω_{F}^{m}M[X^{2}] → Ω_{N(p)}^{m}.

(3.2) Lemma. a) If p ∈ M[X^{2}]

ker(Ω_{F}^{m}M[X^{2}] → Ω_{N(p)}^{m}) = pΩ_{F}^{m}M[X^{2}] + Ω_{F}^{m−1}M[X^{2}] ∧ dp

b) If p ∉ M[X^{2}]

ker(Ω_{F}^{m}M[X^{2}] → Ω_{N(p)}^{m}) = p^{2}Ω_{F}^{m}M[X^{2}]

Proof. a) Since p ∈ M[X^{2}], there is some b_{k} ∈ B such that \overline{D_{b_{k}}(p)} ≠ 0 in N(p), because otherwise one would infer that p is a square in M[X]. But in N(p) we have 0 = dp = ∑_{b_{j}} \overline{D_{b_{j}}(p)}db_{j}, where \overline{D_{b_{k}}(p)} ≠ 0 is the coefficient of db_{k}. Choose Δ ∈ M[X^{2}] with Δ · D_{b_{k}}(p) = 1 + p · r in M[X^{2}] and let Δ̄ be the image in N(p). Then

db_{k} = Δ̄ ∑_{i≠k} \overline{D_{b_{i}}(p)}db_{i} in Ω_{N(p)}

Let w ∈ Ω_{F}^{m}M[X^{2}] be in the kernel of Ω_{F}^{m}M[X^{2}] → Ω_{N(p)}^{m}, i.e. w = 0 in Ω_{N(p)}. We have by definition of Ω_{F}^{m}M[X^{2}]

w = ∑_{γ} a_{γ}db_{γ} + (∑_{δ} c_{δ}db_{δ}) ∧ db_{k}

where γ runs over all ∑_{m}(I) with k ∉ Im(γ) and δ runs over ∑_{m−1}(I) with k ∉ Im(δ) and a_{γ}, c_{γ} ∈ M[X^{2}]. Recall that ∑_{m}(I) are all maps γ : {1, …, m} → I with γ(1) < ⋯ < γ(m) in a fixed ordering of I. Then in Ω_{N(p)}^{m}

∑ a_{γ}db_{γ} + (∑ c_{δ}db_{δ}) ∧ Δ ∑_{i≠k} D_{bi}(p)db_{i} = 0

(we omit the bars for simplicity).

Thus

∑_{k∉γ}(a_{γ} + Δ ∑_{δ∪j=γ} c_{δ}D_{b_{j}}(p))db_{γ} = 0

Here k ∉ γ means γ ∈ ∑_{m}(I) with k ∉ Im(γ) and δ ∪ j = γ means that δ ∈ ∑_{m−1}(I) can be extended to γ with j ∈ Im(γ).

Since B \ {b_{k}} is part of a 2-basis of N(p) it follows that the db_{γ}, k ∉ γ are linear independent over N(p) and hence in N(p)

a_{γ} = Δ ∑_{δ∪j=γ} c_{δ}D_{b_{j}}(p).

Then in N[X] we have

a_{γ} = Δ ∑_{δ∪j=γ} c_{γ}D_{b_{j}}(p) + t_{γ} · p

for each γ ∈ ∑_{m}, k ∉ γ, with t_{γ} ∈ N[X] . It follows easily that t_{γ} ∈ M[X^{2}] for all γ, i.e. the above relation holds in M[X^{2}]. Inserting in w we obtain

w = ∑_{k∉γ}(∑_{δ∪j=γ} Δc_{δ}D_{b_{j}}(p) + pt_{γ})db_{γ} + (∑_{k∉δ} c_{δ}db_{δ}) ∧ db_{k}
= ∑_{k∉δ}(c_{δ}db_{k} + Δ ∑_{j≠k} c_{δ}D_{b_{j}}(p)db_{j}) ∧ db_{δ} + p ∑_{k∉γ} t_{γ}db_{γ}
= ∑_{k∉δ} c_{δ}(db_{k} + Δ ∑_{j≠k} D_{b_{j}}(p)db_{j}) ∧ db_{δ} + p ∑_{k∉γ} t_{γ}db_{γ}

Replacing the coefficient 1 of db_{k} by 1 = ΔD_{b_{k}}(p) + pr, we get

w = ∑_{k∉δ} c_{δ}Δdp ∧ db + p(∑_{k∉γ} t_{γ}db_{γ} + r ∑_{k∉δ} c_{δ}db_{k} ∧ db_{δ}) = w_{1} ∧ dp + pw_{2}

with w_{1}, w_{2} in Ω_{F} M[X^{2}].

b) Let us assume p ∉ M[X^{2}]. Then B is part of a 2-basis of N(p). Let w ∈ ker(Ω_{F}^{m}M[X^{2}] → Ω_{N(p)}^{m}) and set w = ∑_{γ} a_{γ}db_{γ} ∈ Ω_{F}^{m}M[X^{2}], γ ∈ ∑_{m}, a_{γ} ∈ M[X^{2}]. Thus ∑ ā_{γ}db_{γ} = 0 in Ω_{N}^{m}(p), and by the remark above, we have ā_{γ} = 0 in N(p) for all γ. Thus a_{γ} = p · t_{γ}, t_{γ} ∈ N[X].

But a_{γ} ∈ M[X^{2}] implies D_{X_{μ}}(a_{γ}) = 0 for all μ ≠ 1, …, s − 1. Thus

D_{X_{μ}}(p) · t_{γ} + pD_{X_{μ}}(t_{γ}) = 0.

We have p ∉ M[X^{2}], so that there is some μ ≠ 1, …, s − 1 with D_{X_{μ}}(p) ≠ 0. Choose this μ in the equation above. Then it follows p|D_{X_{μ}}(p)t_{γ} in N[X] for all γ. But p does not divide D_{X_{μ}}(p), and hence t_{γ} = ps_{γ}, s_{γ} ∈ N[X]. Thus a_{γ} = p^{2}s_{γ} for all γ, and since a_{γ} ∈ M[X^{2}], it follows s_{γ} ∈ M[X^{2}] for all γ. Therefore w = p^{2}s with some s ∈ Ω_{F}^{m}M[X^{2}]. This proves the lemma. □

Let u ∈ Ω_{F} M[X^{2}] be a form generated over M[X^{2}] by forms defined over F. The choice of any 2-basis B of F enable us to define u^{[2]} (resp. ℘(u)) and this form is uniquely determined module dΩ_{F} M[X^{2}] (see remark (1.17)). We are interested in the behavior of u^{[2]} under the reduction homomorphism Ω_{F} M[X^{2}] → Ω_{N(p)}, where p is any irreducible polynomial in N[X]. In particular we want to compare \overline{u^{[2]}} with \overline{u}^{[2]}, where this last square is taken with respect to the 2-basis of N(p) as defined at the beginning of this section. In this case we have

(3.3) Lemma.

\overline{u^{[2]}} − \overline{u}^{[2]} ∈ \overline{dΩ_{F} M[X^{2}]}

(3.4) Lemma. Let u, dv, λ ∈ Ω_{F}^{·} M[X^{2}] and T ∈ M[X^{2}], defined by T = ∑_{μ∈S_{n},μ≠1,... ,s−1} b^{μ}X_{μ}^{2}.

Assume

u^{[2]} + dv = λ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄

in Ω_{N(p)}, where p ∈ N[X] is irreducible and monic and b_{1}, …, b_{r} ∈ F. Then

a) If p ∈ M[X^{2}], there exists δ, u_{1}, u_{2} ∈ Ω_{F} M[X^{2}] such that

u = δ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT + pu_{1} + u_{2} ∧ dp.

b) If p ∉ M[X^{2}], there exists δ, u_{1} ∈ Ω_{F} M[X^{2}] such that

u = δ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT + p^{2}u_{1}.

Proof. Since u^{[2]} + dv is closed, we can apply the Cartier-operator to this form and we get C(u^{[2]} + dv) = u. Thus in Ω_{N(p)}

u = C(λ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄).

If db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄ = 0, then u = 0 and the lemma follows from (3.2). Thus we can assume db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄ ≠ 0, in Ω_{N(p)}, and therefore we can take {b_{1}, …, b_{r}, T̄} as part of a 2-basis of N(p). Then (1.12) implies

u = δ̄ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄

with some δ̄ ∈ Ω_{N(p)}. We will show that δ̄ is contained in the image Ω̄_{F} M[X^{2}] of Ω_{F} M[X^{2}] in Ω_{N(p)}. It is clear that once we have this, the lemma follows from (3.2).

a) Assume p ∈ M[X^{2}]. Then there is some i_{0} ∈ I such that B\{b_{i_{0}}}∪{X̄_{μ}, μ ≠ 1, …, s−1} is a 2-basis of N(p) (we have chosen a 2-basis of F including b_{1}, …, b_{r}). We write in Ω_{N(p)}

δ̄ = δ̄_{0} + δ̄_{1} ∧ dX̄

where δ̄_{0}, δ̄_{1} are forms not containing dX̄ = dX̄_{s}. Then

u = δ_{0} ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄ + δ_{1} ∧ dX̄ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄

Since u ∈ Ω_{F} M[X^{2}] can not contain dX̄ in its expansion in this 2-basis of N(p), we conclude that

u = δ_{0} ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄

(Notice that the expansion of db_{i_{0}} coming from dp = 0 in Ω_{N(p)} does not contain dX̄, because p ∈ M[X^{2}]).

We can proceed in the same way with the other variables X_{μ}, μ ≠ 1, …, s− 1, and we finally obtain that δ_{0} is free from all differentials dX_{μ}, μ ≠ 1, …, s − 1. Thus δ_{0} is generated over N(p) by the differentials db_{i}, i ∈ I \ {i_{0}}. We write now the coefficients of δ_{0} in the 2-basis expansion. First we set δ_{0} = δ′_{0} + X̄δ′′_{0} where X̄ appears in the coefficients of δ′_{0}, δ′′_{0} only in even powers. Then

u = δ′_{0} ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄ + X̄δ′′_{0}db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄

The fact that u is in Ω̄_{F} M[X^{2}] implies that the coefficients of u (in the 2basis expansion) do not contain odd power of X̄. Comparing coefficients we obtain

u = δ′_{0} ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄

with δ′_{0} free from odd powers of X̄. Doing the same with the other variables we finally conclude u = δ̄ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄ with δ̄ ∈ Ω̄_{F} M[X^{2}]. From (3.2) follows the claim. The case (b) i.e. p ∉ M[X^{2}] can be treated in a similar way and we omit the proof. □

(3.5) Lemma. Let p ∈ M[X^{2}] be irreducible and monic (in N[X]). If

pu = v ∧ dp

in Ω_{F} M[X^{2}] with u, v ∈ Ω_{F} M[X^{2}], then there exist v_{1}, v_{2} ∈ Ω_{F} M[X^{2}] with

v = pv_{1} + v_{2} ∧ dp.

Proof. Since p ∈ M[X^{2}], we have D_{X_{i}}(p) = 0 for all i ≠ 1, …, s − 1. Thus

dp = ∑_{i∈I} D_{b_{i}}(p)db_{i}.

The fact that p is irreducible implies dp ≠ 0 and hence there is some i_{0} ∈ I with D_{b_{i_{0}}}(p) ≠ 0. Let us write p(X) = X^{2n} +⋯ = p_{0} +b_{i_{0}}p_{1}, where p_{0}, p_{1} are polynomials whose coefficients do not contain odd powers of b_{i_{0}} in the 2-basis expansion. Hence D_{b_{i_{0}}}(p) = p_{1} and deg_{X} p_{1} < deg p. In particular p |̸p_{1}. Therefore one can find polynomials Δ, t ∈ M[X^{2}] with D_{b_{i_{0}}}(p)Δ = 1 + p · t. Let us set u = u_{0} + u_{1} ∧ db_{i_{0}}, v = v_{0} + v_{1} ∧ db_{i_{0}} with u_{0}, u_{1}, v_{0}, v_{1} free from db_{i_{0}}. Thus

p(u_{0} + u_{1} ∧ db_{i_{0}}) = (v_{0} + v_{1} ∧ db_{i_{0}}) ∧ (D_{b_{i_{0}}}(p)db_{i_{0}} + ∑_{i≠i_{0}} D_{b_{i}}(p)db_{i})

implies

pu_{0} = v_{0} ∧ ∑_{i≠i_{0}} D_{b_{i}}(p)db_{i}
pu_{1} = D_{b_{i_{0}}}(p)v_{0} + v_{1} ∧ ∑_{i≠i_{0}} D_{b_{i}}(p)db_{i}.

Taking modulo p this equations, we obtain in Ω_{N(p)}

(D_{b_{i_{0}}}(p)v_{0} = v_{1} ∧ ∑_{i≠i_{0}} D_{b_{i}}(p)db_{i}

and since Δ̄D_{b_{i_{0}}}(p) = 1 in N(p), it follows

v_{0} = Δ̄v_{1} ∧ ∑_{i≠i_{0}} D_{b_{i}}(p)db_{i}.

But all these forms are contained in Ω_{F} M[X^{2}] so that (3.2) (a) implies

v_{0} = Δv_{1} ∧ ∑_{i≠i_{0}} D_{b_{i}}(p)db_{i}) + pv_{3} + v_{4} ∧ dp

with v_{3}, v_{4} ∈ Ω_{F} M[X^{2}]. Inserting v_{0} in v = v_{0} + v_{1} ∧ db_{i_{0}} we get

v = Δv_{1} ∧ ∑_{i≠i_{0}} D_{b_{i}}(p)db_{i} + pv_{3} + v_{4} ∧ dp + v_{1} ∧ db_{i_{0}}
= v_{1} ∧ (Δ ∑_{i≠i_{0}} D_{b_{i}}(p)db_{i} + db_{i_{0}}) + pv_{3} + v_{4} ∧ dp.

Since 1 = ΔDb_{i_{0}}(p) + p · t in M[X^{2}], we get

v = v_{1} ∧ (Δdp + ptdb_{i_{0}}) + pv_{3} + v_{4} ∧ dp
= p(v_{1} ∧ tdb_{i_{0}} + v_{3}) + (Δv_{1} + v_{4}) ∧ dp

which shows that v has the desired form. □ (3.6) Lemma. Let dv, λ be forms in Ω_{F} M[X^{2}], b_{1}, …, b_{r} ∈ F be 2independent and T as before. Assume

dv = λ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT

in Ω_{N(p)}, where p is monic and irreducible. Then

a) If p ∈ M[X^{2}], there exist δ, v_{1}, v_{2} ∈ Ω_{F} M[X^{2}] such that dv = δ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT + d(pv_{1} + v_{2} ∧ dp)

b) If p ∉ M[X^{2}], there exist δ, v_{1} ∈ Ω_{F} M[X^{2}] such that dv = δ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT + p^{2}dv_{1} Proof. If db_{1} ∧ ⋯ ∧ db_{r} ∧ dT = 0 in Ω_{N(p)}, and therefore dv = 0, we may use (3.2) to prove the lemma. Hence we will assume db_{1} ∧ ⋯ ∧ db_{r} ∧ dT ≠ 0 in Ω_{N(p)}, and for the time being we set T = b_{r+1}. Thus b_{1}, …, b_{r+1} can be chosen as part of a 2-basis of N(p). From (1.13) and (1.14) we infer in N(p)

v = ∑_{μ} b_{i}b^{μ}z_{μ} ∧ db_{1} ∧ ⋯ ∧ d̂b_{i} ∧ ⋯ db_{r+1} + b1u ∧ db_{1} ∧ ⋯ ∧ db_{r+1} + z

with z_{μ}, z closed forms in Ω_{N(p)}, μ running over the index set indicated by (1.13). We shall next prove that the form z_{μ}, u, z can be chosen in Ω̄_{F} M[X^{2}]. Let us first assume p ∈ M[X^{2}]. We take a 2-basis of N(p) of the form B \ {b_{i_{0}}} ∪ {X_{μ}} where X_{μ} are all variable involved (X = X_{s}). Let us fix some variable X_{μ} which we denote by Y. Thus in N(p) we have

z_{μ} = e_{μ,1} + Ȳ e_{μ,2} + (e_{μ,3} + Ȳ e_{μ,4}) ∧ dȲ

with forms e_{μ,i} free from dȲ and whose coefficients are free from odd powers of Y. There are similar decompositions for u, z. Since z_{μ} is closed we get

0 = de_{μ,1} + Ȳ de_{μ,2} + e_{μ,2} ∧ dȲ + de_{μ,3} ∧ dȲ + Ȳ de_{μ,4} ∧ dȲ .

Thus we obtain

de_{μ,1} = de_{μ,4} = 0, e_{μ,2} = de_{μ,3}.

Inserting these expressions for z_{μ}, u and z in the equation for v, we get

v = ∑_{μ} b_{i}b^{μ}z_{μ,1} ∧ db_{1} ∧ ⋯ ∧ d̂b_{i} ∧ ⋯ ∧ db_{r+1}
+b1u_{1} ∧ db_{1} ∧ ⋯ ∧ db_{r+1} + z_{1}

when all z_{μ,1} and z_{1} are closed and moreover z_{μ,1}, z_{1} and u_{1} are free from dȲ and Ȳ. Doing the same with the other variables, we finally conclude that the forms z_{μ}, u and z can be taken in \overline{Ω_{F}} M[X^{2}].

Therefore we have v − ∑_{μ} b_{i}b^{μ}z_{μ} ∧ db_{1} ∧ ⋯ ∧ d̂b_{i} ∧ ⋯ ∧ db_{r+1} − b1u ∧ db_{1} ∧ ⋯ ∧ db_{r+1} − z ∈ ker(Ω_{F}^{m}M[X^{2}] → Ω_{N(p)}^{m}) and from (3.2) it follows

v − ∑_{μ} b_{i}b^{μ}z_{μ} ∧ db_{1} ∧ ⋯ ∧ d̂b_{i} ∧ ⋯ ∧ db_{r+1} − b1u ∧ db_{1} ∧ ⋯ ∧ db_{r+1} − z
= pv_{1} + v_{2} ∧ dp

with some v_{1}, v_{2} ∈ Ω_{F} M[X^{2}]. We apply now d to this relation and obtain (a).

Let us now assume p ∉ M[X^{2}]. Then these is some index μ_{0} such that D_{X_{μ_{0}}}(p) ≠ 0 and B∪{X_{μ}, μ ≠ μ_{0}}∪{y}, where y is the image of X in N(p), is a 2-basis of N(p). All variables X_{μ}, μ ≠ μ_{0} as well as y can be handled in the same way as in the first case, so that we are led to consider only the variable X_{μ_{0}}. Via the relation dp = 0 we express dX_{μ_{0}} in terms of the other differentials, so that we may assume that all z_{μ}, u, z do not contain dX_{μ_{0}} too. Thus X̄_{μ_{0}} may appear in odd powers in the coefficients of z_{μ}, u, z. Let us write p = p_{0} + X_{μ_{0}}p_{1}with p_{0}, p_{1} not containing X_{μ_{0}} in odd powers. Thus in N(p) we have X̄_{μ_{0}} = p̄_{0}/p̄_{1}, so that replacing X̄_{μ_{0}} by p̄_{0}/p̄_{1} in these coefficients, we get rid of the odd powers of X̄_{μ_{0}}, but there appear again the variables X_{μ}, μ ≠ μ_{0}, in these coefficients. We apply again the above procedure to get rid of the odd powers of these variable in the forms z_{μ}, u, z. Therefore we may assume z_{μ}, u, z ∈ Ω_{F} M[X^{2}]. The assertion (b) follows again from (3.2). □

(3.7) Lemma. Let b_{1}, …, b_{r} ∈ F be 2-independent in F and let T be as before. Assume for a form λ ∈ Ω_{F} M[X^{2}]

λ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT = 0

in Ω_{N(p)}. Then

a) If p ∈ M[X^{2}] and db_{1} ∧ ⋯ ∧ db_{r} ∧ dT ≠ 0 in Ω_{N(p)}, there exist λ_{1}, λ_{2} ∈ Ω_{F} M[X^{2}] with

λ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT = (pλ_{1} + λ_{2} ∧ dp) ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT

in Ω_{F} M[X^{2}].

b) If p ∉ M[X^{2}] and db_{1} ∧ ⋯ ∧ db_{r} ∧ dT ≠ 0 in Ω_{N(p)}, there exist λ_{1} ∈ Ω_{F} M[X^{2}] with

λ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT = p^{2}λ_{1} ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT

in Ω_{F} M[X^{2}].

c) If db_{1} ∧ ⋯ ∧ db_{r} ∧ dT = 0 in N(p), there exists t ∈ M[X^{2}] with deg_{X} t < deg_{X} p and db_{1} ∧ ⋯ ∧ db_{r} ∧ dT = db_{1} ∧ ⋯ ∧ db_{r} ∧ d(pt) or = db_{1} ∧ ⋯ ∧ d(pt) ∧ ⋯ ∧ db_{r} ∧ dT for some 1 ≤ i ≤ r, this case only occurs for p ∈ M[X^{2}].

Proof. Let us first assume db_{1}∧⋯∧db_{r}∧dT ≠ 0 in Ω_{N(p)}. Then b_{1}, …, b_{r}, T̄ can be chosen as part of a 2-basis of N(p) and the assumption λ ∧ db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄ = 0 implies (see (1.1))

λ = ∑_{1}^{r} δ_{i} ∧ db_{i} + δ ∧ dT̄

with some forms δ_{i}, δ ∈ Ω_{N(p)}. Since λ ∈ Ω̄M[X^{2}], one easily shows that the forms δ_{i}, δ can be chosen in Ω̄_{F} M[X^{2}] too. Then (3.2) implies

λ = ∑_{i=1}^{r} δ_{i} ∧ db_{i} + δ ∧ dT + pλ_{1} + λ_{2} ∧ dp

if p ∈ M[X^{2}] and λ = ∑ δ_{i} ∧ db_{i} + δ ∧ dT + p^{2}λ_{2} if p ∉ M[X^{2}], where λ_{1}, λ_{2} ∈ Ω_{F} M[X^{2}]. Taking the product with db_{1} ∧ ⋯ ∧ db_{r} ∧ dT we obtain a) and b). Let us assume now db_{1} ∧ ⋯ ∧ db_{r} ∧ dT̄ = 0 in Ω_{N(p)}. We write b_{r+1} = \overline{T} for the time being, and we choose (after reordering) a maximal 2-independent subset {b_{1}, …, b_{j_{0}}} of {b_{1}, …, b_{r+1}}. For example one could have a relation \overline{T} = b_{r+1} = ∑_{μ} p_{μ}^{2}b_{μ}, where μ runs over the set of map μ : {1, …, j_{0}} → {0, 1} and p_{μ} ∈ N(p), b_{μ} = ∏_{i=1}^{j_{0}} b_{i}^{μ(i)}. Then we can write T = ∑ p_{μ}^{2}b_{μ} + p · t in N[X], with p_{μ}, p, t ∈ N[X]. Since T, p_{μ}^{2} ∈ M[X^{2}], it follows p · t ∈ M[X^{2}], and (3.1) implies p, t ∈ M[X^{2}], or t = p · ℓ with ℓ ∈ M[X^{2}]. Thus we have

T = ∑ p_{μ}^{2}b_{μ} + p · t, p, t ∈ M[X^{2}]

or

T = ∑ p_{μ}^{2}b_{μ} + p^{2} · ℓ, ℓ ∈ M[X^{2}]

and p ∉ M[X^{2}]. But in the last case we see that deg_{X}(p^{2}l) > deg_{X} T and deg_{X} p_{μ}^{2}b_{μ} (since p_{μ} can be chosen with deg_{X} p_{μ} < deg_{X} p). This means that p ∉ M[X^{2}] never happen. Therefore

dT = ∑ p_{μ}^{2}db_{μ} + d(pt)

with t ∈ M[X^{2}] and deg_{X} t < deg_{X} p. It follows

db_{1} ∧ ⋯ ∧ db_{r} ∧ dT = db_{1} ∧ ⋯ ∧ db_{r} ∧ d(pt).

The same argument applies for the case that some b_{k} (k ≠ r + 1) is 2dependent of {b_{1}, …, b_{j_{0}}}. We omit the details. □

(3.8) Lemma. Let p ∈ M[X^{2}] be a monic irreducible polynomial in N[X]. Let u, v, λ ∈ Ω_{F} M[X^{2}] be such that

℘u + dv = λ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT in Ω_{N(p)}

where b_{1}, …, b_{s−1} ∈ F^{∗} and T = b_{s}X^{2} + T′ as before (deg_{X} T′ = 0). Then there exist forms u_{1}, u_{2}, δ ∈ Ω_{F} M[X^{2}], f_{ij}, g_{ij} ∈ M[X^{2}] with deg(f_{ij}) < deg(p) and f_{ij}g_{ij} ≡ 1 ( mod p), such that

u = ∑_{i}(g_{i1}df_{i1}) ∧ ⋯ ∧ (g_{im}df_{im}) + pu_{1} + u_{2} ∧ dp
+ δ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

holds in Ω_{F} M[X^{2}].

Proof. If db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = 0 in Ω_{N(p)}^{s}, then ℘u = dv, and we conclude from (2.15)

u = ∑_{i} \frac{df_{i1}}{f_{i1}} ∧ ⋯ ∧ \frac{df_{im}}{f_{im}}

with certain f_{ij} ∈ N(p)^{∗}. Taking g_{ij} ∈ N[X] with g_{ij}f_{ij} ≡ 1 ( mod p), we obtain u = ∑_{i} g_{i1}df_{i1} ∧ ⋯ ∧ g_{is}df_{is}. Since u ∈ Ω_{F} M[X^{2}] one can show that the f_{ij} and g_{ij} can be chosen in M[X^{2}], and we can now apply (3.2). Thus we can assume db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT ≠ 0 in Ω_{N(p)}, and hence db_{1}, …, db_{s−1}, T are 2-independent in N(p). We choose a 2-basis of F which contains {b_{1}, …, b_{s−1}} and we take the constructed 2-basis of N(p) for the case p ∈ M[X^{2}] as indicated after the proof of lemma (3.1). The excluded index i_{0} can be chosen ≠ 1, …, s − 1. Moreover we replace the next element b_{s} by the image T̄ of T in N(p) and we write b_{s} for T̄. Thus the new 2-basis of N(p) is now {b_{1}, …, b_{s−1}, b_{s} = T̄, …, b̂_{i_{0}}, …, b_{N}, X_{μ}, μ ≠ 1, …, s − 1}. We order this basis such that all b_{i} < X_{μ} for all μ and all i. In Σ_{m,N(p)} we choose the lexicographic ordering and with respect to this ordering choose α minimal with α > γ for all γ with γ(1) = 1, …, γ(s) = s. Hence if u ∈ Ω_{N(p),<α}^{m} then there exists some δ with u = δ ∧ db_{1} ∧ ⋯ ∧ db_{s} and δ in the image of Ω_{F} M[X^{2}]. We choose now β ∈ Σ_{m,N(p)} minimal with the property u ∈ Ω_{N(p),β}. This means that in the representation of u with respect to the above 2-basis of N(p), β corresponds to the leading index of this representation. Since u comes from Ω_{F} M[X^{2}] we see that b_{β(i)} ∈ M[X^{2}] for all i. Now we conclude that N(p)_{β(i)} ⊂ M̄[X^{2}].

If β ≤ α we have u = δ∧db_{1}∧⋯∧db_{s} with δ in the image of Ω_{F} M[X^{2}], and we are done by (3.2). Assume now α < β. Then we have λ ∧ db_{1} ∧ ⋯ ∧ db_{s} ∈ Ω_{N(p),<β} and hence

℘(u) ∈ Ω_{N(p),<β} + dΩ_{N(p)}.

Applying Kato’s lemma one gets

u = \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{m}}{a_{m}} + u′

in Ω_{N(p)} with u′ ∈ Ω_{N(p),<β}, a_{i} ∈ N(p)_{β(i)} ⊂ M[X^{2}]. The form u′ is contained in \overline{Ω_{F}} M[X^{2}] because u and also \frac{da_{1}}{a_{1}} ∧⋯∧ \frac{da_{n}}{a_{n}} belong to \overline{Ω_{F}} M[X^{2}]. We apply now the same procedure to u′ until we get

u = ∑ \frac{df_{i,1}}{f_{i,1}} ∧ ⋯ ∧ \frac{df_{i,m}}{f_{i,m}} + v

with all f_{i,j} ∈ M̄[X^{2}] and v ∈ Ω_{N(p),<α}. By the remark above we conclude v = δ ∧ db_{1} ∧ ⋯ ∧ db_{s} with δ in the image of Ω_{F} M[X^{2}]. All f_{i,j} ∈ M[X^{2}] can be chosen with deg_{X} f_{i,j} < deg_{X} p. Let g_{i,j} ∈ M[X^{2}] with f_{i,j}g_{i,j} ≡ 1( mod p). Inserting in the above equation and applying again lemma (3.2) we finally get the desired result. □

4 The kernel H^{n+1}(F (ϕ)/F)

In this section we will prove the main result of this paper, namely

(4.1) Theorem. Let ϕ =≪ b_{1}, …, b_{n} ≫ be an anisotropic bilinear Pfister form over F. Then for m ≥ n

H^{m+1}(F (ϕ)/F) = \overline{Ω_{F}^{m−n} ∧ db_{1} ∧ ⋯ ∧ db_{n}}.

If m < n, then H^{m+1}(F (ϕ)/F) = 0.

Let us write during this section K for the function field F (ϕ) of the conic ϕ = 0. In section 2 we have shown that w ∈ H^{m+1}(F (ϕ)/F) holds if and only if w satisfies the following equation in Ω_{L}^{m}

w = ℘u + dv + λ ∧ dT (4.2)

with u, v, λ ∈ Ω_{F} M′ ⊂ Ω_{L}, where M′ is the subfield F (X_{μ}^{2}|μ ∈ S_{n}) of L = F (X_{μ}|μ ∈ S_{n}), and Ω_{F} M′ denotes the subspace of Ω_{L} generated by the forms db, b ∈ F over the field M′. Recall that T = ∑_{μ∈S_{n}} b^{μ}X_{μ}^{2} and S_{n} is the set of all maps u : {1, …, n} → {0, 1} with at least one value μ(i) = 1. We will develop a descent procedure, which starting from (4.2) will led us to an equation w = ℘u + dv + λ ∧ db_{1} ∧ ⋯ ∧ db_{n} with u, v, λ ∈ Ω_{F} in Ω_{F}.

Let us fix some integer s with 1 ≤ s ≤ n. Set M = F (X_{μ}^{2}|μ ∈ S_{n}, μ ≠ 1, …, s), X = X_{s} and let us consider the equations in Ω_{L}

w = ℘u + dv + λ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT (4.3)

with w ∈ Ω_{F}, u, v, λ ∈ Ω_{F} M(X^{2}) and T = b_{s}X^{2} + T′, deg_{X} T′ = 0 and T′ is a polynomial in X_{μ}^{2}, μ ≠ 1, …, s over F. The equation (4.3) for s = 1 is just (4.2). Our strategy is to start with (4.2) and to push up the index s until we get the factor db_{1} ∧ ⋯ ∧ db_{n} and then to eliminate the rest of the variables until we get the desired equation w = ℘u_{0} + dv_{0} + λ_{0} ∧ db_{1} ∧ ⋯ ∧ db_{n} in Ω_{F} , which is obviously the assertion of (4.1).

Any u ∈ Ω_{F} M(X^{2}) can be written in the form

u = u_{0} + ∑_{p} u_{p} (4.4)

with u_{0} ∈ Ω_{F} M[X^{2}], u_{p} ∈ p^{−∞}Ω_{F} M[X^{2}], p running over all irreducible monic polynomials in N[X], where N = F (X_{μ}, μ ≠ 1, …, s). Recall that p^{−∞}Ω_{F} M[X^{2}] denotes the space of forms u/p^{r} with u ∈ Ω_{F} M[X^{2}] and deg_{X} u < deg_{X}(p^{r}) (see section 2). Fixing a 2-basis of F and N we have seen that the operators ℘ and d leave invariant the spaces Ω_{F} M[X^{2}], p^{−∞}Ω_{F} M[X^{2}] (see (1.21)). Let us now insert in (4.3) the decompositions u = u_{0}+∑ u_{p}, v = v_{0} + ∑ v_{p} and λ = λ_{0} + ∑ λ_{p}. The terms λ_{p} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT can contribute eventually with an integral form (i.e. from Ω_{F} M[X^{2}]) , which we will denote by E_{p}. Thus λ_{p} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT + E_{p} ∈ p^{−∞}Ω_{F} M[X^{2}]. Thus we conclude from (4.3), (1.20) and (1.21)

w = u_{0}^{[2]} + u_{0} + dv_{0} + λ_{0} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT + ∑_{p} E_{p} (4.5)

0 = u_{p}^{[2]} + u_{p} + dv_{p} + λ_{p} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT + E_{p}. (4.6)

(4.7) Remark. As noticed in lemma (3.3), under the natural homomorphism

Ω_{F} M[X^{2}] ⟶ Ω_{N(p)}

(p ∈ N[X] an irreducible polynomial), the operation u^{[2]} behaves well, i.e., \overline{u^{[2]}}−\overline{u}^{[2]} is contained in \overline{dΩ_{F} M[X^{2}]} provided the computations are done with respect to the 2-basis indicated there. In what follows we will frequently reduce modulo p expressions of the form u^{[2]} + dv and then we will lift back to Ω_{F} M[X^{2}]. By the above remark, the differential form dv will eventually change, but this will be of no importance for further computations. Because of this reason we will not explicitely mention these changes.

Our next goal is to study the forms E_{p} ∈ Ω_{F} M[X^{2}]. To this end we will distinguish three types of monic irreducible polynomials in N[X], namely

a) p ∉ M[X^{2}] and db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT ≠ 0 in Ω_{N(p)}

b) p ∈ M[X^{2}] and db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT ≠ 0 in Ω_{N(p)}

c) db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = 0 in Ω_{N(p)}.

Case (a): We can write (see (4.6))

E_{p} = \frac{ū^{[2]}}{p^{4r}} + \frac{ū}{p^{2r}} + \frac{dv̄}{p^{4r}} + \frac{λ̄}{p^{4r}} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

with integral forms ū, v̄, λ̄ ∈ Ω_{F} M[X^{2}] , and r ≥ 0. For the time being, let us write u, v, λ instead of ū, v̄, λ̄. Then in Ω_{F} M[X^{2}] we get

p^{4r}E_{p} = u^{[2]} + p^{2r}u + dv + λ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT. (4.8)

Next we show that the form E_{p} can be absorbed by the first sum u_{0}^{[2]} + u_{0} + dv_{0} + λ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT in (4.5). Of course we can assume r ≥ 1 in (4.8). Taking (4.8) modulo p we obtain in Ω_{N(p)}

u^{[2]} + dv + λ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = 0. (4.9)

Since p ∉ M[X^{2}], lemma (3.4) (b) implies

u = δ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT + p^{2}u_{1}

in Ω_{F} M[X^{2}]. Inserting this expression in (4.8) we get

p^{4r}E_{p} = p^{4}u_{1}^{[2]} + p^{2r+2}u_{1} + dv + λ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT (4.10)

with some new forms v and λ ∈ Ω_{F} M[X^{2}]. It follows dv = λ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT in Ω_{N(p)} and lemma (3.5) implies

dv = δ_{1} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT + p^{2}dv_{1}

in Ω_{F} M[X^{2}]. Thus (4.10) reads now

p^{4r}E_{p} = p^{4}u_{1}^{[2]} + p^{2r+2}u_{1} + p^{2}dv_{1} + δ′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

and hence in Ω_{N(p)} we have

δ′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = 0.

From (3.6) (b) we conclude

δ′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = p^{2}δ′′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

with δ′′ ∈ Ω_{F} M[X^{2}], since db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT ≠ 0 in Ω_{N(p)}. In this case we get

\begin{aligned}p^{4r}E_{p} & = & p^{4}u_{1}^{[2]} + p^{2r+2}u_{1} + p^{2}dv_{1} + p^{2}δ′′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT \\ p^{4r−2}E_{p} & = & p^{2}u_{1}^{[2]} + p^{2r}u_{1} + dv_{1} + δ′′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT. \end{aligned}

Again one obtains in Ω_{N(p)}

dv_{1} = δ′′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

and it follows dv_{1} = δ′′′∧db_{1}∧⋯∧db_{s−1}∧dT +p^{2}dv_{2} with δ′′′, v_{2} ∈ Ω_{F} M[X^{2}]. Repeating the last argument we finally obtain a relation

p^{4r−4}E_{p} = u_{1}^{[2]} + p^{2r−2}u_{1} + dv_{1} + λ_{1} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

in Ω_{F} M[X^{2}], i.e.

E_{p} = ℘(\frac{u_{1}}{p^{2r−2}} ) + d(\frac{v_{1}}{p^{4r−4}} ) + \frac{λ_{1}}{p^{4r−4}} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT.

Thus we have reduced the number r by one in (4.8). Iterating this process we finally arrive at a relation with r = 0, i.e.

E_{p} = ℘(u′_{0}) + dv′_{0} + λ′_{0} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

with u′_{0}, v′_{0}, λ′_{0} ∈ Ω_{F} M[X^{2}]. This expression can be absorbed by the first integral part of (4.5) and hence we have eliminated E_{p} from this sum.

Case (b): Thus we assume p ∈ M[X^{2}] and db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT ≠ 0 in Ω_{N(p)}. We can write

E_{p} = \frac{u^{[2]}}{p^{2r}} + \frac{u}{p^{r}} + \frac{dv}{p^{2r}} + \frac{λ}{p^{2r}} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

with some integer r ≥ 1. Thus

p^{2r}E_{p} = u^{[2]} + p^{r}u + dv + λ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT (4.11)

with some forms u, v, λ ∈ Ω_{F} M[X^{2}]. In Ω_{N(p)} we get

0 = u^{[2]} + dv + λ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT.

From lemma (3.4) (a) we conclude

u = δ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT + pu_{1} + u_{2} ∧ dp

with δ, u_{1}, u_{2} ∈ Ω_{F} M[X^{2}].

We insert now this expression in E_{p} and obtain

p^{2r}E_{p} = (δ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT)^{[2]} + p^{r}δ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT
+p^{2}u_{1}^{[2]} + pu_{2}^{[2]} ∧ dp + p^{r+1}u_{1} + p^{r}u_{2} ∧ dp
+dv + λ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT
= p^{2}u_{1}^{[2]} + pu_{2}^{[2]} ∧ dp + p^{r+1}u_{1} + p^{r}u_{2} ∧ dp
+dv + λ′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

with λ′ = λ + p^{r}δ + δ′, where δ′ is some form in Ω_{F} M[X^{2}] with (δ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT)^{[2]} = δ′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT. Thus in Ω_{N(p)} it holds

dv = λ′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

and lemma (3.5) implies

dv = λ′′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT + d(pv_{1} + v_{2} ∧ dp)

with λ′′, v_{1}, v_{2} ∈ Ω_{F} M[X^{2}]. Therefore

p^{2r}E_{p} = p^{2}u_{1}^{[2]} + pu_{2}^{[2]} ∧ dp + p^{r}u_{2} ∧ dp + p^{r+1}u_{1}
+pdv_{3} + v_{3} ∧ dp + λ′′′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

where v_{3} = v_{1} + dv_{2} and λ′′′ is some form in Ω_{F} M[X^{2}]. Taking this equation modulo p we obtain λ′′′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = 0 in Ω_{N(p)}. Since db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT ≠ 0 in Ω_{N(p)}, we obtain from lemma (3.6) (a)

λ′′′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = (pλ_{1} + λ_{2} ∧ dp) ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

with λ_{1}, λ_{2} ∈ Ω_{F} M[X^{2}]. Therefore

p[p^{2r−1}E_{p} + pu_{1}^{[2]} + (u_{2}^{[2]} + p^{r−1}u_{2}) ∧ dp
+p^{r}u_{1} + dv_{3} + λ_{1} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT)]
= (v_{3} + λ_{2} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT) ∧ dp.

Lemma (3.4) implies

v_{3} + λ_{2} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = pv_{4} + v_{5} ∧ dp

with v_{4}, v_{5} ∈ Ω_{F} M[X^{2}]. It follows

p^{2r−1}E_{p} + pu_{1}^{[2]} + u_{2}^{[2]} ∧ dp + p^{r−1}u_{2} ∧ dp + p^{r}u_{1}
+ pdv_{4} + dv_{5} ∧ dp + λ_{3} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = 0

where λ_{3} = λ_{1} + dλ_{2}. We get again in Ω_{N(p)}

λ_{3} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = 0

and (3.6) (a) implies λ_{3} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = (pλ_{4} + λ_{5} ∧ dp) ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT in Ω_{F} M[X^{2}]. Inserting this expression in the above equation we obtain

\begin{aligned}p[p^{2r−2}E_{p} & + & u_{1}^{[2]} + p^{r−1}u_{1} + λ_{4} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT + dv_{4}] \\ & = & (u_{2}^{[2]} + p^{r−1}u_{2} + dv_{5} + λ_{5} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT) ∧ dp \end{aligned} (4.12)

in Ω_{F} M[X^{2}]. Lemma (3.4) implies

u_{2}^{[2]} + p^{r−1}u_{2} + dv_{5} + λ_{5} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = 0

in Ω_{N(p)}.

We consider now two cases.

Case 1. r > 1. Then it holds in Ω_{N(p)},

u_{2}^{[2]} + dv_{5} + λ_{5} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = 0

and we can apply (3.4) (a) to obtain

u_{2} = λ_{6} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT + pu_{3} + u_{4} ∧ dp

in Ω_{F} M[X^{2}]. Inserting this value of u_{2} in the above equation and using a similar argument as in case (a) we easily see that the exponent of p in (4.10) can be lowered. Therefore we are led to consider the next case.

Case 2. r = 1. Then we have Ω_{N(p)},

u_{2}^{[2]} + u_{2} + dv_{5} + λ_{5} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = 0.

For the time being we will set w_{s} = db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT. From lemma (3.8) we conclude

u_{2} = ∑_{i} ⋀_{j} g_{ij}df_{ij} + pu_{3} + u_{4} ∧ dp + δ ∧ w_{s}

with f_{ij}, g_{ij} ∈ M[X^{2}], u_{3}, u_{4}, δ ∈ Ω_{F} M[X^{2}] as indicated by the lemma. Therefore the right hand side of the equation (4.12) reads now

[∑_{i} ⋀_{j}(g_{ij}df_{ij})^{2} \frac{df_{ij}}{f_{ij}} + ∑_{i} ⋀_{j} g_{ij}f_{ij} \frac{df_{ij}}{f_{ij}}
+ p^{2}u_{3}^{[2]} + pu_{3} + dv_{5} + λ_{0} ∧w_{s}] ∧ dp

with certain form λ_{0} ∈ Ω_{F} M[X^{2}].

Since g_{ij}f_{ij} ≡ 1( mod p), it follows ∏_{j} g_{ij}f_{ij} = 1 + ph_{i} with some h_{i} ∈ M[X^{2}] for all i. Thus (4.12) implies

p[E_{p} + u_{1}^{[2]} + u_{1} + λ_{4} ∧ w_{s} + dv_{4}] = [(∑_{i} ph_{i} \frac{df_{i1}}{f_{i1}} ∧ ⋯ ∧ \frac{df_{im}}{f_{im}})^{[2]}
+ ∑ ph_{i} \frac{df_{i1}}{f_{i1}} ∧ ⋯ ∧ \frac{df_{im}}{f_{im}} + ℘(pu_{3}) + dv_{5} + λ_{0} ∧ w_{s}] ∧ dp.

But one easily sees that

℘(pu_{3}) ∧ dp ≡ p℘(u_{3} ∧ dp)

℘ [∑_{i} ph_{i} ∧ \frac{df_{i1}}{f_{i1}} ∧ ⋯ ∧ \frac{df_{im}}{f_{im}} ] ∧ dp ≡ p℘ [dp ∧ ∑_{i} h_{i} \frac{df_{i1}}{f_{i1}} ∧ ⋯ ∧ \frac{df_{im}}{f_{im}} ]

( mod dΩ_{F} M[X^{2}]).

We bring both terms to the left side of the above equation and we get

p[E_{p} + ℘(u_{1} + u_{3} ∧ dp + ∑_{i} h_{i} \frac{df_{i1}}{f_{i1}} ∧ ⋯ ∧ \frac{df_{im}}{f_{im}} ∧ dp)
+ λ_{4} ∧ w_{s} + dv_{4}] = [dv_{5} + λ_{0} ∧ w_{s}] ∧ dp.

Thus

E_{p} = ℘(u_{1} + u_{3} ∧ dp + ∑_{i} h_{i} \frac{df_{i1}}{f_{i1}} ∧ ⋯ ∧ \frac{df_{im}}{f_{im}} ∧ dp)

+d(v_{4} + v_{5} ∧ \frac{dp}{p}) + (λ_{4} + λ_{0} ∧ \frac{dp}{p}) ∧ w_{s}.

Notice that u_{1} + u_{3} ∧ dp + ∑_{i} h_{i} \frac{df_{i1}}{f_{i1}} ∧ ⋯ ∧ \frac{df_{im}}{f_{im}} ∧ dp contains denominators all of whose prime factors in M[X^{2}] (and in N[X]) are of degree < deg(p). The other involved forms have at most p in the denominator. Using a similar argument as done at the beginning of this section we get

E_{p} = ℘(u′_{0}) + dv′_{0} + λ′_{0} ∧ w_{s} + G_{p} + ∑_{q deg(q)<deg(p)} G_{q}

with u′_{0}, v′_{0}, λ′_{0} ∈ Ω_{F} M[X^{2}] and integral forms G_{p}, G_{q} of the type

G_{p} = dv_{p} + λ_{p} ∧ w_{s}
G_{q} = ℘(u_{q}) + dv_{q} + λ_{q} ∧ w_{s}

where v_{p}, λ_{p} ∈ \frac{1}{p}Ω_{F} M[X^{2}] and u_{q}, v_{q}, λ_{q} ∈ q^{−∞}Ω_{F} M[X^{2}] if deg(q) < deg(p).

Since the forms G_{q} have denominators of degree < deg(p) and are of the same type as E_{q}, we can add them to the E_{q}’s, so that we forget then now. Let us consider G_{p}. Let us write

v_{p} = \frac{v′}{p^{2}} ,  λ_{p} = \frac{λ′}{p}  with  v′, λ′ ∈ Ω_{F} M[X^{2}].

Thus G_{p} = \frac{dv′}{p^{2}} + \frac{λ′}{p} ∧ w_{s} and hence

p^{2}G_{p} = dv′ + pλ ∧ w_{s}

holds in Ω_{F} M[X^{2}]. Thus dv′ = 0 in Ω_{N(p)}. Using Cartier’s theorem (see [ Ca] or section 1) we can write v′ = A^{[2]} + dB in Ω_{N(p)}. Since v′ ∈ Ω̄_{F} M[X^{2}] = Image of Ω_{F} M[X^{2}] in Ω_{N(p)}, we easily see that B also can be chosen in Ω̄_{F} M[X^{2}]. Therefore in Ω_{F} M[X^{2}] (see (3.2))

v′ = A^{[2]} + dB + pv_{1} + v_{2} ∧ dp

with v_{1}, v_{2} ∈ Ω_{F} M[X^{2}]. It follows dv′ = d(pv_{1} + v_{2} ∧ dp) = d(p(v_{1} + dv_{2})) = d(pv′_{1}) where v′_{1} = v_{1} + dv_{2}. Therefore in Ω_{F} M[X^{2}]

p[pG_{p} + dv′_{1} + λ ∧ w_{s}] = v′_{1} ∧ dp

Lemma (3.4) implies v′_{1} = pv_{2} + v_{3} ∧ dp in Ω_{F} M[X^{2}] and hence after replacing v′_{1} in the above equation it follows

pG_{p} = pdv_{2} + dv_{3} ∧ dp + λ ∧ w_{s}.

Thus λ∧w_{s} = 0 in Ω_{N(p)}. Lemma (3.6) implies λ∧w_{s} = (pλ_{1}+λ_{2}∧dp)∧w_{s} in Ω_{F} M[X^{2}] and therefore

pG_{p} = pdv_{2} + dv_{3} ∧ dp + pλ_{1} ∧ w_{s} + λ_{2} ∧ dp ∧ w_{s}
p[G_{p} + dv_{2} + λ_{1} ∧ w_{s}] = (dv_{3} + λ_{2} ∧ w_{s}) ∧ dp.

We apply again (3.4) and obtain dv_{3} + λ_{2} ∧ w_{s} = pv_{4} + v_{5} ∧ dp in Ω_{F} M[X^{2}]. Hence in Ω_{N(p)} it holds dv_{3} = λ_{2} ∧ w_{s}. Applying (3.5) we obtain dv_{3} = δ ∧ w_{s} + d(pv_{6} + v_{7} ∧ dp). This implies

p[G_{p} + dv_{2} + λ_{1} ∧ w_{s}] = pdv_{6} ∧ dp + δ′ ∧ w_{s} ∧ dp

with δ′ = λ_{2} + δ. Lemma (3.4) implies δ′ ∧ w_{s} = pλ_{3} + λ_{4} ∧ dp in Ω_{p}M[X^{2}]. Hence δ′∧w_{s}∧dp = pλ_{3}∧dp. But we are assuming that w_{s} ≠ 0 in Ω_{N(p)}. Thus we easily conclude that there exist some λ′_{3} ∈ Ω_{F} M[X^{2}] with δ′ ∧ w_{s} ∧ dp = pλ′_{3} ∧ w_{s} ∧ dp. Inserting this in the above relation we get

p[G_{p} + dv_{2} + λ_{1} ∧ w_{s}] = pdv_{6} ∧ dp + pλ′_{3} ∧ w_{s} ∧ dp
G_{p} = d(v_{2} + v_{6} ∧ dp) + (λ_{1} + λ′_{3} ∧ dp) ∧ w_{s}

in Ω_{F} M[X^{2}]. Thus we also get rid of G_{p} in the equation for E_{p}.

Case (c). We assume db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT = 0 in Ω_{N(p)}. Thus we assume now p ∈ M[X^{2}]. In Ω_{F} M[X^{2}] we can write

p^{2h}E_{p} = u^{[2]} + p^{h}u + dv + λ ∧ w_{s} (4.13)

with h ≥ 1, with u, v, λ ∈ Ω_{F} M[X^{2}] and w_{s} = db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT. We claim that there is t ∈ M[X^{2}] , deg_{X} t < deg_{X} p with

t^{2}p^{2h−2}E_{p} = u′^{[2]} + tp^{h−1}u′ + dv′ + λ′ ∧ w_{s} (4.14)

with u′, v′, λ′ ∈ Ω_{F} M[X^{2}].

From (4.13) it is clear that we have reduced the exponent of p in the denominators of E_{p} at the cost that we increase the number of terms of type E_{q} but with deg_{X}(q) < deg_{X}(p). Hence, iterating this process and using partial fractions decompositions, we can finally eliminate E_{p} from the original equation (4.5). Of course we have during this process to modify the other terms E_{q} for irreducibles q with deg_{X}(q) < deg_{X}(p).

The assumption w_{s} = db_{1} ∧⋯∧db_{s−1} ∧dT = 0 in Ω_{N(p)} implies (see (3.7) (c))

w_{s} = db_{1} ∧ ⋯ ∧ db_{s−1} ∧ d(pt)

or

w_{s} = db_{1} ∧ ⋯ ∧ db_{i−1} ∧ d(pt) ∧ db_{i+1} ∧ ⋯ ∧ db_{s−1} ∧ dT

with t ∈ M[X^{2}], deg_{X} t < deg_{X} p, and some i, 1 ≤ i ≤ s − 1. Let us write w_{s} = w_{s−1} ∧ d(pt) where w_{s−1} denotes db_{1} ∧ ⋯ ∧ db_{s−1} or db_{1} ∧ ⋯ ∧ î∧ ⋯ ∧ db_{s−1} ∧ dT. We will assume w_{s−1} ≠ 0 in Ω_{N(p)}, and we will omit the proof in the case w_{s−1} = 0, which can be treated similarly. Since we can assume h ≥ 1, we get from (4.13)

u^{[2]} + dv = 0

in Ω_{N(p)}. It follows u = 0 in Ω_{N(p)} and hence u = pu_{1} + u_{2} ∧ dp with u_{1}, u_{2} ∈ Ω_{F} M[X^{2}] (see (3.2)). Then

p^{2h}E_{p} = p^{2}u_{1}^{[2]} + pu_{2}^{[2]} ∧ dp + p^{h+1}u_{1} + p^{h}u_{1} ∧ dp + dv + λ ∧ w_{s}.

This implies dv = 0 in Ω_{N(p)} and hence dv = d(pv_{1}) with some v_{1} ∈ Ω_{F} M[X^{2}] (see proof of case (2) above). Replacing this value of dv in the above equation it follows

p[p^{2h−1}E_{p} + pu_{1}^{[2]} + u_{2}^{[2]} ∧ dp + p^{h}u_{1} + p^{h−1}u_{2} ∧ dp
+dv_{1} + λ ∧ w_{s−1} ∧ dt] = (v_{1} + tλ ∧ w_{s−1}) ∧ dp.

Lemma (3.4) implies

v_{1} = tλ ∧ w_{s−1} + pv_{2} + v_{3} ∧ dp

with v_{2}, v_{3} ∈ Ω_{F} M[X^{2}]. Thus

p[p^{2h−2}E_{p} + u_{1}^{[2]} + p^{h−1}u_{1} + dv_{2}] =
tdλ ∧ w_{s−1} + [u_{2}^{[2]} + p^{h−1}u_{2} + dv_{3}] ∧ dp.

Hence

tdλ ∧ w_{s−1} = 0

in Ω_{N(p)}. Since deg_{X} t < deg_{X} p, it follows t ≠ 0 in N(p) and we get

dλ ∧ w_{s−1} = 0.

This implies dλ ∧ w_{s−1} = d(pλ_{1}) ∧ w_{s−1} in Ω_{F} [X^{2}]. Therefore

p[p^{2h−2}E_{p} + u_{1}^{[2]} + p^{h−1}u_{1} + dv_{2} + tdλ_{1} ∧ w_{s−1}]

= [u_{2}^{[2]} + p^{h−1}u_{2} + dv_{3} + tλ_{1} ∧ w_{s−1}] ∧ dp.

Lemma (3.4) implies u_{2}^{[2]} + p^{h−1}u_{2} + dv_{3} = tλ_{1} ∧ w_{s−1} (4.15) in Ω_{N(p)}. Thus we are led to consider two cases

a) h > 1. Then

u_{2}^{[2]} + dv_{3} = tλ_{1} ∧ w_{s−1}

implies (see (3.4))

u_{2} = δ ∧ w_{s−1} + pu_{3} + u_{4} ∧ dp

with δ, u_{3}, u_{4} ∈ Ω_{F} M[X^{2}]. Inserting this expression for u_{2} in the equation for E_{p} we obtain p[p^{2h−2}E_{p} + u_{1}^{[2]} + p^{h−1}u_{1} + dv_{2} + tdλ_{1} ∧ w_{s−1} + pu_{3}^{[2]} ∧ dp + p^{h−1}u_{3} ∧ dp]

= [(δ ∧ w_{s−1})^{[2]} + p^{h−1}δ ∧ w_{s−1} + dv_{3} + tλ_{1} ∧ w_{s−1}] ∧ dp.

It follows

(δ ∧ w_{s−1})^{[2]} + dv_{3} + tλ_{1} ∧ w_{s−1} ∈ ker[Ω_{F}^{m}M[X^{2}] → Ω_{N(p)}^{m}]

which means that

dv_{3} = z ∧ w_{s−1} + pu_{4} + u_{5} ∧ dp

in Ω_{F}^{m}M[X^{2}]. This relation says dv_{3} ∈ Ω_{N(p)}^{m−s+1} ∧ w_{s−1}.

We apply now lemma (1.13) and remark (1.14), and we obtain in Ω_{F}^{m}M[X^{2}]

dv_{3} = (∑_{μ≠1} b^{μ}z_{μ} + b^{1}dB) ∧ w_{s−1} + d(pu_{4} + u_{5} ∧ dp)

with z_{μ}, B ∈ Ω_{F} M[X^{2}], dz_{μ} = 0 for all μ. Notice that the form (∑_{μ≠1} b^{μ}z_{μ} + b^{1}dB) ∧ w_{s−1} is exact, i.e. equal to some dH with H ∈ Ω_{F} M[X^{2}]. Inserting this in the above equation it follows

p[p^{2h−2}E_{p} + u_{1}^{[2]} + p^{h−1}u_{1} + dv_{2} + tdλ_{1} ∧ w_{s−1} + (pu_{3}^{[2]} + p^{h−1}u_{3} + du_{4}) ∧ dp]

= [(δ ∧ w_{s−1})^{[2]} + p^{h−1}δ ∧ w_{s−1} + tλ_{1} ∧ w_{s−1} + dH] ∧ dp.

Let us denote by A the expression inside the parenthesis of the right hand side of this equation, which is of the form A = B ∧ w_{s−1}, with some B ∈ Ω_{F} M[X^{2}]. The equation p[ ] = B ∧ w_{s−1} ∧ dp implies by lemma (3.4) B ∧ w_{s−1} = 0 in Ω_{N(p)}. Now we apply lemma (3.6) to conclude B ∧ w_{s−1} = (pμ_{1} +μ_{2} ∧dp)∧w_{s−1} in Ω_{F} M[X^{2}]. Inserting in the above relation we obtain then that we can assume B ∧ w_{s−1} = pμ_{1} ∧ w_{s−1} with μ_{1} ∈ Ω_{F} M[X^{2}]. Thus let us write pB ∧ w_{s−1} instead of B ∧ w_{s−1}. Multiplying the above equation by t and using tdp = d(pt) + pdt as well as w_{s−1} ∧ d(pt) = w_{s}, we get

p^{2h−2}E_{p} + u_{1}^{[2]} + p^{h−1}u_{1} + dv_{2} + tdλ_{1} ∧ w_{s−1} + (pu_{3}^{[2]} + p^{h−1}u_{3} + du_{4}) ∧ dp

= \frac{pA ∧ dt}{pt} + \frac{pB ∧ w_{s−1}}{pt} = A ∧ \frac{dt}{t} + \frac{B}{t} ∧ w_{s−1}.

This equation implies

E_{p} = ℘ (\frac{u_{1}}{p^{h−1}} ) + ℘ (\frac{u_{3} ∧ dp}{p^{h−1}}) + d (\frac{v_{2} + u_{2} ∧ dp}{p^{2(h−1)}})

+℘ (\frac{δ ∧ w_{s−1}}{p^{h−1}} ∧ \frac{dt}{t}) + d (\frac{H}{p^{2(h−1)}} ∧ \frac{dt}{t})

+\frac{tdλ_{1} ∧ w_{s−1} + tλ_{1} ∧ w_{s−1} ∧ \frac{dt}{t}}{p^{2h−2}} + \frac{1}{p^{2h−2}t}B ∧ w_{s}.

Notice tdλ_{1} ∧ w_{s−1} + λ_{1} ∧ w_{s−1} ∧ dt = d(tλ_{1} ∧ w_{s−1}), so that

E_{p} = ℘ (\frac{u_{1}}{p^{h−1}} + \frac{u_{3} ∧ dp}{p^{h−1}} + \frac{δ ∧ w_{s−1} ∧ dt}{p^{h−1}t})

+d (\frac{v_{2} + u_{4} ∧ dp}{p^{2h−2}} + \frac{(tλ_{1}) ∧ w_{s−1}}{p^{2h−2}} + \frac{H ∧ dt}{p^{2h−2}t})

+\frac{B}{p^{2h−2}t} ∧ w_{s}.

Thus

E_{p} = ℘ (\frac{u′}{p^{h−1}t}) + d (\frac{v′}{p^{2(h−1)}t}) + \frac{λ′}{tp^{2h−2}} ∧ w_{s}

with forms u′, v′, λ′ ∈ Ω_{F} M[X^{2}]. Using partial fraction decomposition of the forms u′/(p^{h−1}t), v′/(p^{2(h−1)}t), λ′/(p^{2h−2}t), we see that the exponent h of p in equation (4.13) can be reduced by one, although expressions for polynomials q of lower degree of the same type can appear, which will be absorbed by the corresponding E_{q}. Thus we are led to consider the case

(b) h = 1. Then we have (see (4.15))

u_{2}^{[2]} + u_{2} + dv_{3} = tλ_{1} ∧ w_{s−1} (4.16)

in Ω_{N(p)}. We can now apply lemma (3.7) to conclude

u_{2} = ∑_{i} ⋀_{j} g_{ij}df_{ij} + pv_{1} + v_{2} ∧ dp + δ ∧ w_{s−1}

with v_{1}, v_{2}, δ ∈ Ω_{F} M[X^{2}], f_{ij}, g_{ij} ∈ M[X^{2}], deg_{X} f_{ij} < deg_{X} p, f_{ij}g_{ij} = 1 + h_{ij}p in M[X^{2}]. Next we compute ℘(∑_{i} ⋀_{j} g_{ij}df_{ij}). We have

℘(∑_{i} ⋀_{j} g_{ij}df_{ij}) ≡ [∑_{i} ⋀_{j} g_{ij}f_{ij})\frac{df_{ij}}{f_{ij}} ]^{[2]} + ∑_{i} ⋀_{j} g_{ij}f_{ij} \frac{df_{ij}}{f_{ij}}

≡ ∑_{i}(∏_{j}(g_{ij}f_{ij})^{2} + ∏_{j} g_{ij}f_{ij}) ⋀_{j} \frac{df_{ij}}{f_{ij}}

≡ ∑_{i}(∏_{j} g_{ij}^{2}f_{ij} + ∏_{j} g_{ij}) ⋀_{j} df_{ij}

≡ ∑_{i} ∏_{j} g_{ij}(1 + ∏_{j}(1 + ph_{ij})) ⋀_{j} df_{ij}

≡ ∑_{i} p ∏_{j} g_{ij} · h_{i} ⋀_{j} df_{ij}

mod dΩ_{F} M[X^{2}], with certain h_{i} ∈ M[X^{2}]. Thus we have obtained

℘(∑_{i} ⋀_{j} g_{ij}df_{ij}) = p ∑_{i} g_{i} ⋀_{j} df_{ij} mod dΩ_{F} M[X^{2}]

with g_{i} ∈ M[X^{2}]. Let us insert u_{2} in (4.15). According to the above computation we obtain in Ω_{F} M[X^{2}]

℘(δ ∧ w_{s−1}) + dv_{3} + tλ_{1} ∧ w_{s−1} = pu_{4} + u_{5} ∧ dp

and this implies

dv_{3} = λ ∧ w_{s−1}

in Ω_{N(p)}, with some form λ ∈ Ω̄_{F} M[X^{2}] ⊂ Ω_{N(p)}. From lemma (1.13) and remark (1.14) we obtain in Ω_{F} M[X^{2}]

dv_{3} = (∑_{μ≠1} b^{μ}z_{μ} + b^{1}dB) ∧ w_{s−1} + d(pu_{4})

with z_{μ}, B ∈ Ω_{F} M[X^{2}], dz_{μ} = 0 for all μ. Notice that the form (∑_{μ≠1} b^{μ}z_{μ} + b^{1}dB) ∧ w_{s−1} is exact, i.e. equal to some dH with H ∈ Ω_{F} M[X^{2}]. Therefore

p[E_{p} + ℘u_{1} + dv_{2} + tdλ_{1} ∧ w_{s−1}] = [℘(∑_{i} ⋀_{j} g_{ij}df_{ij} + pv_{1} + δ ∧ w_{s−1}) +

(∑_{μ≠1} b^{μ}z_{μ} + b^{1}dB) ∧ w_{s−1} + d(pu_{4}) + tλ_{1} ∧ w_{s−1}] ∧ dp.

After dividing by p, we are led to consider the following expressions.

a) ℘(pv_{1}) ∧ \frac{dp}{p} = ℘(pv_{1} ∧ \frac{dp}{p}) = ℘(v_{1} ∧ dp) mod dΩ_{F} M[X^{2}] b)

℘ ∑_{i} ⋀_{j} g_{ij}df_{ij} ∧ \frac{dp}{p} = ℘ (∑_{i} ⋀_{j} g_{ij}f_{ij} \frac{df_{ij}}{f_{ij}} ∧ \frac{dp}{p})
= ℘ (∑_{i} ⋀_{j}(1 + ph_{ij})\frac{df_{ij}}{f_{ij}} ∧ \frac{dp}{p})
= ℘ (∑_{i} ⋀_{j} \frac{df_{ij}}{f_{ij}} ∧ \frac{dp}{p} + ∑_{i} H_{i} ⋀_{j} \frac{df_{ij}}{f_{ij}} ∧ dp)
= ℘ (∑_{i} H_{i} ⋀_{j} \frac{df_{ij}}{f_{ij}} ∧ dp) mod dΩ_{F} M[X^{2}]

with certain polynomials H_{i} ∈ M[X^{2}].

c) ℘(δ∧w_{s−1})∧\frac{dp}{p} = ℘(δ∧w_{s−1}∧\frac{dt}{t})+℘(δ∧w_{s−1}∧\frac{d(pt)}{pt}) mod dΩ_{F} M[X^{2}]

d) tλ_{1} ∧ w_{s−1} ∧ \frac{dp}{p} = λ_{1} ∧ w_{s−1} ∧ \frac{tdp}{p} = λ_{1} ∧ w_{s−1} ∧ \frac{d(tp)+pdt}{p} = tλ_{1} ∧ w_{s−1} ∧ \frac{d(tp)}{tp} + λ_{1} ∧ w_{s−1} ∧ dt (recall w_{s} = w_{s−1} ∧ d(pt))

e) (∑_{μ≠1} b^{μ}z_{μ} + b^{1}dB)∧w_{s−1} ∧ \frac{dp}{p} = (∑_{μ≠1} b^{μ}z_{μ} + b^{1}dB)∧w_{s−1} ∧ \frac{d(pt)}{pt} + (∑_{μ≠1} b^{μ}z_{μ} + b^{1}dB) ∧ w_{s−1} ∧ \frac{d(t)}{t}

Thus we get

E_{p} = ℘(u_{1} + v_{1} ∧ dp) + d(v_{2} + u_{4} ∧ w_{s−1} ∧ dp)

+℘(∑_{i} H_{i} ⋀_{j} \frac{df_{ij}}{f_{ij}} ∧ dp) + d(tλ_{1} ∧ w_{s−1})

+℘(δ ∧ w_{s−1}) ∧ \frac{dt}{t} + ℘(δ ∧ w_{s−1}) ∧ \frac{d(pt)}{pt} + \frac{λ_{1}}{p} ∧ w_{s}

E_{p} = ℘ (u_{1} + v_{1} ∧ dp + ∑_{i} H_{i} ⋀_{j} \frac{df_{ij}}{f_{ij}} ∧ dp + δ ∧ w_{s−1} ∧ \frac{dt}{t})

+d(v_{2} + u_{4} ∧ w_{s−1} ∧ dp + tλ_{1} ∧ w_{s−1}) + α_{3} ∧ w_{s} where ℘(δ ∧ w_{s−1}) ∧ \frac{d(pt)}{pt} + \frac{λ_{1}}{p} ∧ w_{s} = α_{3} ∧ w_{s}, α_{3} being a form of the type μ/pt with μ ∈ Ω_{F} M[X^{2}], i.e. α_{3} has a denominator containing p at most in the first power, and all other prime factors are of degree (in X) less than deg(p). Thus we have

E_{p} = ℘(α_{1}) + d(α_{2}) + α_{3} ∧ w_{s}

where α_{1}, α_{2} have numerator in Ω_{F} M[X^{2}] and denominators with prime factors of degree less than deg(p). In particular the above equation shows that α_{3} ∧ w_{s} is also a form of the type β/q where q is a product of prime polynomials of degree less than p in M[X^{2}]. We claim

α_{3} ∧ w_{s} = α ∧ w_{s} (4.17)

where α is a form of the type γ/q where q is a product of irreducible polynomials of degree < deg(p) contained in M[X^{2}]. Once we have shown this claim, we see that we can get rid of E_{p} in the equation (4.5). Thus we must show (after scaling (4.16) with a convenient polynomial q).

(4.18) Lemma. Let λ be a form in \frac{1}{p}Ω_{F} M[X^{2}] such that λ∧w_{s} ∈ Ω_{F} M[X^{2}]. Then there is some form λ′ ∈ Ω_{F} M[X^{2}] with

λ ∧ w_{s} = λ′ ∧ w_{s}.

Proof. Set λ = p^{−1}λ_{0} with λ_{0} ∈ Ω_{F} M[X^{2}]. We can obviously assume that deg_{X} λ_{0} < deg_{X} p. We write dT = k_{s}db_{s}+dT′, with deg_{X} k_{s} = 2, deg_{X} T′ = 0 and dT′ not containing db_{s}. Since w_{s} = db_{1} ∧⋯∧db_{s−1} ∧dT, we may assume that λ_{0} is only generated by forms db_{j}, j ≥ s. Set λ_{0} = λ_{s} ∧ db_{s} + λ′_{s} with λ_{s}, λ′_{s} generated by forms db_{j}, j > s. Then

λ ∧ w_{s} = p^{−1}λ_{0} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ (k_{s}db_{s} + dT′)
= p^{−1}db_{1} ∧ ⋯ ∧ db_{s−1} ∧ db_{s} ∧ (λ_{s} ∧ dT′ + k_{s}λ′_{s})
+p^{−1}db_{1} ∧ ⋯ ∧ db_{s−1} ∧ λ′_{s} ∧ dT′.

Because of the above choices, both summands do not interfere with each other, so that both must be integral, since λ ∧ w_{s} is integral. In particular

p^{−1}db_{1} ∧ ⋯ ∧ db_{s−1} ∧ λ′_{s} ∧ dT′ ∈ Ω_{F} M[X^{2}].

Since deg_{X} λ′_{s} < deg_{X} p and deg_{X} T′ = 0, it follows

db_{1} ∧ ⋯ ∧ db_{s−1} ∧ λ′_{s} ∧ dT′ = 0.

But λ′_{s}∧dT′ is generated only by differentials db_{j}, j ≥ s, so that we obtain λ′_{s} ∧ dT′ = 0. It follows λ′_{s} = δ ∧ dT′ with some form δ ∈ Ω_{F} M[X^{2}], and deg_{X} δ < deg_{X} p. Therefore

λ ∧ w_{s} = p^{−1}db_{1} ∧ ⋯ ∧ db_{s} ∧ (λ_{s} ∧ dT′ + k_{s}δ ∧ dT′)
= p^{−1}db_{1} ∧ ⋯ ∧ db_{s} ∧ (λ_{s} + k_{s}δ) ∧ dT′

Since db_{1} ∧ ⋯ ∧ db_{s} ∧ dT′ does not contain X we easily see that p divides λ_{s} + k_{s}δ i.e. λ_{s} + k_{s}δ = p · μ with μ ∈ Ω_{F} M[X^{2}]. Hence

λ_{0} = (k_{s}δ + pμ) ∧ db_{s} + δ ∧ dT′
= δ ∧ dT + pμ ∧ db_{s}.

This implies

λ ∧ w_{s} = p^{−1}λ_{0} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT
= p^{−1}pμ ∧ db_{s} ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT
= μ ∧ w_{s}

with μ ∈ Ω_{F} M[X^{2}]. This proves the claim. □

Therefore, using this descent procedure we have shown the following result.

(4.19) Proposition. If w ∈ Ω_{F}^{n} satisfies an equation

w = ℘u + dv + λ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

with u, v, λ ∈ Ω_{F} M(X^{2}) and T as before, then w satisfies an equation

w = ℘u′ + dv′ + λ′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT

with u′, v′, λ′ ∈ Ω_{F} M[X^{2}].

(4.20) Lemma. Let w ∈ Ω_{F}^{n} be such that there exist u, v, λ ∈ Ω_{F} M[X^{2}] with w = ℘u + dv + λ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT, T = b_{s}X^{2} + T′, deg_{X} T′ = 0. Then there exist u′, v′, λ′ ∈ Ω_{F} M with

w = ℘u′ + dv′ + λ′ ∧ db_{1} ∧ ⋯ ∧ db_{s−1} ∧ db_{s} ∧ dT′.

Proof. Set deg_{X} u = 2h, deg_{X} v = 2k, deg_{X} λ = 2ℓ. We consider the following cases:

a) 4h ≥ 2k. Then the coefficient of X^{4h} in ℘u + dv + λ ∧ w_{s} (w_{s} = db_{1} ∧ ⋯ ∧ db_{s−1} ∧ dT ) is the form u_{2h}^{[2]} + dv_{4h} + λ_{4h−2} ∧ w_{s−1} ∧ db_{s}, where w_{s−1} = db_{1} ∧ ⋯ ∧ db_{s−1}. Then we must have

u_{2h}^{[2]} + dv_{4h} + λ_{4h−2} ∧ w_{s−1} ∧ db_{s} = 0

if h ≥ 1. Here we have used the following notation: u = u_{0} + u_{2}X^{2} + ⋯ + u_{2h}X^{2h}, with u_{i} ∈ Ω_{F} M. Then applying the Cartier operator

0 = u_{2h} + C(λ_{4h−2} ∧ w_{s−1} ∧ db_{s}).

From lemma (1.12) we obtain in Ω_{F (X_{μ})}

u_{2h} = μ ∧ w_{s−1} ∧ db_{s}.

Since u_{2h} ∈ Ω_{F} M, it is easy to conclude that we can choose μ ∈ Ω_{F} M. Therefore

u = u_{0} + u_{2}X^{2} + ⋯ + u_{2h}X^{2h}
= u_{0} + ⋯ + μ ∧ w_{s−1} ∧ db_{s}X^{2h}
= u_{0} + ⋯ + μ ∧ w_{s−1} ∧ (dT + dT′)X^{2h−2}
= u_{0} + ⋯ + u′_{2h−2}X^{2h−2} + μX^{2h−2} ∧ w_{s−1} ∧ dT

where u′_{2h−2} = u_{2h−2} + μ ∧ dT′. The term μX^{2h−2} ∧ w_{s−1} ∧ dT can be added to λ ∧ s_{s−1} ∧ dT replacing λ by λ + μX^{2h−2}. Thus we have lowered the degree of u. Iterating this procedure we are led to consider the following case

b) 4h < 2k. It follows 2ℓ ≤ 2k − 2, and the coefficient of X^{2k} is dv_{2k} + λ_{2k−2} ∧ w_{s−1} ∧ db_{s}. Therefore if k ≥ 1, we must have

dv_{2k} + λ_{2k−2} ∧ w_{s−1} ∧ db_{s} = 0.

Then according (1.13) there exist forms z_{μ} ∈ Ω_{F} M and B ∈ Ω_{F} M such that

dv_{2k} = ∑_{μ≠1_{s−1}}[b^{μ}(z_{μ,2} + b_{s}z_{μ,3}) + b^{1_{s−1}}(z_{1} + b_{s}dB)] ∧ w_{s−1} ∧ db_{s}

with all z_{μ} closed, and z_{μ,2}, z_{μ,3}, z_{1}, B not containing b_{s} in the 2-expansion of their coefficients. The term b^{μ}z_{μ,2} ∧ w_{s−1} is exact for all μ ≠ 1_{s−1} i.e. b^{μ}z_{μ,2} ∧ w_{s−1} ∈ dΩ_{F} M. Also b_{s}b^{μ} ∧ w_{s−1} ∈ b_{s}dΩ_{F} M. The form b^{1_{s−1}}z_{1} ∧ w_{s−1} is closed and hence contained in dΩ_{F} M + (Ω_{F} M)^{[2]} and b_{s}b^{1_{s−1}}dB ∧ w_{s−1} ∈ b_{s}dΩ_{F} M.

Putting all this together, we obtain

dv_{2k} = (A^{[2]} + dC + b_{s}dE) ∧ db_{s}

with forms A, C, E ∈ Ω_{F} M, which moreover are multiples of w_{s−1}. From

λ_{2k−2} ∧ w_{s−1} ∧ db_{s} = (A^{[2]} + dC + b_{s}dE) ∧ db_{s}

we infer

λ_{2k−2} ∧ w_{s−1} = A^{[2]} + dC + b_{s}dE + F ∧ db_{s}

with some F ∈ Ω_{F} M, which can be chosen as a multiple of w_{s−1}, since F does not have terms containing db_{s} and F ∧ db_{s} ∈ ⟨w_{s−1}⟩. Therefore

λ ∧ w_{s−1} = (λ_{0} + λ_{2}X^{2} + ⋯ + λ_{2k−2}X^{2k−2}) ∧ w_{s−1}
λ ∧ w_{s−1} ∧ dT = (λ_{0} + λ_{2}X^{2} + ⋯ + λ_{2k−4}X^{2k−4}) ∧ w_{s−1} ∧ dT
+X^{2k−2}(A^{[2]} + dC + b_{s}dE + F ∧ db_{s}) ∧ dT.

The terms X^{2k−2}A^{[2]} ∧ dT and X^{2k−2}dC ∧ dT are exact, and hence can be absorbed by dv without increasing the degree of v. Now

X^{2k−2}b_{s}dE ∧ dT = X^{2k−4}(X^{2}b_{s})dE ∧ dT
= X^{2k−4}(T + T′)dE ∧ dT
= d(X^{2k−4}T E ∧ dT) + X^{2k−4}T′dE ∧ dT.

The first summand can be added to dv without increasing the degree of v. Since dE is a multiple of w_{s−1}, it follows that the second summand in this expression is contained in X^{2k−4}Ω_{F} M ∧ w_{s−1} ∧ dT and hence it can be added to the term λ_{2k−4} · X^{2k−4} of degree 2k − 4 in λ ∧ w_{s−1} ∧ dT. Finally the term X^{2k−2}F ∧ db_{s} ∧ dT = X^{2k−4}F ∧ dT′ ∧ dT is contained in X^{2k−4}Ω_{F} M ∧ w_{s−1} ∧ dT = X^{2k−4}Ω_{F} M ∧ w_{s}, because F is a multiple of w_{s−1}, and therefore this term is absorbed by the term of degree 2k − 4 in λ ∧ w_{s}.

This shows that can assume deg λ ≤ 2k − 4. But this implies dv_{2k} = 0, i.e. deg v < 2k. Thus we have lowered the degree of v, and iterating this procedure we come again to the case a).

Combining the procedures of cases a) and b) we finally arrive at an equation

w = u^{[2]} + u + dv + λ ∧ w_{s−1} ∧ dT

with u, v, λ ∈ Ω_{F} M. Since λ∧w_{s−1}∧dT = X^{2}λ∧w_{s−1}∧db_{s}+λ∧w_{s−1}∧dT′, we see that X^{2}λ∧w_{s−1}∧db_{s} ∈ Ω_{F} M, and this is possible only if λ∧w_{s−1}∧db_{s} = 0. We have assumed that λ does not contain terms with db_{1}, …, db_{s−1}, so that it followsλ = λ′ ∧ db_{s} in Ω_{F} M. Thus

w = ℘u + dv + λ′ ∧ db_{1} ∧ ⋯ ∧ db_{s} ∧ dT

and this proves the lemma. □

We are now ready to prove the main result of this paper.

Proof of Theorem (4.1) Let us introduce the following subfields of L = F (X_{μ}|μ ∈ S_{n}) and polynomials

L_{0} = L = F (X_{μ}|μ ∈ S_{n})
T_{0} = T = ∑_{μ∈S_{n}} b^{μ}X_{μ}^{2}

M_{0} = M = F (X_{μ}^{2}|μ ∈ S_{n})
L_{1} = F (X_{μ}|μ ≠ e_{1}), e_{1} = (1, 0, …, 0)
T_{1} = ∑_{μ≠e_{1}} b^{μ}X_{μ}^{2}

M_{1} = F (X_{μ}^{2}|μ ≠ e_{1})

⋮
L_{j} = F (X_{μ}|μ ≠ e_{1}, …, e_{j}), e_{i} = (0, … 1, …, 0)
T_{j} = ∑_{μ≠e_{1},... ,e_{j}} b^{μ}X_{μ}^{2}

M_{j} = F (X_{μ}^{2}|μ ≠ e_{1}, …, e_{j})

where j = 1, 2, …, n. We have L_{j−1} = L_{j}(X_{j}),T_{j+1} = T_{j} +b_{j+1}X_{j+1}^{2} ∈ M_{j+1}.

Thus equation (4.2) corresponds to (4.3) with s = 1, i.e. w = ℘u + dv + λ∧dT_{0} with w ∈ Ω_{F} , u, v, λ ∈ Ω_{F} M_{0}. The above process implies an equation w = ℘u′ + dv′ + λ′ ∧ dT_{0} with u′, v′, λ′ ∈ Ω_{F} M_{1}[X_{1}^{2}]. Lemma (4.19) implies now

w = ℘u′′ + dv′′ + λ′′ ∧ db_{1} ∧ dT_{1}

with u′′, v′′, λ′′ ∈ Ω_{F} M_{1}. We continue with this process for s = 2, …, n − 1 until we get forms ū, v̄, λ̄ ∈ Ω_{F} M_{n−1} such that

w = ℘ū + dv̄ + λ̄ ∧ db_{1} ∧ ⋯ ∧ db_{n−1} ∧ dT_{n−1}.

But we have

db_{1} ∧ ⋯ ∧ db_{n−1} ∧ dT_{n−1} = db_{1} ∧ ⋯ ∧ db_{n−1} ∧ k_{n}db_{n}

where k_{n} = b_{n}^{−1} ∑_{μ(n)=1} b^{μ}X_{μ}^{2}. Therefore we obtain an equation

w = ℘ū + dv̄ + λ̄′ ∧ db_{1} ∧ ⋯ ∧ db_{n} (4.21)

in Ω_{F} M_{n−1}. We can now get rid of the remaining variables just by comparing coefficients. Suppose namely that k is a field, K = k(X) a pure transcendental extension in one variable of k, w ∈ Ω_{k} and that there is a relation

w = ℘u + dv + λ ∧ w_{n} (4.22)

in Ω_{k}(X^{2}), where w_{n} is now defined over k. We decompose u, v, λ in partial fractions

u = u_{0} + ∑ u_{p}
v = v_{0} + ∑ v_{p}
λ = λ_{0} + ∑ λ_{p}

with u_{0}, v_{0}, λ_{0} ∈ Ω_{k}[X^{2}], u_{p}, v_{p}, λ_{p} ∈ p^{−∞}Ω_{k}[X^{2}]. Since the operators ℘ and d respect these decomposition, we conclude from (4.21) that

w = ℘u_{0} + dv_{0} + λ_{0} ∧ db_{1} ∧ ⋯ ∧ db_{n}

in Ω_{k}[X^{2}]. Letting X = 0, we obtain w = ℘ū_{0} + dv̄_{0} + λ̄_{0} ∧ db_{1} ∧ ⋯ ∧ db_{n} in Ω_{k}. Applying this argument to the equation (4.21) we conclude

w = ℘α + dβ + γ ∧ db_{1} ∧ ⋯ ∧ db_{n}

with α, β, γ ∈ Ω_{F}. This finishes the proof of the theorem. □

5 Quadratic forms and differential forms

We review briefly in this section for the sake of completeness some basic notations and results from quadratic form theory and its relations with differential forms over fields of characteristic two. Our basic references will be [ A-Ba 1], [ Ba 1], [ Ka 1] and [ Sa]. Let F be a field of characteristic two. We denote by [a, b] the binary nonsingular quadratic form ax^{2} +xy +by^{2} (a, b ∈ F). Any non singular quadratic form over F is of the form ⊥_{i=1}^{n} [a_{i}, b_{i}], (a_{i}, b_{i} ∈ F), where ⊥ means orthogonal sum. The form [0, 0] is the hyperbolic plane and any orthogonal sum ⊥ [0, 0] is called a hyperbolic space. Two quadratic forms q_{1}, q_{2} are called equivalent (q_{1} ∼ q_{2}) if H_{1} ⊥ q_{1} ≃ H_{2} ⊥ q_{2}, where H_{1}, H_{2} are hyperbolic spaces. A form q is called isotropic if there is a nonzero vector x with q(x) = 0, otherwise q is called anisotropic. The set of equivalence classes of anisotropic quadratic forms over F form the Witt-group W_{q}(F) (with respect to orthogonal sums). Respectively, we denote by ⟨a⟩ the one dimensional symmetric bilinear form axy (a ∈ F^{∗}) and by ⟨a_{1}, …, a_{n}⟩ the orthogonal sum ⟨a_{1}⟩ ⊥ … ⊥ ⟨a_{n}⟩. Let W (F) be the Witt ring of F, i.e. the ring of classes of non singular symmetric bilinear forms over F. Then W_{q}(F) is a W (F)-module via the operation b ⊗ q(x ⊗ y) = b(x, x) · q(y) (see [ Ba 1], [ Sa]). The maximal ideal I ⊂ W (F) of even dimensional bilinear forms is additively generated by the 1-Pfister forms ⟨1, a⟩, a ∈ F^{∗}, so that the n-power I^{n} is additively generated by the n-fold Pfister forms ≪ a_{1}, …, a_{n} ≫= ⟨1, a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{n}⟩. We get the submodules I^{n}W_{q}(F) of W_{q}(F), n ≥ 0, which are additively generated by the quadratic n-fold Pfister forms ≪ a_{1}, …, a_{n}; b]] =≪ a_{1}, …, a_{n} ≫ ⊗[1, b] , where [1, b] = x^{2} +xy +by^{2} is a 0-fold Pfister form. Thus we have the filtration W (F) ⊃ I ⊃ I^{2} ⊃ ⋯ and W_{q}(F) ⊃ IW_{q}(F) ⊃ I^{2}W_{q}(F) ⊃ · · ·. In [ Ka 1] it is shown that there are a natural isomorphisms (see section 2 for the definition of ν_{F} (n) and H^{n+1}(F))

α : ν_{F} (n) →^{∼} I_{F}^{n}/I_{F}^{n+1} (5.1)

β : H^{n+1}(F) →^{∼} I^{n}W_{q}(F)/I^{n+1}W_{q}(F) (5.2)

given on generators by

\begin{aligned}α (\frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}} ) & = & ≪ x_{1}, …, x_{n} ≫ mod I^{n} \\ β (\overline{b\frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}}} ) & = & ≪ x_{1}, …, x_{n}; b]] mod I^{n}W_{q}(F). \end{aligned}

If L/F is any field extension, any quadratic form q (or bilinear form b) over F can be viewed as a form over L, which we denote by q_{L} ( or b_{L}). The natural homomorphisms W_{q}(F) → W_{q}(L), resp. W (F) → W (L) are compatible with the above isomorphisms, i.e.

[figure: ν_{L}(n); I_{L}^{n}/I_{L}^{n+1}; ⟶^{∼}; ν_{F} (n); I_{F}^{n}/I_{F}^{n+1}; ⟶^{∼}]

resp.

[figure: H^{n+1}(L); I^{n}W_{q}(L)/I^{n+1}W_{q}(L); ⟶^{∼}; H^{n+1}(F); I^{n}W_{q}(F)/I^{n+1}W_{q}(F); ⟶^{∼}]

In particular the main isomorphism (4.1) can be restated in terms of quadratic forms as follows.

(5.3) Theorem. Let ϕ =≪ b_{1}, …, b_{n} ≫ be an anisotropic bilinear n-fold Pfister form over F. Then

ker [\frac{I^{n}W_{q}(F)}{I^{n+1}W_{q}(F)} → \frac{I^{n}W_{q}(F (ϕ))}{I^{n+1}W_{q}(F (ϕ))}] = \{\overline{ϕ ⊗ [1, b]}|b ∈ F \}.

Let p =≪ b_{1}, …, b_{n}; b]] be an anisotropic quadratic n -fold Pfister form over F. Let F (p) be the function field of the quadric {p = 0} over F. In [ A-Ba 2] we have shown that (5.3) implies the following result, whose proof will be given here for the sake of completeness.

(5.4) Theorem. Let p be as above. Then

ker [\frac{I^{n}W_{q}(F)}{I^{n+1}W_{q}(F)} → \frac{I^{n}W_{q}(F (p))}{I^{n+1}W_{q}(F (p))}] = \{0, p̄\}

Proof. Let q ∈ I^{n}W_{q}(F) be such that q_{F (p)} ∈ I^{n+1}W_{q}(F (p)). Set q = ∑_{i=1}^{r} ϕ_{i}[1, a_{i}], ϕ_{i} an n-fold bilinear Pfister form over F, 1 ≤ i ≤ r. If r = 1, i.e. q = ϕ[1, a], then the above assumption implies that ϕ[1, a] is hyperbolic over F (p) (see [ Ba 2]) and then by the norm theorem ( see [ Ba 3]) we conclude ϕ[1, a] ≃ p over F, i.e. q̄ = p̄. Now assume r > 1, and we will prove the assertion by induction on r. Thus we assume the assertion true for any field and any form of length less than r. Thus without restriction ϕ_{r}[1, a_{r}] is anisotropic. Set ψ = ϕ_{r}, and let F (ψ) be its function field. Then q_{F (ψ)} = ∑_{1}^{r−1} ϕ_{i}[1, a_{i}] ∈ I^{n}W_{q}(F (ψ)) and over F (ψ)(p) we get q_{F (ψ)(p)} ∈ I^{n+1}W_{q}(F (ψ)(p)).

Therefore by induction we obtain

q_{F (ψ)} ≡ ϵp_{F (ψ)} mod I^{n+1}W_{q}(F (ψ))

with ϵ = 0 or 1. Thus

(q ⊥ ϵp)_{F (ψ)} ∈ I^{n+1}W_{q}(F (ψ)).

From (5.1) we conclude q ⊥ ϵp ≡ ψ[1, c] mod I^{n+1}W_{q}(F) with some c ∈ F. Since q_{F (p)} ∈ I^{n+1}W_{q}(F (p)), it follows ψ[1, c]_{F (p)} ∈ I^{n+1}W_{q}(F (p)), i.e. ψ[1, c] is hyperbolic over F (p), and hence by the norm theorem (see loc. cit. ) ψ[1, c] ≡ ηp mod I^{n+1}W_{q}(F) with η = 0 or 1. Therefore q ≡ (ϵ + η)p mod I^{n+1}W_{q}(F). This conclude the proof. □

Using the isomorphism (5.2) we can restate the above result as follows.

(5.5) Theorem. Let p =≪ b_{1}, …, b_{n}; b]] be an anisotropic quadratic n-fold Pfister form over F. Then

H^{n+1}(F (p)/F) = \{0, \overline{b\frac{db_{1}}{b_{1}} ∧ ⋯ ∧ \frac{db_{n}}{b_{n}}} \} .

(5.6) Remark. The analogue of (5.3) for bilinear forms is the following result

(5.7) Proposition. Let ϕ =≪ b_{1}, …, b_{n} ≫ be an anisotropic bilinear n-fold Pfister form. Then for m ≥ n

ker [\frac{I_{F}^{m}}{I_{F}^{m+1}} → \frac{I_{F (ϕ)}^{m}}{I_{F (ϕ)}^{m+1}} ] = \{∑ I^{m−n}\overline{≪ x_{1}, …, x_{n} ≫}|x_{1}, …, x_{n} ∈ F^{2}(b_{1}, …, b_{n})\} .

In the case m = n, we have

ker [\frac{I_{F}^{n}}{I_{F}^{n+1}} → \frac{I_{F (ϕ)}^{n}}{I_{F (ϕ)}^{n+1}} ] = \{\overline{≪ x_{1}, …, x_{n} ≫}|x_{1}, …, x_{n} ∈ F^{2}(b_{1}, …, b_{n})\} .

We will show the special case m = n for simplicity.

If x_{1}, …, x_{n} ∈ F^{2}(b_{1}, …, b_{n}), then dx_{i} ∈ F db_{1} ⊕ ⋯ ⊕ F db_{n} and hence

\frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}} = a\frac{db_{1}}{b_{1}} ∧ ⋯ ∧ \frac{db_{n}}{b_{n}}

with some a ∈ F^{2}(b_{1}, …, b_{n}). Since a\frac{db_{1}}{b_{1}} ∧ ⋯ ∧ \frac{db_{n}}{b_{n}} ∈ ker(Ω_{F}^{n} → Ω_{F (ϕ)}^{n}), it follows \frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}} ∈ ker[ν(n)_{F} → ν(n)_{F (ϕ)}]. Thus

\overline{≪ x_{1}, …, x_{n} ≫} = α(\frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}}) ∈ ker [\frac{I_{F}^{n}}{I_{F}^{n+1}} → \frac{I_{F (ϕ)}^{n}}{I_{F (ϕ)}^{n+1}} ] .

Conversely take any \overline{ψ} = α(w) ∈ ker [\frac{I_{F}^{n}}{I_{F}^{n+1}} → \frac{I_{F (ϕ)}^{n}}{I_{F (ϕ)}^{n+1}} ], with w ∈ ker[ν(n)_{F} →

ν(n)_{F (ϕ)}], i.e. w = a\frac{db_{1}}{b_{1}} ∧ ⋯ ∧ \frac{db_{n}}{b_{n}}. Using Kato’s lemma (see (2.15)) one immediately obtains w = \frac{dx_{1}}{x_{1}} ∧⋯∧ \frac{dx_{n}}{x_{n}} with x_{1}, …, x_{n} ∈ F^{2}(b_{1}, …, b_{n}) and this implies \overline{ψ} = \overline{≪ x_{1}, …, x_{n} ≫}.

In particular, since this kernel is a group, we obtain

(5.8) Corollary. Given x_{1}, …, x_{n}, y_{1}, …, y_{n} ∈ F^{2}(b_{1}, …, b_{n}), then there exist z_{1}, …, z_{n} ∈ F^{2}(b_{1}, …, b_{n}) with

≪ x_{1}, …, x_{n} ≫ + ≪ y_{1}, …, y_{n} ≫=≪ z_{1}, …, z_{n} ≫ mod I_{F}^{n+1}

This corollary says that the field F^{2}(b_{1}, …, b_{n}) is n-linked relative to F. In particular if {b_{1}, …, b_{N}} is a 2-basis of the field F, i.e. F = F^{2}(b_{1}, …, b_{N}),

then I_{F}^{N+1} = 0 and it follows

I_{F}^{N} = {≪ x_{1}, …, x_{N} ≫ |x_{i} ∈ F^{∗}} .

6 Generic splitting of quadratic forms and the degree conjecture

Since Knebusch’s seminal papers on generic splitting of quadratic forms over fields of characteristic ≠ 2 (see [ Kn 1], [ Kn 2]) appeared, few work has been done on the subject (see [ Ar-Kn], [ F], ... ). In particular Knebusch’s degree conjecture I^{n} = J_{n}, where J_{n} is the ideal of W (F) of forms of degree ≥ n remains still open. In what follows we will briefly develop the analog of Knebusch’s theory over fields of characteristic 2 and using the results of section 5 we will show that in this case the corresponding degree conjecture is true.

Our main reference will be Knebusch’s paper [ Kn] on reduction theory of quadratic and bilinear forms, which holds true for fields of any characteristic, as well as his generic splitting papers cited above. Many of the definitions and results of Knebusch’s theory can be extended (using [ Ba 2], [ Ba 3]) mutatis mutandis to the case 2 = 0, so that we will often refer to the above papers for proofs.

From now on all fields have characteristic 2. The most basic notion in this theory is that of generic zero field of a quadratic form q over F. A field extension L/F is a generic zero field of q if q_{L} is isotropic and if E/F is any extension with q_{E} isotropic, then there exists a F -place λ : L → E ∪ ∞ (see [ La]). One easily checks that the function field F (q) of q is a generic field of q. If q = ⟨a_{1}⟩[1, b_{1}] ⊥ ⋯ ⊥ ⟨a_{n}⟩[1, b_{n}], then F (q) = F (x_{1}, …, x_{n}, y_{1}, …, y_{n}) with the single relation ∑_{1}^{n} a_{i}(x_{i}^{2} + x_{i}y_{i} + b_{i}y_{i}^{2}) = 0. Also the field F (q)_{0} = F (u_{1}, …, u_{n−1}, v_{1}, …, v_{n−1}, z) with a_{n}(z^{2} + z + b_{n}) + ∑_{1}^{n−1} a_{i}(u_{i}^{2} + u_{i}v_{i} + b_{i}v_{i}^{2}) = 0 is a generic zero field of q. F (q)_{0} is purely transcendental over F if and only if q is isotropic over F. Starting with a non singular quadratic form q over F we can define a field tower F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{h} and forms q = q_{0}, q_{1}, …, , q_{h} defined over F_{0}, F_{1}, …, F_{h} respectively such that q_{r} is anisotropic over F_{r}, 0 ≤ r ≤ h−1, q_{h} is hyperbolic over F_{h}, and q_{r−1}⊗F_{r} ≃ q_{r} ⊥ i_{r} × [0, 0] with some integer i_{r}, F_{r} being a generic zero field of q_{r−1} over F_{r−1} . The sequence (F_{r}, q_{r}, i_{r}, 0 ≤ r ≤ h) is called a generic splitting tower of q. Recall that two field extensions L_{1}/F, L_{2}/F are called F -equivalent if there exist F -places λ_{1} : L_{1} → L_{2} ∪ ∞, λ_{2} : L_{2} → L_{1} ∪ ∞. Then any field L which is F -equivalent to F_{h} is called a generic splitting field of q and any field F -equivalent to F_{h−1} is called a leading field of q. A generic splitting tower of q is essentially unique in the sense that if (F_{r}, q_{r}, i_{r}, 0 ≤ r ≤ h), (F′_{s}, q′_{s}, i′_{s}, 0 ≤ s ≤ h′) are two generic splitting towers of q, then h = h′ , i_{r} = i′_{r}, 0 ≤ r ≤ h, F_{h} is equivalent with F′_{h}. The number h = h(q) is called the height of q. Obviously any form ⟨a⟩p, a ∈ F^{∗} and p a Pfister form, has height 1. Conversely if h(q) = 1, then q_{F (q)} is hyperbolic and the norm theorem proved in [ Ba 2] implies immediately that q ≃ ⟨a⟩p with a ∈ F^{∗} and p a Pfister form. In particular for any form q, the form q_{h−1} is similar to a nfold Pfister form over F_{h−1}. The degree n of this form is uniquely determined and we will call it the degree of q and we denote it by deg(q). If q is hyperbolic we set deg(q) = ∞. For any extension L/F we have deg(q_{L}) ≥ deg(q) and deg is a well defined function on W_{q}(F), deg : W_{q}(F) → N ∪{∞}. We define for any n ≥ 0

J(n) = {q̄ ∈ W_{q}(F)| deg(q) ≥ n}. (6.1)

One easily checks that J(0) = W_{q}(F), J(1) = IW_{q}(F), J(2) = I^{2}W_{q}(F) (see [ Ba 1], [ A-Ba 1] ). First we show that J(n) is a W (F) submodule of W_{q}(F) and that I^{n}W_{q}(F) ⊆ J(n). The key fact is the following result (compare [ Kn 1]).

(6.2) Proposition. Let q = ⟨a⟩p ⊥ q′ be a quadratic form over F, where p is an anisotropic Pfister form of degree n ≥ 1, a ∈ F^{∗} and deg(q′) ≥ n + 1. Let L be a leading field of q. Then

i) deg(q) = n

ii) p_{L} is a leading form of q.

iii) If deg(q′) ≥ n + 2, then p_{L} is anisotropic and q_{L} is Witt-equivalent to ⟨a⟩p_{L} with some a ∈ L^{∗}.

Proof. We may assume that q′ is not hyperbolic. We will show that deg(q) = n. Let (L_{i}, q′_{i}, 0 ≤ i ≤ e) be a generic splitting tower of q′. Then p_{L_{e}} is anisotropic. Otherwise p_{L_{e}} is hyperbolic and we can choose 0 ≤ m ≤ e maximal with p_{L_{m}} anisotropic. Then p is hyperbolic over L_{m}(q′_{m}) and the norm theorem (see [ Ba 2]) together with the sub form theorem (see [ Ba 3]) show that ⟨a⟩q′_{m} is a sub form of p_{L_{m}} for some b ∈ L_{m}^{∗}. In particular deg(q′) = deg(q′_{m}) ≤ n which is a contradiction. Thus p_{L_{e}} is anisotropic, and therefore ⟨a⟩p_{L_{e}} is the kernel form of q_{L_{e}}. This shows deg(q) ≤ n. If deg(q) = m < n, let L be a leading field of q with ker(q_{L}) = ⟨c⟩r, c ∈ L^{∗} and r a m-fold Pfister form over L. Thus q′_{L} ∼ ⟨c⟩r ⊥ ⟨a⟩p_{L}, and since dim(⟨c⟩r ⊥ ⟨a⟩p_{L}) = 2^{m} + 2^{n} < 2^{n+1}, it follows q′_{L} ∼ 0 because deg(q′) ≥ n + 1. Thus ⟨c⟩r ∼ ⟨a⟩p_{L} over L, and this implies p_{L} hyperbolic and hence r is hyperbolic, which is a contradiction. Thus deg(q) = n. The rest of the proposition follows easily and we omit the proof. □

(6.3) Corollary. For any two forms q_{1}, q_{2} over F

deg(q_{1} ⊥ q_{2}) ≥ min{deg(q_{1}), deg(q_{2})}

and if deg(q_{1}) ≠ deg(q_{2}), the equality holds.

From these results we immediately obtain

(6.4) Theorem.

(i) J(n) is a W (F)-sub-module of W_{q}(F)

(ii) I^{n}W_{q}(F) ⊂ J(n)

(iii) I^{m}J(n) ⊂ J(m + n).

Proof. (i) The above corollary shows that J(n) is subgroup of W_{q}(F). Since for any a ∈ F^{∗}, deg(⟨a⟩q) = deg(q), again the same corollary implies deg(⟨a_{1}, …, a_{m}⟩q) ≥ deg(q) for any a_{1}, …, a_{m} ∈ F^{∗}, i.e. J(n) is a W (F)-submodule of W_{q}(F). This shows (i). Since I^{n}W_{q}(F) is additively generated by the n-fold Pfister forms ≪ a_{1}, …, a_{n}; a]] of degree n, (ii) follows from (i), (iii) is also an immediate corollary of (i). □

If F is a field of characteristic different from 2, then one of the major conjectures of Knebusch’s generic splitting theory is the equality J_{n} = I^{n} in W (F). In [ A-Ba 2] Theorem 1, we have shown that the equality I^{n}W_{q}(F) = J(n) for any field F is equivalent with the equality

ker[I^{n}W_{q}(F)/I^{n+1}W_{q}(F) → I^{n}W_{q}(F (p))/I^{n+1}W_{q}(F (p))] = {0, p̄}

for any quadratic n-fold Pfister form p over F. Now if F is a field of characteristic 2, theorem (5.4) asserts that this equality is true. Thus combining these results we have the following.

(6.5) Main theorem. Let F be a field of characteristic 2. Then for any n ≥ 0

I^{n}W_{q}(F) = J(n).

(6.6) Remark. Of course (6.5) implies (6.4), but the above outlined proof of (6.4) is much more elementary.

References

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[ 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.: A note on generic splitting of quadratic forms. Comm. Algebra, 27(7), 3473-3477 (1999).

[ Ba 1] Baeza, R.: Quadratic forms over semi-local rings . Lecture Notes in Math. Vol. 655, Springer-Verlag (1978).

[ Ba 2] Baeza, R.: The norm theorem for quadratic forms over a field of characteristic two. Comm. Algebra. 18(5), 1337-1348 (1990).

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[ Ka 2] Kato, K.: Galois cohomology of complete discrete valuation fields. Algebraic K-theory. Proc. Oberwolfach, Part II (R.K. Dennis, Ed.) Lecture Notes in Math. Vol. 967, Springer-Verlag (1982).

[ Kn] Knebusch, M.: Specialization of quadratic and symmetric bilinear forms, and a norm theorem. Acta Arith. 24, 279-299 (1973).

[ Kn 1] Knebusch, M.: Generic splitting of quadratic forms I. Proc. London Math. Soc. (3) 33, 65-93 (1976).

[ Kn 2] Knebusch, M.: Generic splitting of quadratic forms II. Proc. London Math. Soc. (3) 34, 1-31 (1977).

[ La] Lang, S.: Algebra. (Third Edition) Addison Wesley (1983).

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

[ OVV] Orlov, D., Vishik, A., Voevodsky, V.: Motivic cohomology of Pfister quadrics and Milnor’s conjecture on quadratic forms . Preprint, (1998).

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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.