Annihilators of quadratic and bilinear forms over fields of characteristic two

R. Aravire^{∗} Departamento de Cs. Físicas y Matemáticas Universidad Arturo Prat Casilla 121, Iquique, Chile e-mail: raravire@unap.cl

R. Baeza^{†}

Instituto de Matemáticas, Universidad de Talca, Casilla 721, Talca, Chile e-mail: rbaeza@inst-mat.utalca.cl

Abstract

Let F be a field with 2 = 0, W (F) the Witt ring of symmetric bilinear forms over F and W_{q}(F) the W (F)-module of quadratic forms over F. Let I_{F} ⊂ W (F) be the maximal ideal. We compute explicitly in I_{F}^{m} and I^{m}W_{q}(F) the annihilators of n-fold bilinear and quadratic Pfister forms, thereby answering positively, in the case 2 = 0, certain conjectures stated by Krüskemper in [Kr].

1 Introduction

Let F be a field with 2 = 0. We denote by W (F) the Witt ring of symmetric non singular bilinear forms over F and by W_{q}(F) the W (F)-module of non singular quadratic forms over F (see [Sa], [Ba-1], [Ba-2]).

For a_{i} ∈ F^{∗} = F − {0} , 1 ≤ i ≤ n , we denote by ⟨a_{1}, …, a_{n}⟩ the bilinear form with diagonal Gramm matrix and entries a_{i} on the diagonal. The quadratic form x^{2} + xy + ay^{2} , a ∈ F , is denoted by [1, a]. The maximal ideal I_{F} of W (F) is additively generated by the forms ⟨1, a⟩ =≪ a ≫, a ∈ F^{∗} , so that the powers I_{F}^{n}, n ≥ 1, are additively generated by the n-fold bilinear forms ≪ a_{1}, …, a_{n} ≫= ⟨1, a_{1}⟩ ⋯ ⟨1, a_{n}⟩, a_{i} ∈ F^{∗}. The submodules I^{n}W_{q}(F), n ≥ 1, are generated by the n-fold quadratic Pfister forms ≪ a_{1}, …, a_{n}; a|] =≪ a_{1}, …, a_{n} ≫ ·[1, a], a_{i} ∈ F^{∗}, a ∈ F.

∗Partially supported by Proyecto Fondecyt #198 0681 and #1020516.

†Partially supported by Proyecto Fondecyt #1030218 and Programa Reticulados, Universidad de Talca.

2000 Mathematical Subject Classification. Primary 11E04; Secondary 11E81.

Key words and phrases. Quadratic Form, bilinear form, Pfister form, Witt ring, differential form.

We have the filtrations W (F) ⊃ I_{F} ⊃ I_{F}^{2} ⊃ ⋯ and W_{q}(F) ⊃ IW_{q}(F) ⊃ · · ·. The graded objects I_{F}^{n}/I_{F}^{n+1} and I^{n}W_{q}(F)/I^{n+1}W_{q}(F) are denoted by \overline{I}_{F}^{n} resp. \overline{I}^{n}W_{q}(F).

In this paper we will study annihilators of n-fold Pfister forms. Let x =≪ a_{1}, …, a_{n} ≫ be an n-fold bilinear Pfister form. For any m ≥ 0 we set

annb_{m}(x) = {y ∈ I_{F}^{m} | xy = 0}
annq_{m}(x) = {y ∈ I^{m}W_{q}(F) | xy = 0}

\overline{ann}b_{m}(x) = \{\overline{y} ∈ \overline{I}_{F}^{m} | x\overline{y} = 0\}

\overline{ann}q_{m}(x) = \{\overline{y} ∈ \overline{I}^{m}W_{q}(F) | x\overline{y} = 0\}.

If x =≪ a_{1}, …, a_{n}; a|] is a quadratic n-fold Pfister form, we set

annb_{m}(x) = {y ∈ I_{F}^{m} | yx = 0}

\overline{ann}b_{m}(x) = \{\overline{y} ∈ \overline{I}_{F}^{m} | \overline{y}x = 0\}.

The main results of this paper are contained in the following two theorems.

(1.1) Theorem. ( i ) Let x =≪ a_{1}, …, a_{n} ≫ be a bilinear n-fold Pfister form over F with x ≠ 0 in W (F). Then for any m ≥ 1

\overline{ann}b_{m}(x) = \overline{ann}b_{1}(x)\overline{I}_{F}^{m−1}

\overline{ann}q_{m}(x) = \overline{I}_{F}^{m} · \overline{ann}q_{0}(x) + \overline{ann}b_{1}(x)\overline{I}^{m−1}W_{q}(F)

( ii ) Let x =≪ a_{1}, …, a_{n}; a|] be a quadratic n-fold Pfister form over F with x ≠ 0 in W_{q}(F). Then for m ≥ 1

\overline{ann}b_{m}(x) = \overline{ann}b_{1}(x)\overline{I}_{F}^{m−1}.

and the much stronger

(1.2) Theorem. ( i ) Let x =≪ a_{1}, …, a_{n} ≫ be a bilinear n-fold Pfister form over F with x ≠ 0 in W (F). Then for any m ≥ 1

annb_{m}(x) = annb_{1}(x)I_{F}^{m−1}

annq_{m}(x) = I_{F}^{m} · annq_{0}(x) + annb_{1}(x)I^{m−1}W_{q}(F)

( ii ) Let x =≪ a_{1}, …, a_{n}; a|] be a quadratic n-fold Pfister form over F with x ≠ 0 in W_{q}(F). Then for m ≥ 1

annb_{m}(x) = annb_{1}(x)I_{F}^{m−1}.

These results were conjectured by M. Krüskemper in [Kr] for fields of characteristic different from 2. The proof of theorem (1.1) will be given in section 4 and it is based on Kato’s correspondence between quadratic or symmetric bilinear forms and differential forms over F. We will shortly explain this correspondence en section 3 (see [Ka], [Ba-2]) and prove there some technical results needed in the proof of (1.1). In section 2 we show that theorem (1.2) follows from theorem (1.1).

The terminology used in this paper is standard and we refer to [Ba-2], [Mi] and [Sa] for details on basic facts needed in the paper. In any case let us mention that for a_{1}, …, a_{n} ∈ F^{∗} the form ≪ a_{1}, …, a_{n} ≫ is anisotropic over F if and only if a_{1}, …, a_{n} are part of a 2-basis of F and the subfield F^{2}(a_{1}, …, a_{n}) of F consists of all elements of F represented by the form ≪ a_{1}, …, a_{n} ≫. The elements of F represented by the pure part ≪ a_{1}, …, a_{n} ≫′ of ≪ a_{1}, …, a_{n} ≫ form a subgroup denoted by F^{2}(a_{1}, …, a_{n})′. Recall that ≪ a_{1}, …, a_{n} ≫′ is defined by ≪ a_{1}, …, a_{n} ≫=< 1 >⊥≪ a_{1}, …, a_{n} ≫′.

2 Proof of theorem (1.2)

We will assume theorem (1.1) and derive from it theorem (1.2). Recall that a 2-basis of a field F of characteristic 2 is a set B = {b_{i} | i ∈ I} ⊂ F such that the elements ∏_{i∈I} b_{i}^{ε_{i}}, ε_{i} ∈ {0, 1} and only finitely many ε_{i} ≠ 0, form a basis

of F over F^{2}. An n-fold bilinear Pfister form ≪ a_{1}, …, a_{n} ≫ over F is ≠ 0 in W (F) if and only if {a_{1}, …, a_{n}} are part of a 2-basis of F (i.e. 2-independent). Moreover if F has a finite 2-basis {b_{1}, …, b_{N}} then I_{F}^{m} = 0 for all m ≥ N + 1 (see [Mi]).

We will need the following

(2.1) Lemma. ( i ) Let x be an n-fold bilinear Pfister form, x ≠ 0, and z ∈ I_{F} such that zx ∈ I_{F}^{n+2}, i.e. z ∈ \overline{ann}b_{1}(x). Then

z = z_{0} + w

with z_{0} ∈ I, z_{0}x = 0 and w ∈ I_{F}^{2} .

( ii ) Let x be an n-fold bilinear Pfister form, x ≠ 0, and z ∈ W_{q}(F) with xz ∈ I^{n+1}W_{q}(F). Then

z = z_{0} + w

with z_{0} ∈ W_{q}(F), xz_{0} = 0 and w ∈ IW_{q}(F).

Proof: ( i ) For any z ∈ I_{F} we can write z =< 1, d > +w with d = det(z) and w ∈ I_{F}^{2} . Then xz ∈ I_{F}^{n+2} implies < 1, d > x ∈ I_{F}^{n+2}, and since < 1, d > x is (n + 1)-fold Pfister form, it follows < 1, d > x = 0 in W (F).

( ii ) Any z ∈ W_{q}(F) can be written as

z = [1, d] + w

with d = Arf(z) ∈ F and w ∈ IW_{q}(F) (see [Sa]). From xz, xw ∈ I^{n+1}W_{q}(F), it follows x[1, d] ∈ I^{n+1}W_{q}(F) and hence x[1, d] = 0. □

Let us now prove (1.2). We assume first that F has a finite 2-basis, i.e. I_{F}^{N+1} = 0 for some integer N. Let x ≠ 0 (in W (F)) be an n-fold bilinear Pfister form. The contentions ⊇ in ( i ) (and ( ii )) are obvious. Let y ∈ annb_{m}(x), i.e. y ∈ I_{F}^{m}, yx = 0. Hence \overline{y} ∈ \overline{ann}b_{m}(x) and (1.1) implies y = ∑ \overline{z}_{i}\overline{y}_{i,0} with \overline{z}_{i} ∈ \overline{ann}b_{1}(x), y_{i,0} ∈ I_{F}^{m−1}. Then y − ∑ z_{i}y_{i,0} ∈ I_{F}^{m+1}. Using (2.1) (i) we can write z_{i} = z_{i,0} +w_{i} with z_{i,0} ∈ annb_{1}(x) and w_{i} ∈ I_{F}^{2} . Then y_{1} = y − ∑ z_{i}y_{i,0} ∈ I_{F}^{m+1} and moreover y_{1}x = 0. The same argument implies y_{1} − ∑ z_{i,1}y_{i,1} ∈ I_{F}^{m+2} with elements z_{i,1} ∈ annb_{1}(x), y_{i,1} ∈ I_{F}^{m}. Iterating this process we obtain, for any k ≥ 0, elements z_{i,l} ∈ annb_{1}(x) and y_{i,l} ∈ I_{F}^{m+l−1}, 0 ≤ l ≤ k such that y − ∑_{i,l} z_{i,l}y_{i,l} ∈ I_{F}^{m+k}. Choosing k ≥ N + 1 − m we obtain y = ∑_{i,l} z_{i,1}y_{i,1} ∈ annb_{1}(x)I_{F}^{m−1}, since I^{N+1} = 0.

Let now y ∈ annq_{m}(x), i.e. y ∈ I^{m}W_{q}(F) with xy = 0. Theorem (1.1) implies \overline{y} = ∑ \overline{y_{i}}\overline{z_{i}} + ∑ \overline{u_{j}}\overline{v_{j}} with \overline{y_{i}} ∈ \overline{I}_{F}^{m}, \overline{z_{i}} ∈ \overline{ann}q_{0}(x), \overline{u_{j}} ∈ \overline{ann}b_{1}(x), \overline{v_{i}} ∈ \overline{I}^{m−1}W_{q}(F). Hence y − ∑ y_{i}z_{i} − ∑ u_{j}v_{j} ∈ I^{m+1}W_{q}(F). Using lemma (2.1) we can find z_{i,0} ∈ annq_{0}(x), u_{j,0} ∈ annb_{1}(x) such that z_{i} = z_{i,0} + w_{i}, w_{i} ∈ IW_{q}(F) and u_{i} = u_{j,0} + t_{j}, t_{j} ∈ I_{F}^{2} . We obtain

y_{1} = y − ∑ y_{i}z_{i,0} − ∑ u_{j,0}v_{j} ∈ I^{m+1}W_{q}(F)

with y_{1}x = 0. Iterating this procedure we obtain after k ≥ N + 1 − m steps that

y ∈ I_{F}^{m}annq_{0}(x) + annb_{1}(x)I^{m−1}W_{q}(F).

The proof of part ( ii ) of (1.2) is similar and we omit the details. Thus we have proved (1.2) in the case I^{N+1} = 0 for some N. Let us now consider the general case.

Let B be a 2-basis of F, x a bilinear n-fold Pfister form over F, x ≠ 0 in W (F). Take y ∈ annb_{m}(x), i.e. y ∈ I_{F}^{m} with yx = 0. This relation involves only finitely many elements {a_{1}, …, a_{N}} ⊂ B of the 2-basis. We define F_{0} = F^{2}(a_{1}, …, a_{N}) ⊂ F. Then there exist an n-fold bilinear Pfister form x_{0} over F_{0} and y_{0} ∈ I_{F_{0}}^{m} such that x = x_{0} ⊗ F, y = y_{0} ⊗ F and y_{0}x_{0} = 0 in W (F_{0}). From the first part of the proof of (1.2) we obtain y_{0} ∈ annb_{1}(x_{0})I_{F_{0}}^{m−1} and hence y ∈ annb_{1}(x)I_{F}^{m−1}. The same argument applies for the other assertions in (1.2) and this concludes the proof of theorem (1.2). □

(2.2) Remark. If x is a bilinear n-fold Pfister form over F, then one can describe explicitely the annihilators annb_{1}(x) ⊂ W (F) and annq_{0}(x) ⊂ W_{q}(F) as follows

(2.3) annb_{1}(x) = ∑_{d∈D_{F} (x)^{∗}} W (F) ⟨1, d⟩

(2.4) annq_{0}(x) = ∑_{d∈D_{F} (x)} W (F)[1, d].

Here D_{F} (z) denotes the set in F of elements represented by the form z. The result (2.3) is shown in [Ho] and (2.4) in [Ba-Kn]. If x denotes now a quadratic n-fold Pfister form over F, x ≠ 0 in W_{q}(F), then (see [Kn])

(2.5) annb_{1}(x) = ∑_{d∈D_{F} (x)^{∗}} W (F) ⟨1, d⟩ .

In section 4 we will give an independent proof of these facts based on Kato’s correspondence (see 3.3) and on the arguments used in this section.

3 Quadratic, symmetric bilinear and differential forms

In this section we will briefly describe Kato’s correspondence between quadratic, bilinear and differential forms over a field F with 2 = 0 and prove a technical result needed in the proof of theorem (1.1) (see [Ka], [Ba-2], [A-Ba]).

Let Ω_{F}^{1} = F d F be the F -space of 1-differential forms generated over F by the symbols d a, a ∈ F, with d(a + b) = d a + d b, d(ab) = a d b + b d a.

For any n ≥ 1 set Ω_{F}^{n} = ⋀^{n} Ω_{F}^{1} and let d : Ω_{F}^{n} → Ω_{F}^{n+1} be the differential operator d(x d x_{1} ∧ ⋯ ∧ d x_{n}) = d x ∧ d x_{1} ∧ ⋯ ∧ d x_{n}, where ∧ denotes exterior multiplication.

Let ℘ : Ω_{F}^{n} → Ω_{F}^{n} / d Ω_{F}^{n−1} be the Artin-Schreier operator defined on generators by

℘ (x\frac{d x_{1}}{x_{1}} ∧ ⋯ ∧ \frac{d x_{n}}{x_{n}} ) = (x^{2} − x) \frac{d x_{1}}{x_{1}} ∧ ⋯ ∧ \frac{d x_{n}}{x_{n}} mod d Ω_{F}^{n−1}

and denote by ν_{F} (n) its kernel and by H^{n+1}(F) its cokernel (see loc. cit.). In [Ka] it is shown that there are natural isomorphisms α : ν_{F} (n) ≃ \overline{I}_{F}^{n} and β : H^{n+1}(F) ≃ \overline{I}^{n}W_{q}(F) given on generators by α(\frac{d x_{1}}{x_{1}} ∧ ⋯ ∧ \frac{d x_{n}}{x_{n}}) =≪

x_{1}, …, x_{n} ≫ mod I_{F}^{n+1} and β (\overline{x\frac{d x_{1}}{x_{1}} ∧ ⋯ ∧ \frac{d x_{n}}{x_{n}}} ) =≪ x_{1}, …, x_{n}; x|] mod

I^{n+1}W_{q}(F). The fact that ν_{F} (n) is additively generated by the pure logarithmic forms \frac{d x_{1}}{x_{1}} ∧ ⋯ ∧ \frac{d x_{n}}{x_{n}} follows from a result of Kato which we explain now. Let us fix a 2-basis B of F, B = {b_{i} | i ∈ I}, and endow I with a total ordering. For any j ∈ I, let F_{j}, resp. F_{<j}, be the subfields of F generated over F^{2} by b_{i}, i ≤ j, resp. b_{i}, i < j. For any n ≥ 1 let Σ_{n} be the set of maps α : {1, …, n} → I such that α(i) < α(j) whenever 1 ≤ i < j ≤ n, and endow Σ_{n} with the lexicographic ordering.

We obtain a filtration of Ω_{F}^{n} given by the subspaces Ω_{F,α}^{n}, resp. Ω_{F,<α}^{n},

which are generated by the elements \frac{d b_{β}}{b_{β}} = \frac{d b_{β(1)}}{b_{β(1)}} ∧ ⋯ ∧ \frac{d b_{β(n)}}{b_{β(n)}} with β ≤ α, resp. β < α. An important result of Kato, named here as Kato’s lemma, asserts that for any α ∈ Σ_{n}, y ∈ F, if ℘ (y \frac{d b_{α}}{b_{α}} ) ∈ Ω_{F,<α}^{n} + d Ω_{F}^{n−1}, then there exist

v ∈ Ω_{F,<α}^{n} and a_{i} ∈ F_{α(i)}^{∗}, 1 ≤ i ≤ n, such that y \frac{d b_{α}}{b_{α}} = v + \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} (see [Ka]). This implies that any u ∈ Ω_{F,α}^{n} satisfying ℘(α) ∈ d Ω_{F}^{n−1}, can be written as

(3.1) u = ∑_{γ≤α} \frac{d a_{γ(1)}}{a_{γ(1)}} ∧ ⋯ ∧ \frac{d a_{γ(n)}}{a_{γ(n)}}

with a_{γ(i)} ∈ F_{γ(i)}\F_{<γ(i)}. Then the following result will be used in section 4 during the proof of theorem (1.1).

(3.2) Lemma. Let B = {b_{i} | i ∈ I} be a 2-basis of F with a given ordering on I. Let α ∈ Σ_{n} and ∑_{γ≤α} c_{γ} \frac{d b_{γ}}{b_{γ}} be a differential form with c_{α} ≠ 0

such that ∑_{γ≤α} c_{γ} \frac{d b_{γ}}{b_{γ}} ∈ d Ω_{F}^{n−1} . Then there exist elements M_{i} ∈ F_{<α(i)}, 1 ≤ i ≤ n, such that

c_{α} = b_{α(1)}M_{1} + ⋯ + b_{α(n)}M_{n}

Proof: Let k ∈ I be the index with c_{α} ∈ F_{k}\F_{<k}. We claim that k = α(i) for some 1 ≤ i ≤ n. Otherwise we have k > α(n) or k < α(1) or α(j) < k < α(j + 1) for some 1 ≤ j ≤ n. From the choice of k we have c_{α} = b_{k}A + B with A, B ∈ F_{<k}, A ≠ 0. Then

d t = (b_{k}A + B) \frac{d b_{α}}{b_{α}} + ∑_{γ<α} c_{γ} \frac{d b_{γ}}{b_{γ}}

and applying the differential operator to this form, we get

b_{k}A\frac{d b_{α}}{b_{α}} ∧ \frac{d b_{k}}{b_{k}} +b_{k}A\frac{d b_{α}}{b_{α}} ∧ \frac{d A}{A} +B \frac{d b_{α}}{b_{α}} ∧ \frac{d B}{B} + ∑_{γ<α} ∑_{i∈I} b_{i}D_{i}(c_{γ})\frac{d b_{γ}}{b_{γ}} ∧ \frac{d b_{i}}{b_{i}} = 0

where D_{i}(c_{γ}) is the derivative of c_{γ} with respect to b_{i} (see [A-Ba]). Looking at the coefficient of \frac{d b_{α}}{b_{α}} ∧ \frac{d b_{k}}{b_{k}} we obtain

b_{k}A = ∑_{(α,k)=(γ_{i},i)} b_{i}D_{i}(c_{γ})

where (α, k) resp. (γ_{i}, i) denotes the unique λ ∈ Σ_{n+1} with Im(λ) =Im(α)∪ {k} resp. Im(λ) =Im(γ_{i}) ∪ {i}. Since for those i we have i > k, A ∈ F_{<k} and D_{i}(D_{i}(c_{γ_{i}})) = 0, we conclude A = 0, which is a contradiction. Thus k = α(i) for some 1 ≤ i ≤ n.

Let c_{α} = b_{α(i)}M_{i} + B with M_{i}, B ∈ F_{<α(i)}. Then

d t = (b_{α(i)}M_{i} + B) \frac{d b_{α}}{b_{α}} + ∑_{γ<α} c_{γ} \frac{d b_{γ}}{b_{γ}} .

But

b_{α(i)}M_{i} \frac{d b_{α}}{b_{α}} = b_{α(i)}M_{i} \frac{d b_{α(1)}}{b_{α(1)}} ∧ ⋯ ∧ \frac{d b_{α(i)}}{b_{α(i)}} ∧ ⋯ ∧ \frac{d b_{α(n)}}{b_{α(n)}}

= d(b_{α(i)}M_{i}) ∧ \frac{d b_{α(1)}}{b_{α(1)}} ∧ ⋯ ∧ \frac{d b_{α(i−1)}}{b_{α(i−1)}} ∧ \frac{d b_{α(i+1)}}{b_{α(i+1)}} ∧ ⋯ ∧ \frac{d b_{α(n)}}{b_{α(n)}}

+ b_{α(i)}M_{i} \frac{d b_{α(1)}}{b_{α(1)}} ∧ ⋯ ∧ \frac{d M_{i}}{M_{i}} ∧ ⋯ ∧ \frac{d b_{α(n)}}{b_{α(n)}}

so that replacing t by t′ = t+b_{α(i)}M_{i} \frac{d b_{α(1)}}{b_{α(1)}} ∧⋯∧ \frac{d b_{α(i−1)}}{b_{α(i−1)}} ∧ \frac{d b_{α(i+1)}}{b_{α(i+1)}} ∧⋯∧ \frac{d b_{α(n)}}{b_{α(n)}} ,

and since b_{α(i)}M_{i} \frac{d b_{α(1)}}{b_{α(1)}} ∧ ⋯ ∧ \frac{d M_{i}}{M_{i}} ∧ ⋯ ∧ \frac{d b_{α(n)}}{b_{α(n)}} ∈ Ω_{<α}^{n}, we get

d t′ = B \frac{d b_{α}}{b_{α}} + ∑_{γ<α} c′_{γ} \frac{d b_{γ}}{b_{γ}}

with certain c′_{γ} ∈ F and B ∈ F_{<α(i)}. We proceed again as before with B instead of c_{α} and the lemma follows by induction. □

An immediate generalization of (3.2) is

(3.3) Proposition. Let

∑_{γ≤α} c_{γ} \frac{d b_{γ}}{b_{γ}} = d(t) + ℘(w)

with c_{α} ≠ 0, where B = {b_{i} | i ∈ I} is a given 2-basis of F (and a fixed ordering in I ) and t ∈ Ω_{F}^{n−1}, w ∈ Ω_{F}^{n}. Then there exist elements u ∈ F, M_{i} ∈ F_{<α(i)}, 1 ≤ i ≤ n, such that

c_{α} = ℘u + b_{α(1)}M_{1} + ⋯ + b_{α(n)}M_{n}.

4 Annihilators of differential forms in ν_{F} (m) and H^{m+1}(F)

The groups ν_{F} (m) act on the groups H^{n+1}(F) through exterior multiplication

∧ : ν_{F} (m) × H^{n+1}(F) ⟶ H^{m+n+1}(F)
∧ : ν_{F} (m) × ν_{F} (n) ⟶ ν_{F} (m + n)

and we can define for any x ∈ ν_{F} (n) the annihilators

annb_{m}(x) = {y ∈ ν_{F} (m) | xy = 0 in ν_{F} (m + n)}

annq_{m}(x) = {y ∈ H^{m+1}(F) | xy = 0 in H^{n+m+1}(F)}.

Also if x ∈ H^{n+1}(F), we define

annb_{m}(x) = {y ∈ ν_{F} (m) | yx = 0 in H^{n+m+1}(F)}.

Through Kato’s isomorphisms (see § 3) these annihilators are isomorphic to the corresponding graded annihilators of bilinear and quadratic forms, namely, if x ∈ ν_{F} (n)

α : annb_{m}(x) ≃ \overline{annb}_{m}(α(x))
β : annq_{m}(x) ≃ \overline{annq}_{m}(α(x))

and if x ∈ H^{n+1}(F),

α : annb_{m}(x) ≃ \overline{annb}_{m}(β(x)).

Thus, theorem (1.1) is equivalent to the following

(4.1) Theorem. (i) Let x = \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ∈ ν_{F} (n) be a pure logarithmic differential form, x ≠ 0. Then for any m ≥ 1

annb_{m}(x) = annb_{1}(x) ∧ ν_{F} (m − 1)
annq_{m}(x) = ν_{F} (m) ∧ annq_{0}(x) + annb_{1}(x) ∧ H^{m}(F).

(ii) If x = \overline{a\frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}}} ≠ 0 in H^{n+1}(F), then in ν_{F} (m)

annb_{m}(x) = annb_{1}(x) ∧ ν_{F} (m − 1).

Proof: Let B = {b_{i} | i ∈ I} be a 2-basis of F such that a_{1}, …, a_{n} ∈ B are the first elements in some ordering of I. Let y ∈annb_{m}(x). Using Kato’s lemma we can write

y = ∑_{γ∈Σ_{m}} ε_{γ} \frac{d a_{γ(1)}}{a_{γ(1)}} ∧ ⋯ ∧ \frac{d a_{γ(m)}}{a_{γ(m)}}

with a_{γ(i)} ∈ F_{γ(i)}\F_{<γ(i)}, ε_{γ} ∈ {0, 1}. Let α ∈ Σ_{m} be maximal with ε_{α} ≠ 0. Then

y ≡ \frac{d a_{α}}{a_{α}} mod Ω_{F,<α}^{m}.

The assumption xy = 0 means

(\frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ) ∧ \frac{d a_{α}}{a_{α}} + (\frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ) ∧ ∑_{γ<α} ε_{γ} \frac{d a_{γ}}{a_{γ}} = 0.

Assume first α(1) > n and define δ = (1, …, n, α(1), …, α(m)) ∈ Σ_{n+m}. It follows δ > (1, …, n, γ) for all γ ∈ Σ_{m} with γ < α. From the last relation we conclude

d a_{1} ∧ ⋯ ∧ d a_{n} ∧ d a_{α(1)} ∧ ⋯ ∧ d a_{α(m)} = 0

which is a contradiction to the fact that a_{1}, …, a_{n}, a_{α(1)}, …, a_{α(m)} are 2independent. Thus we have α(1) ≤ n, and this implies x ∧ \frac{d a_{α(1)}}{a_{α(1)}} = 0, i.e.

\frac{d a_{α(1)}}{a_{α(1)}} ∈ annb_{1}(x). Hence y− \frac{d a_{α}}{a_{α}} ∈ annb_{m}(x) and moreover y− \frac{d a_{α}}{a_{α}} ∈ Ω_{F,<α}^{m}. Proceeding by induction on α we get the first assertion in ( i ).

Take now \overline{y} ∈ annq_{m}(x) ⊂ H^{m+1}(F). Then

y ≡ ∑_{γ∈Σ_{m}} c_{γ} \frac{d b_{γ}}{b_{γ}} mod ℘Ω_{F}^{m} + d Ω_{F}^{m−1}

with x ∧ y ∈ ℘Ω_{F}^{m+n} + d Ω_{F}^{m+n−1}, i.e.

(4.2) ∑_{γ∈Σ_{m}} c_{γ} \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ∧ \frac{d b_{γ}}{b_{γ}} ∈ ℘Ω_{F}^{m+n} + d Ω_{F}^{m+n−1}.

(Here the elements b_{γ(i)} belong to B). Let α ∈ Σ_{m} be maximal with c_{α} ≠ 0. If α(1) ≤ n, then \frac{d b_{α(1)}}{b_{α(1)}} ∈ annb_{1}(x) and c_{α} \frac{d b_{α}}{b_{α}} = \frac{d b_{α(1)}}{b_{α(1)}} ∧c_{α} \frac{d b_{α(2)}}{b_{α(2)}} ∧

⋯ ∧ \frac{d b_{α(m)}}{b_{α(m)}} ∈ annb_{1}(x) ∧ H^{m}(F), and y − c_{α} \frac{d b_{α}}{b_{α}} ∈ Ω_{F,<α}^{m}. Hence we may proceed by induction on α. Thus we can assume α(1) > n and we define δ = (1, …, n, α(1), …, α(m)) ∈ Σ_{n+m}. We see in (4.2) that δ is the maximal multi-index with coefficient c_{α} ≠ 0. Using now proposition (3.3), we conclude from (4.2) that

c_{α} = ℘(u) + E_{α}

with E_{α} = ∑_{i=1}^{n} a_{i}M_{i} + ∑_{j=1}^{m} b_{α(j)}M_{α(j)} and M_{k} ∈ F_{<k}. Here we have chosen the ordering of B such that a_{1}, …, a_{n} are the first elements.

Inserting c_{α} in y we get

y ≡ c_{α} \frac{d b_{α}}{b_{α}} mod ℘Ω_{F}^{m} + d Ω_{F}^{m−1} + Ω_{F,<α}^{m}

y ≡ [℘(u) + ∑_{i=1}^{n} a_{i}M_{i} + ∑_{j=1}^{m} b_{α(j)}M_{α(j)}] \frac{d b_{α}}{b_{α}}

y ≡ [∑_{i=1}^{n} a_{i}M_{i}] \frac{d b_{α}}{b_{α}} + [∑_{j=1}^{m} b_{α(j)}M_{α(j)}] \frac{d b_{α}}{b_{α}} .

Since M_{k} ∈ F_{<k}, we have a_{i}M_{i} \frac{d b_{α}}{b_{α}} ∈ ν_{F} (m)∧ annq_{0}(x) because a_{i}M_{i} \frac{d a_{1}}{a_{1}} ∧

⋯ ∧ \frac{d a_{n}}{a_{n}} = d (a_{i}M_{i} \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧^{i} ⋯ ∧ \frac{d a_{n}}{a_{n}} ) ∈ d Ω_{F}^{n−1} implies a_{i}M_{i} ∈ annq_{0}(x)

(we have used d M_{i}∧x = 0). The same argument shows, since M_{α(j)} ∈ F_{<α(j)}, that

b_{α(j)}M_{α(j)} \frac{d b_{α(j)}}{b_{α(j)}} = d (b_{α(j)}M_{α(j)}) + b_{α(j)}M_{α(j)} \frac{d M_{α(j)}}{M_{α(j)}} ∈ d F + Ω_{F,<α(j)}^{1}

and hence

(∑_{j=1}^{m} b_{α(j)}M_{α(j)}) \frac{d b_{α}}{b_{α}} ∈ d Ω_{F}^{m−1} + Ω_{F,<α}^{m}.

Thus we have

y = y′ + z mod ℘Ω_{F}^{m} + d Ω_{F}^{m−1}

with y′ ∈ Ω_{F,<α}^{m}, y′ ∈ annq_{m}(x) and z ∈ ν_{F} (m)∧ annq_{0}(x). Applying now the above procedure to y′ we get our second assertion by induction on α. This proves ( i ).

( ii ) Let x = \overline{a\frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}}} ∈ H^{n+1}(F) be a pure element, x ≠ 0. We fix as before a 2-basis B = {b_{i} | i ∈ I} of F such that a_{1}, …, a_{n} are the first elements in B in some ordering of I. Let y ∈ annb_{m}(x) ⊂ ν_{F} (m). From Kato’s lemma we have y = ∑_{γ∈Σ_{m}} ε_{γ} \frac{d a_{γ}}{a_{γ}} with ε_{γ} ∈ \{0, 1\} and a_{γ(i)} ∈ F_{γ(i)}\F_{<γ(i)}, 1 ≤ i ≤ m. We write

y = ∑_{{γ∈Σm \atop γ(1)≤n}} ε_{γ} \frac{d a_{γ}}{a_{γ}} + ∑_{{γ∈Σm \atop γ(1)>n}} ε_{γ} \frac{d a_{γ}}{a_{γ}} .

For γ ∈ Σ_{m} with γ(1) ≤ n we have \frac{d a_{γ}}{a_{γ}} ∈ annb_{1}(x) since a_{γ(1)} ∈ F_{n} = F^{2}(a_{1}, …, a_{n}) and hence the first summand in this decomposition is in annb_{1}(x) ∧ ν_{F} (m − 1). Thus the second summand is in annb_{m}(x) and we can assume y = ∑_{γ∈Σ_{m}} ε_{γ} \frac{d a_{γ}}{a_{γ}} with all γ such that γ(1) > n. Let α be maximal in this sum with ε_{α} ≠ 0. We can replace B by a new 2-basis B′ = {c_{i} | i ∈ I} such that c_{α(j)} = a_{α(j)}, 1 ≤ j ≤ m and c_{i} = b_{i} for all i ∉ {α(1), …, α(m)}. Let δ = (1, …, n, α(1), …, α(m)) ∈ Σ_{n+m}. Hence

0 ≡ y ∧ x ≡ a\frac{d c_{δ}}{c_{δ}} mod ℘Ω_{F}^{n+m} + d Ω_{F}^{n+m−1} + Ω_{F,<δ}^{n+m}.

Then proposition (3.3) implies

a = ℘(u) + ∑_{i=1}^{n} c_{i}M_{i} + ∑_{j=1}^{m} c_{α(j)}M_{α(j)}

with M_{k} ∈ F_{<k}. Let s ∈ {1, …, m} be maximal with M_{α(s)} ≠ 0 and set Q = a + ℘u + ∑_{i=1}^{n} c_{i}M_{i} i.e. Q = ∑_{j=1}^{m} c_{α(j)}M_{α(j)}. Then c_{α(s)} =

M_{α(s)}^{−1} (Q + ∑_{j=1}^{s−1} c_{α(j)}M_{α(j)}). Inserting in y we get modulo ν_{F,<α}(m)

y ≡ \frac{d c_{α(1)}}{c_{α(1)}} ∧ ⋯ ∧ \frac{d c_{α(s)}}{c_{α(s)}} ∧ ⋯ ∧ \frac{d c_{α(m)}}{c_{α(m)}} mod ν_{F,<α}(m)

≡ \frac{d c_{α(1)}}{c_{α(1)}} ∧ ⋯ ∧ \frac{d M_{α(s)}^{−1} (Q + ∑_{j=1}^{s−1} c_{α(j)}M_{α(j)})}{M_{α(s)}^{−1} (Q + ∑_{j=1}^{s−1} c_{α(j)}M_{α(j)})} ∧ ⋯ ∧ \frac{d c_{α(m)}}{c_{α(m)}}

≡ \frac{d (c_{α(1)}M_{α(1)})}{(c_{α(1)}M_{α(1)})} ∧ ⋯ ∧ \frac{d (Q + ∑_{j=1}^{s−1} c_{α(j)}M_{α(j)})}{Q + ∑_{j=1}^{s−1} c_{α(j)}M_{α(j)}} ∧ ⋯ ∧ \frac{d c_{α(m)}}{c_{α(m)}} .

Here we have inserted M_{α(j)} whenever it is ≠ 0, without altering the congruence modulo ν_{F,<α}(m). Use now the relation \frac{d a}{a} ∧ \frac{d b}{b} = \frac{d(ab)}{ab} ∧ \frac{d(a+b)}{a+b} to conclude

y ≡ \frac{d (c_{α(1)}M_{α(1)})}{(c_{α(1)}M_{α(1)})} ∧ ⋯ ∧ \frac{d (Q + ∑_{j=1}^{s−1} c_{α(j)}M_{α(j)})}{Q + ∑_{j=1}^{s−1} c_{α(j)}M_{α(j)}} ∧ ⋯ ∧ \frac{d c_{α(m)}}{c_{α(m)}} mod ν_{F,<α}(m)

≡ \frac{d f_{1}}{f_{1}} ∧ ⋯ ∧ \frac{d Q}{Q} ∧ ⋯ ∧ \frac{d c_{α(m)}}{c_{α(m)}}

with certain f_{1}, …, f_{s−1} ∈ F. Since \frac{d Q}{Q} ∈ annb_{1}(x) (we can assume a ∈ F^{2}

without restriction), we get \frac{d f_{1}}{f_{1}} ∧⋯∧ \frac{d Q}{Q} ∧⋯∧ \frac{d c_{α(m)}}{c_{α(m)}} ∈ annb_{1}(x)∧ν_{F} (m−1). Thus we have shown y ∈ annb_{1}(x) ∧ ν_{F} (m − 1) + ν_{F,<α}(m). We apply now induction on α to conclude the proof of ( ii ). □

Let us briefly compute the annihilators annb_{1}(x) and annq_{0}(x) for x = \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ∈ ν_{F} (n) and annb_{1}(x) for x = \overline{a\frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}}} ∈ H^{n+1}(F).

(4.3) Proposition. ( i ) Let x = \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ∈ ν_{F} (n), x ≠ 0. Then

annb_{1}(x) = \{\frac{d z}{z} | z ∈ F^{2}(a_{1}, …, a_{n})^{∗}\}

annq_{0}(x) = \{\overline{z} ∈ F/℘F | z ∈ F^{2}(a_{1}, …, a_{n})′\}

where F^{2}(a_{1}, …, a_{n})′ are the pure elements in F^{2}(a_{1}, …, a_{n}) (notice that H^{1}(F) = F/℘F).

( ii ) Let x = \overline{a\frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}}} ∈ H^{n+1}(F), x ≠ 0. Then

annb_{1}(x) = \{\frac{d z}{z} | z ∈ D_{F} (≪ a_{1}, …, a_{n}; |])^{∗}\}

where D_{F} (q) denotes the elements represented in F by the quadratic form q.

Proof: ( i ) Let x = \frac{d a_{1}}{a_{1}} ∧⋯∧ \frac{d a_{n}}{a_{n}} ≠ 0 in ν_{F} (n). If \frac{d z}{z} ∈ annb_{1}(x) ⊂ ν_{F} (1), then

\frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ∧ \frac{d z}{z} = 0

in ν_{F} (n + 1), which means that a_{1}, …, a_{n}, z are 2-dependent, and since a_{1}, …, a_{n} are 2-independent, this means z ∈ F^{2}(a_{1}, …, a_{n})^{∗} (which is the set in F^{∗} of elements represented by the n-fold Pfister form ≪ a_{1}, …, a_{n} ≫).

Let now \overline{y} ∈ H^{1}(F) = F/℘F be in annq_{0}(x). Then \overline{y \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}}} = 0 in H^{n+1}(F), and this means

y \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ∈ ℘Ω_{F}^{n} + d Ω_{F}^{n−1}.

Taking a 2-basis of F so that a_{1}, …, a_{n} are the first elements of it (in some ordering), we conclude from proposition (3.3)

y = ℘u + b

with u ∈ F and b ∈ F^{2}(a_{1}, …, a_{n})′. This proves ( i ).

( ii ) Let x = \overline{a\frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}}} ∈ H^{n+1}(F), x ≠ 0 and take \frac{d z}{z} ∈ annb_{1}(x) ⊂ ν_{F} (1). This means

a\frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ∧ \frac{d z}{z} ∈ ℘Ω_{F}^{n+1} + d Ω_{F}^{n} .

If \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ∧ \frac{d z}{z} = 0, then we get as before z ∈ F^{2}(a_{1}, …, a_{n})^{∗} ⊂ D_{F} (≪ a_{1}, …, a_{n}; |])^{∗}. Assume \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ∧ \frac{d z}{z} ≠ 0. Then we can assume that a_{1}, …, a_{n}, z are the first elements of some 2-basis of F (in some ordering), and applying now proposition (3.3) we obtain a = ℘u + b with b ∈ F^{2}(a_{1}, …, a_{n}, z)′, i.e. b = z · h + g with h ∈ F^{2}(a_{1}, …, a_{n})^{∗} and g ∈ F^{2}(a_{1}, …, a_{n})′.

Thus z = h^{−1} (℘u + a + g) ∈ D_{F} (≪ a_{1}, …, a_{n}; |])^{∗}. This proves ( ii ). □

The isomorphisms ν_{F} (m) ≃ \overline{I}_{F}^{m} and H^{m+1}(F) ≃ \overline{I}^{m}W_{q}(F) enable us to translate this result into the language of bilinear and quadratic forms.

Let x =≪ a_{1}, …, a_{n} ≫ be a bilinear anisotropic n-fold Pfister form. Then we have

\overline{ann}b_{1}(x) = \{\overline{≪ z ≫} | z ∈ D_{F} (x)^{∗}\}

\overline{ann}q_{0}(x) = \{\overline{z} ∈ F/℘F | z ∈ D_{F} (x′)^{∗}\}

where we identify \overline{I}^{0}W_{q}(F) with F/℘F through the Arf-invariant.

If x =≪ a_{1}, …, a_{n}; a|] is a quadratic anisotropic n-fold Pfister form, then

\overline{ann}b_{1}(x) = \{\overline{≪ z ≫} | z ∈ D_{F} (x)^{∗}\}

Now the technique used in section 2 enables us to compute the full annihilators annb_{1}(x), annq_{0}(x) if x =≪ a_{1}, …, a_{n} ≫ and annb_{1}(x) if x =≪ a_{1}, …, a_{n}; a|], thereby obtaining the results (2.3), (2.4) and (2.5). Let us prove for example (2.3) (the others cases are left as exercises). Let x =≪ a_{1}, …, a_{n} ≫ and take y ∈ annb_{1}(x) ⊂ I_{F}. Then \overline{y} ∈ \overline{ann}b_{1}(x) and hence \overline{y} = \overline{≪ z ≫} for some z ∈ D_{F} (x)^{∗}. Thus y− ≪ z ≫∈ I^{2} and (y− ≪ z ≫) x = 0 i.e. y− ≪ z ≫∈ annb_{2}(x) = annb_{1}(x) · I_{F}. Write y− ≪ z ≫= ∑ y_{i}v_{i} with y_{i} ∈ annb_{1}(x), v_{i} ∈ I_{F}. Then y_{i}− ≪ z_{i} ≫∈ I_{F}^{2} for some z_{i} ∈ D_{F} (x)^{∗} and hence

y− ≪ z ≫ − ∑ ≪ z_{i} ≫ v_{i} ∈ I_{F}^{3} .

Iterating this procedure and assuming I_{F}^{N+1} = 0 for some N, we get (2.3). The general case can be reduced to the assumption I_{F}^{N+1} = 0 using the trick of section 2. This proves (2.3). The same argument applies for (2.4) and (2.5). Thus we have a complete description of the annihilators of Pfister forms over a field F with 2 = 0.

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