Witt kernels of quadratic forms for purely inseparable multiquadratic extensions in characteristic 2

Ahmed Laghribi

Abstract. The aim of this note is to give a complete answer to the hyperbolicity of nonsingular quadratic forms over purely inseparable multiquadratic extensions in characteristic 2. This completes partial computations in [MM].

1. Introduction

Let F denote a field of characteristic 2, and let W_{q}(F) denote the Witt group of nonsingular quadratic forms over F [B2]. For a field extension K/F, there exists a group homomorphism W_{q}(F) ⟶ W_{q}(K) induced by the inclusion F ⊂ K. An important problem in the algebraic theory of quadratic forms is to compute the kernel W_{q}(K/F) of this homomorphism.

Let us recall some examples of field extensions K/F where W_{q}(K/F) was computed. It is well-known that W_{q}(K/F) is trivial for K purely transcendental over F. Recently in [L2] some general results have been proved on W_{q}(K/F) when K is given by the function field of a quadratic form. Moreover, very few results are known for K/F algebraic of finite degree, and the case of multiquadratic extensions aroused a lot of interest. In characteristic 2, it is well-known that a quadratic extension K/F is either separable and thus it is given by K = F (℘^{−1}(α)) where ℘ : K ⟶ K is the homomorphism defined by ℘(x) = x^{2} + x, or inseparable and thus it is given by K = F (\sqrt{α}) for some α ∈ F.

For α_{1}, α_{2} ∈ F, we know the kernels for bi-quadratic extensions:

(1)  W_{q} (F (℘^{−1}(α_{1}), ℘^{−1}(α_{2}))/F ) = W (F) ⊗ [1, α_{1}] + W (F) ⊗ [1, α_{2}]

(2)  W_{q} (F (℘^{−1}(α_{1}), \sqrt{α_{2}})/F ) = W (F) ⊗ [1, α_{1}] + ⟨1, α_{2}⟩ ⊗ W_{q}(F)

(3)  W_{q} (F (\sqrt{α_{1}}, \sqrt{α_{2}})/F) = ⟨1, α_{1}⟩ ⊗ W_{q}(F) + ⟨1, α_{2}⟩ ⊗ W_{q}(F)

Key words and phrases. Witt groups, nonsingular quadratic forms, multiquadratic extensions, differential forms.

The author was supported by the European research network HPRN-CT-2002-00287 “Algebraic K-Theory, Linear Algebraic Groups and Related Structures”.

where [α, β] denotes the quadratic form αx^{2} + xy + βy^{2}; W (F) is the Witt ring of regular symmetric bilinear forms over F; ⊗ is the action of W (F) on W_{q}(F) [B2]; and ⟨α_{1}, ⋯, α_{n}⟩ is the bilinear form ∑_{i=1}^{n} α_{i}x_{i}y_{i} for α_{1}, ⋯, α_{n} ∈ F^{∗}.

The kernel (1) is due to Baeza [B2, Cor. 4.16, Page 128]. The “mixed” kernel (2) is due to Ahmad [A]. The kernel (3) is also due to Baeza in the case of quadratic extensions [B2, Lem. 4.3, Page 182], and due to Mammone and Moresi in the case of bi-quadratic extensions [MM, Th. 2(i)]. Moreover, Mammone and Moresi proved that, in general, the kernel (1) does not generalize to separable tri-quadratic extensions [MM, Proposition 1], and the kernel (3) generalizes to inseparable multiquadratic extensions provided that the W (F)-submodule I^{3}W_{q}(F) of W_{q}(F) is trivial [MM, Th. 2(ii)] (see below for the definition of this submodule).

Our aim in this note is to prove that the kernel (3) generalizes to inseparable multiquadratic extensions without additional hypothesis on the ground field F. More precisely, we will prove the following theorem:

Theorem 1. Let F be a field of characteristic 2. Then for any scalars α_{1}, ⋯, α_{n} ∈ F (n ≥ 1), we have:

W_{q} (F (\sqrt{α_{1}}, ⋯, \sqrt{α_{n}})/F) = ⟨1, α_{1}⟩ ⊗ W_{q}(F) + ⋯ + ⟨1, α_{n}⟩ ⊗ W_{q}(F).

Let us make a comment on the situation when the characteristic is different from 2. Obviously, in this case a quadratic extension is of the form F (\sqrt{α}) for some α ∈ F^{∗}. An analogue result of the kernel (1) has been proved by Elman, Lam and Wadsworth [ELW, Th. 2.10, page 137]. Still in this case, Elman, Lam, Tignol and Wadsworth constructed an example of a field for which the kernel (1) does not generalize to the case of tri-quadratic extensions [ELTW, § 5, Page 1142].

For the proof of Theorem 1, we will begin by a generalization to the case of purely inseparable multiquadratic extensions of a recent result by Aravire and Baeza [AB] concerning the behaviour of differential forms under inseparable quadratic extensions (Proposition 1). We also use a result by Kato [K] which establishes the connection between nonsingular quadratic forms and differential forms of F over F^{2}. We first reduce the proof of Theorem 1 to the case of a field which admits a finite 2-basis, and then give a proof in this case.

We suppose that the reader is familiar with the algebraic theory of quadratic forms in characteristic 2. For any unexplained notation and terminology we refer to [B2] or [HL]. However, some definitions used here are taken from [HL], and some of them differ from those introduced in [B2]. For this reason, we fix again some rappels: A nonsingular quadratic form over F is just an orthogonal sum of binary quadratic forms of type [a, b] with a, b ∈ F. The hyperbolic plane, denoted by ℍ, is the quadratic form [0, 0]. For n ≥ 1, an n-fold Pfister form is a nonsingular quadratic form of type ⟨1, a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{n−1}⟩ ⊗ [1, a_{n}] for a_{1} ≠ 0, ⋯, a_{n−1} ≠ 0, a_{n} ∈ F. We denote it by ⟨⟨a_{1}, ⋯, a_{n}]]. Let P_{n}F (resp. GP_{n}F) denote the set of n-fold Pfister forms up to isometry (resp. the set {απ | α ∈ F^{∗}, π ∈ P_{n}F }). We denote by I^{n}W_{q}(F) the W (F)-submodule of W_{q}(F) generated by GP_{n}F.

Let Ω_{F}^{n} denote the vector space of n-differential forms of F over F^{2} (with Ω_{F}^{0} = F), and let d : Ω_{F}^{n} ⟶ Ω_{F}^{n+1} denote the differential operator defined by:

d (αda_{1} ∧ da_{2} ∧ ⋯ ∧ da_{n}) = dα ∧ da_{1} ∧ da_{2} ∧ ⋯ ∧ da_{n}.

For {b_{i} | i ∈ I} a 2-basis of F, and after choosing an ordering on the set I, we get that \{\frac{db_{i_{1}}}{b_{i_{1}}} ∧ ⋯ ∧ \frac{db_{in}}{b_{in}} | i_{1} < ⋯ < i_{n}\} is an F -basis of Ω_{F}^{n}. A result by Cartier

[C] asserts the existence of a homomorphism ℘_{n} : Ω_{F}^{n} ⟶ Ω_{F}^{n} /dΩ_{F}^{n−1} given on generators by:

℘_{n} (x\frac{db_{i_{1}}}{b_{i_{1}}} ∧ ⋯ ∧ \frac{db_{i_{n}}}{b_{i_{n}}} ) = \overline{(x^{2} − x)\frac{db_{i_{1}}}{b_{i_{1}}} ∧ ⋯ ∧ \frac{db_{i_{n}}}{b_{i_{n}}}} .

We denote by H^{n}F (resp. Q_{n}(F)) the cokernel of ℘_{n} (resp. the quotient I^{n}W_{q}(F)/I^{n+1}W_{q}(F)).

An important result that we will use in this note is due to Kato [K], and asserts that for any integer n ≥ 1, there exists an isomorphism k_{n} : Q_{n}(F) ⟶ H^{n−1}F given on generators by:

k_{n} (\overline{⟨⟨a_{1}, ⋯, a_{n}]]}) = \overline{a_{n} \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n−1}}{a_{n−1}}} .

2. A preliminary result

To prepare the proof of Theorem 1, we begin by a generalization of [AB, Lem. 2.18] to the case of purely inseparable multiquadratic extensions:

Proposition 1. Let F be a field of characteristic 2 and α_{1}, ⋯, α_{m} ∈ F^{∗} (m ≥ 1). Then we have:

Ker (H^{n}F ⟶ H^{n}F (\sqrt{α_{1}}, ⋯, \sqrt{α_{m}})) = ∑_{i=1}^{m} \overline{Ω_{F}^{n−1} ∧ dα_{i}}.

Proof. Without loss of generality we may suppose that [F (\sqrt{α_{1}}, ⋯, \sqrt{α_{m}}) : F] = 2^{m}. We proceed by induction on m. The case m = 1 has been proved by Aravire and Baeza [AB, Lem. 2.18]. Suppose m ≥ 2 and that the proposition is true as soon as we have a purely inseparable multiquadratic extension of degree < 2^{m} over a field of characteristic 2. Let ω ∈ Ω_{F}^{n} be such that \overline{ω} ∈ Ker(H^{n}F ⟶ H^{n}F (\sqrt{α_{1}}, ⋯, \sqrt{α_{m}})). Since \overline{ω}_{F (\sqrt{α_{1}})} ∈ Ker(H^{n}F (\sqrt{α_{1}}) ⟶ H^{n}F (\sqrt{α_{1}}, ⋯, \sqrt{α_{m}})), it follows from induction hypothesis that

(4)  \overline{ω}_{F (\sqrt{α_{1}})} = ∑_{i=2}^{m} \overline{ω_{i} ∧ dα_{i}}

for suitable ω_{2}, ⋯, ω_{m} ∈ Ω_{F (\sqrt{α_{1}})}^{n−1}.

Claim: In the equality (4), we may suppose ω_{2}, ⋯, ω_{m} in Ω_{F}^{n−1}, and thus again by induction hypothesis applied to the field F (\sqrt{α_{1}}) we may conclude.

Hence, it suffices to justify the claim. Since α_{1} ∉ F^{∗2}, we may choose a 2-basis B of F such that B ∪ \{\sqrt{α_{1}}\} is a 2-basis of F (\sqrt{α_{1}}).

Let us now fix i ∈ \{2, ⋯, m\}. We have ω_{i} = ∑_{finite} x_{j}^{i}ω_{j}^{i} with x_{j}^{i} ∈ F (\sqrt{α_{1}})^{∗},

and

ω_{j}^{i} = dc_{j_{1}}^{i} ∧ ⋯ ∧ dc_{j_{n−1}}^{i}

not zero with c_{j_{1}}^{i}, ⋯, c_{j_{n−1}}^{i} ∈ B ∪ \{\sqrt{α_{1}}\} (j_{1} < ⋯ < j_{n−1}).

• On the one hand, since the equality (4) is taken modulo ℘_{n}(Ω_{F (\sqrt{α_{1}})}^{n}), we may suppose, after changing if necessary x_{j}^{i}ω_{j}^{i} by (x_{j}^{i})^{2}ω_{j}^{i}, that x_{j}^{i} ∈ F^{∗}.

• One the other hand, if there exists an indice j_{k} such that in the expression of ω_{j}^{i} (see above) we have c_{j_{k}}^{i} = \sqrt{α_{1}}, says j_{k} = j_{1}, then c_{j_{2}}^{i}, ⋯, c_{j_{n−1}}^{i} ∈ B since ω_{j}^{i} is not zero. But by using the differential operator d we get:

\begin{aligned}d [(x_{j}^{i}\sqrt{α_{1}}) dc_{j_{2}}^{i} ∧ ⋯ ∧ dc_{j_{n−1}}^{i} ∧ dα_{i}] = & x_{j}^{i}ω_{j}^{i} ∧ dα_{i} + \\ & \sqrt{α_{1}} (dx_{j}^{i} ∧ dc_{j_{2}}^{i} ∧ ⋯ ∧ dc_{j_{n−1}}^{i} ∧ dα_{i}) . \end{aligned}

Hence, in H^{n}F (\sqrt{α_{1}}) we get

\begin{aligned}\overline{x_{j}^{i}ω_{j}^{i} ∧ dα_{i}} & =^{(1)} & \overline{(\sqrt{α_{1}}) (dx_{j}^{i} ∧ dc_{j_{2}}^{i} ∧ ⋯ ∧ dc_{j_{n−1}}^{i} ∧ dα_{i})} \\ & =^{(2)} & \overline{α_{1} (dx_{j}^{i} ∧ dc_{j_{2}}^{i} ∧ ⋯ ∧ dc_{j_{n−1}}^{i} ∧ dα_{i})} \end{aligned}

where for (1) (resp. for (2)) we proceed modulo dΩ_{F (\sqrt{α_{1}})}^{n−1} (resp. modulo ℘_{n}(Ω_{F (\sqrt{α_{1}})}^{n})). Hence the claim. □

As a corollary of Kato’s result (cited before) and Proposition 1 we get the following:

Corollary 1. With the same notation as in Proposition 1, we have:

Ker (Q_{n}(F) ⟶ Q_{n} (F (\sqrt{α_{1}}, ⋯, \sqrt{α_{m}}))) = ∑_{i=1}^{m} \overline{⟨1, α_{i}⟩ I^{n−1}W_{q}(F)}.

Proof. Set L = F (\sqrt{α_{1}}, ⋯, \sqrt{α_{m}}). We have a commutative diagram
Q_{n}(F) ⟶ Q_{n}(L)
↓  ↓
H^{n−1}(F) ⟶ H^{n−1}(L)

where the vertical arrows are given by the isomorphism k_{n} (k_{n} as before), and the horizontal arrows are induced by the inclusion F ⊂ L. Hence, x ∈ Ker(Q_{n}(F) ⟶ Q_{n}(L)) implies that k_{n}(x) ∈ Ker(H^{n−1}F ⟶ H^{n−1}L). By Proposition 1 we de-

duce that k_{n}(x) ∈ ∑_{i=1}^{m} \overline{Ω_{F}^{n−2} ∧ dα_{i}}, and by the isomorphism k_{n} it is clear that

x ∈ ∑_{i=1}^{m} \overline{⟨1, α_{i}⟩ I^{n−1}W_{q}(F)}. Obviously, ∑_{i=1}^{m} \overline{⟨1, α_{i}⟩ I^{n−1}W_{q}(F)} ⊂ Ker(Q_{n}(F) ⟶

Q_{n}(L)). □

3. Proof of Theorem 1

With the same notation as in the theorem, let φ ∈ W_{q} (F (\sqrt{α_{1}}, ⋯, \sqrt{α_{n}})/F ). Set L = F (\sqrt{α_{1}}, ⋯, \sqrt{α_{n}}) and φ = [a_{1}, b_{1}] ⊥ ⋯ ⊥ [a_{m}, b_{m}]. We will give the proof in two steps.

1) Reduction to the case of a field with a finite 2-basis: Let 𝔽_{2} be the finite field with two elements and F_{0} = 𝔽_{2}(α_{1}, ⋯, α_{n}; a_{1}, b_{1}, ⋯, a_{m}, b_{m}). The quadratic form φ is defined over the subfield F_{0} of F. Since φ_{L} is hyperbolic, and after viewing L^{2m} as the underlying vector space of φ_{L}, there exist vectors v_{1}, ⋯, v_{m} ∈ L^{2m}, linearly independent over L, such that φ_{L}(v_{i}) = 0 and B_{φ_{L}}(v_{i}, v_{j}) = 0 where B_{φ_{L}} is the symmetric bilinear form associated to φ_{L}. Set v_{i} = (w_{1}^{i}, ⋯, w_{2m}^{i}) with w_{1}^{i}, ⋯, w_{2m}^{i} ∈ L (1 ≤ i ≤ m). For any (i, k) ∈ {1, ⋯, m} × {1, ⋯, 2m} set

w_{k}^{i} = ∑_{finite} c_{i_{1},··· ,i_{n}}^{i,k}(\sqrt{α_{1}})^{i_{1}} … (\sqrt{α_{n}})^{i_{n}}

with i_{j} ∈ {0, 1} and c_{i_{1},··· ,i_{n}}^{i,k} ∈ F. Now let

K = F_{0} (c_{i_{1},··· ,i_{n}}^{i,k} | i = 1, ⋯, m; k = 1, ⋯, 2m)

and

K′ = K(\sqrt{α_{1}}, ⋯, \sqrt{α_{n}}).

Since K′ ⊂ L, the vectors v_{1}, ⋯, v_{m} are linearly independent over K′, and thus φ_{K′} is hyperbolic since v_{1}, ⋯, v_{m} ∈ (K′)^{2m}. Clearly, K has a finite 2-basis since it is finitely generated over 𝔽_{2}. Moreover, we may suppose that φ is not hyperbolic over F, and thus it is not hyperbolic over K. Since an element of ∑_{i=1}^{n} ⟨1, α_{i}⟩ W_{q}(K) can be considered as an element of ∑_{i=1}^{n} ⟨1, α_{i}⟩ W_{q}(F) after extending scalars to F, we see that the proof can be reduced to the case of a field with a finite 2-basis.

2) Case where F has a finite 2-basis: If n_{0} is the number of elements of a 2-basis, we deduce that the space Ω_{F}^{i} is trivial for i > n_{0}. By Kato’s result and the Hauptsatz of Arason and Pfister [B1, Satz 4.1], the group I^{i}W_{q}(F) is also trivial for i > n_{0} + 1. Let r ≥ 1 be such that φ ∈ I^{r}W_{q}(F). Since φ_{L} is hyperbolic, we deduce that \overline{φ} ∈ Ker(Q_{r}(F) ⟶ Q_{r}(L)). It follows from Corollary 1 that

\overline{φ} ∈ ∑_{i=1}^{n} \overline{⟨1, α_{i}⟩ I^{r−1}W_{q}(F)}. Let ρ_{1}, ⋯, ρ_{n} ∈ I^{r−1}W_{q}(F) be such that

(5)  φ ⊥ ∑_{i=1}^{n} ⟨1, α_{i}⟩ ⊗ ρ_{i} ∈ I^{r+1}W_{q}(F)

It is clear that the form φ′ := φ ⊥ ∑_{i=1}^{n} ⟨1, α_{i}⟩ ⊗ ρ_{i} is hyperbolic over L. If φ′ is

hyperbolic over F, then we are done. If not, we reproduce the same argument as

above for φ′ to get φ′ ⊥ ∑_{i=1}^{n} ⟨1, α_{i}⟩ ⊗ ρ′_{i} ∈ I^{r+2}W_{q}(F) for some quadratic forms

ρ′_{1}, ⋯, ρ′_{n} ∈ I^{r}W_{q}(F), i.e. φ′′ := φ ⊥ ∑_{i=1}^{n} ⟨1, α_{i}⟩ ⊗ (ρ′_{i} ⊥ ρ_{i}) ∈ I^{r+2}W_{q}(F). If

φ′′ is hyperbolic over F, then φ ∈ ∑_{i=1}^{n} ⟨1, α_{i}⟩ ⊗ W_{q}(F) and we are done. If not, we

continue the process in order to get a quadratic form φ ⊥ ∑_{i=1}^{n} ⟨1, α_{i}⟩⊗δ_{i} ∈ I^{k}W_{q}(F)

for some δ_{1}, ⋯, δ_{n} ∈ W_{q}(F) and k > n_{0} +1, and thus to get a hyperbolic quadratic form. □

4. A question

Before we formulate a general question, let us recall that the function field of a symmetric bilinear form B, denoted by F (B), is the function field of the quadratic form B̃ defined by B̃(v) = B(v, v) for v ∈ V, where V denotes the underlying vector space of B (B̃ is uniquely determined by the isometry class of B). In particular, the function field of the bilinear form ⟨1, α⟩, α ∈ F^{∗}, is the field F (x)(\sqrt{α}) for x a variable over F. Hence, Theorem 1 describes the kernels W_{q} (F (B_{1}) ⋯ (B_{n})/F) for B_{i} = ⟨1, α_{i}⟩ (1 ≤ i ≤ n). So in view of this it is natural to ask the following question:

Question 1. Let F be a field of characteristic 2, and let B_{1}, ⋯, B_{n} be symmetric bilinear Pfister forms of dimension ≥ 2 (n ≥ 1). Is it true that the kernel W_{q}(F (B_{1}) ⋯ (B_{n})/F) equals B_{1} ⊗ W_{q}(F) + ⋯ + B_{n} ⊗ W_{q}(F)?

Acknowledgment. I am grateful to Pasquale Mammone who drew my attention on the problem studied in this note, and for some discussions that I have had with him during the preparation of this work.

References

[A] H. Ahmad, Witt kernels of bi-quadratic extensions in characteristic 2, preprint.

[B1] R. Baeza, Ein Teilformensatz für quadratische Formen in Charakteristik 2, Math. Z. 135 (1974), 175–184.

[B2] R. Baeza, Quadratic forms over semilocal rings, Lect. Notes Math. vol. 665, Berlin, Heidelberg, New York: Springer 1978.

[AB] R. Aravire, R. Baeza, The behavior of quadratic and differential forms under function field extensions in characteristic 2, J. Algebra 259 (2003), 361–414.

[C] P. Cartier, Questions de rationalité en géométrie algébrique, Bull. Soc. Math. France 86 (1958), 177–251.

[ELTW] R. Elman, T. Y. Lam, J.-P. Tignol, A. Wadsworth, Witt rings and Brauer groups under multiquadratic extensions I, Am. J. Math. 105 (1983), 1119– 1170.

[ELW] R. Elman, T. Y. Lam, A. Wadsworth, Quadratic forms under multiquadratic extensions, Indag. Math. 42 (1980), 131–145.

[HL] D. W. Hoffmann, A. Laghribi, Quadratic forms and Pfister neighbors in characteristic 2, Trans. Amer. Math. Soc. 356 (2004), 4019-4053.

[K] K. Kato, Symmetric bilinear forms, quadratic forms and Milnor K-theory in characteristic 2, Inventiones Math. 66 (1982), 493–510.

[L1] A. Laghribi, Certaines formes quadratiques de dimension au plus 6 et corps des fonctions en caractéristique 2, Israel J. Math. 129 (2002), 317-361.

[L2] A. Laghribi, Witt kernels of function field extensions in characteristic 2, preprint 2004.

[MM] P. Mammone, R. Moresi, Formes quadratiques, algèbres à division et extensions multiquadratiques inséparables, Bull. Belg. Math. Soc. 2 (1995), 311–319.

Faculté des Sciences Jean Perrin, rue Jean Souvraz - SP18, F-62307 Lens.

    E-mail address: laghribi@euler.univ-artois.fr