A DESCENT PROPERTY FOR PFISTER FORMS

MARKUS ROST

Summary

The Rosenberg-Ware theorem states that for a Galois extension K/F of odd degree the natural map of Witt rings of quadratic forms

W (F) → W (K)^{Gal(K/F)}

is an isomorphism. We extend this result to arbitrary field extensions K/F of odd degree. Basically we show that (Proposition 1)

0 → W (F) ⟶^{r_{K/F}} W (K) ⟶^{i_{1}−i_{2}} W (K ⊗ K)

is exact, where i_{1}, i_{2} are induced from the two natural maps K → K ⊗K. Further it is shown that an element of the graded Witt ring is represented by a Pfister form if this is true after an extension of odd degree (Proposition 2). We apply this to trace forms of exceptional Jordan algebras (Proposition 3). In the last section similar questions for symbols in Milnor’s K-theory and Galois cohomology are considered.

The final version of this article has been prepared during a visit at ETH Zurich. I thank the FIM at ETH Zurich for its hospitality.

1. The transfer map

For generalities on quadratic forms we refer to [7, 9].

Let F be a field of characteristic different from 2 and let W (F) be the Witt group of quadratic forms over F. Due to the Witt cancelation theorem one may identify W (F) with the set of isomorphism classes of anisotropic quadratic forms over F.

For a field extension K/F we denote by

r_{K/F} : W (F) → W (K),
r_{K/F} ([φ]) = [φ_{K}]

the homomorphism given by extension of scalars.

Let s: K → F be a nontrivial F -linear map. According to Scharlau [9, Chap. 2, § 5] there is an associated transfer map

s_{∗}: W (K) → W (F),
s_{∗}([ψ]) = [s ∘ ψ]

such that

s_{∗} ∘ r_{K/F} (x) = s_{∗}(1)x.

Date: January 6, 1998.

If K/F has odd degree, then s can be chosen so that

s_{∗}(1) = 1,

cf. [9, Chap. 2, Lemma 5.8]. In this case s_{∗} is a left inverse to r_{K/F}. It follows that the restriction map r_{K/F} is injective for extensions K/F of odd degree.

If K/F is purely inseparable (necessarily of odd degree since char F ≠ 2), then K and F have the same square class groups. Hence every quadratic form over K is extended from a form over F, as can be seen via diagonalization. Therefore W (F) = W (K) in the purely inseparable case.

Let I(F) ⊂ W (F) be the fundamental ideal consisting of the classes of even dimensional quadratic forms and let I^{n}(F) denote its n-th power. The transfer maps respect the filtration of the Witt ring by the powers of its fundamental ideal, i. e.,

s_{∗}(I^{n}(K)) ⊂ I^{n}(F),

cf. [1, Lemma 3.2]. It follows that

I^{n}(F) = r_{K/F}^{−1} (I^{n}(K))

for an extension K/F of odd degree.

2. Descent for extension of odd degree

Let K/F be a finite field extension. For each prime ideal 𝔪 of K ⊗_{F} K let

H_{𝔪} = (K ⊗_{F} K)/𝔪,

and let

i_{𝔪}^{1} , i_{𝔪}^{2} : W (K) → W (H_{𝔪} )

be the restriction maps induced from the homomorphisms

K → H_{𝔪} , a ↦ a ⊗ 1 mod 𝔪, a ↦ 1 ⊗ a mod 𝔪,

respectively. We put

δ: W (K) → ∏_{𝔪}W (H_{𝔪} ),

δ(x) = ((i_{𝔪}^{1} − i_{𝔪}^{2} )(x))_{𝔪} .

Proposition 1. Let K/F be an extension of odd degree. Then the sequence

0 → W (F) ⟶^{r_{K/F}} W (K) ⟶^{δ} ∏_{𝔪}W (H_{𝔪} )

is exact.

Proof. The injectivity of r_{K/F} has been discussed already. Further, using the bijectivity of r_{K/F} in the purely inseparable case, one easily reduces to separable extensions K/F. In this case one has

K ⊗_{F} K = ⨁_{𝔪}H_{𝔪} .

Let s: K → F be F -linear with s_{∗}(1) = 1 and let s_{𝔪} : H_{𝔪} → K be the components of s ⊗ id_{K}, i. e.,

s(a)b = ∑_{𝔪}s_{𝔪} (a ⊗ b mod 𝔪).

With these settings one has for x ∈ W (K)

∑_{𝔪}(s_{𝔪} )_{∗}(i_{𝔪}^{1} (x)) = r_{K/F} ∘ s_{∗}(x),
∑_{𝔪}(s_{𝔪} )_{∗}(i_{𝔪}^{2} (x)) = s_{∗}(1)x = x,

as may be verified on the level of forms.

If x ∈ ker δ, then i_{𝔪}^{1} (x) = i_{𝔪}^{2} (x) for all 𝔪, whence x = r_{K/F} ∘ s_{∗}(x).

The Rosenberg-Ware theorem appears as a special case of Proposition 1. Namely if K/F is a Galois extension with Galois group G, the map δ can be identified with the homomorphism

W (K) → ∏_{σ∈G} W (K),

x ↦ ((1 − σ)(x))_{σ}.

Therefore ker δ equals the subgroup of Galois invariants.

3. Descent of Pfister forms

An n-fold Pfister form is a quadratic form of type

⟨⟨a_{1}, …, a_{n}⟩⟩ = ⨂_{i=1}^{n}⟨1, −a_{i}⟩

with a_{i} ∈ F^{∗}.

The classes of n-fold Pfister forms generate I^{n}(F). If a Pfister form is isotropic, then it is hyperbolic [7, Chap. Ten].

A basic theorem of Pfister [9, Chap. 4, Theorem 4.4] asserts that an anisotropic quadratic form φ in indeterminates X = (x_{1}, …, x_{m}) is isomorphic to a Pfister form if and only if

φ(X)φ_{F (X)} ≃ φ_{F (X)}. (1)

If α and β are n-fold Pfister forms such that

α = β mod I^{n+1}(F)

in W (F)/I^{n+1}(F), then α and β are isomorphic [7, Chap. Ten, Corollary 3.4].

Lemma. Let φ be a quadratic form over F, let K/F be an extension of odd degree, and suppose that φ_{K} is isomorphic to a Pfister form. Then φ is isomorphic to a Pfister form.

Proof. Since φ_{K} is a Pfister form, the dimension of φ is a 2-power. If φ_{K} is isotropic, it is hyperbolic and therefore φ is hyperbolic. It follows that φ is a Pfister form.

Assume that φ_{K} is anisotropic. Then φ is anisotropic and the claim follows from the criterion (1) and the injectivity of W (F (X)) → W (K(X)).

Proposition 2. Let φ be a quadratic form over F, let K/F be an extension of odd degree, and suppose that there exists an n-fold Pfister form β over K such that

φ_{K} = β mod I^{n+1}(K).

Then there exists an n-fold Pfister form α over F such that

φ = α mod I^{n+1}(F).

Proof. Let H/F be a field extension and let f, g: K → H be two homomorphisms over F. We denote the by r_{f} resp. r_{g} the extension of scalars via f resp. g. Obviously

r_{f}(β) = r_{g}(β) mod I^{n+1}(H).

Hence r_{f}(β) = r_{g}(β) in W (H).

Using the last equality for the fields H_{𝔪} , it follows from Proposition 1 that β ∈ r_{K/F} (W (F)). The Lemma shows β = α_{K} for some Pfister form α. Then (φ − α)_{K} ∈ I^{n+1}(K) and therefore φ − α ∈ I^{n+1}(F).

Corollary 1. Let 0 ≤ n_{1} < n_{2} < ⋯ < n_{r} be integers and let c_{1}, …, c_{r} ∈ F^{∗}. Let further φ be a quadratic form over F, let K/F be an extension of odd degree, and suppose that there exist n_{i}-fold Pfister forms β_{i} over K (i = 1, …, r) such that

φ_{K} = c_{1}β_{1} + c_{2}β_{2} + ⋯ + c_{r}β_{r}

in W (K). Then there exist n_{i}-fold Pfister forms α_{i} over F such that β_{i} = (α_{i})_{K} (i = 1, …, r) and

φ = c_{1}α_{1} + c_{2}α_{2} + ⋯ + c_{r}α_{r}

in W (F). Moreover, if for some j > i one has β_{j} = ⟨⟨b_{1}, …, b_{s}⟩⟩β_{i} for some b_{k} ∈ K^{∗}, then α_{j} = ⟨⟨a_{1}, …, a_{s}⟩⟩α_{i} for some a_{k} ∈ F^{∗}.

Proof. For the first statement we use induction on r ≥ 0. One has φ_{K} = β_{1} mod I^{n_{1}+1}(K). By Proposition 2 there exists an n_{1}-fold Pfister form α over F such that φ = α mod I^{n_{1}+1}(F). Then necessarily β_{1} = α_{K}. The claim follows by applying the induction hypothesis to the form φ ⊥ −c_{1}α.

For the second statement first note that α_{i} is a subform of α_{j} by Springer’s theorem [9, Chap. 2, Theorem 5.3]. The claim follows from [3, Theorem 2.7].

Quadratic forms φ of the type as in Corollary 1 appear when studying trace forms of various algebras. An example is considered in the next section.

4. Serre’s (mod 2) invariants for F_{4}

It has been noticed by Serre that there are cohomological invariants

f_{3} : H^{1}(F, F_{4}) → H^{3}(F, Z/2),
f_{5} : H^{1}(F, F_{4}) → H^{5}(F, Z/2),

cf. [10, III. Annexe, § 3.4] or [11, III. Appendix 2, 3.4] and [6, § 40]. The construction of these invariants is based on the interpretation of H^{1}(F, F_{4}) as the set of isomorphism classes of exceptional Jordan algebras (cf. [5, 6, 12]) and the following description of their trace forms:

Proposition 3. Let J be an exceptional Jordan algebra over F and let

q_{J}: J → F,
q_{J}(x) = T_{J}(x^{2}),

where T_{J} denotes the trace map of J. Then there exist elements a_{1}, …, a_{5} ∈ F^{∗} such that

q_{J} ≃ ⟨1, 1, 1⟩ ⊥ 2⟨⟨a_{1}, a_{2}, a_{3}⟩⟩⟨−a_{4}, −a_{5}, a_{4}a_{5}⟩. (2)

In terms of this description, Serre’s invariants are given by

f_{3}([J]) = (a_{1})(a_{2})(a_{3}), f_{5}([J]) = (a_{1})(a_{2})(a_{3})(a_{4})(a_{5}).

For a proof of Proposition 3 using an analysis of the Tits constructions of exceptional Jordan algebras see [6, Lemma 40.1].

Alternatively one can prove Proposition 3 using Corollary 1 as follows. Let φ = 2(q_{J} ⊥ −⟨1, 1, 1⟩). After passing to an appropriate cube extension K/F, the Jordan algebra J has zero divisors. In this case (the so called “reduced” case), the description (2) of the trace form can be read off a presentation of J in terms of 3×3 matrices over an octonion algebra [5, 6]. Hence φ_{K} = ⟨⟨b_{1}, b_{2}, b_{3}, b_{4}, b_{5}⟩⟩−⟨⟨b_{1}, b_{2}, b_{3}⟩⟩ in W (K) for some b_{i} ∈ K^{∗}. By Corollary 1 one has φ = ⟨⟨a_{1}, a_{2}, a_{3}, a_{4}, a_{5}⟩⟩ − ⟨⟨a_{1}, a_{2}, a_{3}⟩⟩ in W (F) for some a_{i} ∈ F^{∗}, whence (2).

5. Descent for (mod 2) symbols in Milnor’s K-ring

Let K_{n}^{M}F be Milnor’s K-group of F [2, 8]. For a_{1}, …, a_{n} ∈ F^{∗} one denotes by {a_{1}, …, a_{n}} the image of a_{1} ⊗ ⋯ ⊗ a_{n} in K_{n}^{M}F.

Let p ≠ char F be a prime. By a (mod p)-symbol we understand an element in K_{n}^{M}F/p of the form {a_{1}, …, a_{n}} mod pK_{n}^{M}F. Let us call an element x ∈ K_{n}^{M}F/p a weak (mod p)-symbol, if there exists a finite field extension K/F of degree prime to p such that x_{K} is a (mod p)-symbol.

Is a weak (mod p)-symbol always a (mod p)-symbol?

I don’t know any counterexample. For p = 2 one has:

Corollary 2. Every weak (mod 2)-symbol is a (mod 2)-symbol.

Proof. Milnor [8] defined a homomorphism

s_{n}: K_{n}^{M}F/2 → I^{n}(F)/I^{n+1}(F),
s_{n}({a_{1}, …, a_{n}}) = ⟨⟨a_{1}, …, a_{n}⟩⟩ mod I^{n+1}(F).

The map s_{n} is surjective and it is injective on symbols [3, Prop. 2.1].

Let x ∈ K_{n}^{M}F/2, let K/F be of odd degree, and suppose that

x_{K} = {b_{1}, …, b_{n}}

with b_{i} ∈ K^{∗}. Let φ be a quadratic form such that

s_{n}(x) = φ mod I^{n+1}(F).

Applying Milnor’s homomorphism over K yields

⟨⟨b_{1}, …, b_{n}⟩⟩ = φ_{K} mod I^{n+1}(K).

By Proposition 2 there exist a_{i} ∈ F^{∗} with

s_{n}(x) = s_{n}({a_{1}, …, a_{n}}).

Since s_{n} is injective on symbols one has

x_{K} = {a_{1}, …, a_{n}}_{K} mod 2K_{n}^{M}K.

Applying the transfer map in Milnor’s K-theory yields

x = {a_{1}, …, a_{n}} mod 2K_{n}^{M}F

since K/F is of odd degree.

If n = 2 and μ_{p} ⊂ F, the question whether a weak symbol is a symbol is equivalent to the question whether an algebra of prime degree p is cyclic. This is known for p ≤ 3 and unsettled otherwise.

Using the results of [4] one can show that every weak symbol in K_{2}^{M}F/3 is a symbol for any field F of characteristic different from 2 and 3.

Beyond the cases p = 2 and n = 2, p = 3 not much seems to be known, even for instance for the following question: Let K/F be a quadratic extension, let x ∈ K_{3}^{M}F/3 and suppose that x_{K} is a symbol. Is then x itself a symbol? Similar for x ∈ K_{2}^{M}F/5.

References

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[7] T. Y. Lam, The algebraic theory of quadratic forms, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1980, Revised second printing, Mathematics Lecture Note Series.

[8] J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1969/1970), 318–344.

[9] W. Scharlau, Quadratic and Hermitian forms, Grundlehren der mathematischen Wissenschaften, vol. 270, Springer-Verlag, Berlin, 1985.

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[11] , Galois cohomology, Springer-Verlag, Berlin, 1997, Translated from the French by Patrick Ion and revised by the author.

[12] T. A. Springer, Jordan algebras and algebraic groups, Springer-Verlag, Berlin, 1998, Reprint of the 1973 edition.

NWF I - Mathematik, Universität Regensburg, D-93040 Regensburg, Germany

E-mail address: markus.rost@mathematik.uni-regensburg.de
URL: http://www.physik.uni-regensburg.de/~rom03516