Excellent Special Orthogonal Groups

Oleg H. Izhboldin and Ina Kersten

Received: September 16, 2001

Communicated by Ulf Rehmann

Abstract. In this paper we give a complete classification of excellent special orthogonal groups SO(q) where q is a regular quadratic form over a field of characteristic 0.

2000 Mathematics Subject Classification: 11E81, 11E04

Keywords and Phrases: Quadratic forms, Pfister forms and neighbors, quasi-excellent forms and sequences

Contents

0. Introduction 385 1. Notation and background material 388 2. Elementary properties of quasi-excellent forms and sequences 391 3. Inductive properties of quasi-excellent sequences 393 4. Classification theorem for sequences of the first type 396 5. Quasi-excellent sequences modulo some ideals 397 6. Five classical conjectures 398 7. Classification theorem for sequences of the second type 403 8. Classification theorem for sequences of the third type 405 9. Quasi-excellent sequences of type 3 and height 3 408 References 410

0. Introduction

Let F be a field of characteristic ≠ 2 and let φ be a regular quadratic form over F. Then φ is said to be excellent if, for any field extension E/F, the anisotropic part of φ_{E} := φ ⊗_{F} E is defined over F. This notion was introduced by M. Knebusch in [Kn1, Kn2]. In [KR], a similar notion for semisimple algebraic groups was introduced and studied for special linear and special orthogonal groups. Let us recall that the main result of [KR] says that the following conditions are equivalent.

(i) The special orthogonal group 𝕊𝕆(φ) is excellent. (ii) For every field extension E/F there is an element a ∈ E^{∗} and a form ψ

over F such that the anisotropic part of φ_{E} is isomorphic to aψ_{E} . In general, if φ is excellent 𝕊𝕆(φ) is also excellent. The converse holds for odddimensional forms (see [KR]). For even-dimensional forms there are examples of non-excellent forms φ such that the group 𝕊𝕆(φ) is excellent.

We say that the form φ is quasi-excellent if the group 𝕊𝕆(φ) is excellent. Taking into account the criterion mentioned above, we can rewrite the definition as follows: φ is quasi-excellent if for any field extension E/F there exists a form ψ over F such that (φ_{E})_{an} is similar to ψ_{E} . In this case we write (φ_{E})_{an} ∼ ψ_{E} .

To study even-dimensional quasi-excellent forms, it is very convenient to give another definition.

Definition 0.1. We say that a sequence of quadratic forms φ_{0}, φ_{1}, …, φ_{h} over F is quasi-excellent if the following conditions hold:

• the forms φ_{0}, …, φ_{h−1} are regular and of dimension > 0 ;

• the form φ_{0} is anisotropic and the form φ_{h} is zero;

• for i = 1, …, h, we have ((φ_{0})_{F_{i}})_{an} ∼ (φ_{i})_{F_{i}} where F_{i} = F (φ_{0}, …, φ_{i−1}). Then the number h is called the height of the sequence. (It coincides with the height of φ_{0} defined by Knebusch in [Kn1], 5.4.)

It is not difficult to show that we have a surjective map (see Lemma 2.2 and Corollary 2.4 below):

{quasi-excellent sequences} → {even-dim. quasi-excellent anisotropic forms}
(φ_{0}, φ_{1}, …, φ_{h}) ↦ φ_{0}

Any regular quadratic form of dimension n > 0 over F is isomorphic to a diagonal form ⟨a_{1}, …, a_{n}⟩ := a_{1}X_{1}^{2} + … + a_{n}X_{n}^{2} with a_{1}, …, a_{n} ∈ F^{∗} and variables X_{1}, …, X_{n}. A d-fold Pfister form is a form of the type

⟨ ; ⟨a_{1}, …, a_{d}⟩ ; ⟩ := ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{d}⟩ .

Let (φ_{0}, φ_{1}, …, φ_{h}) be a quasi-excellent sequence. We prove in Lemma 2.5 that φ_{h−1} is similar to some d-fold Pfister form, and then say that d is the degree of the sequence.

Example 0.2. Let a_{1}, a_{2}, …, a_{d}, k_{0}, k_{1}, k_{2}, u, v, c ∈ F^{∗}. Set

ϕ_{0} = k_{0} ⟨ ; ⟨a_{1}, a_{2}, …, a_{d−1}⟩ ; ⟩ ⊗ (⟨ ; ⟨u, v⟩ ; ⟩ ⊥ −c ⟨ ; ⟨a_{d}⟩ ; ⟩) ,
ϕ_{1} = k_{1} ⟨ ; ⟨a_{1}, a_{2}, …, a_{d−1}⟩ ; ⟩ ⊗ ⟨−u, −v, uv, a_{d}⟩ ,
ϕ_{2} = k_{2} ⟨ ; ⟨a_{1}, a_{2}, …, a_{d}⟩ ; ⟩ ,
ϕ_{3} = 0 .

Suppose that ϕ_{0} , ϕ_{1} and ϕ_{2} are anisotropic. Then the sequence (ϕ_{0}, ϕ_{1}, ϕ_{2}, ϕ_{3}) is quasi-excellent of degree d (see Lemma 9.1). We notice that dim ϕ_{h−1} = 2^{d}, dim ϕ_{h−2} = 2^{d+1} and dim ϕ_{h−3} = 3 · 2^{d}.

Clearly, the sequences (ϕ_{1}, ϕ_{2}, ϕ_{3}) and (ϕ_{2}, ϕ_{3}) are also quasi-excellent. In particular, the forms ϕ_{0}, ϕ_{1}, ϕ_{2}, and ϕ_{3} are quasi-excellent.

Definition 0.3. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence of degree d. We say that the sequence is of the

• “first type” if dim φ_{h−2} ≠ 2^{d+1} or h = 1

• “second type” if dim φ_{h−2} = 2^{d+1} and, if h ≥ 3 , dim φ_{h−3} ≠ 3 · 2^{d}

• “third type” if dim φ_{h−2} = 2^{d+1} and dim φ_{h−3} = 3 · 2^{d}, (here h ≥ 3) .

Example 0.4. Let (ϕ_{0}, ϕ_{1}, ϕ_{2}, ϕ_{3}) be the sequence constructed in Example 0.2. Assume that ϕ_{0} , ϕ_{1} and ϕ_{2} are anisotropic.

• The sequence (ϕ_{2}, ϕ_{3}) is of the first type,

• The sequence (ϕ_{1}, ϕ_{2}, ϕ_{3}) is of the second type,

• The sequence (ϕ_{0}, ϕ_{1}, ϕ_{2}, ϕ_{3}) is of the third type.

According to Knebusch [Kn2], 7.4, a regular quadratic form ψ is called a Pfister neighbor, if there exist a Pfister form π, some a ∈ F^{∗}, and a form η with dim η < dim ψ, such that ψ⊥η ≃ aπ. The form η is called the complementary form of the Pfister neighbor ψ.

Example 0.5. Let (φ_{1}, …, φ_{h}) be a quasi-excellent sequence. Let φ_{0} be an anisotropic Pfister neighbor whose complementary form is similar to φ_{1}. Then the sequence (φ_{0}, φ_{1}, …, φ_{h}) is quasi-excellent. Moreover, this sequence is of the same type as the sequence (φ_{1}, …, φ_{h}). (Note that ((φ_{0})_{F_{1}})_{an} ∼ (φ_{1})_{F_{1}} by [Kn2], p. 3.)

Clearly, Examples 0.4 and 0.5 give rise to the construction of many examples of quasi-excellent sequences of prescribed type: We start with a quasi-excellent sequence given in Example 0.4. We can then apply the construction presented in Example 0.5 to obtain a new quasi-excellent sequence. Since we can apply the construction in Example 0.5 many times, we get quasi excellent sequences of arbitrary height.

The main goal of this paper is to prove (under certain assumptions) that all quasi-excellent sequences can be constructed by using this recursive procedure. To be more accurate, for sequences of the first type, we prove the following classification result:

Theorem 0.6. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence of the first type. Then for any i < h the form φ_{i} is a Pfister neighbor whose complementary form is similar to φ_{i+1}.

For sequences of the second and the third type, we state our classification results as conjectures which we will prove for sequences of degree 1 . For sequences of arbitrary degree we will deduce our conjectures from some classical conjectures which now seem to be settled for all fields of characteristic 0 , cf. [Vo, OVV].

Conjecture 0.7. Let (φ_{0}, φ_{1}, …, φ_{h}) be a quasi-excellent sequence of the second type. Then for any i < h − 2 the form φ_{i} is a Pfister neighbor whose complementary form is similar to φ_{i+1}. Besides, the forms φ_{h−2} and φ_{h−1} look as follows:

φ_{h−2} ∼ ⟨ ; ⟨a_{1}, …, a_{d−1}⟩ ; ⟩ ⊗ ⟨−u, −v, uv, a_{d}⟩ ,
φ_{h−1} ∼ ⟨ ; ⟨a_{1}, …, a_{d−1}, a_{d}⟩ ; ⟩ .

(For d = 1 we put ⟨ ; ⟨a_{1}, …, a_{d−1}⟩ ; ⟩ = ⟨1⟩.)

Conjecture 0.8. Let (φ_{0}, φ_{1}, …, φ_{h}) be a quasi-excellent sequence of the third type. Then for any i < h − 3 the form φ_{i} is a Pfister neighbor whose complementary form is similar to φ_{i+1}. Besides, the forms φ_{h−3} , φ_{h−2} and φ_{h−1} look as follows:

φ_{h−3} ∼ ⟨ ; ⟨a_{1}, …, a_{d−1}⟩ ; ⟩ ⊗ (⟨ ; ⟨u, v⟩ ; ⟩ ⊥ −c ⟨ ; ⟨a_{d}⟩ ; ⟩),
φ_{h−2} ∼ ⟨ ; ⟨a_{1}, …, a_{d−1}⟩ ; ⟩ ⊗ ⟨−u, −v, uv, a_{d}⟩ ,
φ_{h−1} ∼ ⟨ ; ⟨a_{1}, …, a_{d−1}, a_{d}⟩ ; ⟩ .

The main results of this paper are Theorem 0.6 and the following two theorems.

Theorem 0.9. Conjectures 0.7 and 0.8 are true for quasi-excellent sequences of degree 1. The well-known so far unpublished result by Rost, that the Milnor invariant e^{4} is bijective, implies that 0.7 and 0.8 are also true for sequences of degree 2 .

Theorem 0.10. Modulo results proved in [Vo, OVV] both Conjectures 0.7 and 0.8 are true over any field of characteristic 0.

All results of this paper are due to the first-named author Oleg Izhboldin. The second-named author is responsible for a final version of Oleg’s beautiful draft which he could not complete because of his sudden death on April 17, 2000.

1. Notation and background material

We fix a ground field F of characteristic different from 2 and set F^{∗} = F \{0}. If two quadratic forms φ and ψ are isomorphic we write φ ≃ ψ. We say that φ and ψ are similar if φ ≃ aψ for some a ∈ F^{∗}, and write φ ∼ ψ. A regular quadratic form φ of dimension dim φ > 0 is said to be isotropic if there is a non-zero vector v in the underlying vector space of φ such that φ(v) = 0 , and anisotropic otherwise. The zero form 0 is assumed to be anisotropic. As has been shown by Witt [W], any regular quadratic form φ has a decomposition

φ ≃ i × ⟨1, −1⟩ ⊥ φ_{an}

where φ_{an} is anisotropic and i ≥ 0 . Moreover, the number i =: i(φ) and, up to isomorphism, the form φ_{an} are uniquely determined by φ. We call i(φ) the Witt index of φ. If i(φ) > 0 then φ is isotropic. A form φ ≠ 0 is said to be hyperbolic if φ_{an} = 0 . We have a Witt equivalence relation φ ∼_{w} ψ defined by

φ ∼_{w} ψ ⟺ φ_{an} ≃ ψ_{an}

The Witt equivalence classes [φ] of regular quadratic forms φ over F form a commutative ring W (F) with zero element [0] and unit element [⟨1⟩] . The operations in this Witt ring W (F) are induced by:

⟨a_{1}, …, a_{m}⟩ ⊥ ⟨b_{1}, …, b_{n}⟩ = ⟨a_{1}, …, a_{m}, b_{1}, …, b_{n}⟩

⟨a_{1}, …, a_{m}⟩ ⊗ ⟨b_{1}, …, b_{n}⟩ = ⟨a_{1}b_{1}, …, a_{1}b_{n}, …, a_{m}b_{1}, …, a_{m}b_{n}⟩.

In particular, there is a surjective ring homomorphism

e^{0} : W (F) → ℤ/2ℤ , [φ] ↦ (dim φ) mod 2ℤ .

Its kernel I(F) := ker(e^{0}) is called the fundamental ideal of W (F). Since ⟨a, b⟩ ∼_{w} ⟨ ; ⟨−a⟩ ; ⟩ ⊥ − ⟨ ; ⟨b⟩ ; ⟩ the ideal I(F) is generated by the classes of the 1-fold Pfister forms ⟨ ; ⟨a⟩ ; ⟩ = ⟨1, −a⟩ with a ∈ F^{∗}. Consequently, the nth power ideal I^{n}(F) of I(F) is generated by the classes of n-fold Pfister forms ⟨ ; ⟨a_{1}, …, a_{n}⟩ ; ⟩ = ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩ . We will use the Arason-Pfister Hauptsatz [AP]:

Theorem 1.1. (Arason-Pfister) If [φ] ∈ I^{n}(F) and dim φ_{an} < 2^{n} then φ ∼_{w} 0 .

Put F^{∗}^{2} = \{x^{2} ∈ F^{∗} | x ∈ F^{∗}\} and d(φ) := (−1)^{\binom{m}{2}} det(φ) with m = dim φ. Then there is a surjective group homomorphism

e^{1} : I(F) → F^{∗}/F^{∗2}, [φ] ↦ d(φ) F^{∗}^{2},

satisfying ker(e^{1}) = I^{2}(F) , see [Pf1], 2.3.6. For any ideal I in W (F) we write φ ≡ ψ mod I when [φ⊥−ψ] ∈ I. Let μ = ⟨a_{1}, …, a_{m}⟩ with a_{1}, …, a_{m} ∈ F^{∗}. If m is odd then d(⟨a_{1}, …, a_{m}, −d(μ)⟩) = ∏_{i=1}^{m} a_{i}^{2} ∈ F^{∗}^{2}, and we obtain the following remark which will be used for the classification of quasi-excellent sequences of the first type.

Remark 1.2. If dim μ is odd then μ ≡ ⟨ d(μ) ⟩ mod I^{2}(F) .

Of special interest for us is the function field F (φ) of a regular quadratic form φ. Assuming that dim φ ≥ 2 and φ ≄ ⟨1, −1⟩ we let F (φ) be the function field of the projective variety defined by φ. Its transcendence degree is (dim φ) − 2 and φ_{F (φ)} is isotropic. Moreover, F (φ) is purely transcendental over F iff φ is isotropic (cf. [Kn1], 3.8). We denote by F (φ, ψ) the function field of the product of the varieties defined by the forms φ and ψ.

We say that φ is a subform of ψ, and write φ ⊂ ψ, if φ is isomorphic to an orthogonal summand of ψ. We will use the following two consequences of the Cassels-Pfister Subform Theorem [Pf1], 1.3.4:

Theorem 1.3. ([Kn1], 4.4, and [S], 4.5.4 (ii)) Let λ be an anisotropic form and ρ be a Pfister form. Then the following conditions are equivalent:

• there exists a form μ such that λ ≃ ρ ⊗ μ,

• there exists a form ν such that λ ∼_{w} ρ ⊗ ν ,

• λ_{F (ρ)} is hyperbolic. Moreover, in these cases kρ ⊂ λ for any k ∈ D(λ) := {a ∈ F^{∗} | ⟨a⟩ ⊂ φ}.

Theorem 1.4. ([Kn1], 4.5) Let φ and ψ be forms of dimension ≥ 2 satisfying φ ≄ ⟨1, −1⟩ and ψ ≁_{w} 0 . If ψ_{F (φ)} ∼_{w} 0 then φ is similar to a subform of ψ, hence dim φ ≤ dim ψ. Consequently, if dim φ > dim ψ then ψ_{F (φ)} is not hyperbolic.

If, in addition, φ and ψ are anisotropic and the dimensions of φ and ψ are separated by a 2-power, then ψ_{F (φ)} is not isotropic as Hoffmann has shown.

Theorem 1.5. (Hoffmann ([H1], Theorem 1) Let φ and ψ be anisotropic forms with dim ψ ≤ 2^{n} < dim φ for some n > 0 . Then ψ_{F (φ)} is anisotropic.

In accordance with the definition given in the introduction, a form φ is a Pfister neighbor of a d-fold Pfister form π if dim φ > 2^{d−1} and φ is similar to a subform of π.

Theorem 1.6. (Hoffmann [H1], Corollaries 1, 2) Let φ be an anisotropic form of dimension 2^{n} + m with 0 < m ≤ 2^{n}. Then dim(φ_{F (φ)})_{an} ≥ 2^{n} − m. If, in addition, φ is a Pfister neighbor then dim(φ_{F (φ)})_{an} = 2^{n} − m.

The following theorem is a result by Izhboldin on “virtual Pfister neighbors”, cf. [Izh], Theorem 3.5.

Theorem 1.7. (Izhboldin) Let φ be an anisotropic form of dimension 2^{n} + m with 0 < m ≤ 2^{n}. Assume that there is a field extension E/F such that φ_{E} is an anisotropic Pfister neighbor. Then either dim(φ_{F (φ)})_{an} ≥ 2^{n} or dim(φ_{F (φ)})_{an} = 2^{n} − m.

Theorem 1.8. (Knebusch [Kn2], 7.13) Let φ and ψ be anisotropic forms such that (φ_{F (φ)})_{an} ≃ ψ_{F (φ)}. Then φ is a Pfister neighbor and −ψ is the complementary form of φ.

The following theorem is a special case of the Knebusch-Wadsworth Theorem [Kn1], 5.8. It will be used in Lemma 2.5 below.

Theorem 1.9. (Knebusch-Wadsworth) Let φ be an anisotropic form such that φ_{F (φ)} is hyperbolic. Then φ is similar to a Pfister form.

Knebusch introduced in [Kn1] a generic splitting tower K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ K_{h} of a form ψ ≁_{w} 0 which is easily described as follows. Let K_{0} = F and ψ_{0} ≃ ψ_{an} and proceed inductively by letting K_{i} = K_{i−1}(ψ_{i−1}) and ψ_{i} ≃ ((ψ_{i−1})_{K_{i}})_{an} . Then h is the height of ψ, that is the smallest number such that dim ψ_{h} ≤ 1 .

The form ψ is excellent iff all forms ψ_{i} are defined over F (that is, for each i there exists a form η_{i} over F such that ψ_{i} ≃ (η_{i})_{K_{i}} ), cf. [Kn2], 7.14.

Now assume that dim ψ is even. Then ψ_{h−1} ≃ aπ for some a ∈ K_{h−1}^{∗} and some d-fold Pfister form π over K_{h−1} by Theorem 1.9. The form π is called the leading form of ψ and the number d =: deg ψ the degree of ψ. We say that ψ is a good form if π is defined over F . Then there is, up to isomorphism, a unique d-fold Pfister form τ over F such that π ≃ τ_{K_{h−1}} , cf. [Kn2], 9.2, and we will refer to this Pfister form over F as the leading form of a good form.

Saying that all odd-dimensional forms have degree 0 and that the zero form has degree ∞ we get a degree function, cf. [Kn1], p. 88:

deg : W (F) → ℕ ∪ {0} ∪ {∞} , [ψ] ↦ [deg ψ] .

For every n ≥ 0 let J_{n}(F) := { [ψ] ∈ W (F) | deg ψ ≥ n }. Then J_{n}(F) is an ideal in the Witt ring W (F) and J_{1}(F) = I(F) is the fundamental ideal. We are now prepared to formulate the next result we will need later.

Theorem 1.10. (Knebusch [Kn2], 9.6, 7.14, and 10.1; Hoffmann [H2]) Let ψ be an anisotropic good form of degree d ≥ 1 with leading form τ. Then

ψ ≡ τ mod J_{d+1}(F) .

If, in addition, ψ is of height 2 then one of the following conditions holds.

• The form ψ is excellent. In this case, ψ is a Pfister neighbor whose complementary form is similar to τ. In particular, dim ψ = 2^{N} − 2^{d} with N ≥ d + 2 , and ψ_{F (τ)} is hyperbolic.

• The form ψ is non-excellent and good. In this case dim ψ = 2^{d+1} and ψ_{F (τ)} is similar to an anisotropic (d + 1)-fold Pfister form.

We denote by P_{d}(F) (resp. GP_{d}(F)) the set of all quadratic forms over F which are isomorphic (resp. similar) to d-fold Pfister forms.

Finally, we mention the following well-known facts (e.g., [L], IX.1.1, X.1.6).

Remark 1.11. (i) Anisotropic forms over F remain anisotropic over purely transcendental extensions of F.

(ii) Isotropic Pfister forms are hyperbolic.

2. Elementary properties of quasi-excellent forms and sequences

Lemma 2.1. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence. Then • all forms φ_{i} are forms of even dimension,

• all forms φ_{i} are anisotropic,

• dim φ_{0} > dim φ_{1} > ⋯ > dim φ_{h} = 0,

• for all s = 1, …, h, we have

((φ_{0})_{F_{s}})_{an} ∼ ((φ_{1})_{F_{s}})_{an} ∼ ⋯ ∼ ((φ_{s−1})_{F_{s}})_{an} ∼ (φ_{s})_{F_{s}}

where F_{s} = F (φ_{0}, …, φ_{s−1}).

Proof. Obvious from Definition 0.1.

Lemma 2.2. Let φ be an anisotropic even-dimensional quasi-excellent form over F. Then there exists a quasi-excellent sequence (φ_{0}, φ_{1}, …, φ_{h}) such that φ_{0} = φ.

Proof. Let us define the forms φ_{i} recursively. We set φ_{0} = φ. Now, we suppose that i > 0 and that all forms φ_{0}, …, φ_{i−1} are already defined. Also, we can suppose that these forms are of dimension > 0. Put F_{i} = F (φ_{0}, …, φ_{i−1}). Since φ is quasi-excellent, there exists a form ψ over F such that (φ_{F_{i}})_{an} is similar to ψ_{F_{i}} . We put φ_{i} = ψ. If φ_{i} = 0 then we are done by setting h = i. If φ_{i} ≠ 0 we repeat the above procedure.

Lemma 2.3. Let (φ_{0}, φ_{1}, …, φ_{h}) be a quasi-excellent sequence. Let E/F be a field extension such that (φ_{0})_{E} is isotropic, and let i be the maximal integer such that all forms (φ_{0})_{E}, …, (φ_{i−1})_{E} are isotropic. Then ((φ_{0})_{E})_{an} ∼ (φ_{i})_{E} .

Proof. Since the forms (φ_{0})_{E}, …, (φ_{i−1})_{E} are isotropic, the field extension E_{i} := E(φ_{0}, …, φ_{i−1}) is purely transcendental over E. Since F_{i} ⊂ E_{i} and ((φ_{0})_{F_{i}})_{an} ∼ (φ_{i})_{F_{i}} , it follows that ((φ_{0})_{E_{i}})_{an} ∼ ((φ_{i})_{E_{i}})_{an} . Since E_{i}/E is purely transcendental we can use Springer’s theorem (e.g., [L], 6.1.7) to obtain ((φ_{0})_{E})_{an} ∼ ((φ_{i})_{E})_{an} . By definition of i, the form (φ_{i})_{E} is anisotropic.

Corollary 2.4. Let (φ_{0}, φ_{1}, …, φ_{h}) be a quasi-excellent sequence. Then the form φ_{0} is a quasi-excellent even-dimensional form.

Proof. Obvious from Lemmas 2.1 and 2.3.

Lemma 2.5. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence. Then the form φ_{h−1} is similar to a Pfister form.

Proof. By Definition 0.1, we have φ_{h} = 0 and ((φ_{h−1})_{F_{h}})_{an} ∼ (φ_{h})_{F_{h}}, where F_{h} = F (φ_{0}, …, φ_{h−1}). Therefore, (φ_{h−1})_{F_{h}} is hyperbolic. Note that F_{h} ≃ F (φ_{h−1})(φ_{0}, …, φ_{h−2}) . Since the dimensions of the forms φ_{0}, …, φ_{h−2} are strictly greater than dim φ_{h−1} and (φ_{h−1})_{F (φ_{h−1})(φ_{0},...,φ_{h−2})} is hyperbolic, it follows from Theorem 1.4 that (φ_{h−1})_{F (φ_{h−1})} is hyperbolic. By Theorem 1.9, φ_{h−1} is similar to a Pfister form.

Definition 2.6. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence.

• By Lemma 2.5, the form φ_{h−1} is similar to some Pfister form τ ∈ P_{d}(F) . We say that τ is the leading form and d is the degree of the sequence. Besides, we say that h is the height of the sequence.

• The form φ_{h−2} is called the pre-leading form of the sequence. Clearly, here we assume that h ≥ 2.

Remark 2.7. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence. Then its leading form is the leading form of φ_{0} as well. In particular, φ_{0} is a good form whose height and degree coincide with the height and degree of the sequence.

We finish this section with a lemma which we will need for the classification of quasi-excellence sequences of the second and third type.

Lemma 2.8. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence with h ≥ 2. Let E/F be an extension such that (φ_{0})_{E} is an anisotropic Pfister neighbor whose complementary form is similar to (φ_{2})_{E} . Then dim φ_{1} is a power of 2 and dim φ_{0} = 2 dim φ_{1} − dim φ_{2}.

Proof. Let us write dim φ_{0} in the form dim φ_{0} = 2^{n} + m with 0 < m ≤ 2^{n}. Since (φ_{2})_{E} is similar to the complementary form of (φ_{0})_{E}, we have dim φ_{2} = 2^{n+1} −dim φ_{0} = 2^{n} −m. Since dim φ_{1} = dim((φ_{0})_{F (φ_{0})})_{an}, Theorem 1.7 shows that either dim φ_{1} ≥ 2^{n} or dim φ_{1} = 2^{n} − m. The equality dim φ_{1} = 2^{n} − m is obviously false because dim φ_{2} = 2^{n} − m. Therefore, dim φ_{1} ≥ 2^{n}. If dim φ_{1} = 2^{n} then dim φ_{0} = 2^{n} + m = 2 · 2^{n} − (2^{n} − m) = 2 dim φ_{1} − dim φ_{2} and the proof is complete. Hence, we can assume that dim φ_{1} > 2^{n}. Then 2^{n} < dim φ_{1} < dim φ_{0} = 2^{n} + m. Therefore dim φ_{1} can be written in the form 2^{n} + m_{1} with 0 < m_{1} < m ≤ 2^{n}. Let K = F (φ_{0}). Then Lemma 2.1 shows that ((φ_{1})_{K(φ_{1})})_{an} is similar to (φ_{2})_{K(φ_{1})}. Hence, dim((φ_{1})_{K(φ_{1})})_{an} = dim φ_{2}. Since dim φ_{1} = 2^{n} +m_{1}, Theorem 1.6 shows that dim φ_{2} = dim((φ_{1})_{K(φ_{1})})_{an} ≥ 2^{n} − m_{1}. Since dim φ_{2} = 2^{n} − m, we get m_{1} ≥ m. This contradicts to the inequality m_{1} < m proved earlier.

3. Inductive properties of quasi-excellent sequences

In this section we study further properties of a quasi-excellent sequence (φ_{0}, …, φ_{h}) of degree d with leading form τ . Then we derive some results on its pre-leading form γ := φ_{h−2}. In particular, we show that dim γ is either 2^{d+1} or 2^{N} − 2^{d} with N ≥ d + 2 and that (φ_{i})_{F (γ,τ)} is hyperbolic for all i = 0, …, h − 1 .

Lemma 3.1. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence and let E = F (φ_{1}) .

• If (φ_{0})_{E} is isotropic then (φ_{1}, φ_{2}, …, φ_{h}) is a quasi-excellent sequence and ((φ_{0})_{E})_{an} ∼ (φ_{2})_{E} .

• If (φ_{0})_{E} is anisotropic then ((φ_{0})_{E}, (φ_{2})_{E}, (φ_{3})_{E}, …, (φ_{h})_{E}) is a quasiexcellent sequence.

Proof. Let F_{i} = F (φ_{0}, …, φ_{i−1}) and F_{0,i} = F (φ_{1}, …, φ_{i−1}). Assume that φ_{0} is isotropic over F (φ_{1}) . Then the extension F_{i}/F_{0,i} is purely transcendental for all i ≥ 2. By Lemma 2.1, we have ((φ_{1})_{F_{i}})_{an} ∼ (φ_{i})_{F_{i}} for all i ≥ 1. Since F_{i}/F_{0,i} is purely transcendental for i ≥ 2, we have ((φ_{1})_{F_{0,i}})_{an} ∼ (φ_{i})_{F_{0,i}} for all i ≥ 2. This means that the sequence (φ_{1}, φ_{2}, …, φ_{h}) is quasi-excellent. Now Lemma 2.3 implies that ((φ_{0})_{E})_{an} ∼ (φ_{2})_{E} . The last statement is obvious from Definition 0.1.

Lemma 3.2. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence. Suppose that φ_{0} is a Pfister neighbor whose complementary form is similar to φ_{1} . Then the sequence (φ_{1}, φ_{2}, …, φ_{h}) is quasi-excellent.

Proof. By Lemma 3.1, it suffices to show that (φ_{0})_{F (φ_{1})} is isotropic. By assumption, there is a form η ∼ φ_{1} and a Pfister form π such that φ_{0}⊥η ∼ π. Since η_{F (φ_{1})} is isotropic, the form π_{F (φ_{1})} must be hyperbolic. Since dim φ_{0} > dim η it follows that (φ_{0})_{F (φ_{1})} is isotropic.

Lemma 3.3. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence of height h ≥ 2 with leading form τ ∈ P_{d}(F) and let F_{i} = F (φ_{0}, …, φ_{i−1}) . Then the sequence

((φ_{i})_{F_{i}}, (φ_{i+1})_{F_{i}} , …, (φ_{h})_{F_{i}})

is quasi-excellent of height h − i with leading form τ_{F_{i}} ∈ P_{d}(F_{i}) for 1 ≤ i < h.

Proof. By Lemma 2.1, the forms (φ_{s})_{F_{s}} are anisotropic for s = 1, …, h. Thus (φ_{s})_{F_{i}} is anisotropic for fixed i < h and s = i, …, h. In particular, τ_{F_{i}} is anisotropic since τ_{F_{i}} ∼ (φ_{h−1})_{F_{i}} by Definition 2.6. Now the result is obvious.

Lemma 3.4. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence of degree d with leading form τ . Then for all i = 0, …, h − 1, we have φ_{i} ≡ τ mod J_{d+1}(F) . In particular, deg(φ_{i}) = deg(τ) = d for all i = 0, …, h − 1 .

Proof. By Remark 2.7 and Theorem 1.10, we have φ_{0} ≡ τ mod J_{d+1}(F) . Using Lemma 3.3 we obtain from Remark 2.7 and Theorem 1.10 that

(φ_{i})_{F_{i}} ≡ τ_{F_{i}} mod J_{d+1}(F_{i})

for all i = 1, …, h − 1 . Since dim φ_{0} > … > dim φ_{h−2} > 2^{d} = dim φ_{h−1} , the canonical map J_{d}(F_{i−1})/J_{d+1}(F_{i−1}) → J_{d}(F_{i})/J_{d+1}(F_{i}) is injective for i = 1, …, h − 1 and F_{0} = F as Knebusch [Kn1], 6.11, has shown. Thus the composed map J_{d}(F)/J_{d+1}(F) → J_{d}(F_{i})/J_{d+1}(F_{i}) is also injective. Hence φ_{i} ≡ τ mod J_{d+1}(F) for i = 1, …, h − 1 . The second statement follows from the first since J_{d}(F) is an ideal in the Witt ring W (F) .

Lemma 3.5. Let γ and τ be anisotropic forms. Suppose that dim γ = 2^{d+1} and τ ∈ P_{d}(F) for suitable d. Suppose also that γ_{F (γ)} is not hyperbolic and γ_{F (γ,τ)} is hyperbolic. Then the form (γ_{F (γ)})_{an} is similar to τ_{F (γ)} .

Proof. Let K = F (γ). By assumption, the form γ_{K(τ)} is hyperbolic. Thus Theorem 1.3 implies that there exists a K-form μ such that (γ_{K})_{an} ≃ τ_{K} ⊗ μ. Since dim τ = 2^{d} and dim(γ_{K})_{an} = dim(γ_{F (γ)})_{an} < dim γ = 2^{d+1}, it follows that dim μ < 2^{d+1}/2^{d} = 2 . Hence, dim μ = 0 or 1.

If dim μ = 0 then (γ_{K})_{an} = 0 . Then γ_{F (γ)} = γ_{K} is hyperbolic. We get contradiction to the hypothesis of the lemma.

If dim μ = 1, then the isomorphism (γ_{K})_{an} ≃ τ_{K} ⊗ μ shows that (γ_{K})_{an} is similar to τ_{K}. The lemma is proved.

Proposition 3.6. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence with leading form τ ∈ P_{d}(F) and pre-leading form γ = φ_{h−2} . Then

(1) dim γ = 2^{d+1} or dim γ = 2^{N} − 2^{d} with N ≥ d + 2.

(2) If dim γ = 2^{d+1} then γ is a good non-excellent form of height 2 and degree d with leading form τ.

(3) If dim γ ≠ 2^{d+1} then γ is excellent and γ_{F (τ)} is hyperbolic.

Proof. (1). Let E = F_{h−2} = F (φ_{0}, …, φ_{h−3}). By Lemma 3.3 with i = h − 2, the sequence (γ_{E}, (φ_{h−1})_{E}, (φ_{h})_{E}) is quasi-excellent of height 2 with leading form τ_{E} ∈ P_{d}(E) . Thus Remark 2.7 implies that γ_{E} is a good form of height 2 , degree d, and leading form τ_{E} . By Theorem 1.10, there are two types of good forms of height 2 , non-excellent and excellent.

If γ_{E} is good non-excellent of height 2 and degree d, then dim γ = 2^{d+1}. If γ_{E} is excellent form of height 2 and degree d, then dim γ = 2^{N} − 2^{d} with

N ≥ d + 2 .

(2). Assume that dim γ = 2^{d+1}. We have to prove that (γ_{F (γ)})_{an} is similar to τ_{F (γ)}. By Lemma 3.5, it suffices to verify that γ_{F (γ)} is not hyperbolic and γ_{F (γ,τ)} is hyperbolic.

Since E(γ) = F_{h−1} , we have (γ_{E(γ)})_{an} ∼ (φ_{h−1})_{E(γ)} ∼ τ_{E(γ)} by Lemma 2.1 and Definition 2.6. This shows that γ_{F (γ)} is not hyperbolic and that γ_{E(γ,τ)} is hyperbolic. Since E = F (φ_{0}, …, φ_{h−3}) is the function field of forms of dimension > dim φ_{h−2} = dim γ and γ_{E(γ,τ)} is hyperbolic, it follows from Theorem 1.4 that γ_{F (γ,τ)} is also hyperbolic. By Lemma 3.5, we are done.

(3). If dim γ ≠ 2^{d+1} then γ_{E} is an excellent form of height 2 and degree d with the leading form τ_{E}. In this case γ_{E(τ)} is hyperbolic by Theorem 1.10. Hence γ_{F (τ)} is also hyperbolic by Theorem 1.4.

Proposition 3.7. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence with leading form τ ∈ P_{d}(F) and pre-leading form γ = φ_{h−2} . Then (φ_{i})_{F (γ,τ)} is hyperbolic for all i = 0, …, h − 1.

Proof. If h = 1 then φ_{0} ∼ τ by Definition 2.6, hence (φ_{0})_{F (τ)} is hyperbolic. If h = 2 then the statement is obvious as well. Thus, we can assume that h ≥ 3. We use induction on h.

Let E = F (φ_{0}). By Lemma 3.3, ((φ_{1})_{E}, (φ_{2})_{E}, …, (φ_{h})_{E}) is quasi-excellent. By induction assumption, (φ_{i})_{E(γ,τ)} is hyperbolic for all i = 1, …, h − 1. Since E(γ, τ) = F (γ, τ, φ_{0}) and dim φ_{0} is strictly greater than the dimensions of all forms φ_{1}, …, φ_{h−1}, Theorem 1.4 shows that the forms (φ_{i})_{F (γ,τ)} are hyperbolic for all i = 1, …, h − 1.

Now, it suffices to prove that (φ_{0})_{F (γ,τ)} is hyperbolic. We consider three cases and use the following observation. Since (φ_{1})_{F (γ,τ)} is hyperbolic, hence isotropic, it follows that F (γ, τ, φ_{1}) is purely transcendental over F (γ, τ).

Case 1. The form (φ_{0})_{F (φ_{1})} is isotropic.

Then (φ_{0})_{F (γ,τ,φ_{1})} is isotropic. Thus (φ_{0})_{F (γ,τ)} is isotropic too by the above observation. Since the forms (φ_{i})_{F (γ,τ)} are hyperbolic for all i = 1, …, h − 1 , Lemma 2.3 applies with i = h so that ((φ_{0})_{F (γ,τ)})_{an} ∼ (φ_{h})_{F (γ,τ)} = 0 .

Case 2. The form (φ_{0})_{F (φ_{1})} is anisotropic and h = 3.

In this case γ = φ_{1} . Let E = F (φ_{1}) = F (γ) . By Lemma 3.1, the sequence ((φ_{0})_{E}, (φ_{2})_{E}, 0) is quasi-excellent of height 2.

Clearly, dim φ_{2} = dim τ = 2^{d}, and dim φ_{0} > dim φ_{1} = dim γ ≥ 2^{d+1} by Proposition 3.6. Since dim(φ_{0})_{E} ≠ 2^{d+1} it follows that (φ_{0})_{E} is excellent of height 2 with leading form τ_{E}. Hence (φ_{0})_{E(τ)} is hyperbolic (see Theorem 1.10). Since E(τ) = F (γ, τ), we are done.

Case 3. The form (φ_{0})_{F (φ_{1})} is anisotropic and h ≥ 4.

Let E = F (φ_{1}). By Lemma 3.1, the sequence

((φ_{0})_{E}, (φ_{2})_{E}, …, (φ_{h−2})_{E}, (φ_{h−1})_{E}, 0)

is a quasi-excellent of height h − 1. Clearly, τ_{E} is the leading form and γ_{E} = (φ_{h−2})_{E} is the pre-leading form of this sequence (we note, that here we use the condition h ≥ 4). Applying the induction hypothesis, we see that the form (φ_{0})_{E(γ,τ)} is hyperbolic. Therefore, (φ_{0})_{F (γ,τ)} is hyperbolic by the above observation.

4. Classification theorem for sequences of the first type

Recall that a quasi-excellent sequence (φ_{0}, …, φ_{h}) of degree d is of the first type if dim φ_{h−2} ≠ 2^{d+1} or if h = 1 .

Lemma 4.1. Let (φ_{0}, …, φ_{h}) be a quasi-excellent sequence of the first type with leading form τ ∈ P_{d}(F) . Then

(i) the form (φ_{i})_{F (τ)} is hyperbolic for all i = 0, …, h − 1, (ii) for every i = 0, …, h − 1 there exists an odd-dimensional form μ_{i} such

that φ_{i} ≃ μ_{i} ⊗ τ , (iii) φ_{0} is a Pfister neighbor, whose complementary form is similar to φ_{1} .

Proof. (i). For h = 1 the statement follows from Remark 1.11. Now assume that h ≥ 2 , and put γ = φ_{h−2} . By Proposition 3.6 (3), the form γ_{F (τ)} is isotropic. Hence, the extension F (γ, τ)/F (τ) is purely transcendental. This implies, since (φ_{i})_{F (γ,τ)} is hyperbolic by 3.7, that (φ_{i})_{F (τ)} is hyperbolic.

(ii). By Theorem 1.3 and (i), there exists a form μ_{i} such that φ_{i} ≃ μ_{i} ⊗ τ . Thus it suffices to prove that dim μ_{i} is odd. If we assume that μ_{i} is an evendimensional form, then we get [φ_{i}] ∈ I(F) · I^{d}(F) = I^{d+1}(F). This contradicts to Lemma 3.4, where we have proved that deg(φ_{i}) = d for all i = 0, …, h − 1 , since I^{d+1}(F) ⊂ J_{d+1}(F) by [Kn1], 6.6.

(iii). Let K = F (φ_{0}). By Definition 0.1, there exists x ∈ K^{∗} such that ((φ_{0})_{K})_{an} ≃ x (φ_{1})_{K} . By (ii), this implies

(∗) (μ_{0} ⊗ τ)_{K} ∼_{w} x (μ_{1} ⊗ τ)_{K} .

Let s_{0} = d(μ_{0}) and s_{1} = d(μ_{1}). Since μ_{0} and μ_{1} are both of odd dimension, we have μ_{0} ≡ ⟨s_{0}⟩ mod I^{2}(F) and μ_{1} ≡ ⟨s_{1}⟩ mod I^{2}(F) by Remark 1.2. Thus s_{0}τ_{K} ≡ (μ_{0} ⊗τ)_{K} mod I^{d+2}(K) and x(μ_{1} ⊗τ)_{K} ≡ xs_{1}τ_{K} mod I^{d+2}(K) , since τ ∈ I^{d}(F). This yields s_{0}τ_{K} ≡ xs_{1}τ_{K} mod I^{d+2}(K) by (∗). Setting s = s_{0}s_{1} we obtain sτ_{K} ≡ xτ_{K} mod I^{d+2}(K) . Theorem 1.1 now shows that sτ_{K} ≃ xτ_{K} .

Therefore, (ii) and the above yield ((φ_{0})_{K})_{an} ≃ x(φ_{1})_{K} ≃ x(μ_{1} ⊗ τ)_{K} ≃ (μ_{1})_{K} ⊗ xτ_{K} ≃ s(μ_{1} ⊗ τ)_{K} ≃ (sφ_{1})_{K}. Theorem 1.8 now shows that φ_{0} is a Pfister neighbor whose complementary form is isomorphic to −sφ_{1} .

The following theorem proves Theorem 0.6.

Theorem 4.2. Let (φ_{0}, …, φ_{h}) be a sequence of anisotropic forms. Then this sequence is a quasi-excellent sequence of the first type if and only if the following two conditions hold.

• For any i = 0, …, h − 1, the form φ_{i} is a Pfister neighbor whose complementary form is similar to φ_{i+1} .

• The form φ_{h} is zero.

Proof. Obvious induction by using Lemma 3.2 and Lemma 4.1 (iii). The converse direction follows from Example 0.5 starting with the sequence {φ_{h−1}, φ_{h} = 0}.

5. Quasi-excellent sequences modulo some ideals

Let I(F) be the ideal of classes of even-dimensional forms in the Witt ring W (F) , and let I^{n}(F) denote the nth power of I(F) . We need the following proposition for the classification of sequences of the second and third type.

Proposition 5.1. Let (φ_{0}, φ_{1}, …, φ_{h}) be a quasi-excellent sequence. Suppose that there exists an integer k such that 1 ≤ k < h with the following property: for all i = 0, …, k − 1, there exists f_{i} ∈ F^{∗} such that φ_{i} ≡ f_{i} φ_{k} mod I^{m+1}(F) where m is a minimal integer such that dim φ_{k} < 2^{m}.

Then φ_{0} is a Pfister neighbor whose complementary form is similar to φ_{1} .

Proof. For convenience, we will include in our consideration also the case where k = 0 and prove, by induction, the following two properties:

(a) If k ≥ 1 then φ_{0} is a Pfister neighbor whose complementary form is similar to φ_{1} .

(b) If k ≥ 0 then for any x ∈ F^{∗} the conditions φ_{0} ≡ xφ_{0} mod I^{m+1}(F) and dim φ_{k} < 2^{m} imply that φ_{0} ≃ xφ_{0} .

For a given k we denote properties (a) and (b) by (a)_{k} and (b)_{k} correspondingly. The plan of the proof of Proposition 5.1 will be the following.

• We start with the proof of property (b)_{0} .

• For k ≥ 1 , we prove that (b)_{k−1} ⇒ (a)_{k}.

• For k ≥ 1, we prove that (b)_{k−1} ⇒ (b)_{k}.

Lemma 5.2. Condition (b) holds in the case k = 0 .

Proof. If k = 0 we have in (b) the conditions [φ_{0} ⊥ −xφ_{0}] ∈ I^{m+1}(F) and dim φ_{0} < 2^{m}, in particular dim(φ_{0} ⊥ −xφ_{0}) < 2^{m+1}. By Theorem 1.1, the form φ_{0} ⊥ −xφ_{0} is hyperbolic. Therefore φ_{0} ≃ xφ_{0} .

Lemma 5.3. Let k ≥ 1. Then property (b)_{k−1} (stated for all quasi-excellent sequences over all fields of characteristic ≠ 2) implies property (a)_{k}.

Proof. Consider a sequence (φ_{0}, …, φ_{h}) as in Proposition 5.1. Then, by assumption, there exist f_{0}, f_{1} ∈ F^{∗} such that φ_{0} ≡ f_{0}φ_{k} mod I^{m+1}(F) and φ_{1} ≡ f_{1}φ_{k} mod I^{m+1}(F) . Hence φ_{0} ≡ f_{0}f_{1}φ_{1} mod I^{m+1}(F) .

Let E = F (φ_{0}). By Definition 0.1, there exists x ∈ E^{∗} such that ((φ_{0})_{E})_{an} ≃ x(φ_{1})_{E} . Hence x(φ_{1})_{E} ≡ (φ_{0})_{E} ≡ f_{0}f_{1}(φ_{1})_{E} mod I^{m+1}(E) . Property (b)_{k−1} stated for the quasi-excellent sequence ((φ_{1})_{E}, (φ_{2})_{E}, …, (φ_{k})_{E}, …, (φ_{h})_{E}) shows that x(φ_{1})_{E} ≃ f_{0}f_{1}(φ_{1})_{E} . Hence ((φ_{0})_{E})_{an} ≃ x(φ_{1})_{E} ≃ (f_{0}f_{1}φ_{1})_{E} . By Theorem 1.8, φ_{0} is a Pfister neighbor whose complementary form is similar to φ_{1} .

Lemma 5.4. Let k ≥ 1 . Then property (b)_{k−1} (stated for all quasi-excellent sequences over all fields of characteristic ≠ 2) implies property (b)_{k}.

Proof. Consider a sequence (φ_{0}, …, φ_{h}) as in Proposition 5.1. We assume that property (b)_{k−1} holds. Then property (a)_{k} also holds (see previous lemma). This means that there exist a, s ∈ F^{∗}, an integer n > 0 , and an n-fold Pfister form π such that aπ ≃ φ_{0} ⊥ −sφ_{1} and dim φ_{1} < 2^{n−1}.

Since 2^{n−1} > dim φ_{1} ≥ dim φ_{k}, the definition of m yields n − 1 ≥ m. Therefore [aπ] ∈ I^{n}(F) ⊂ I^{m+1}(F) . Hence, φ_{0} ≡ sφ_{1} mod I^{m+1}(F) . Now, let x ∈ F^{∗} be as in (b)_{k}. In other words, φ_{0} ≡ xφ_{0} mod I^{m+1}(F) . Then sφ_{1} ≡ φ_{0} ≡ xφ_{0} ≡ sxφ_{1} mod I^{m+1}(F) . Hence, φ_{1} ≡ xφ_{1} mod I^{m+1}(F) . Property (b)_{k−1}, applied to the quasi-excellent sequence

((φ_{1})_{F (φ_{0})}, (φ_{2})_{F (φ_{0})}, …, (φ_{k})_{F (φ_{0})}, …, (φ_{h})_{F (φ_{0})})

shows that (φ_{1})_{F (φ_{0})} ≃ x(φ_{1})_{F (φ_{0})} . Hence the form φ_{1} ⊗ ⟨ ; ⟨x⟩ ; ⟩ is hyperbolic over F (φ_{0}) . Since φ_{0} is a Pfister neighbor of π, it follows that F (φ_{0}, π)/F (π) is purely transcendental. Thus φ_{1} ⊗ ⟨ ; ⟨x⟩ ; ⟩ is also hyperbolic over F (π) . Since dim(φ_{1} ⊗ ⟨ ; ⟨x⟩ ; ⟩) < 2^{n−1} · 2 = dim π, Theorem 1.4 shows that φ_{1} ⊗ ⟨ ; ⟨x⟩ ; ⟩ is hyperbolic, hence φ_{1} ≃ xφ_{1} . Moreover, aπ ⊗ ⟨ ; ⟨x⟩ ; ⟩ ≃ (φ_{0}⊥ − sφ_{1}) ⊗ ⟨ ; ⟨x⟩ ; ⟩ is isotropic and therefore hyperbolic (since π is a Pfister form). Hence aπ ≃ xaπ. Since aπ ≃ φ_{0} ⊥ −sφ_{1} and φ_{1} ≃ xφ_{1} , it follows that φ_{0} ≃ xφ_{0} .

Clearly, the three lemmas complete the proof of Proposition 5.1.

6. Five classical conjectures

Let H^{n}(F) := H^{n}(F, ℤ/2ℤ) be the nth Galois cohomology group. Let I^{0}(F) := W (F) be the Witt ring and I^{1}(F) := I(F) be the fundamental ideal in W (F) of classes of even-dimensional forms. In §1, we have considered the homomorphisms

e^{0} : I^{0}(F) → ℤ/2ℤ ≃ H^{0}(F) and e^{1} : I^{1}(F) → F^{∗}/F^{∗}^{2} ≃ H^{1}(F)

defined by the dimension and the discriminant respectively. Denoting by 2Br(F) the 2-torsion part of the Brauer group of F we obtain a homomorphism e^{2} : I^{2}(F) → _{2}Br(F) ≃ H^{2}(F) defined by the Clifford algebra. We have ker(e^{n}) = I^{n+1}(F) for n = 0, 1, 2 .

For each integer n > 0 let (a_{1}) · . . . · (a_{n}) denote the cup-product where (a_{i}) is the class of a_{i} ∈ F^{∗} in H^{1}(F) for i = 1, …, n. The following five conjectures are believed to be true for all fields F of characteristic ≠ 2 .

Conjecture 6.1. (Milnor conjecture). Let n ≥ 0 be an integer. Then there exists a homomorphism

e^{n} : I^{n}(F) → H^{n}(F)

such that ⟨ ; ⟨a_{1}, …, a_{n}⟩ ; ⟩ ↦ (a_{1}) · . . . · (a_{n}) . Moreover, the homomorphism e^{n} induces an isomorphism

e^{n} : I^{n}(F)/I^{n+1}(F) ≃ H^{n}(F) .

Conjecture 6.2. For any π ∈ P_{m}(F) and all integers n ≥ m ≥ 0 , we have

ker(H^{n}(F) → H^{n}(F (π))) = e^{m}(π) H^{n−m}(F) .

Conjecture 6.3. We have J_{n}(F) = I^{n}(F) for all integers n ≥ 0 .

Conjecture 6.4. Let φ be an anisotropic form over F. If [φ] ∈ I^{n}(F) and 2^{n} ≤ dim φ < 2^{n} + 2^{n−1} then dim φ = 2^{n}.

Conjecture 6.5. Let γ be an even-dimensional anisotropic form. Assume that γ is a good non-excellent form of height 2 with leading form τ ∈ P_{n}(F) . Then there exists τ_{0} ∈ P_{n−1}(F) , (τ_{0} = ⟨1⟩ if n = 1), and a, b, c ∈ F^{∗} such that

• γ is similar to τ_{0} ⊗ ⟨−a, −b, ab, c⟩,

• τ ≃ τ_{0} ⊗ ⟨ ; ⟨c⟩ ; ⟩.

Remark 6.6. For proving Conjecture 6.5 it suffices to show that there exists τ_{0} ∈ P_{n−1}(F) such that γ ≃ τ_{0} ⊗ γ′ where dim γ′ = 4 .

Proof. Write γ′ = ⟨r, s, t, u⟩ with r, s, t, u ∈ F^{∗}. Then setting d = rst, we obtain dγ′ ≃ ⟨−a, −b, ab, c⟩ with a = −st , b = −rt, and c = du. This shows γ ∼ τ_{0} ⊗ ψ where ψ = ⟨−a, −b, ab, c⟩. Since ⟨d(ψ)⟩ ≃ ⟨c⟩ by definition of d(ψ) = (−1)^{\binom{4}{2}} det(ψ) and since τ is the leading form of γ, it follows from

[Kn1], 6.12, that τ ≃ τ_{0} ⊗ ⟨ ; ⟨c⟩ ; ⟩.

Remark 6.7. Recent results of Voevodsky [Vo] and Orlov-Vishik-Voevodsky [OVV] show that Conjectures 6.1, 6.2, and 6.3 hold for all fields of characteristic 0. These three conjectures were proved earlier in the cases n ≤ 4 and characteristic ≠ 2, (cf. [Pf2], [KRS], and [Kahn]).

Conjecture 6.4 is proved for all fields of characteristic 0 by Vishik [Vi]. In the case n ≤ 4, it is proved for all fields of characteristic ≠ 2 (see [H4]).

Conjecture 6.5 is proved in the case n ≤ 3 for all fields of characteristic ≠ 2, (see Remark 6.6 and [Kahn]). Moreover, it follows from Proposition 3.6 and from [Kahn], Proposition 4.3 (b), that Conjectures 6.3 and 6.4 for degree n + 1 imply Conjecture 6.5 for degree n.

Definition 6.8. Let d ≥ 0 be an integer, and let F be a field. We say that condition A_{d} holds for F if F is of characteristic ≠ 2 and if for all field extensions F′/F the following conjectures hold:

• Conjecture 6.1 for all n ≤ d + 2 ,

• Conjecture 6.2 for n ≤ d + 2 ,

• Conjecture 6.3 for n = d + 1 ,

• Conjecture 6.4 for n = d + 2 ,

• Conjecture 6.5 for n = d > 0 .

Theorem 6.9. Let F be a field of characteristic ≠ 2 . If d = 0 or d = 1 then condition A_{d} holds for F. If d = 2 then condition A_{d} holds for F, possibly with the exception of the bijectivity of the homomorphism e^{4} : I^{4}(F)/I^{5}(F) → H^{4}(F) .

Proof. Conjecture 6.1 holds for n = 0 by definition of the ideal I(F) . It has been proved by Pfister for n = 1, cf. [Pf1], 2.3.6, and by Merkurjev for n = 2, cf. [M1]. The existence of e^{3} has been proved by Arason [Ara], Satz 5.7, and the bijectivity of e^{3} by Rost [R] and independently by Merkurjev-Suslin [MS]. The existence of e^{4} has been proved by Jacob-Rost [JR] and independently by Szyjewsi [Sz]. The bijectivity of e^{4} was claimed by Rost (unpublished).

Conjecture 6.2 holds n ≤ 4 , cf. [KRS].

Conjecture 6.3. For n = 1, 2 see [Kn1], 6.2; for n = 3 (and n = 4), see [Kahn], Théorème 2.8.

Conjecture 6.4 is trivial for n = 2. For n = 3 it is due to Pfister and for n = 4 to Hoffmann, see [H4], Main Theorem for n = 4, and 2.9 for n = 3.

Conjecture 6.5. For n = 1 see [Kn1], 5.10. For for n = 2 see Remark 6.6 and [Kahn], Corollaire 2.1.

Remark 6.7 gives rise to the following theorem.

Theorem 6.10. Let d ≥ 0 be an integer. Modulo results proved in [Vo, OVV] condition A_{d} holds for all fields of characteristic 0 .

We are going to prove some consequences of the above conjectures. Let GP_{n}(F) denote the set of quadratics forms over F which are similar to n-fold Pfister forms, and let H^{n}(K/F) = ker(H^{n}(F) → H^{n}(K)).

Lemma 6.11. Let F be a field of characteristic ≠ 2 and let π ∈ GP_{r}(F) for some integer r ≥ 0 . Suppose that Conjecture 6.2 holds for n = m = r and for all field extensions of F . Then

(1) for any extension K/F, we have H^{r}(K(π)/F) = H^{r}(K/F) + H^{r}(F (π)/F) ;

(2) for any form φ over F, we have H^{r}(F (φ, π)/F) = H^{r}(F (φ)/F) + H^{r}(F (π)/F) .

Proof. (1) Let u ∈ H^{r}(K(π)/F). Then u_{K} ∈ H^{r}(K(π)/K). Conjecture 6.2 applied with n = m = r shows that u_{K} = ℓ · e^{r}(π_{K}) with ℓ ∈ H^{0}(K) ≃ ℤ/2ℤ.

Hence (u − ℓ · e^{r}(π)) ∈ H^{r}(K/F). Therefore, u ∈ H^{r}(K/F) + ℓ · e^{r}(π) ⊂ H^{r}(K/F) + H^{r}(F (π)/F).

(2) It suffices to set K = F (φ) in (1).

Lemma 6.12. Let d > 0 be an integer and F be a field such that condition A_{d} holds for F. Let γ be an even-dimensional anisotropic form which is good non-excellent of degree d and height 2. Then

H^{d+2}(F (γ)/F) = {e^{d+2}(γ ⊗ ⟨ ; ⟨f⟩ ; ⟩) | f ∈ F^{∗} with γ ⊗ ⟨ ; ⟨f⟩ ; ⟩ ∈ GP_{d+2}(F)} .

Proof. Let τ, τ_{0} and a, b, c ∈ F^{∗} be as in Conjecture 6.5. We can assume that

γ = τ_{0} ⊗ ⟨−a, −b, ab, c⟩. Let π = τ_{0} ⊗ ⟨ ; ⟨a, b⟩ ; ⟩.

Clearly, γ ∼_{w} π⊥ − τ. Hence (γ_{F (π)})_{an} ≃ (−τ_{F (π)})_{an}. Since dim γ > dim τ, it follows that γ_{F (π)} is isotropic. Hence F (γ, π)/F (π) is purely transcendental, forcing H^{d+2}(F (γ)/F) ⊂ H^{d+2}(F (π)/F). Conjecture 6.2 shows that H^{d+2}(F (π)/F) = e^{d+1}(π)H^{1}(F). Hence, an arbitrary element of the group

H^{d+2}(F (γ)/F) is of the form e^{d+1}(π) · (s) = e^{d+2}(π ⊗ ⟨ ; ⟨s⟩ ; ⟩) with s ∈ F^{∗}. Let

ρ = π ⊗ ⟨ ; ⟨s⟩ ; ⟩. Then ρ ∈ P_{d+2}(F) and e^{d+2}(ρ_{F (γ)}) = 0. By Conjecture 6.1, the form ρ_{F (γ)} is hyperbolic. Hence γ is similar to a subform of ρ of by Theorem 1.4. Let γ^{∗} and t ∈ F^{∗} be such that tγ ⊥ −γ^{∗} ≃ ρ. The forms γ and γ^{∗} are half-neighbors. By [H3, Prop. 2.8], there exists k ∈ F^{∗} such that γ^{∗} ≃ kγ. Then γ ⊗ ⟨ ; ⟨tk⟩ ; ⟩ ≃ tρ ∈ GP_{d+2}(F) . To complete the proof, it suffices to notice that e^{d+2}(tρ) = e^{d+2}(ρ).

Lemma 6.13. Let d > 0 be an integer and F be a field such that condition A_{d} holds for F . Assume that γ is an even-dimensional anisotropic form which is good non-excellent of height 2 with leading form τ ∈ P_{d}(F) . Now, let φ be a form such that φ ≡ τ mod I^{d+1}(F) and φ_{F (γ,τ)} is hyperbolic. Also assume that there exists an extension E/F such that dim(φ_{E})_{an} = 2^{d+1}. Then the following is true.

(1) There exists f ∈ F^{∗} such that

φ ≡ fγ mod I^{d+2}(F) .

(2) If we suppose additionally that dim(φ_{F (γ)})_{an} < dim γ, then there exists f ∈ F^{∗} such that

φ ≡ fγ mod I^{d+3}(F) .

Proof. (1). Let ψ = φ ⊥ −τ . By assumption, we have [ψ] ∈ I^{d+1}(F) . Hence, we can consider the element e^{d+1}(ψ) ∈ H^{d+1}(F) . Since φ_{F (γ,τ)} and τ_{F (τ)} are hyperbolic, it follows that ψ_{F (γ,τ)} is also hyperbolic. Hence, e^{d+1}(ψ) ∈ H^{d+1}(F (γ, τ)/F). By Conjecture 6.5, we can assume that γ =

τ_{0} ⊗ ⟨−a, −b, ab, c⟩ and τ = τ_{0} ⊗ ⟨ ; ⟨c⟩ ; ⟩ where τ_{0} ∈ P_{d−1}(F) .

Let π = τ_{0} ⊗ ⟨ ; ⟨a, b⟩ ; ⟩ ∈ P_{d+1}(F) . Then γ ∼_{w} π⊥ − τ. Hence γ_{F (τ)} ∼_{w} π_{F (τ)} .

Therefore, by Lemma 6.11(2) and Conjecture 6.2 we have

H^{d+1}(F (γ, τ)/F) = H^{d+1}(F (π, τ)/F)
= H^{d+1}(F (π)/F) + H^{d+1}(F (τ)/F)
= e^{d+1}(π)H^{0}(F) + e^{d}(τ)H^{1}(F) .

Since e^{d}(τ)H^{1}(F) = {e^{d+1}(τ ⊗ ⟨ ; ⟨s⟩ ; ⟩) | s ∈ F^{∗}}, it follows that any element of the group H^{d+1}(F (γ, τ)/F) has one of the following forms:

• either e^{d+1}[π⊥(τ ⊗ ⟨ ; ⟨s⟩ ; ⟩)]

• or e^{d+1}(τ ⊗ ⟨ ; ⟨s⟩ ; ⟩) .

Since e^{d+1}(ψ) ∈ H^{d+1}(F (γ, τ)/F) and ψ = φ⊥ − τ , Conjecture 6.1 shows that

• either φ⊥ − τ ≡ π⊥(τ ⊗ ⟨ ; ⟨s⟩ ; ⟩) mod I^{d+2}(F)

• or φ⊥ − τ ≡ τ ⊗ ⟨ ; ⟨s⟩ ; ⟩ mod I^{d+2}(F) .

Consider the first case where φ⊥ − τ ≡ π⊥(τ ⊗ ⟨ ; ⟨s⟩ ; ⟩) mod I^{d+2}(F) . Clearly, we can compute [φ] modulo I^{d+2}(F). In our computation, we note that [π] and [τ ⊗ ⟨ ; ⟨s⟩ ; ⟩] belong to I^{d+1}(F) . Hence for any x ∈ F^{∗}, we have xπ ≡ π and xτ ⊗ ⟨ ; ⟨s⟩ ; ⟩ ≡ τ ⊗ ⟨ ; ⟨s⟩ ; ⟩ mod I^{d+2}(F) . Besides, we recall that γ ∼_{w} π⊥ − τ. Now, we have the following calculation φ ≡ τ⊥π⊥(τ ⊗⟨ ; ⟨s⟩ ; ⟩) ≡ τ⊥−π⊥−(τ ⊗⟨ ; ⟨s⟩ ; ⟩) ≡

−γ⊥((γ⊥ − π) ⊗ ⟨ ; ⟨s⟩ ; ⟩) ≡ sπ⊥ − π⊥ − sγ ≡ −sγ mod I^{d+2}(F) . Hence φ ≡ fγ

mod I^{d+2}(F) with f = −s.

Now we consider the second case where φ⊥ − τ ≡ τ ⊗ ⟨ ; ⟨s⟩ ; ⟩ mod I^{d+1}(F) .

Here, we get φ ≡ τ⊥(τ ⊗ ⟨ ; ⟨s⟩ ; ⟩) ≡ τ⊥ − (τ ⊗ ⟨ ; ⟨s⟩ ; ⟩) ≡ sτ mod I^{d+2}(F) .

By the assumption of the lemma there exists a field extension E/F such that dim(φ_{E})_{an} = 2^{d+1}. Since φ ≡ sτ mod I^{d+2}(F), we have (φ_{E})_{an} ≡ sτ_{E} mod I^{d+2}(E) . Since dim(φ_{E})_{an} + dim τ_{E} = 2^{d+1} + 2^{d} < 2^{d+2}, Theorem 1.1 shows that (φ_{E})_{an} ≃ s(τ_{E})_{an} . This contradicts to the inequality dim(φ_{E})_{an} = 2^{d+1} > 2^{d} ≥ dim(τ_{E})_{an} .

(2). Let f be as in (1). Set ψ = φ ⊥ −fγ. We have ψ ∈ I^{d+2}(F) . Since dim(φ_{F (γ)})_{an} < dim γ, we have dim(ψ_{F (γ)})_{an} < 2 dim γ = 2 · 2^{d+1} = 2^{d+2}. By Theorem 1.1, the form ψ_{F (γ)} is hyperbolic. Hence e^{d+2}(ψ) ∈ H^{d+2}(F (γ)/F). By Lemma 6.12, there exists s ∈ F^{∗} such that γ ⊗ ⟨ ; ⟨s⟩ ; ⟩ ∈ GP_{d+2}(F) and e^{d+2}(ψ) = e^{d+2}(γ ⊗ ⟨ ; ⟨s⟩ ; ⟩) . Thus ψ ≡ γ ⊗ ⟨ ; ⟨s⟩ ; ⟩ mod I^{d+3}(F) by Conjecture 6.1.

Since γ⊗⟨ ; ⟨s⟩ ; ⟩ ∈ GP_{d+2}(F), we have γ⊗⟨ ; ⟨s⟩ ; ⟩ ≡ −fγ⊗⟨ ; ⟨s⟩ ; ⟩ mod I^{d+3}(F). There-

fore, φ ≡ ψ⊥fγ ≡ (γ⊗⟨ ; ⟨s⟩ ; ⟩)⊥fγ ≡ −f(γ⊗⟨ ; ⟨s⟩ ; ⟩)⊥fγ ≡ fsγ mod I^{d+3}(F) .

Proposition 6.14. Let d > 0 be an integer and F be a field such that condition A_{d} holds for F . Let (φ_{0}, φ_{1}, …, φ_{h}) be a quasi-excellent sequence with leading form τ ∈ P_{d}(F) and pre-leading form γ = φ_{h−2} . Suppose that this sequence is of the second or third type (in particular h ≥ 2). Let φ = φ_{i} with i ≤ h − 2 .

(1) We have φ ≡ τ mod I^{d+1}(F) .

(2) There exists f ∈ F^{∗} such that φ ≡ fγ mod I^{d+2}(F) .

(3) If dim(φ_{F (γ)})_{an} < dim γ then there exists f ∈ F^{∗} such that φ ≡ fγ mod I^{d+3}(F) .

Proof. (1). Follows from Lemma 3.4 and Conjecture 6.3.

(2) and (3). Definition 0.3 shows that dim γ = 2^{d+1}. By Proposition 3.6(2), γ is a good non-excellent form of height 2 and degree d. By Proposition 3.7, φ_{F (γ,τ)} is hyperbolic. Lemma 2.1 yields dim(φ_{E})_{an} = dim φ_{h−2} = 2^{d+1} where E = F_{h−2} = F (φ_{0}, …, φ_{h−3}) . Now, Lemma 6.13 completes the proof.

Corollary 6.15. Let (φ, γ, τ, 0) be a quasi-excellent sequence with τ ∈ P_{d}(F) and dim γ = 2^{d+1}. Then dim φ ≥ 3 · 2^{d}. Moreover, if dim φ = 3 · 2^{d} then φ_{F (γ)} is anisotropic.

Proof. Let E = F (γ) . If φ_{E} is anisotropic, then the sequence (φ_{E}, τ_{E}, 0) is quasi-excellent by Lemma 3.1. Since dim φ_{E} = dim φ > dim γ = 2^{d+1}, it follows from Proposition 3.6 that dim φ = 2^{N} − 2^{d} for some N ≥ d + 2 . Therefore, dim φ ≥ 2^{d+2} − 2^{d} = 3 · 2^{d}.

Now, we assume that φ_{E} is isotropic. Then (φ_{E})_{an} ∼ τ_{E} by Lemma 3.1 and hence dim(φ_{E})_{an} = 2^{d} < dim γ. By Proposition 6.14, there exists f ∈ F^{∗} such that φ ≡ fγ mod I^{d+3}(F). Suppose that dim φ ≤ 3·2^{d}. Then dim φ+dim γ ≤ 3 · 2^{d} + 2^{d+1} < 2^{d+3}. By Theorem 1.1, we get φ ≃ fγ. This is a contradiction because dim φ > dim γ.

7. Classification theorem for sequences of the second type

In Definition 6.8 we formulated the condition A_{d} for a field F . We showed that A_{d} is true for d = 0, 1 and char(F) ≠ 2 and that (based on results in [Vo, OVV]) A_{d} is true for all d ≥ 0 and all fields of characteristic 0 , cf. Theorems 6.9 and 6.10.

The main purpose of this section is to prove the following

Theorem 7.1. Let d > 0 be an integer and F be a field such that condition A_{d} holds for F . Let (φ_{0}, φ_{1}, …, φ_{h}) be a quasi-excellent sequence of the second type and of degree d. Then

• for all i < h − 2 the form φ_{i} is a Pfister neighbor whose complementary form is similar to φ_{i+1} ,

• the sequence (φ_{h−2}, φ_{h−1}, φ_{h}) is quasi-excellent of the second type.

First of all, we state the following corollary (which proves Theorems 0.9 and 0.10 for the quasi-excellent sequences of second type).

Corollary 7.2. Let d > 0 be an integer and F be a field such that condition A_{d} holds for F . Then Conjecture 0.7 holds for all quasi-excellent sequences of degree d over F .

Proof. By Theorem 7.1, it suffices to consider the case h = 2 . In this case, the required result follows immediately from Proposition 3.6(2) and Conjecture 6.5.

Now, we return to Theorem 7.1. We will prove this theorem by using induction on h. In the case where h = 2 the statement is obvious. Thus we can assume that h ≥ 3. In what follows we will suppose that h ≥ 3 and Theorem 7.1 holds for all quasi-excellent sequences of height < h.

We start with the following lemma.

Lemma 7.3. If (φ_{0})_{F (φ_{1})} is anisotropic then (φ_{0})_{F (φ_{1})} is a Pfister neighbor whose complementary form is similar to (φ_{2})_{F (φ_{1})} .

Proof. Let E = F (φ_{1}). By Lemma 3.1, the sequence ((φ_{0})_{E}, (φ_{2})_{E}, …, 0) is quasi-excellent of height h − 1. Let us consider two cases, h ≥ 4 and h = 3.

If h ≥ 4 then the sequence ((φ_{0})_{E}, (φ_{2})_{E}, …, (φ_{h})_{E}) is of the second type. Then Theorem 7.1 (stated for sequences of height < h) completes the proof.

If h = 3 then the quasi-excellent sequence ((φ_{0})_{E}, (φ_{2})_{E}, 0) is of the first type because dim φ_{0} > dim φ_{1} = 2^{d+1}. In this case, Theorem 4.2 completes the proof.

The following lemma shows that the situation described in Lemma 7.3 is actually impossible.

Lemma 7.4. The form (φ_{0})_{F (φ_{1})} is isotropic.

Proof. Assume the contrary, (φ_{0})_{F (φ_{1})} is anisotropic. Then Lemmas 7.3 and 2.8 show that dim φ_{1} is a power of 2 and dim φ_{0} = 2 dim φ_{1} − dim φ_{2} . Let K = F (φ_{0}). The sequence ((φ_{1})_{K}, …, (φ_{h})_{K}) is quasi-excellent of height h−1 by Lemma 3.3. Clearly, this sequence is of the second type. We consider the two cases h ≥ 4 and h = 3. If h ≥ 4 then Theorem 7.1 (stated for sequences of height < h) shows that (φ_{1})_{K} is a Pfister neighbor whose complementary form is similar to the non-zero form (φ_{2})_{K} . This in particular shows that dim φ_{1} is not a power of 2 . We get a contradiction. Now, we assume that h = 3 . In other words, we have the sequence (φ_{0}, φ_{1}, φ_{2}, φ_{3}) of the second type. By Definitions 0.3 and 2.6, we have dim φ_{1} = 2^{d+1}, dim φ_{2} = 2^{d} and dim φ_{0} ≠ 3 · 2^{d}. On the other hand, dim φ_{0} = 2 dim φ_{1} − dim φ_{2} = 2 · 2^{d+1} − 2^{d} = 3 · 2^{d}. We get a contradiction.

Corollary 7.5. The sequence (φ_{1}, φ_{2}, …, φ_{h}) is a quasi-excellent sequence of the second type and of degree d.

Proof. Follows from Lemmas 7.4, and 3.1.

Lemma 7.6. Let γ = φ_{h−2} . Then dim((φ_{i})_{F (γ)})_{an} ≤ 2^{d} for all i = 0, …, h.

Proof. Using induction and Corollary 7.5 we see that dim((φ_{i})_{F (γ)})_{an} ≤ 2^{d} for all i ≥ 1. Now, it suffices to prove that dim((φ_{0})_{F (γ)})_{an} ≤ 2^{d}. Since h ≥ 3, we have dim((φ_{1})_{F (γ)})_{an} ≤ 2^{d} = dim φ_{h−1} < dim φ_{1}. Hence (φ_{1})_{F (γ)} is isotropic forcing that F (γ, φ_{1})/F (γ) is purely transcendental. Thus 7.4 yields that (φ_{0})_{F (γ)} is isotropic. By Lemma 2.3, there exists i > 0 such that ((φ_{0})_{F (γ)})_{an} ∼ (φ_{i})_{F (γ)} = ((φ_{i})_{F (γ)})_{an} . Hence dim((φ_{0})_{F (γ)})_{an} = dim((φ_{i})_{F (γ)})_{an} ≤ 2^{d}.

Corollary 7.7. There exists f_{i} ∈ F^{∗} such that φ_{i} ≡ f_{i}γ mod I^{d+3}(F) for all i = 0, …, h − 2 .

Proof. Obvious consequence of Proposition 6.14 and Lemma 7.6.

Corollary 7.8. If h ≥ 3 then φ_{0} is a Pfister neighbor whose complementary form is similar to φ_{1} .

Proof. Corollary 7.7 shows that the condition of Proposition 5.1 holds in the case k = h − 2 and m = d + 2.

Proof of Theorem 7.1. If h ≥ 3 , Corollaries 7.8 and 7.5 show that • φ_{0} is a Pfister neighbor whose complementary form is similar to φ_{1} ,

• the sequence (φ_{1}, φ_{2}, …, φ_{h}) is quasi-excellent of the second type. After that, an evident induction completes the proof.

8. Classification theorem for sequences of the third type

We proceed similarly as in the previous section. The main purpose is to prove the following theorem.

Theorem 8.1. Let d > 0 be an integer and F be a field such that condition A_{d} holds for F . Let (φ_{0}, φ_{1}, …, φ_{h}) be a quasi-excellent sequence of the third type and of degree d. Then

• for all i < h − 3 the form φ_{i} is a Pfister neighbor whose complementary form is similar to φ_{i+1} ,

• the sequence (φ_{h−3}, φ_{h−2}, φ_{h−1}, 0) is quasi-excellent of the third type.

We will prove this theorem by using induction on h. In the case where h = 3 the statement is obvious. Thus we can assume that h ≥ 4 . In what follows we will suppose that h ≥ 4 and Theorem 8.1 holds for all quasi-excellent sequences of height < h.

Lemma 8.2. If (φ_{0})_{F (φ_{1})} is anisotropic then (φ_{0})_{F (φ_{1})} is a Pfister neighbor whose complementary form is similar to (φ_{2})_{F (φ_{1})} .

Proof. Let E = F (φ_{1}) . By Lemma 3.1, the sequence ((φ_{0})_{E}, (φ_{2})_{E}, …, 0) is quasi-excellent of height h − 1 . Let us consider two cases, h ≥ 5 and h = 4.

If h ≥ 5 then the sequence ((φ_{0})_{E}, (φ_{2})_{E}, …, (φ_{h})_{E}) is of the third type. Then Theorem 8.1 (stated for sequences of height < h) completes the proof.

If h = 4 then the quasi-excellent sequence ((φ_{0})_{E}, (φ_{2})_{E}, (φ_{3})_{E}, 0) has the second type because dim φ_{0} > dim φ_{1} = 3 · 2^{d}. In this case, Theorem 7.1 completes the proof.

Lemma 8.3. The form (φ_{0})_{F (φ_{1})} is isotropic.

Proof. Assume the contrary, (φ_{0})_{F (φ_{1})} is anisotropic. Then Lemmas 8.2 and 2.8 show that dim φ_{1} is a power of 2 . Let K = F (φ_{0}) . The sequence ((φ_{1})_{K}, …, (φ_{h})_{K}) is quasi-excellent of height h − 1 by Lemma 3.3. Clearly, this sequence is of the third type. We consider two cases, h ≥ 5 and h = 4. If h ≥ 5 then Theorem 8.1 (stated for sequences of height < h) shows that (φ_{1})_{K} is a Pfister neighbor whose complementary form is similar to the non-zero form (φ_{2})_{K} . This in particular shows that dim φ_{1} is not a power of 2. We get a contradiction. Now, we assume that h = 4. Then dim φ_{1} = dim φ_{h−3} = 3 · 2^{d} is not a power of 2, a contradiction.

Corollary 8.4. The sequence (φ_{1}, φ_{2}, …, φ_{h}) is a quasi-excellent sequence of the third type and of degree d.

Proof. Obvious in view of Lemmas 8.3, and 3.1.

In what follows we use the following notation:

τ is the leading form. Clearly, we can assume that φ_{h−1} = τ ;

γ = φ_{h−2} is the pre-leading form;

λ = φ_{h−3} is “pre-pre-leading” form.

Thus, our quasi-excellent sequence looks as follows: (φ_{0}, …, φ_{h−4}, λ, γ, τ, 0)

Lemma 8.5. For all i = 0, …, h − 1 , we have dim((φ_{i})_{F (λ,γ)})_{an} = 2^{d}.

Proof. It follows from Lemma 2.1 that (τ)_{F (λ,γ)} is anisotropic.

Using induction and Corollary 8.4, we see that dim((φ_{i})_{F (λ,γ)})_{an} = 2^{d} for all i = 1, …, h−1. In particular, (φ_{1})_{F (λ,γ)} is isotropic. Hence F (λ, γ, φ_{1})/F (λ, γ) is purely transcendental. Since (φ_{0})_{F (φ_{1})} is isotropic by Lemma 8.3, it follows that (φ_{0})_{F (λ,γ)} is also isotropic. By Lemma 2.3, there exists i > 0 such that ((φ_{0})_{F (λ,γ)})_{an} ∼ (φ_{i})_{F (λ,γ)} = ((φ_{i})_{F (λ,γ)})_{an} .

Hence dim((φ_{0})_{F (λ,γ)})_{an} = dim((φ_{i})_{F (λ,γ)})_{an} = 2^{d}.

Proposition 8.6. For any i = 0, …, h − 3 there exists f_{i} ∈ F^{∗} such that φ_{i} ≡ f_{i}λ mod I^{d+3}(F) .

Proof. There is s_{i} ∈ F^{∗} such that φ_{i} ≡ s_{i}γ mod I^{d+2}(F) for i = 0, …, h − 3 by Proposition 6.14(2).

Changing notation φ_{i} := s_{i}φ_{i}, we can assume that φ_{i} ≡ γ mod I^{d+2}(F) for all i = 0, …, h − 3 . In particular, λ = φ_{h−3} ≡ γ mod I^{d+2}(F) . Hence, we get the element e^{d+2}(λ⊥ − γ) ∈ H^{d+2}(F) .

Now, we fix an integer i ≤ h − 3 and set φ = φ_{i}. We have φ ≡ γ ≡ λ mod I^{d+2}(F) . Hence, we get the elements e^{d+2}(φ⊥ − γ) and e^{d+2}(φ⊥ − λ) in H^{d+2}(F) . Recall that H^{n}(F′/F) := ker(H^{n}(F) → H^{n}(F′)) .

Lemma 8.7. (1) e^{d+2}(φ⊥ − γ) ∈ H^{d+2}(F (λ, γ)/F) ,

(2) e^{d+2}(φ⊥ − γ) ∉ H^{d+2}(F (γ)/F) ,

(3) e^{d+2}(φ⊥ − λ) ∈ H^{d+2}(F (γ)/F) .

Proof. To prove item (1), it suffices to verify that (φ ⊥ −γ)_{F (λ,γ)} is hyperbolic. By Lemma 8.5, we have dim(φ_{F (λ,γ)})_{an} ≤ 2^{d} and dim(γ_{F (λ,γ)})_{an} ≤ 2^{d}. Hence, dim((φ ⊥ −γ)_{F (λ,γ)})_{an} ≤ 2^{d+1} < 2^{d+2}. Since [φ⊥ − γ] ∈ I^{d+2}(F), Theorem 1.1 shows that (φ ⊥ −γ)_{F (λ,γ)} is hyperbolic.

(2) Assume that e^{d+2}(φ⊥−γ) ∈ H^{d+2}(F (γ)/F) . Let E = F (φ_{0}, …, φ_{h−4}) . Then (φ_{E})_{an} ∼ (φ_{h−3})_{E} = λ_{E} by Lemma 2.1. Hence, there exists s ∈ E^{∗} such that (φ_{E})_{an} ≃ sλ_{E} . Therefore, e^{d+2}(sλ_{E}⊥ − γ_{E}) ∈ H^{d+2}(E(γ)/E) . Hence e^{d+2}(sλ_{E(γ)}⊥ − γ_{E(γ)}) = 0 .

Then Conjecture 6.1 implies that [sλ_{E(γ)} ⊥ −γ_{E(γ)}] ∈ I^{d+3}(E(γ)) . Since dim λ + dim γ = 3 · 2^{d} + 2^{d+1} < 2^{d+3}, Theorem 1.1 shows that sλ_{E(γ)} ⊥ −γ_{E(γ)} is hyperbolic. Thus dim(λ_{E(γ)})_{an} ≤ dim γ < dim λ. Hence, λ_{E(γ)} is isotropic. By Lemma 3.3, the sequence (λ_{E}, γ_{E}, τ_{E}, 0) is quasi-excellent. By Corollary 6.15, the form λ_{E(γ)} is anisotropic. We get a contradiction.

(3). Set K = F (γ) . Then we have a non-zero element e^{d+2}(φ_{K} ⊥ −γ_{K}) in the group H^{d+2}(K(λ)/K) by (1) and (2).

Since (γ_{K})_{an} is similar to τ_{K} by Lemma 3.5 and Proposition 3.7, there exists s ∈ K^{∗} such that (γ_{K})_{an} ≃ sτ_{K}. Since [λ ⊥ −γ] ∈ I^{d+2}(F) we obtain that [λ_{K} ⊥ −sτ_{K}] ∈ I^{d+2}(K) . Computing dim λ + dim τ = 3 · 2^{d} + 2^{d} = 2^{d+2}, we conclude from the Arason-Pfister Hauptsatz that there is a form π ∈ GP_{d+2}(K) such that π ≃ λ_{K} ⊥ −sτ_{K} , (see Theorem 1.1, and [AP], p. 174, Korollar 3). It follows that λ_{K(π)} is isotropic, since dim λ > dim τ , and π_{K(π)} is hyperbolic. Hence, K(π, λ)/K(π) is purely transcendental. Since by (1) and (2) we have 0 ≠ e^{d+2}(φ_{K} ⊥ −γ_{K}) ∈ H^{d+2}(K(π, λ)/K) we see that

0 ≠ e^{d+2}(φ_{K} ⊥ −γ_{K}) ∈ H^{d+2}(K(π)/K) .

Thus Conjecture 6.2 shows that e^{d+2}(φ_{K} ⊥ −γ_{K}) = e^{d+2}(π) . Clearly, this yields e^{d+2}(φ_{K} ⊥ −γ_{K} ⊥ −π) = 0. Since π ≃ λ_{K} ⊥ −sτ_{K} ∼_{w} λ_{K} ⊥ −γ_{K} we have φ_{K} ⊥ −γ_{K} ⊥ −π ∼_{w} (φ ⊥ −λ)_{K} . Hence, e^{d+2}(φ ⊥ −λ)_{K} = 0 . Therefore, e^{d+2}(φ ⊥ −λ) ∈ H^{d+2}(K/F).

By Proposition 3.6(2), γ is a good non-excellent form of height 2 and degree d. Now, Lemma 6.12 and item (3) of Lemma 8.7 show that there exists f ∈ F^{∗} such that e^{d+2}(λ ⊥ −φ) = e^{d+2}(γ ⊗ ⟨ ; ⟨f⟩ ; ⟩) . By Conjecture 6.1, we have

λ ⊥ −φ ≡ γ ⊗ ⟨ ; ⟨f⟩ ; ⟩ mod I^{d+3}(F) .

Since γ ≡ λ mod I^{d+2}(F) , it follows that λ ⊥ −φ ≡ λ ⊗ ⟨ ; ⟨f⟩ ; ⟩ ≃ λ ⊥ −fλ

(mod I^{d+3}(F)) . Therefore, φ ≡ fλ mod I^{d+3}(F) . This completes the proof of Proposition 8.6.

Corollary 8.8. If h ≥ 3 then φ_{0} is a Pfister neighbor whose complementary form is similar to φ_{1}.

Proof. Proposition 8.6 shows that the condition of Proposition 5.1 holds in the case k = h − 3 and m = d + 2.

Proof of Theorem 8.1. In case h ≥ 4 , Corollaries 8.8 and 8.4 show that • φ_{0} is a Pfister neighbor whose complementary form is similar to φ_{1} ,

• the sequence (φ_{1}, φ_{2}, …, φ_{h}) is quasi-excellent of the third type. After that, an evident induction completes the proof.

9. Quasi-excellent sequences of type 3 and height 3

The main purpose of this section is to complete the classification of quasiexcellent sequences of the third type. The results of the previous section show that it suffices to consider only quasi-excellent sequences of height 3.

Lemma 9.1. The sequence (ϕ_{0}, ϕ_{1}, ϕ_{2}, ϕ_{3}) in Example 0.2 is quasi-excellent.

Proof. Set ϱ = ⟨ ; ⟨a_{1}, …, a_{d−1}⟩ ; ⟩ and K = F (ϕ_{0}) . Then (ϕ_{1})_{K} is anisotropic by Theorem 1.5. We have i((ϕ_{0})_{K}) ≥ dim ρ = 2^{d−1} by [HR], Lemma 2.5 ii, hence

dim((ϕ_{0})_{K})_{an} ≤ 2^{d+1} = dim(ϕ_{1})_{K} . Set η = k_{0}ρ ⊗ ⟨ ; ⟨u, v, c⟩ ; ⟩. Then

ϕ_{0} ⊥ −c k_{0}k_{1}ϕ_{1} ≃ k_{0}ρ ⊗ (⟨ ; ⟨u, v⟩ ; ⟩ ⊥ −c ⟨ ; ⟨a_{d}⟩ ; ⟩ ⊥ −c ⟨−u, −v, uv, a_{d}⟩)

≃ k_{0}ρ ⊗ (⟨ ; ⟨u, v, c⟩ ; ⟩ ⊥ ⟨c a_{d}, −c a_{d}⟩)

≃ η ⊥ ρ ⊗ ⟨1, −1⟩ .

The form η_{F (η)} is hyperbolic. But (ϕ_{1})_{F (η)} is anisotropic by Theorem 1.5, hence ((ϕ_{0})_{F (η)})_{an} ∼ (ϕ_{1})_{F (η)} . Since there is an F -place K → F (η) ∪ {∞} it follows that dim((ϕ_{0})_{K})_{an} ≥ dim(ϕ_{1})_{F (η)} = 2^{d+1}, cf [Kn1], Proposition 3.1. Thus dim((ϕ_{0})_{K})_{an} = 2^{d+1} = dim(ϕ_{1})_{K} .

If η_{K} is hyperbolic then it follows that ((ϕ_{0})_{K})_{an} ∼ (ϕ_{1})_{K} . Otherwise, η_{K} is anisotropic, and ((ϕ_{0})_{K})_{an} ⊥ −c k_{0}k_{1}(ϕ_{1})_{K} ≃ η_{K} . This shows that for every x ∈ K^{∗}, the forms (x(ϕ_{0})_{K})_{an} and (ϕ_{1})_{K} are half-neighbors in the sense of [H3], p. 258. Since (ϕ_{0})_{K} is isotropic there is an x ∈ K^{∗} such that x(ϕ_{0})_{K} ≃ (ρ ⊗ ⟨ ; ⟨u, v⟩ ; ⟩)_{K} ⊥ (−ρ ⊗ ⟨ ; ⟨a_{d}⟩ ; ⟩)_{K} , e.g., [HR], Lemma 2.5 i. Thus (x(ϕ_{0})_{K})_{an} is a (2^{d+1}, 2^{d})-Pfister form in the sense of [H3], p. 262. Now, [H3], Proposition 2.8, shows that ((ϕ_{0})_{K})_{an} ∼ (ϕ_{1})_{K} .

By Theorem 1.5, the forms ϕ_{1}, ϕ_{2} remain anisotropic over K = F (ϕ_{0}). Thus we consider ϕ_{1}, ϕ_{2} as forms over K and show that ((ϕ_{1})_{K(ϕ_{1})})_{an}) ∼ (ϕ_{2})_{K(ϕ_{1})}. We have

ϕ_{1} ⊥ k_{1} k_{2} ϕ_{2} ≃ k_{1}ρ ⊗ (⟨−u, −v, uv, a_{d}⟩ ⊥ ⟨ ; ⟨a_{d}⟩ ; ⟩)

∼_{w} k_{1}ρ ⊗ ⟨ ; ⟨u, v⟩ ; ⟩ .

Set ψ = k_{1}ρ ⊗ ⟨ ; ⟨u, v⟩ ; ⟩. Then dim ψ = 2^{d+1} = dim ϕ_{1} . Since (ϕ_{2})_{K(ψ)} is anisotropic by Theorem 1.5 and since ψ_{K(ψ)} is hyperbolic, the form (ϕ_{1})_{K(ψ)} is not hyperbolic. There is a K-place K(ϕ_{1}) → K(ψ) ∪ {∞} forcing that (ϕ_{1})_{K(ϕ_{1})} is not hyperbolic. But the form (ϕ_{1})_{K(ϕ_{1},ϕ_{2})} is hyperbolic, for otherwise, since (ϕ_{2})_{K(ϕ_{1},ϕ_{2})} is hyperbolic, we would have the contradiction dim((ϕ_{1})_{(K(ϕ_{1},ϕ_{2})})_{an} = dim ψ = 2^{d+1} = dim(ϕ_{1})_{K(ϕ_{1},ϕ_{2})} . Now Lemma 3.5 yields that ((ϕ_{1})_{K(ϕ_{1})})_{an}) ∼ (ϕ_{2})_{K(ϕ_{1})} .

The main result of this section is the following Proposition 9.2. Let d > 0 be an integer and F be a field such that condition A_{d} holds for F . Let (φ_{0}, φ_{1}, φ_{2}, φ_{3}) be a quasi-excellent sequence of degree d and of the third type. Then this sequence looks as in Example 0.2.

Clearly, this proposition together with the results of the previous sections completes the proof of Theorems 0.9 and 0.10.

We say that a form φ is divisible by a form ρ if there is a form χ such that φ ≃ ρ ⊗ χ.

Lemma 9.3. Let φ and ψ be anisotropic forms. Suppose that φ and ψ are divisible by a Pfister form ρ (including the case ρ = ⟨1⟩). Then there exist forms φ_{0}, ψ_{0}, μ such that

• φ_{0}, ψ_{0} and μ are divisible by ρ,

• φ ≃ φ_{0} ⊥ μ and ψ ≃ ψ_{0} ⊥ μ,

• (φ ⊥ −ψ)_{an} ≃ φ_{0} ⊥ −ψ_{0} .

Proof. Let μ be a form of maximal dimension satisfying the following conditions:

(a) μ is divisible by ρ,

(b) μ ⊂ φ and μ ⊂ ψ. Then there exist forms φ_{0} and ψ_{0} such that φ ≃ φ_{0} ⊥ μ and ψ ≃ ψ_{0} ⊥ μ. Since φ, ψ and μ are divisible by ρ, it follows from Theorem 1.3 that φ_{0} and ψ_{0} are also divisible by ρ. Now, it suffices to prove that (φ ⊥ −ψ)_{an} ≃ φ_{0} ⊥ −ψ_{0} .

Since φ ⊥ −ψ ≃ (φ_{0} ⊥ μ) ⊥ −(ψ_{0} ⊥ μ) ∼_{w} φ_{0} ⊥ −ψ_{0} , it suffices to prove that the form φ_{0} ⊥ −ψ_{0} is anisotropic. Suppose the contrary. Then the forms φ_{0} and ψ_{0} have a common value, say ℓ ∈ F^{∗}. By Theorem 1.3, we have ℓρ ⊂ φ_{0} and ℓρ ⊂ ψ_{0} . Setting μ̃ = μ ⊥ ℓρ, we see that μ̃ satisfies conditions (a) and (b). Since dim μ̃ > dim μ, we get a contradiction to the definition of μ.

Lemma 9.4. Let φ and ψ be anisotropic forms being divisible by a Pfister form ρ ∈ P_{d−1}(F) (where ρ = ⟨1⟩ if d = 1 ). Suppose that dim φ = 3 · 2^{d}, dim ψ = 2^{d+1}, and (φ ⊥ −ψ)_{an} ∈ GP_{d+2}(F) . Then there exist u, v, a_{d}, c ∈ F^{∗} such that

φ ∼ ρ ⊗ (⟨ ; ⟨u, v⟩ ; ⟩ ⊥ −c ⟨ ; ⟨a_{d}⟩ ; ⟩) and ψ ∼ ρ ⊗ ⟨−u, −v, uv, a_{d}⟩ .

Proof. Let φ_{0}, ψ_{0} and μ be as in Lemma 9.3. We have

2 dim μ = (dim φ + dim ψ) − (dim φ_{0} + dim ψ_{0})
= dim φ + dim ψ − dim(φ ⊥ −ψ)_{an}
= 3 · 2^{d} + 2^{d+1} − 2^{d+2} = 2^{d}.

Hence, dim μ = 2^{d−1}. Since μ is divisible by ρ, there exists s ∈ F^{∗} such that μ ≃ sρ. Clearly, dim ψ_{0} = dim ψ − dim μ = 2^{d+1} − 2^{d−1} = 3 · 2^{d−1}. Since ψ_{0} is divisible by ρ ∈ P_{d−1}(F) there exist k, u, v ∈ F^{∗} such that

ψ_{0} ≃ kρ ⊗ ⟨1, −u, −v⟩ .

Then ψ ≃ (kρ ⊗ ⟨1, −u, −v⟩) ⊥ sρ ≃ kuvρ ⊗ ⟨uv, −v, −u, a_{d}⟩ with a_{d} = skuv. Thus ψ ∼ ρ ⊗ ⟨−u, −v, uv, a_{d}⟩ .

Put π := ρ ⊗ ⟨ ; ⟨u, v⟩ ; ⟩ ∈ P_{d+1}(F) . Since ψ_{0} ≃ kρ ⊗ ⟨1, −u, −v⟩, it is easily

checked that kπ ≃ ψ_{0} ⊥ kuvρ. Hence, (ψ_{0})_{F (π)} is isotropic.

Let η := −k(φ ⊥ −ψ)_{an} ≃ −k(φ_{0} ⊥ −ψ_{0}) . By the hypotheses of the lemma, η ∈ GP_{d+2}(F) . Since kψ_{0} ≃ ρ ⊗ ⟨1, −u, −v⟩ represents 1 , it follows that η represents 1 . Hence, η ∈ P_{d+2}(F) . Since kψ_{0} ⊂ η and (ψ_{0})_{F (π)} is isotropic, it follows that η_{F (π)} is isotropic. Thus η ≃ π ⊗ η_{0} for some form η_{0} by Theorem 1.3. Since η ∈ P_{d+2}(F) and π ∈ P_{d+1}(F), it follows that η ≃ π ⊗ ⟨ ; ⟨c⟩ ; ⟩ for suitable c ∈ F^{∗}. Hence, η ≃ ρ ⊗ ⟨ ; ⟨u, v, c⟩ ; ⟩. Clearly, uv ∈ D(η) . Since η is a Pfister form, we obtain η ≃ uvη, (see [S], 2.10.4). By definition of η we have kη ∼_{w} ψ ⊥ −φ. Hence,

φ ∼_{w} ψ ⊥ −kη ≃ ψ ⊥ −kuvη ≃ kuvρ ⊗ ⟨uv, −v, −u, a_{d}⟩ ⊥ −kuvρ ⊗ ⟨ ; ⟨u, v, c⟩ ; ⟩
∼_{w} kuvρ ⊗ (⟨ ; ⟨u, v⟩ ; ⟩ ⊥ − ⟨ ; ⟨a_{d}⟩ ; ⟩) ⊥ −kuvρ ⊗ (⟨ ; ⟨u, v⟩ ; ⟩ ⊥ −c ⟨ ; ⟨u, v⟩ ; ⟩)
∼_{w} kuvρ ⊗ (c ⟨ ; ⟨u, v⟩ ; ⟩ ⊥ − ⟨ ; ⟨a_{d}⟩ ; ⟩) ≃ kuvcρ ⊗ (⟨ ; ⟨u, v⟩ ; ⟩ ⊥ −c ⟨ ; ⟨a_{d}⟩ ; ⟩) .

Since dim φ = 3 · 2^{d} = dim ρ ⊗ (⟨ ; ⟨u, v⟩ ; ⟩ ⊥ −c ⟨ ; ⟨a_{d}⟩ ; ⟩), we get

φ ≃ kuvcρ ⊗ (⟨ ; ⟨u, v⟩ ; ⟩ ⊥ −c ⟨ ; ⟨a_{d}⟩ ; ⟩) .

Proof of Proposition 9.2. Let τ ∈ P_{d}(F) be the leading form. Clearly, we can assume that φ_{2} = τ. In other words, we have a quasi-excellent sequence of the form (λ, γ, τ, 0) with dim λ = 3 · 2^{d}, dim γ = 2^{d+1} and dim τ = 2^{d}. By Proposition 6.14, there exists s ∈ F^{∗} such that λ ≡ sγ mod I^{d+2}(F) . Replacing λ by sλ, we can assume that λ ≡ γ mod I^{d+2}(F) . Let ξ := (λ ⊥ −γ)_{an} . Clearly, ξ ≁_{w} 0 and [ξ] ∈ I^{d+2}(F) . By Theorem 1.1, we have dim ξ ≥ 2^{d+2}. Since dim ξ ≤ dim λ + dim γ = 3 · 2^{d} + 2^{d+1} = 5 · 2^{d} < 3 · 2^{d+1} , Conjecture 6.4 yields dim ξ = 2^{d+2}. Hence (λ ⊥ −γ)_{an} = ξ ∈ GP_{d+2}(F) , cf. [AP], Kor. 3.

By Proposition 3.6, γ is a good non-excellent form of height 2 with leading form τ . By Conjecture 6.5, there exists ρ ∈ P_{d−1}(F) such that γ and τ are divisible by ρ. Then γ_{F (ρ)} and τ_{F (ρ)} are isotropic. Hence F (ρ, γ, τ)/F (ρ) is purely transcendental. Since the form λ_{F (γ,τ)} is hyperbolic by Proposition 3.7, it follows that λ_{F (ρ)} is hyperbolic. Therefore λ is divisible by ρ (see Theorem 1.3). Applying Lemma 9.4 to the forms ρ, φ = λ, and ψ = γ, we see that there exists u, v, a_{d} ∈ F^{∗} such that

λ ∼ ρ ⊗ (⟨ ; ⟨u, v⟩ ; ⟩ ⊥ −c ⟨ ; ⟨a_{d}⟩ ; ⟩) and γ ∼ ρ ⊗ ⟨−u, −v, uv, a_{d}⟩ .

It follows that τ ∼ ρ ⊗ ⟨ ; ⟨a_{d}⟩ ; ⟩ by [Kn1], 6.12. The proof is complete.

References

[Ara] Arason, J. Kr. Cohomologische Invarianten quadratischer Formen. J. Algebra 36 (1975), 448–491.

[AP] Arason, J. Kr.; Pfister A. Beweis des Krullschen Durchschnittsatzes für den Wittring. Invent. math. 12 (1971), 173–176.

[H1] Hoffmann, D. W. Isotropy of quadratic forms over the function field of a quadric. Math. Z. 220 (1995), 461–476.

[H2] Hoffmann, D. W. Sur les dimensions des formes quadratiques de hauteur 2. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 1, 11–14.

[H3] Hoffmann, D. W. Similarity of quadratic forms and half-neighbors. J. Algebra 204 (1998), no. 1, 255–280.

[H4] Hoffmann, D. W. On the dimensions of anisotropic quadratic forms in I^{4}. Invent. math. 131 (1998), no. 1, 185–198.

[HR] Hurrelbrink, J.; Rehmann U. Splitting patterns and linear combinations of two Pfister forms. J. Reine Angew. Math. 495 (1998), 163–174.

[Izh] Izhboldin, O. T. Virtual Pfister neighbors and first Witt index. Preprint 2000.

http://wwwmath.uni-muenster.de/math/u/karpenko/IZHBOLDIN/quadratic.html

[JR] Jacob, W.; Rost, M. Degree four cohomological invariants for quadratic forms. Invent. math. 96 (1989), 551–570.

[Kahn] Kahn, B. Formes quadratiques de hauteur et de degré 2. Indag. Math. (N.S.) 7 (1996), no. 1, 47–66.

[KRS] Kahn, B.; Rost, M; Sujatha R.J. Unramified cohomology of quadrics I. Amer. J. Math. 120 (1998), no. 4, 841–891.

[KR] Kersten, I.; Rehmann, U. Excellent algebraic groups, I. J. Algebra 200 (1998), no. 1, 334–346.

[Kn1] Knebusch, M. Generic splitting of quadratic forms, I. Proc. London Math. Soc. (3) 33 (1976), no. 1, 65–93.

[Kn2] Knebusch, M. Generic splitting of quadratic forms, II. Proc. London Math. Soc. (3) 34 (1977), no. 1, 1–31.

[L] Lam, T. Y. The algebraic Theory of Quadratic Forms. Massachusetts: Benjamin 1973 (revised printing 1980).

[M1] Merkurjev, A. S. On the norm residue symbol of degree 2. Dokl. Akad. Nauk SSSR 261 (1981), 542–547. Engl. transl.: Soviet Math. Dokl. 24 (1981), 546–551.

[MS] Merkurjev, A. S.; Suslin, A. A. The norm residue homomorphismus of degree 3. Izv. Akad. Nauk SSSR Ser. Mat. (Russian) 54 (1990), no. 3, 339–356. Engl. transl.: Math. USSR, Izv. 36 (1991), no.3, 349–368.

[Pf1] Pfister, A. Quadratic forms with applications to algebraic geometry and topology. Cambridge Univ. Press 1995.

[Pf2] Pfister, A. On the Milnor conjectures: history, influence, applications. Jber. d. Dt. Math.-Verein. 102 (2000), 15–41.

[R] Rost, M. On Hilbert Satz 90 for K_{3} for quadratic extensions extensions. Preprint (1988).

http://www.math.ohio-state.edu/~rost/K3-88.html

[S] Scharlau W. Quadratic and hermitian forms. Springer-Verlag 1985.

[Sz] Szyjewskii, M. The fifth invariant of quadratic forms. (Russian) Algebra An. 2 (1990), 213–234. English transl.: Leningrad Math. J. 2 (1991), 179–198.

[Vi] Vishik, A. On the dimension of anisotropic forms. (Russian) Dokl. Akad. Nauk SSSR 373 (2000), no. 4, 445–447.

[Vo] Voevodsky, V. On 2-torsion in motivic cohomology. K-theory Preprint Archives 502, July 15, 2001.

        http://www.math.uiuc.edu/K-theory/0502/index.html
        http://www.mathematik.uni-osnabrueck.de/K-theory/0502/index.html

[OVV] Orlov, D.; Vishik, A.; Voevodsky, V. An exact sequence for Milnor’s K-theory with applications to quadratic forms. K-theory Preprint Archives 454, December 6, 2000.

        http://www.math.uiuc.edu/K-theory/0454/index.html
        http://www.mathematik.uni-osnabrueck.de/K-theory/0454/index.html

[W] Witt, E. Theorie der quadratischen Formen in beliebigen Körpern. J. reine angew. Math. 176 (1937), 31–44. (Collected Papers, Springer-Verlag 1998, 2–18).

Oleg H. Izhboldin († April 17, 2000) http://www.mathematik.unibielefeld.de/˜ oleg/ Ina Kersten Mathematisches Institut Bunsenstr. 3–5 D-37073 Göttingen Germany kersten@mathematik.uni-bielefeld.de