ON “HORIZONTAL” INVARIANTS ATTACHED TO QUADRATIC FORMS

BRUNO KAHN

Abstract. We introduce series of invariants related to the dimension for quadratic forms over a field, study relationships between them and prove a few results about them.

This is the TEX-ing of a manuscript from 1993 entitled Quadratic forms and simple algebras of exponent two. The original manuscript contained an appendix that has appeared in [K3]: I removed it and replaced references to it by references to [K3]. I also extended Section 2 somewhat, improved Proposition 3.3 a bit and removed a section that did not look too useful. Finally I changed the title to a better-suited one. These are essentially the only changes to the original manuscript.

The main reasons I have to exhume it are that 1) the notion of dimension modulo I^{n+1} has recently been used very conceptually by Vishik (e.g. [Vi], to which the reader is referred for lots of highly nontrivial computations) and 2) Question 1.1 has been answered positively by Parimala and Suresh [P-S]. I have included a proof that is different from theirs (see Corollary 2.1).

Vishik has suggested that the invariant λ which is studied here might be replaced by a finer one: the “geometric length”, where one takes transfers into account. Namely, if x ∈ I^{n}F/I^{n+1}F, its geometric length is the smallest integer ℓ such that there exists an étale F -algebra E of degree ℓ and a Pfister form y = ⟨⟨u_{1}, …, u_{n}⟩⟩ ∈ I^{n}E/I^{n+1}E such that x = T r_{E/F} (y). This invariant should definitely be investigated as well.

Everything here is anterior to Voevodsky’s proof of the Milnor conjecture, which is not used.

Introduction

Let F be a field of characteristic ≠ 2. The u-invariant u(F) of F is the least integer n such that any quadratic form in more than n variables over F is isotropic, or +∞ if no such integer exists. (A finer version exists for formally real fields, but for simplicity we shall not consider it.) In [Me2] (see also [Ti]), Merkurjev disproved a longstanding conjecture of Kaplansky, asserting that the u-invariant should always be a power of 2. In fact, Merkurjev produced for any integer m ≥ 3 examples of fields of u-invariant 2m.

Date: Jan. 9, 2005.

2000 Mathematics Subject Classification. Primary: 11E04, 11E81.

Key words and phrases. Quadratic forms, dimension, u-invariant.

A remarkable feature of Merkurjev’s examples is that they can be contrived to have 2-cohomological dimension 2. This destroys a naïve belief that u(F) would be 2^{ν(F)}, where ν(F), the ν-invariant of F, is the largest integer n such that I^{ν}F ≠ 0, which was the case in all previously known examples (see e.g. [K1, th. 1]). It hints that a good understanding of the u-invariant involves not only the ν-invariant, but also ‘horizontal’ invariants attached to the quotients I^{n}F/I^{n+1}F.

Introducing such invariants and starting their study of is the object of this paper. Given a quadratic form, one may approximate its anisotropic dimension, the dimension of its kernel forms, by its ‘dimension modulo I^{n+1}’ for every n ≥ 1 (the fact that this actually is an approximation is a consequence of the Arason-Pfister theorem). Given an element of I^{n}F/I^{n+1}F, one may study its ‘length’ or ‘linkage index’, the smallest number of classes of Pfister forms necessary to express it. The suprema u_{n}(F) of the former invariants (‘u-invariant modulo I^{n+1}’) approximate the u-invariant; the suprema λ^{n}(F) of the latter help giving upper bounds for the former. More precisely, one can bound u_{n}(F) in terms of λ^{n}(F) and u_{n−1}(F) (Proposition 1.2).

The only case in which I can prove a converse to these bounds is n = 2, where u_{2}(F) = 2λ^{2}(F) + 2. However, it is not impossible that all the λ^{n}(F), as well as u(F) when F is not formally real, can actually be bounded in terms of λ^{2}(F) (and n for λ^{n}(F)). At least this is the case when λ^{2}(F) = 1, by a theorem of Elman-Lam [Lam, th. XI.4.10]. I partially generalize this theorem in one direction (Propositions 3.2 and 3.3), but the general case seems quite open.

In all this paper, we use Lam’s [Lam] notations for Pfister forms, i.e. ⟨⟨a_{1}, …, a_{n}⟩⟩ = ⟨1, a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{n}⟩. We write ≅ (resp. ∼) for isometry (resp. Witt-equivalence) of quadratic forms.

1. A few quadratic invariants

As in Arason [A], for a quadratic form q we denote by diman(q) the rank of the unique anisotropic quadratic form whose class in W (F) equals the class of q (the kernel form of q). As will be seen in Proposition 1.1, Definition 1.1 generalises this definition.

Definition 1.1. a) Let q be a quadratic form over F and J an ideal of W (F). The J-dimension of q is dim_{J}(q) = inf{dim(q′) | q′ ≡ q (mod J)}. b) Let n ≥ 0 be an integer. The n-dimension of q is dim_{n}(q) = dim_{I^{n+1}F} (q).^{1}

Remark 1.1. dim_{n}(q) only depends on the class of q modulo I^{n+1}F. For all n, one has dim_{n}(q) ≤ diman(q).

Definition 1.2. A quadratic form q is anisotropic modulo I^{n+1}F (or n-anisotropic) if dim_{n}(q) = dim(q).

It is clear that for two quadratic forms q, q′, one has dim_{n}(q ⊥ q′) ≤ dim_{n}(q) + dim_{n}(q′). The following lemma strengthens this result when q′ ∈ I^{n}F. Despite its simplicity, it is basic in much of this section.

Lemma 1.1. Let (q, q′) ∈ W (F) × I^{n}F with q, q′ ≠ 0. Then dim_{n}(q ⊥ q′) ≤ dim_{n}(q) + dim_{n}(q′) − 2.

Proof. Without loss of generality, we may assume that q and q′ are anisotropic modulo I^{n+1}F. Since q′ ∈ I^{n}F, q′ ≡ ⟨a⟩q′ (mod I^{n+1}F) for any a ∈ F^{∗}. For suitable a, the form q ⊥ ⟨a⟩q′ is isotropic. Hence

dim_{n}(q ⊥ q′) = dim_{n}(q ⊥ ⟨a⟩q′) ≤ diman(q ⊥ ⟨a⟩q′)

≤ dim(q) + dim(q′) − 2 = dim_{n}(q) + dim_{n}(q′) − 2.

□

Lemma 1.2. Let q ∈ W (F) be anisotropic modulo I^{n+1}F. Then its only subforms belonging to I^{n}F are 0 and possibly q.

Proof. This follows from Lemma 1.1. □

Definition 1.3. Let n ≥ 0 and x ∈ I^{n}F/I^{n+1}F. The length of x is

λ(x) = inf{r | x is a sum of r classes of n-fold Pfister forms}.

If q ∈ I^{n}F, we write λ(q) for λ(x), where x is the image of q in I^{n}F/I^{n+1}F.

Proposition 1.1. a) dim_{0}(q) = \begin{cases}0 & if dim(q) is even \\ 1 & if dim(q) is odd. \end{cases}

b) dim_{1}(q) = \begin{cases}0 & if dim(q) is even and d_{±}(q) = 1 \\ 1 & if dim(q) is odd \\ 2 & if dim(q) is even and d_{±}(q) ≠ 1. \end{cases}

c) If q ∈ I^{n}F − I^{n+1}F, then 2^{n} ≤ dim_{n}(q) ≤ (2^{n} − 2)λ(q) + 2.

d) If q ∉ I^{n}F, then dim_{n}(q) ≤ (2^{n} − 2)λ(q′) + dim_{n−1}(q) for some q′ ∈ I^{n}F.

1One should be careful that Vishik’s notation in [Vi, Def. 6.8] is different: our dim_{n}(q) is his dim_{n+1}(q).

e) If q ∈ I^{2}F − I^{3}F, then dim_{2}(q) = 2λ(q) + 2.

f) If 2^{n} ≥ diman(q), then dim_{n}(q) = diman(q).

Proof. a) is obvious. Let dim(q) be odd. Then, for the right choice of ε = ±1, q ⊥ ⟨εd_{±}(q)⟩ ∈ I^{2}F, so dim_{1}(q) = 1. Let dim(q) be even. Then q ∈ I^{2}F if and only if d_{±}(q) = 1; if q ∈ I^{2}F, then q ⊥ ⟨−1, d_{±}(q)⟩ ∈ I^{2}F, so dim_{1}(q) = 2. This proves b).

In c), the lower bound for dim_{n}(q) is a consequence of the Arason-Pfister theorem [AP]. The upper bounds in c) and d) follow from Lemma 1.1 by induction on λ(q). Let us prove equality in the case n = 2 (cf. [Me1, lemma]). We may assume dim(q) = dim_{2}(q) = 2m. We argue by induction on m. The case m = 1 is impossible. Assume m ≥ 2. We may write q = ⟨a, b, c⟩ ⊥ q′. Then q is Witt-equivalent to ⟨a, b, c, abc⟩ ⊥ (⟨−abc⟩ ⊥ q′). The first summand is ⟨a⟩⟨⟨ab, ac⟩⟩ ≡ ⟨⟨ab, ac⟩⟩ (mod I^{3}F), while the second one q′′ = ⟨−abc⟩ ⊥ q′ has dimension 2m − 2, so that dim_{2}(q′′) ≤ 2m − 2. By induction, 2λ(q′′) + 2 ≤ dim_{2}(q′′), so 2λ(q) + 2 ≤ 2λ(q′′) + 4 ≤ dim_{2}(q′′) + 2 ≤ dim_{2}(q).

Note that this argument fails for n ≥ 3.

Let us prove f). We may assume that q is anisotropic. Assume that dim_{n}(q) < dim(q). Then there exists q′ with q′ ≡ q (mod I^{n+1}F) and dim(q′) < dim(q). Therefore, q ⊥ −q′ ∈ I^{n+1}F. But dim(q ⊥ −q′) < 2 dim(q) ≤ 2^{n+1}: therefore, by the Arason-Pfister theorem [AP], q ⊥ −q′ is hyperbolic. This means that q is Witt-equivalent to q′: this is impossible, since q is anisotropic and dim(q′) < dim(q). □

Definition 1.4. Let n ≥ 1 and k ≤ n. The k-restricted u-invariant modulo I^{n+1} of F is u_{n}^{k}(F) = sup{dim_{n}(q) | q ∈ I^{k}(F)}. If k = 0, we write u_{n}(F) for u_{n}^{k}(F) and call it the u-invariant of F modulo I^{n+1}.

Remark 1.2. When k is fixed, the u_{n}^{k}(F)’s form a non-decreasing sequence for increasing n. In particular the u_{n}(F) form a non-decreasing sequence. Similarly, when n is fixed, the u_{n}^{k}(F) form a non-increasing sequence for increasing k.

Definition 1.5. Let n ≥ 1. The n-th λ-invariant of F is λ^{n}(F) = sup{λ(x) | x ∈ I^{n}F/I^{n+1}F }.

Proposition 1.2. a) For k > 0 and n ≥ k, I^{k}F ≠ 0 ⟺ u_{n}^{k}(F) ≠ 0 ⟺ u_{n}^{k}(F) ≥ 2^{k}. If I^{n}F ≠ 0, u_{n}^{k}(F) ≥ 2^{n}. In particular, if I^{n}F ≠ 0, u_{n}(F) ≥ 2^{n}.

b) For k ≥ 0, sup_{n≥k} u_{n}^{k}(F) = u_{k}(F) := sup{diman(q) | q ∈ I^{k}(F)}; in particular, sup_{n≥0} u_{n}(F) = u(F).

c) u_{n}^{n}(F) ≤ (2^{n} − 2)λ^{n}(F) + 2. d) For any k < n, u_{n}^{k}(F) ≤ (2^{n} − 2)λ^{n}(F) + u_{n−1}^{k}(F).

e) u_{0}(F) = 1.

f) If IF ≠ 0, u_{1}(F) = u_{1}^{1}(F) = 2.

g) If I^{2}F ≠ 0, then u_{2}(F) = u_{2}^{2}(F) = 2λ^{2}(F) + 2.

(I am indebted to O. Gabber for pointing out d).) Proof. It is clear that I^{k}F = 0 ⇒ u_{n}^{k}(F) = 0 for all n ≥ k. By Remark 1.2, u_{n}^{k}(F) ≥ u_{k}^{k}(F) when n ≥ k. Assume I^{k}F = I^{k+1}F. Then any k-fold Pfister form belongs to I^{k+1}F. By [AP], such a form must be hyperbolic, hence I^{k}F = 0. This shows that if I^{k}F ≠ 0, there exists a form q ∈ I^{k}F − I^{k+1}F. By Proposition 1.1 c), we have dim_{k}(q) ≥ 2^{k}, hence u_{k}^{k}(F) ≥ 2^{k} and u_{n}^{k}(F) ≥ 2^{k}. The last two claims of a) follow by Remark 1.2 (u_{n}(F) ≥ u_{n}^{k}(F) ≥ u_{n}^{n}(F) when k ≤ n). This proves a).

To prove b), first assume that u_{k}(F) is finite. Let n be such that 2^{n} ≥ u_{k}(F). By Proposition 1.1 f), dim_{n}(q) = diman(q) for any q ∈ I^{k}F. In particular, u_{n}^{k}(F) = u^{k}(F) for all such n. Assume now that the sequence (u_{n}^{k}(F))_{n≥k} is bounded, say by N. Let q ∈ I^{k}F and choose n such that 2^{n} ≥ dim(q). Applying Proposition 1.1 f) again, we have dim_{n}(q) = diman(q). This shows that diman(q) ≤ N, hence that u^{k}(F) ≤ N.

Part c) is an immediate consequence of Proposition 1.1 c). To prove d), we may assume that IF ≠ 0, otherwise it is trivial. We may further assume that I^{k}F ≠ 0. Let q ∈ I^{k}F. We distinguish two cases:

(i) q ∈ I^{n}F. By Proposition 1.1 d), dim_{n}(q) ≤ (2^{n} − 2)λ^{n}(F) + dim_{n−1}(q) ≤ (2^{n} − 2)λ^{n}(F) + u_{n−1}^{k}(F).

(ii) q ∈ I^{n}F. By Proposition 1.1 c), dim_{n}(q) ≤ (2^{n} − 2)λ^{n}(F) + 2. This is ≤ (2^{n} − 2)λ^{n}(F) + u_{n−1}^{k}(F) provided u_{n−1}^{k}(F) ≥ 2. If k ≥ 1, u_{n−1}^{k}(F) ≥ 2^{k} ≥ 2 by a). If k = 0, u_{n−1}(F) ≥ u_{1}(F) ≥ 2 since IF ≠ 0 (see f)).

e) and f) follow from Proposition 1.1 a) and b). It remains to prove g). First we prove that u_{2}(F) cannot be odd, i.e. u_{2}(F) = u_{2}^{1}(F). This is a consequence of Proposition 1.1 e), Proposition 1.2 e) and the following lemma.

Lemma 1.3. a) u_{2}(F) = ∞ iff λ^{2}(F) = ∞.

b) Assume that u_{2}(F) < ∞ and IF ≠ 0. Then u_{2}(F) = u_{2}^{1}(F).

Proof. a) is a consequence of Proposition 1.1, b) d) and e). For b), let q be such that dim(q) = dim_{2}(q) = u_{2}(F). We show that dim(q) cannot be odd, unless IF = 0. If it is, then as in the proof of Proposition 1.1 b) we choose ε = ±1 such that q′ = q ⊥ ⟨εd_{±}(q)⟩ ∈ I^{2}F. By assumption, dim_{2}(q′) ≤ dim(q), so that q′ ≡ q′′ (mod I^{3}F), where q′′ ∈ I^{2}F is such that dim(q′′) ≤ dim(q). Then q ≡ q′′ ⊥ ⟨−εd_{±}(q)⟩ (mod I^{3}F).

If q′′ = 0, then u_{2}(F) = 1, which implies IF = 0 (Proposition 1.2,

b) and remark 1.2). Otherwise, by Lemma 1.1 we have dim_{2}(q) ≤ dim(q′′) − 1 < dim(q), a contradiction. □

We now prove that u_{2}^{1}(F) = u_{2}^{2}(F). By lemma 1.3 we may assume that u_{2}(F) < ∞. Let q ∈ IF be such that dim_{2}(q) = u_{2}^{1}(F). We may assume that q is anisotropic modulo I^{3}F. Then q′ = q ⊥ ⟨−1, d_{±}(q)⟩ ∈ I^{2}F, with dim(q′) = dim(q) + 2. Since dim(q) = u_{2}^{1}(F) = u_{2}(F), we have q ⊥ ⟨−1⟩ ≡ q′′ (mod I^{3}F), where q′′ is such that dim(q′′) ≤ dim(q). But dim(q′′) ≡ dim(q ⊥ ⟨−1⟩) (mod 2), so that dim(q′′)⟨dim(q). Therefore, q′ ≡ q′′′ (mod I^{3}F), with dim(q′′′) ≤ dim(q) = dim_{2}(q), and q ≡ q′′′ ⊥ ⟨1, −d_{±}(q)⟩ (mod I^{3}F). If q′′′ = 0, u_{2}^{1}(F) = 2, but then I^{2}F = 0. Otherwise, since q′′′ ∈ I^{2}F, dim_{2}(q) = dim_{2}(q′′′ ⊥ ⟨1, −d_{±}(q)⟩) ≤ dim_{2}(q′′′) by Lemma 1.1. Hence dim_{2}(q′′′) = dim_{2}(q) and u_{2}^{1}(F) = u_{2}^{2}(F). □

Remark 1.3. The proof of g) is borrowed from [Lam, ch XI, proof of lemma 4.9].

Remark 1.4. The statement u_{n}(F) = u_{n}^{1}(F) is equivalent to “u_{n}(F) is even”.

Remark 1.5. These proofs prompt the definition of quadratic forms universal modulo I^{n+1}, round modulo I^{n+1}. This is left to the reader.

Corollary 1.1 (cf [Lam, ch. XI, lemma 4.9]). If u_{2}(F) > 1, it is even.

Example 1.1. For all n ≥ 0, u_{n}(R) = 2^{n}.

Question 1.1. For n > 2, can one bound λ^{n}(F) in terms of n and u_{n}(F)?

For n = 3, one would like to prove this by using the following generic argument. Let k be a base field, m a fixed integer, F_{0} = k(T_{1}, …, T_{2m}), Q the quadratic form ⟨T_{1}, …, T_{2m}⟩ over F_{0}, F_{1} = F_{0}(\sqrt{(−1)^{m}T_{1} … T_{2m}}) and F_{2} the function field of the Severi-Brauer variety of the Clifford algebra of Q_{F_{1}}. Then, by Merkurjev’s theorem [Me] Q_{F_{2}} ∈ I^{3}F_{2} and is a ‘generic element of rank 2m in I^{3}’. Show that, for any field F containing k and any q ∈ I^{3}F of rank 2m, one has λ(q) ≤ λ(Q_{F_{2}}).

This has been achieved by Parimala and Suresh [P-S] with a general position argument: we shall give a different argument avoiding general position in the next section (see Cor. 2.1).

Question 1.2. By Elman-Lam [EL2], if F is not formally real and λ^{2}(F) = 1 then u(F) = 1, 2, 4 or 8. Is there a nonformally real field F such that λ^{2}(F) = 2 and u(F) = ∞?

In the next section, we give some evidence that the answer to this question might be ‘no’.

2. Discrete valuations; iterated power series

Let A be a complete discrete (rank 1) valuation ring, E its quotient field anf F its residue field. We assume that char F ≠ 2.

Proposition 2.1. a) For all n ≥ 1, λ^{n}(E) ≤ λ^{n}(F) + λ^{n−1}(F).

b) For all n ≥ 1 and 0 ≤ k ≤ n, u_{n}^{k}(E) ≤ u_{n}^{k−1}(F) + u_{n−1}^{k−1}(F).

Note. In b), one should interpret u_{n}^{−1}(F) as u_{n}(F). Proof. Let π be a prime element of E. Every quadratic form q over E can be written q ≅ q_{1} ⊥ πq_{2}, where q_{1} and q_{2} are classes of unimodular forms over A. Alternatively, q can be written up to Witt equivalence q′_{1} ⊥ ⟨⟨π⟩⟩q_{2}, still with q′_{1} integral. If q ∈ I^{n}E, then q′_{1} ∈ I^{n}A and q_{2} ∈ I^{n−1}A [S]. Since W (A) ⟶^{∼} W (F) is a filtered isomorphism, this proves a).

To see b), let q ∈ I^{n}E and q′_{1}, q_{2} as above. Let q̄_{2} be the residue image of q_{2} over F. Take q̄′_{2} ≡ q̄_{2} (mod I^{n}F) with q̄′_{2} ∈ I^{k−1}F and dim(q̄′_{2}) ≤ u_{n−1}^{k−1}(F). Let q′_{2} be a lift of q̄′_{2} to A. Then q′_{2} ≡ q_{2} (mod I^{n}A) and q ≡ q′_{1} ⊥ ⟨⟨π⟩⟩q′_{2} = q′_{1} ⊥ −q′_{2} ⊥ ⟨π⟩q′_{2} (mod I^{n+1}F). Now let q′′_{1} = q′_{1} ⊥ −q′_{2} ∈ I^{k−1}A; choose q′′′_{1} ∈ I^{k−1}A such that q̄′′′_{1} ≡ q̄′′_{1} (mod I^{n+1}F) and dim(q̄′′′_{1}) ≤ u_{n}^{k−1}(F). Then q′′′_{1} ≡ q′′_{1} (mod I^{n+1}A), q ≡ q′′′_{1} ⊥ ⟨π⟩q′_{2} (mod I^{n+1}F) and dim(q′′′_{1} ⊥ ⟨π⟩q′_{2}) ≤ u_{n}^{k−1}(F) + u_{n−1}^{k−1}(F). □

An example. Start from a field K and set K_{d} = K((t_{1})) … ((t_{d})), a field of iterated power series. Then K_{d} is complete for a discrete valuation, with residue field K_{d−1}. Proposition 2.1 allows one to get upper bounds for the invariants of K_{d} in terms of d and those of K, by induction on d.

However, these inductive bounds are by no means sharp in general. Computing, or at least estimating λ^{n}(K_{d}) and u_{n}(K_{d}) turns out to be “global” in d. To illustrate this, we now consider the case where K is algebraically closed.

Definition 2.1. Let k be a field, V a d-dimensional k-vector space and n an integer ≤ d. Let Λ^{n}(V) be the n-th exterior power of V and let x ∈ Λ^{n}(V) be an n-vector. The length of x is the smallest integer ℓ(x) such that x is the sum of ℓ(x) pure n-vectors. We denote by N(k, d, n) the supremum of ℓ(x) when x runs through V (this is independent of V).

The following proposition is clear.

Proposition 2.2. a) Let V = K_{d}^{∗}/K_{d}^{∗2}, viewed as an F_{2}-vector space with basis t_{1}, …, t_{d}. Then there are canonical isomorphisms

I^{n}K_{d}/I^{n+1}K_{d} ≃ Λ^{n}(V)

mapping Pfister forms to pure n-vectors.

b) For x ∈ I^{n}K_{d}/I^{n+1}K_{d}, with image x′ in Λ^{n}(V), λ(x) = ℓ(x′), where λ(x) is as in Definition 1.3 and ℓ(x′) is as in Definition 2.1.

c) λ^{n}(K_{d}) = N(F_{2}, d, n). □

The following information on N(k, d, n) is collected from [K2].

Proposition 2.3. a) N(k, d, n) = N(k, d, d − n).

b) N(k, d, 0) = N(k, d, 1) = 1.

c) N(k, d, 2) = [d/2].

d) N(k, 6, 3) = 3 for any field k.

e) If k is algebraically closed, N(k, 7, 3) ≤ 4 (probably = 4); N(R, 7, 3) = 5; for any field k, N(k, 7, 3) ≤ 6 (probably ≤ 5).

f) If k is algebraically closed of characteristic 0, N(k, 8, 3) = 5; N(R, 8, 3) ≤ 8; for any k, N(k, 8, 3) ≤ 10.

g) If k is algebraically closed of characteristic 0, N(k, 9, 3) ≤ 9.

h) There exists a polynomial f_{n} of degree ≤ n − 2 such that, for

any field k, N(k, d, n) ≤ \frac{d^{n−1}}{2(n − 1)!} + f_{n}(d). In particular, for any

k, lim sup_{d→∞} \frac{N(k, d, n)}{d^{n−1}} ≤ \frac{1}{2(n − 1)!}.

i) If k is infinite, N(k, d, n) ≥ \frac{\binom{d}{n}}{n(d − n) + 1}. If k is finite with q ele-

ments, N(k, d, n) ≥ \frac{\binom{d}{n}}{n(d − n) + 1 + ε(q)}, where

ε(q) = log_{q}(∏_{i=2}^{∞}(1 − q^{−i}) − 1).

(So ε(2) ≈ 0.75.) In particular, for any field k, lim inf_{d→∞} \frac{N(k, d, n)}{d^{n−1}} ≥

\frac{1}{n.n!}.

This proposition enables us to list values of λ^{n}(K_{d}) for d ≤ 6:

d\n 0 1 2 3 4 5 6
1 1 1
2 1 1 1
3 1 1 1 1
4 1 1 2 1 1
5 1 1 2 2 1 1
6 1 1 3 3 3 1 1

From this table, we see that the inequality

u(K_{d}) ≤ ∑_{n=2}^{d}(2^{n} − 2)λ^{n}(K_{d}) + 2

from Proposition 1.2 d) quickly becomes completely inaccurate.

More generally, let us consider a field F provided with a discrete valuation v of rank d. Let K be the residue field of v, L the value group of v and F̂ the completion of F at v. Assume that char K ≠ 2. Then we have

F̂ ≃ K((t_{1})) … ((t_{d}))

where t_{1}, …, t_{d} ∈ F̂ are elements such that v(t_{1}), …, v(t_{d}) form a basis of L.

Let K̄ be an algebraic closure of K and F̃ = K̄((t_{1})) … ((t_{d})). By weak approximation, the map F^{∗}/F^{∗2} → F̂^{∗}/F̂^{∗2} is surjective, hence so is the composition

I^{n}F/I^{n+1}F → I^{n}F̂/I^{n+1}F̂ → I^{n}F̃/I^{n+1}F̃

for any n ≥ 1. It follows that N(F_{2}, d, n) is a lower bound to λ^{n}(F). I don’t know if it is even true that

λ^{n}(F) ≥ λ^{n}(F_{0}) + N(F_{2}, d, n)?

We also have:

Proposition 2.4. Let μ denote the place from F to F_{0} associated to v. Let Q ∈ I^{n}F have good reduction at μ. Then λ(μ_{∗}Q) ≤ λ(Q).

Proof. Since v is a composition of discrete valuations of rank 1, we may reduce to v of rank 1. Pick a prime element π. Then we have a ring homomorphism (e.g. see [M-H, Ch. IV, §1])

Δ_{v,π} : W (F) → W (F_{0})
⟨π^{r}u⟩ ↦ ⟨ū⟩

where u is a unit and ū is its image in F_{0}^{∗}. Clearly Δ_{v,π} preserves nfold Pfister forms, hence λ(Δ_{v,π}(Q)) ≤ λ(Q) for any Q ∈ I^{n}F, and the claim is a special case. □

Corollary 2.1. The answer to Question 1.1 is positive for n = 3.

Indeed, the strategy outlined just after the question works as follows: given q = ⟨a_{1}, …, a_{2m}⟩ ∈ I^{3}F, consider the place μ_{0} from F_{0} to F sending T_{i} to a_{i}. It is well-known that F_{2} is a generic splitting field for the Clifford algebra of Q_{F_{1}}; hence μ_{0} extends to a place μ from F_{2} to F sending Q_{F_{2}} to q. Since the valuation associated to μ_{0} is discrete, so is the one associated to μ, and we may apply Proposition 2.4.

If any element of I^{n}F/I^{n+1}F has a generic splitting field also for n > 3, then the above argument applies verbatim to answer Question 1.1 positively.

Question 2.1. Does Proposition 2.4 remain true when the valuation v is not discrete?

3. Relationships between the u-invariant, the ν-invariant and the λ-invariants

Lemma 3.1 ([EL1, th. 4.5]). Let φ be an m-fold Pfister form and ψ be an n-fold Pfister form. Assume that φ and ψ are r-linked but not (r + 1)-linked. Then, for any a, b ∈ F^{∗}, ind(⟨a⟩φ ⊥ ⟨b⟩ψ) = 2^{r}. □

Proposition 3.1. Assume that u(F) ≤ 2^{n}. Then λ^{n}(F) ≤ 1.

Proof. Let φ and ψ be two n-fold Pfister forms. Then ψ is universal, so ψ ≡ −ψ, and ind(φ ⊥ ψ) = ind(φ ⊥ −ψ) ≥ 2^{n−1}. By Lemma 3.1, φ and ψ are (n − 1)-linked, so φ ⊥ ψ is isometric to an n-fold Pfister form. □

Using a result of Bloch and independently Kato, we can deduce a nice corollary to this proposition, generalising a well-kown result for global fields:

Corollary 3.1. Let F be a function field in n variables over an algebraically closed field, or in n − 1 variables over a finite field. Then every element of H^{n}(F, Z/2) is a symbol.

Proof. A theorem of Kato [Ka, p. 609, prop. 3] (see also Bloch’s argument in [B, Lecture 5]) shows that, for a field as in the statement, H^{n}(F, Z/2) is generated by symbols. It is then sufficient to prove that every element of K_{n}^{M}(F)/2 is a symbol. Notice that F is C_{n} in the sense of Lang [G], hence u(F) ≤ 2^{n}. In view of Proposition 3.1, it is then sufficient to have:

Lemma 3.2 ([EL1, th. 6.1]). Let F be a field, n ≥ 1 and x, y, z ∈ K_{n}^{M}(F)/2 be three symbols. Assume that ν_{n}(x) + ν_{n}(y) = ν_{n}(z). Then x + y = z. □

In this lemma, ν_{n} : K_{n}^{M}(F)/2 → I^{n}F/I^{n+1}F is the homomorphism defined in [Mi].

The following is no more than [EL2, lemma 2.3 and cor. 2.5].

Proposition 3.2. Assume that λ^{n}(F) ≤ 1. Then λ^{n+1}(F) ≤ 1. If furthermore F is not formally real, then I^{n+2}F = 0. □

The next proposition is the main result of this section.

Proposition 3.3. Assume that λ^{n}(F) = m < ∞ and that F is not formally real. Then I^{n(m+1)+1}F = 0. If −1 is not a square in F, then I^{n(m+1)}F = 0.

In other words, if F is not formally real then ν(F) ≤ n(λ^{n}(F) + 1) for all n. For n = 2, the right hand side is u_{2}(F) by Proposition 1.2 d). When −1 is a square in F, this bound is improved to ν(F) ≤ n(λ^{n}(F) + 1) − 1.

Proof. By Milnor [Mi], there are surjective homomorphisms ν_{n} : K_{n}^{M}(F)/2 → I^{n}F/I^{n+1}F, and ν_{2} is an isomorphism. Let K_{∗}^{′M}(F) = K_{∗}^{M}(F)/{−1}K_{∗}(F): as explained in [K3, Appendix], the commutative ring K_{∗}^{′M}(F)/2 enjoys graded divided power operations x ↦ x^{[i]} which vanish on symbols: by [the argument of the proof of] [K3, Prop. 1 (8)], K_{n(m+1)}^{′M}(F)/2 = 0, hence every element of I^{n(m+1)}F/I^{n(m+1)+1}F is a multiple of (the 1-fold Pfister form) ⟨⟨1⟩⟩.

If −1 is a square in F, then ⟨⟨1⟩⟩ is hyperbolic and I^{n(m+1)}F/I^{n(m+1)+1}F = 0; using the Arason-Pfister theorem, we deduce that I^{n(m+1)}F = 0. Assume now that −1 is not a square in F; let E = F (\sqrt{−1}). By [BT, Cor. 5.3], K_{n(m+1)+1}^{M}(E) is generated by symbols {a_{1}, …, a_{n(m+1)+1}}, with a_{1}, …, a_{n(m+1)} in F^{∗}. Since every element of K_{n(m+1)}^{M}(F)/2 is a multiple of {−1}, every element in K_{n(m+1)+1}^{M}(E)/2 is a multiple of {−1} = 0, i.e. K_{n(m+1)+1}^{M}(E)/2 = 0, hence I^{n(m+1)+1}E/I^{n(m+1)+2}E = 0 and I^{n(m+1)+1}E = 0. If now F is not formally real, [A, Satz 3.6 (ii)] implies that I^{n(m+1)+1}F = 0. □

Remark 3.1. For n = 2 Prop. 3.3 is optimal, at least when −1 is a square in F. For example, let F = C((t_{1})) … ((t_{d})) be the field of iterated formal power series in d variables over C. Then ν(F) = d and, by Section 2, λ^{2}(F) = [d/2] which is also the least integer greater than

\frac{d − 1}{2}.

Question 3.1. Is there a universal bound for λ^{n}(F) in terms of n and λ^{2}(F)? In view of Propositions 1.2 and 3.3, this would provide a negative answer to question 1.2. For example, it seems plausible that if λ^{n}(F) = m, then λ^{nm}(F) ≤ 1. This is true in the test example of Remark 3.1. If one could prove it in general, then the estimate ν(F) ≤ n(λ^{n}(F) + 1) or ν(F) ≤ n(λ^{n}(F) + 1) − 1 of Proposition 3.3 would be improved to ν(F) ≤ nλ^{n}(F) + 1 thanks to Proposition 3.2 (note that this is no improvement if n = 2 and −1 is a square in F).

It is clear that divided power operations have not been used up to their full potential.

References

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