A PFISTER FORM INVARIANT FOR ETALE ALGEBRAS

MARKUS ROST

Let k be a field of characteristic different from 2. We denote by GW (k) the Witt-Grothendieck ring of k.

For a ∈ k^{∗} we denote k_{a} = k[t]/(t^{2} − a). Further we denote

[[a]] = ⟨1⟩ − ⟨a⟩

This is a 0-dimensional element of GW (k), with image ⟨⟨a⟩⟩ in the Witt ring. We fix a quadratic extension L = k_{a}.

The multiplicative norm in GW for quadratic extensions. The conjugation in L is denoted by α ↦ ᾱ. We consider the additive trace map

T_{L/k}: GW (L) → GW (k)

and the multiplicative norm map

N_{L/k}: GW (L) → GW (k)

One has

N_{L/k}(x + y) = N_{L/k}(x) + T_{L/k}(xȳ) + N_{L/k}(y)

and

N_{L/k}(⟨α⟩) = ⟨N_{L/k}(α)⟩

for α ∈ L^{∗}.

Lemma 1. For α ∈ L^{∗} one has

N_{L/k}(⟨⟨α⟩⟩) = ⟨⟨T_{L/k}(α), −aN_{L/k}(α)⟩⟩ + [[a]]

Proof.

N_{L/k}(⟨⟨α⟩⟩) = N_{L/k}(⟨1, −α⟩)
= N_{L/k}(⟨1⟩) + T_{L/k}(⟨−ᾱ⟩) + N_{L/k}(⟨−α⟩)
= ⟨1⟩ + ⟨−T_{L/k}(α)⟩⟨1, aN_{L/k}(α)⟩ + ⟨N_{L/k}(α)⟩
= ⟨1, −T_{L/k}(α)⟩⟨1, aN_{L/k}(α)⟩ + ⟨N_{L/k}(α)⟩[[a]]
□

Corollary 1. Let n > 0 and let f be a n-fold Pfister form over L. Then

N_{L/k}(f) − [[a]]^{n}

is a 2n-fold Pfister form over k.

Proof. For α ∈ L^{∗} one has ⟨⟨−aN_{L/k}(α)⟩⟩[[a]] = 0. Hence Lemma 1 gives

N_{L/k}(⟨⟨α_{1}, …, α_{n}⟩⟩) = ∏_{i}⟨⟨T_{L/k}(α_{i}), −aN_{L/k}(α_{i})⟩⟩ + [[a]]^{n}

and the claim is clear. □

Date: November 23, 2002.

Invariants for etale algebras over a quadratic extension. Let Θ be a cohomological (or Witt group) invariant for etale algebras E over L of some fixed degree n. That is, for each field F over k and each etale extension E of F ⊗ L of degree n we are given an element Θ(E) ∈ H(F), compatible with the restriction maps for morphisms F → F′. We denote by Θ_{L} the induced invariant given by Θ_{L}(E) = Θ(E)_{L} ∈ H(F ⊗ L).

Proposition 1. Suppose Θ_{L} = 0. Then Θ is constant.

Proof. A versal parameter space for our objects E is the Weil restriction Z = R_{L/k}(A^{n} \ Δ) of the versal parameter space A^{n} \ Δ (with Δ the discriminant locus of normed polynomials of degree n). One has R_{L/k}(A^{n}) = A^{2n}. Thus Z is an open subvariety of A^{2n}. Its complement Δ′ is a divisor birational isomorphic to A^{n} × Δ × Spec L.

The invariant Θ defines a versal class Θ_{gen} ∈ H(k(Z)) unramified on Z. If Θ_{L} = 0, then Θ_{gen} is unramified in Δ′ as well, since k(Δ′) contains L.

Thus Θ_{gen} is unramified on all of A^{2n}. □

Serre’s splitting principle. A cohomological (or Witt group) invariant for etale algebras E over k (of some fixed degree) which vanishes on multiquadratic extensions is trivial.

(A multiquadratic extension is a product of extensions of degree 1 or 2.)

The reduced trace form S_{E}. For an etale extension E of degree n let q_{E} be its trace form, let

ω_{n} = \begin{cases}⟨2, 2, ..., 2⟩ (rank m) & if n = 2m \\ ⟨1, 2, ..., 2⟩ (rank m + 1) & if n = 2m + 1 \end{cases}

and let

S_{E} = q_{E} − ω_{n}

to be considered as an element of GW (k). One has dim S_{E} = m.

For extensions E, F of even degree and a ∈ k^{∗} one has

S_{E×k} = S_{E}
S_{E×F} = S_{E} + S_{F}
S_{k_{a}} = ⟨2a⟩

If E is multiquadratic, then S_{E} is given by an honest quadratic form of dimension m. Although this is not true in general, S_{E} has some properties of a m-dimensional quadratic form. Here are two of these properties (due to Serre):

λ^{i}(S_{E}) = 0 if i > m
w_{i}(S_{E}) = 0 if i > m

They are immediate from Serre’s splitting principle. Theorem 1 below yields another one.

The Pfister form P (E). Let

P (E) = P (E/k) = ∑_{i=0}^{m} λ^{i}(S_{E}) = ∑_{i=0}^{∞} λ^{i}(S_{E})

Note that dim P (E) = 2^{m}. If E is multiquadratic, then it is easy to see that P (E) is a m-fold Pfister form.

We first note some elementary properties: Let E, F be of even degree and a ∈ k^{∗}. Then

P (E × k) = P (E)
P (E × F) = P (E)P (F)
P (k_{a}) = ⟨1, 2a⟩ = ⟨⟨−2a⟩⟩

These statements are clear from the definitions and the standard properties of the λ-operations. For the multiplicativity property one may also resort to Serre’s splitting principle and reduce to the trivial case of multiquadratic extensions.

Theorem 1. P (E) is a m-fold Pfister form.

Proof. One may assume that k has no proper field extensions of odd degree. Then, by the multiplicativity property of P, we may further assume that E is a field extension of a quadratic extension L = k_{a}.

Let r be the degree of E/L. The case r = 1 is clear. Otherwise r = 2s.

Let

Θ(E) = P (E/k) − N_{L/k}(P (E/L)) + [[a]]^{s}

By induction on the degree and by Corollary 1 it suffices to show Θ(E) = 0. The class Θ(E) defines an invariant as considered in Proposition 1. One has

Θ(E)_{L} = P (E × Ē) − P (E)P (Ē) = 0

(with ¯ denoting the conjugation of L/k). Thus Θ is constant.

It remains to check that Θ(E) = 0 for some specific E. Thus we may assume that E = (F × ⋯ × F) ⊗ L for any given quadratic extension F of k. Using the multiplicativity properties of P and N_{L/k} and Lemma 1 one reduces to the case E = F ⊗ L.

Take F = k × k. Then S_{E/k} = ⟨2a, 2a⟩ and therefore P (E/k) = ⟨⟨−2a, −2a⟩⟩ = ⟨⟨−1, −a⟩⟩. Furthermore, S_{E/L} = ⟨2⟩, P_{E/L} = ⟨⟨−2⟩⟩. Therefore N_{L/k}(P (E/L)) = ⟨⟨−4, −4a⟩⟩ + [[a]] by Lemma 1. The claim follows.

One may also take F = k_{−1}. In this case the trace forms of E/L and E/k are hyperbolic. Thus S_{E/k} = ⟨−2, −2⟩ and P (E/k) = ⟨⟨2, 2⟩⟩ is hyperbolic. Moreover S_{E/L} = ⟨−2⟩, P_{E/L} = ⟨⟨2⟩⟩ and N_{L/k}(P (E/L)) = ⟨⟨4, −4a⟩⟩ + [[a]] by Lemma 1. Again the claim follows. □

Proposition 2. For the invariant e_{m} of P (E) in H^{m}(k, Z/2) one has

e_{m}(P (E)) = ∑_{i=0}^{m}(−1)^{m−i}w_{i}(S_{E})

Proof. By Serre’s splitting principle one may assume that E is multiquadratic. Both sides are multiplicative, and so one may assume E = k_{a}. In this case q_{E} = ⟨2, 2a⟩, P (E) = ⟨⟨−2a⟩⟩ and w_{1}(S_{E}) = (2a). □

4 MARKUS ROST

Remark. I don’t know any “explanation” of Theorem 1. What is the nature of P (E)? One can play around and base everything on S′_{E} = ⟨−1⟩S_{E} instead of S_{E}. Then

e_{m}(P (E)) = w_{m}(S′_{E})

Let us normalize P (E) to have dimension 0 and write

P′(E) = P (E) − ⟨⟨1⟩⟩^{m}

Then

P′(E) = ∑_{i=0}^{m}(−1)^{i}λ^{i}(S′_{E})

This way the Pfister form invariant appears as a sort of Euler characteristic. Remark. One has w(q_{E}) = w(S_{E})(1 + (2))^{m} = w(S_{E})(1 + m(2)). It follows that

e_{m}(P (E)) = ∑_{i=0}^{m}(−1)^{m−i}w_{i}(q_{E}) + m(2)w_{m−1}(q_{E})

Furthermore:

e_{m}(P (E)) = ∑_{i=0}^{m}(−2)^{m−i}w_{i}^{gal}(q_{E})

I have not made up my mind about the preferred way to look at P (E). Remark. If one develops the material in terms of S′_{E} and P′(E), then it seems natural to consider the normalized Pfister forms

[[a_{1}, …, a_{n}]] = [[a_{1}]] ⋯ [[a_{n}]] = ⟨⟨a_{1}, …, a_{n}⟩⟩ − ⟨⟨1⟩⟩^{n}

of dimension 0. One gets the following variant of Lemma 1: Lemma 2. For α ∈ L^{∗} one has

N_{L/k}([[α]]) = [[T_{L/k}(α), −aN_{L/k}(α)]] + [[2, a]]

□

Note that [[2, a]]^{2} = 0 and [[T_{L/k}(α), −aN_{L/k}(α)]][[2, a]] = 0. Corollary 2. Let n ≥ 2 and let f be a normalized n-fold Pfister form over L. Then N_{L/k}(f) is a normalized 2n-fold Pfister form over k. Example. Let E be of degree 4. Then

q_{E} = ⟨1⟩ + ⟨c⟩⟨−a, −b, ab⟩

for some a, b, c ∈ k^{∗}. One has P (E) = ⟨⟨a, b⟩⟩. Remark. Suppose k is ordered. Then the Pfister form P (E) is positive definite if and only if E is totally real. Remark. One has

∧q_{E} = P (E)⟨⟨−2⟩⟩^{m}

for even n = 2m and

∧q_{E} = P (E)⟨⟨−1⟩⟩^{m+1}

for odd n = 2m + 1. This follows easily from Serre’s splitting principle.

Remark (Serre). The Pfister form P (E) represents 2^{m}d. Here m = [n/2] and d is the discriminant of E (and n = dim E).

Beweis: this amounts to the equation ⟨2^{m}d⟩P (E) = P (E) in W (k), which is true when E is multiquadratic, hence is true in general.

Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA

   E-mail address: rost@math.ohio-state.edu
   URL: http://www.math.ohio-state.edu/~rost