NOTE ON THE COHOMOLOGICAL INVARIANT OF PFISTER FORMS

M.TEZUKA AND N.YAGITA

Abstract. The cohomological invariant ring of the n-Pfister forms is isomorphic to the invariant ring in that of an elementary abelian 2-group of rank n under a GL_{n}(ℤ/2)-action.

1. Introduction

Let G be an algebraic group over k with ch(k) ≠ 2. The cohomological invariant Inv^{∗}(G; ℤ/2) is (roughly speaking) the ring of natural functors H^{1}(F; G) → H^{∗}(F; ℤ/2) for the category of finitely generated field F over k. (For details, see the excellent book [Ga-Me-Se]). Moreover, we can define the cohomological invariant Inv^{∗}(P fister_{n}; ℤ/2) of n-Pfister forms, while there does not exist the corresponding group G for n ≥ 4. In Theorem 18.1 in [Ga-Me-Se], this invariant has been computed.

In this note, we show that this cohomological invariant is isomorphic to the invariant ring in that of an elementary abelian 2-group of rank n under a GL_{n}(ℤ/2)-action, namely,

Inv^{∗}(P fister_{n}; ℤ/2) ≅ Inv^{∗}((ℤ/2)^{n}; ℤ/2)^{GL_{n}(ℤ/2)}.

2. motivic cohomology and cohomological invariant

We recall the motivic cohomology H^{∗,∗′}(X; ℤ/2) for a smooth scheme X over k with ch(k) ≠ 2. By the Milnor conjecture (which is now solved by Voevosky), we know H^{∗,∗′}(X; ℤ/2) ≅ H_{et}^{∗}(X; ℤ/2) for ∗ ≤ ∗′. Take τ ∈ H^{0,1}(Spec; ℤ/2) ≅ ℤ/2 as a nonzero element. It is known that H^{∗,∗′}(X; ℤ/2) = 0 for (∗ − ∗′) > dim(X). Hence we have

H^{∗,∗′}(Spec(k); ℤ/2) ≅ H_{et}^{∗}(Spec(k); ℤ/2)[τ].

Let us write H^{∗,∗′} = H^{∗,∗′}(Spec(k); ℤ/2) and H^{∗} = H_{et}^{∗}(Spec(k); ℤ/2) so that H^{∗,∗′} ≅ H^{∗}[τ].

1991 Mathematics Subject Classification. Primary 11E72, Secondary 12G05.

Key words and phrases. Pfister forms, cohomological invariant, Dickson invariant.

Let BG be a classifying space of G ([To], [Vo2]). Let H^{∗}(X; H_{ℤ/2}^{∗′}) be the sheaf cohomology where H_{ℤ/2}^{n} is the Zarisky sheaf induced from the presheaf H_{et}^{n}(V; ℤ/p) for open subset V of X. Then Totaro proved that

Inv^{∗}(BG; ℤ/2) ≅ H^{0}(BG; H_{ℤ/2}^{∗})

in a letter to Serre [Ga-Me-Se]. The Milnor conjecture implies the Beilinson and Lichtenbaum conjecture (see [Vo2,3]). This fact implies the following long exact sequences of motivic and sheaf cohomology theories (Lemma 3.1 in [Or-Vi-Vo], [Vo3])

→ H^{m,n−1}(X; ℤ/2) →^{×τ} H^{m,n}(X; ℤ/2)

→ H^{m−n}(X; H_{ℤ/2}^{n}) → H^{m+1,n−1}(X; ℤ/2) →^{×τ} .

Thus we have

Theorem 2.1. There is an additive isomorphism

Inv^{∗}(G; ℤ/2) ≅ H^{∗,∗}(BG; ℤ/2)/(τ) ⊕ Ker(τ)|H^{∗+1,∗−1}(BG; ℤ/2).

As an application, we first consider the case G = ℤ/2. The mod(2) motivic cohomology is computed in [Vo1,2].

H^{∗,∗′}(Bℤ/2; ℤ/2) ≅ H^{∗,∗′}[y] ⊗ Δ(x)

with β(x) = y, hence deg(y) = (2, 1) and deg(x) = (1, 1). Here Voevodsky shows ([Vo1,2])

x^{2} = τy + ρx with ρ = (−1) ∈ H^{1} = k^{∗}/(k^{∗})^{2}.

Next consider their product G = (ℤ/2)^{n}. The cohomology H^{∗,∗′}(Bℤ/2; ℤ/2) has the Kunneth formula (also by Voevodsky). Hence the motivic cohomology is given

H^{∗,∗′}(BG; ℤ/2) ≅ H^{∗,∗′}[y_{1}, ..., y_{n}] ⊗ Δ(x_{1}, .., x_{n})

where β(x_{i}) = y_{i} and x_{i}^{2} = τy_{i} + ρx_{i}. Hence from Theorem 2.1, we get (as stated in [Ga-Me-Se])

Lemma 2.2. Let G be an elementary abelian 2-group of rank(G) = n. Then Inv^{∗}(G; ℤ/2) ≅ H^{∗} ⊗ Δ(x_{1}, ..., x_{n}) with x_{i}^{2} = ρx_{i}.

3. Dickson invariants

Recall that the mod 2 (topological) cohomology

H^{∗}(B(ℤ/2)^{n}; ℤ/2) ≅ ℤ/2[x_{1}, ..., x_{n}] |x_{i}| = 1.

It is well known that the invariant ring under the GL_{n}(ℤ/2)-action is the Dickson algebra

H^{∗}(B(ℤ/2)^{n}; ℤ/2)^{GL_{n}(ℤ/2)} ≅ ℤ/2[d_{n,0}, ..., d_{n,n−1}]

where each generator d_{n,i} is given by

w_{t}(x) = Π_{ϵ_{i}=0 or 1}(t + ϵ_{1}x_{1} + ...ϵ_{n}x_{n})
= t^{2^{n}} + d_{n,n−1}t^{2^{n−1}} + d_{n,n−2}t^{2^{n−2}} + ... + d_{n,0}t.

Examples. Let us write by w_{i} the elementary symmetric function for x_{j} in H^{∗}(B(ℤ/2)^{n}; ℤ/2). Then

\begin{cases}d_{2,1} = x_{1}^{2} + x_{1}x_{2} + x_{2}^{2} = w_{1}^{2} + w_{2} \\ d_{2,0} = x_{1}^{2}x_{2} + x_{1}x_{2}^{2} = w_{1}w_{2} \end{cases} .

We want to know the Dickson invariant ring in the cohomological invariant. Let us consider

U = Inv_{k=ℝ}^{∗}((ℤ/2)^{n}; ℤ/2) ≅ H^{∗,∗′}(B(ℤ/2)^{n}|_{ℝ}; ℤ/2)/(τ)
≅ ℤ/2[ρ] ⊗ Δ(x_{1}, ..., x_{n}), x_{i}^{2} = ρx_{i}.

For example, in U, we see d_{2,0} = ρx_{1}x_{2} + x_{1}ρx_{2} = 0, and

d_{2,1} = ρx_{1} + x_{1}x_{2} + ρx_{2} = ρw_{1} + w_{2}.

Lemma 3.1. In U, we have d_{n,i} = 0 for i < n − 1 and

d_{n,n−1} = ∑_{i≥1}^{n} w_{i}ρ^{2^{n−1}−i} = (ρ + x_{1})...(ρ + x_{n})ρ^{2^{n−1}−n} + ρ^{2^{n−1}}.

Proof. Decompose that

w_{t}(x) = Π(t + ϵ_{1}x_{1} + ...ϵ_{n−1}x_{n−1}) × Π(t + x_{n} + ϵ_{1}x_{1} + ...ϵ_{n−1}x_{n−1}).

By induction on n, we assume this element is

(t^{2^{n−1}} + d_{n−1,n−2}t^{2^{n−2}})((t + x_{n})^{2^{n−1}} + d_{n−1,n−2}(t + x_{n})^{2^{n−2}}).

Letting d_{n−1,n−2} = d, t^{2^{n−2}} = T and x^{2^{n−2}} = X, we see that the above formula is

= (T^{2} + dT)(T^{2} + X^{2} + dT + dX)
= T^{4} + (d^{2} + dX + X^{2})T^{2} + (d^{2}X + dX^{2})T.

Here note X^{2} = ρ^{∗}X = ρ^{∗′}x_{n}, d^{2} = ρ^{∗}d (since (ρ + x)^{2} = ρ(ρ + x)). So (d^{2}X + dX^{2}) = 0. Let a = a_{n−1} = (ρ + x_{1})...(ρ + x_{n−1}). Then we have

d^{2} + dX + X^{2} = ρ^{∗}d + ρ^{∗−1}dx_{n} + ρ^{∗′}x_{n}
= ρ^{♯}(a + ρ^{∗′′}) + ρ^{♯−1}(a + ρ^{∗′′})x_{n} + ρ^{∗′}x_{n}
= ρ^{♯−1}a(ρ + x_{n}) + ρ^{♯′′} = ρ^{♯−1}a_{n} + ρ^{♯′′}

as desired. □

Corollary 3.2. Let us write

e_{n} = ρ^{−2^{n−1}+n}d_{n,1} = ∑_{i≥1}^{n} w_{i}ρ^{n−i} = (ρ + x_{1})...(ρ + x_{n}) + ρ^{n}.

Then we have

Inv^{∗}((ℤ/2)^{n}; ℤ/2)^{GL_{n}(ℤ/2)} ≅ H^{∗}{1, e_{n}}.

Proof. By Ideal(ρ), we consider the associated graded algebra

gr(H^{∗} ⊗ Δ(x_{1}, ..., x_{n})) ≅ gr(H^{∗}) ⊗ Λ(x_{1}, ..., x_{n}) (x_{i}^{2} = 0).

Note e_{n} = w_{n} in the above graded algebra. We can see

Λ(x_{1}, ..., x_{n})^{GL_{n}(ℤ/2)} ≅ ℤ/2{1, w_{n}}

as following arguments.

Since x_{i}^{2} = 0, the only possibility of invariant is w_{s}. Suppose s < n. Write

w_{s} = x_{1}(∑_{1≠i_{k}} x_{i_{1}}...x_{i_{s−1}}) + (∑_{1≠i_{k}} x_{i_{1}}...x_{i_{s}}).

Consider the action x_{12} : x_{1} ↦ x_{1} + x_{2} but x_{12} : x_{i} ↦ x_{i} for i > 1. Then

(x_{12}−1)w_{s} = x_{2}(∑_{1≠i_{k}} x_{i_{1}}...x_{i_{s−1}}) = x_{2}x_{3}...x_{s+1}+... ≠ 0 in Λ(x_{1}, ..., x_{n}).

All elements in H^{∗} and e_{n} are really invariants in H^{∗} ⊗ Δ(x_{1}, ..., x_{n}). Thus we have the corollary. □

Let n = 2 and G = SO_{3} or n = 3 and G = G_{2} the exceptional group. Then G has only one conjugacy class A_{n} of maximal elementary abelian 2 groups of rank n. The Weyl group W_{G}(A_{n}) ≅ GL_{n}(ℤ/2). Hence we have the restriction map

Inv^{∗}(G; ℤ/2) → Inv^{∗}(A_{n}; ℤ/2)^{W_{G}(A_{n})} ≅ H^{∗}{1, e_{n}}.

The result in [Ga-Me-Se] shows this map is an isomorphism.

4. Pfister forms

The most important quadratic forms are Pfister forms. Given a = (a_{1}, ..., a_{n}) ∈ (k^{∗}/(k^{∗})^{2})^{×n}, the n-th Pfister form P_{a} is the quadratic form defined as

P_{a} = ⟨⟨a_{1}, ..., a_{n}⟩⟩ = ⟨1, −a_{1}⟩ ⊗ ... ⊗ ⟨1, −a_{n}⟩
= ⊕_{1≤i_{1}<...<i_{s}≤n}⟨(−a_{i_{1}})...(−a_{i_{s}})⟩.

Given a quadratic form q_{a} = ⟨a_{1}, ..., a_{n}⟩, the total Stiefel-Whitney class is given by

w_{t}(q_{a}) = Π(t + a_{i}).

Hence we know

w_{t}(P_{a}) = Π_{ϵ_{i}=0 or 1}(t + ϵ_{1}(ρ + x_{1}) + ... + ϵ_{n}(ρ + x_{n}))

identifying x_{i} = (a_{i}). Hence the following proposition follows the preceding lemma. (Substitute x_{i} + ρ for x_{i} in the right hand side of the euation in Lemma 3.1.)

Proposition 4.1. Let x_{i} = (a_{i}) ∈ k^{∗}/(k^{∗})^{2} and w_{n} = x_{1}...x_{n}. Then

w_{t}(P_{a}) = t^{2^{n}} + (w_{n} + ρ^{n})ρ^{2^{n−1}−n}t^{2^{n−1}}.

Next consider the map from (k^{∗}/(k^{∗})^{2})^{×n} to the set of n-th Pfister forms P fist_{n} defined by

p : a = (a_{1}, ..., a_{n}) ↦ P_{−a} = ⟨⟨ − a_{1}, ..., −a_{n}⟩⟩ = ⟨1, a_{1}⟩ ⊗ ... ⊗ ⟨1, a_{n}⟩.

This map induces the map of cohomological invariants

p^{∗} : Inv^{∗}(P fister_{n}; ℤ/2) → Inv^{∗}((k^{∗}/(k^{∗})^{2})^{×n}; ℤ/2).

Here the last invariant is isomorphic to

Inv^{∗}((ℤ/2)^{n}; ℤ/2) ≅ H^{∗} ⊗ Δ(x_{1}, ..., x_{n}).

On H^{∗} ⊗ Δ(x_{1}, ..., x_{n}), we can define the usual GL_{n}(ℤ/2)-action. This action is also written as follows. Consider the Bruhat decomposition

GL_{n}(ℤ/2) = ∐_{w∈S_{n}} BwB

where B is the Borel group generated by upper triangular matrices, and S_{n} is the n-th symmetric group generated by transition matrices. The group B is generated by x_{ij} = 1+e_{ij} ; the elementary matrix with (i, j) entries 1 with the relations

x_{ij}^{2} = 1, [x_{ij}, x_{kl}] = \begin{cases}x_{il} if j = k \\ 0 otherwise. \end{cases}

Define w(x_{i}) = x_{w(i)} for w ∈ S_{n} and

x_{ij}(x_{i}) = x_{i} + x_{j}, x_{ij}(x_{k}) = x_{k} for i ≠ k.

Then the GL_{n}(ℤ/2)-action is decided on Inv^{∗}((ℤ/2)^{n}; ℤ/2).

Theorem 4.2. The above map p^{∗} induces the isomorphism

Inv^{∗}(P fister_{n}; ℤ/2) ≅ H^{∗} ⊗ Δ(x_{1}, ..., x_{n})^{GL_{n}(ℤ/2)} ≅ H^{∗}{1, e_{n}}.

Proof. On (k^{∗}/(k^{∗})^{2})^{×n}, we can define the GL_{n}(ℤ/2)-action by

x_{ij}(a_{1}, ...a_{n}) = (a_{1}, ..., a_{i−1}, a_{i}a_{j}, a_{i+1}, ..., a_{n}),
w(a_{1}, ..., a_{n}) = (a_{w(1)}, ..., a_{w(n)}).

This induces the action on Inv^{∗}((ℤ/2)^{n}; ℤ/2) by w(x_{i}) = x_{w(i)} and

x_{ij}(x_{i}) = x_{i} + x_{j}, x_{ij}(x_{k}) = x_{k} for i ≠ k.

Define a GL_{n}(ℤ/2) action on P fister_{n} by x_{ij}p(a) = p(x_{ij}(a)). Then this action is invariant, indeed,

x_{12}⟨⟨a_{1}, a_{2}⟩⟩ = px_{12}(−a_{1}, −a_{2}) = p(a_{1}a_{2}, −a_{2})
= ⟨⟨ − a_{1}a_{2}, a_{2}⟩⟩ = ⟨1, a_{1}a_{2}, −a_{2}, −a_{1}a_{2}^{2}⟩
= ⟨1, a_{1}a_{2}, −a_{2}, −a_{1}⟩ = ⟨⟨a_{1}, a_{2}⟩⟩.

Hence we have the map

q^{∗} : Inv^{∗}(P fister_{n}; ℤ/2) → H^{∗} ⊗ Δ(x_{1}, ..., x_{n})^{GL_{n}(ℤ/2)}.

Since P_{−a} ↦ e_{n} = w_{n}(P_{−a}) + ρ^{n} is a cohomology invariant and hence the above map is epic.

For each finitely generated field F over k, the map p : (K^{∗}/(K^{∗})^{2})^{n} → P fister_{n}|_{K} is of course an epimorphism. Hence the induced map

p^{∗}(x) : (K^{∗}/(K^{∗})^{2})^{n} → P fister_{n} →^{x} H^{n}(K; ℤ/2)

is always injective. □

If we consider the map

q : a = (a_{1}, ..., a_{n}) ↦ P_{a} = ⟨⟨a_{1}, ..., a_{n}⟩⟩,

then the map q^{∗} also induces the isomorphism

Inv^{∗}(P fister_{n}; ℤ/2) ≅ H^{∗} ⊗ Δ(x_{1}, ..., x_{n})^{GL_{n}(ℤ/2)_{II}} ≅ H^{∗}{1, w_{n}}

where GL(ℤ/2)_{II} is the unusual action defined by x_{ij}(x_{i}) = ρ+x_{i} +x_{j}.

References

[Ga-Me-Se] S.Garibaldi, A.Merkurjev and J.P.Serre. Cohomological invariants in

Galois cohomology. University lect. series vol(28) AMS (2003).

[Or-Vi-Vo] D.Orlov,A.Vishik and V.Voevodsky. An exact sequence for Milnor’s Ktheory with applications to quadric forms. Ann. of Math. 165 (2007) 1-13.

[To] B. Totaro. The Chow ring of classifying spaces. Proc.of Symposia in Pure Math. ”Algebraic K-theory” (1997:University of Washington,Seattle) 67 (1999), 248-281.

[Vo1] V. Voevodsky. Motivic cohomology with ℤ/2 coefficient. Publ. Math. IHES 98 (2003), 59-104.

[Vo2] V.Voevodsky. Reduced power operations in motivic cohomology. Publ.Math. IHES 98 (2003),1-57.

[Vo3] V.Voevodsky. On motivic cohomology with ℤ/l-coefficients. www.math.uiuc.edu/K-theory/0631 (2003).

Department of mathematics, Faculty of Science, Ryukyu University, Okinawa, Japan, Department of Mathematics, Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan

  E-mail address: tez@sci.u-ryukyu.ac.jp, yagita@mx.ibaraki.ac.jp,