PURITY OF F_{4}–TORSORS WITH TRIVIAL g_{3} INVARIANT

VLADIMIR CHERNOUSOV AND IVAN PANIN

Abstract. We prove that the functor R → 𝒫f _{n}(R) of n-fold Pfister forms satisfies purity for regular local rings containing the field Q of rational numbers. As an application we show that purity holds for F_{4}- torsors with trivial g_{3} invariant.

1. Main results

In the present note we address to the purity conjecture for F_{4}-torsors. The question on purity of torsors was raised in [CTS, Question 6.4, p. 124] by J.-L. Colliot-Thélène and J.-J. Sansuc. Until recently for exceptional groups the answer was known for type G_{2} only (see [ChP]) and our aim is to consider the next open case of groups of type F_{4} with trivial g_{3} invariant.

Let us recall what the purity property for a functor is. Let ℱ be a covariant functor from the category of commutative rings to the category of sets, and let R be a domain with field of fractions K. We say that an element ξ ∈ ℱ (K) is unramified at a prime ideal 𝔓 ⊂ R of height 1 if

ξ ∈ Im [ ℱ (R_{𝔓}) ⟶ ℱ (K) ].

We say that ξ is unramified if it is unramified with respect to all prime ideals in R of height 1. It is clear that

Im [ ℱ (R) → ℱ (K) ] ⊆ ℱ (K)_{ur}

where ℱ (K)_{ur} is the set of all unramified elements. We say that the functor ℱ satisfies purity for a domain R if every ξ ∈ ℱ (K)_{ur} is in the image of ℱ (R), i.e. if

⋂_{ht 𝔓=1}Im [ℱ (R_{𝔓}) → ℱ (K)] = Im [ℱ(R) → ℱ(K)].

In what follows we assume that 2 is invertible in R. We say that a quadratic space over R is an n-fold Pfister space if the corresponding quadratic form is isomorphic to a form

⟨⟨ a_{1}, …, , a_{n} ⟩⟩ = ⟨ 1, −a_{1} ⟩ ⊗ ⋯ ⊗ ⟨ 1, −a_{n} ⟩

Key words and phrases. Linear algebraic group, exceptional groups, torsor, non-abelian cohomology, Pfister forms, purity conjecture.

Supported by the Canada Research Chairs Program, and by NSERC research grant.

Supported by the Presidium of RAS Program “Fundamental Research”, the RFFIgrant, joint DFG-RFBR project 09-01-91333-NNIO-a.

where a_{1}, …, a_{n} are units in R. We will consider the following two functors:

𝒫f_{n} (R) = { isomorphism classes of n-fold Pfister spaces over R }.

and

𝒯 (R) = H_{ét}^{1}(R, G_{0}) = { isomorphism classes of G_{0}-torsors over R }

where G_{0} is a split group of type F_{4}.

Recall that the set of isomorphism classes of group schemes of type F_{4} over R (resp. over K) can be identified in a natural way with H_{ét}^{1}(R, G_{0}) (resp. H^{1}(K, G_{0})). So abusing notation we will identify a group G of type F_{4} with the corresponding cocycle ξ and the isomorphism class [G] with the equivalence class [ξ]. Given such a group G over a field K one can associate [S93], [GMS], [PetRac], [Ro] the cohomological invariants f_{3}(G), f_{5}(G) and g_{3}(G) of G in H^{3}(K, μ_{2}), H^{5}(K, μ_{2}) and H^{3}(K, ℤ/3ℤ) respectively. The group G can be viewed as the automorphism group of a corresponding 27dimensional Jordan algebra J. The invariant g_{3}(G) vanishes if and only if J is reduced, i.e. J has zero divisors.

The main results of this paper are the following purity theorems:

1.1. Theorem. Let R be a regular local ring containing the field Q of rational numbers and let K be its field of fractions. Let [G] ∈ 𝒯 (K)_{ur} be such that g_{3}(G) = 0. Then [G] is in the image of 𝒯 (R) → 𝒯 (K).

1.2. Theorem. The functor R ↦ 𝒫f_{n} (R) satisfies purity for regular local rings containing Q.

1.3. Theorem. Let φ_{m} (resp. φ_{n} ) be an m-fold (resp. an n-fold ) Pfister space over R. If φ_{m,K} is a subform of the form φ_{n,K}, then there exists an (n − m)-fold Pfister space φ_{n−m} over R such that φ_{n} ≅ φ_{m} ⊗ φ_{n−m} over R.

1.4. Remark. For many groups of classical type the purity theorem is known; more precisely it is known for split groups of type A_{n} (unpublished); groups of the form SL_{1,A}, where A is a central simple algebra over a field [CTO]; split groups of type B_{n} [P]; split simply connected groups of type C_{n} (obvious); certain split groups of type D_{n} (like the special orthogonal group of a quadratic form) [P].

1.5. Remark. The characteristic restriction in the theorem is due to the fact that we use the main result in [P] on rationally isotropic quadratic spaces which was proven in characteristic zero only (the resolution of singularities is involved in that proof).

2. Rationally direct summands of a quadratic space

Throughout the paper R denotes a regular local ring containing the field Q of rational numbers. Note that R is a unique factorization domain ([Ma, Theorem 48, page 142]).

PURITY OF F_{4}-TORSORS 3

For the definition and basic properties of quadratic spaces over a commutative ring we refer to [K]. The aim of this section is to establish a criterion when one quadratic space over R can be realized as a direct summand of the second one. That criterion is stated in Proposition 2.2.

Let (V, f) be a quadratic space over R. We denote by f(−, −) a symmetric bilinear form on V corresponding to f.

2.1. Lemma. Let v ∈ V be a vector such that f(v) is a unit in R. Then we have V ≃ ⟨ v ⟩⊥⟨ v ⟩^{⊥}.

Proof. Obviously we have ⟨ v ⟩ ∩ ⟨ v ⟩^{⊥} = 0. So it suffices to show that every vector w ∈ V can be written in the form w = w_{1} + w_{2} where w_{1} ∈ ⟨ v ⟩ and w_{2} ∈ ⟨ v ⟩^{⊥}. Let w_{1} = f(w, v)f(v)^{−1}v. Then w_{2} = w − w_{1} is obviously orthogonal to v, so the result follows. □

2.2. Proposition. Let (V, f) and (W, g) be two quadratic spaces over R. Then g is a direct summand of f if and only if g_{K} is a subform of f_{K}.

Proof. If g is a direct summand of f then obviously g_{K} is a subform of f_{K}. Conversely, assume g_{K} is a subform of f_{K}. Let m = dim g and n = dim f. We argue by induction on m. Consider first the case m = 1. Then g = ⟨ a ⟩ where a ∈ R^{×} and f_{K} represents a. By [P, Cor.2] there exists v ∈ V such that f(v) = a. Clearly that v is unimodular. By Lemma 2.1 we have V = ⟨ v ⟩⊥⟨ v ⟩^{⊥} implying f can be decomposed as f = ⟨ a ⟩⊥f′.

Assume now that our statement is proven for all quadratic spaces of dimension m = i. Let (W, g) be a quadratic space of dimension i + 1. Take a decomposition g = ⟨ a ⟩⊥g′ where g′ is a quadratic form of dimension m − 1 and a is a unit in R. By case m = 1, the quadratic form f can be decomposed as f = ⟨ a ⟩⊥f′. Since g_{K} is a subform of f_{K}, by Witt cancelation we get g′_{K} is a subform of f′_{K}. By induction g′ is a direct summand of f′, so the result follows. □

3. Proofs of Theorems 1.2 and 1.3

Proof of Theorem 1.2. Let K be a quotient field of R and let f_{K} be an n-fold Pfister form over K unramified with respect to every prime ideal in R of height 1. If f_{K} is isotropic, then f_{K} is split over K and there is nothing to prove. So we may assume that f_{K} is anisotropic over K.

By [P, Cor.1] there exists a quadratic space f over R such that its fiber at the generic point of Spec (R) is isomorphic to f_{K}. Our aim is to show by induction that f can be decomposed as f = g_{i}⊥h_{i} where g_{i} = ⟨⟨ −a_{1}, −a_{2}, …, −a_{i} ⟩⟩ is an i-fold Pfister form, h_{i} is a quadratic space over R and i = 1, …, n. In view of dimension argument, taking i = n we have f = g_{n} and Theorem 1.2 follows.

Let first i = 1. Since f_{K} represents 1 over K, by Proposition 2.2 we may write f = ⟨ 1 ⟩⊥f′ for a suitable quadratic form f′ = ⟨ a_{1}, …, a_{n−1} ⟩ over R. Denote g_{1} = ⟨ 1, a_{1} ⟩ = ⟨⟨ −a_{1} ⟩⟩. By our construction g_{1} is a direct summand of f as required. Assume now that f = g_{i}⊥h_{i} where g_{i} = ⟨⟨ −a_{1}, …, −a_{i} ⟩⟩ is an i-fold Pfister form and i < n. Consider a diagonalization h_{i} = ⟨b_{2^{i}+1}, …, b_{2^{n}}⟩. Let g_{i+1} = ⟨⟨ −a_{1}, …, −a_{i}, −a_{i+1} ⟩⟩ where a_{i+1} = b_{2^{i}+1}. Clearly we have f_{K} = g_{i,K} ⊗ q for some (n − i)-fold Pfister form q over K so that h_{i,K} is isomorphic over K to g_{i,K} ⊗ q′ where q′ is the pure subform of q. Since h_{i,K} represents a_{i+1} so is g_{i,K} ⊗ q′. Then by [KMRT, Theorem 1.10], g_{i+1,K} is a subform of f_{K}. By Proposition 2.2 it follows g_{i+1} is a direct summand of f and we are done.

Proof of Theorem 1.3. For m = n there is nothing to prove. Assume that m < n. Write φ_{m} as φ_{m} = ⟨⟨ −a_{1}, …, −a_{m} ⟩⟩. Write φ_{n} as φ_{n} = φ_{m}⊥h_{m}. Consider a diagonalization h_{m} = ⟨b_{2^{m}+1}, …, b_{2^{n}}⟩. Arguing as in the proof of Theorem 1.2 we see that the space φ_{m+1} := ⟨⟨ −a_{1}, …, −a_{m}, −a_{m+1} ⟩⟩ is subspace of the space φ_{n}, where a_{m+1} = b_{2^{m}+1}. Continuing this process we get φ_{n} ≅ ⟨⟨ −a_{1}, …, −a_{m}, …, −a_{n} ⟩⟩ for certain units a_{m+1}, …, a_{n} in R as required.

4. Proof of Theorem 1.1

Let R be a regular local ring containing Q and let K be its field of fractions. Consider a group G of type F_{4} over K unramified with respect to all prime ideals 𝔓 ⊂ R of height 1. It is a twisted form G = ^{ξ}G_{0,K} of a split group G_{0} over R with some cocycle ξ ∈ Z^{1}(K, G_{0,K}).

If T_{0} ⊂ G_{0} is a maximal K-split torus we denote by c ∈ Aut(G_{0}) = G_{0} an element such that c^{2} = 1 and c(t) = t^{−1} for every t ∈ T_{0} (it is known that such an automorphism exists, see e.g. [DG], Exp. XXIV, Prop. 3.16.2, p. 355). Let Σ = Σ(G_{0}, T_{0}) be a root system of G_{0} with respect to T_{0}. Fix its basis { α_{1}, α_{2}, α_{3}, α_{4} }. Denote by α̌_{i} : G_{m} → T_{0}, i = 1, …, 4, the cocharacters dual to α_{1}, …, α_{4}. Clearly, T_{0} can be decomposed as a direct product T_{0} = ∏_{i} Tα̌_{i} where Tα̌_{i} is the image of α̌_{i}.

Let u ∈ R be a unit which is not a square in K. Let S = R(\sqrt{u}) be the corresponding étale quadratic extension of R. Denote the nontrivial involution of S/R by τ.

4.1. Lemma. Let t = ∏_{i} α̌_{i}(u_{i}) ∈ T_{0}(R) where u_{1}, u_{2}, u_{3}, u_{4} are units in R and let a_{τ} = ct. Then λ = (a_{τ}) is a cocycle in Z^{1}(S/R, G_{0}(L)).

Proof. We need to check that a_{τ} τ(a_{τ}) = 1. Indeed, we have

a_{τ} τ(a_{τ}) = ct τ(ct) = ctct = t^{−1}t = 1

as required. □

To complete the proof of Theorem 1.1 we are going to find the parameters u, u_{1}, u_{2}, u_{3}, u_{4} such that the twisted group scheme H = ^{λ}G_{0} has generic fiber isomorphic to G. With this purpose we first remind a criterion when two groups of type F_{4} over K are isomorphic. Note that both H and G have trivial g_{3} invariant. Then by a result of Springer [Sp] (see also [Ch, Theorem 7.1]) we have G and H_{K} are K-isomorphic if and only if f_{3}(G) = f_{3}(H_{K}) and f_{5}(G) = f_{5}(H_{K}). This criterion suggests that the required parameters u, u_{1}, u_{2}, u_{3}, u_{5} must be given in terms of f_{3}(G) and f_{5}(G).

4.2. Proposition. Let G be as above. Then f_{3}(G) and f_{5}(G) are unramified with respect to all prime ideals 𝔓 ⊂ R of height 1.

Proof. The invariants f_{3}, f_{5} are symbols given in terms of the trace quadratic form of the Jordan algebra J corresponding to G and hence we may associate to them the 3-fold and 5-fold Pfister forms. Abusing notation we denote these Pfister forms by the same symbols f_{3}(G) and f_{5}(G).

Let 𝔓 ⊂ R be a prime ideal of height 1 and let v = v_{𝔓} be the corresponding discrete valuation on K. We need to show that f_{3}(G), f_{5}(G) are in the images of 𝒫f_{3} (R_{𝔓})) → 𝒫f_{3} (K) and 𝒫f_{5} (R_{𝔓})) → 𝒫f_{5} (K) where R_{𝔓} is the localization of R at 𝔓. Equivalently, f_{3}(G_{K_{v}}), f_{5}(G_{K_{v}}) are in the images of 𝒫f_{3} (R_{v}) → 𝒫f_{3} (K_{v}) and 𝒫f_{5} (R_{v}) → 𝒫f_{5} (K_{v}) where R_{v} and K_{v} are completions of R and K with respect to v.

To see this, we consider a simple group scheme ℋ_{v} of type F_{4} over R_{v} such that its fiber at the generic point of Spec (R_{v}) is isomorphic to G_{K_{v}}. Note that such a scheme do exists because G is unramified at 𝔓 and hence G_{K_{v}} is also unramified. Since R_{v} is a regular local ring containing Q by [Ch, Theorems 6.1, 6.6, Remark 8.4] we have f_{3}(G_{K_{v}}) and f_{5}(G_{K_{v}}) are of the form (v_{1}) ∪ (v_{2}) ∪ (v_{3}) and (v_{1}) ∪ (v_{2}) ∪ (v_{3}) ∪ (v_{4}) ∪ (u_{5}) respectively where v_{1}, …, v_{5} are units in R_{v}. □

4.3. Proposition. There exist units v_{1}, …, v_{5} ∈ R^{×} such that f_{3}(G) = ⟨⟨ v_{1}, v_{2}, v_{3} ⟩⟩ and f_{5}(G) = ⟨⟨ v_{1}, v_{2}, v_{3}, v_{4}, v_{5} ⟩⟩.

Proof. By Theorem 1.2 and Proposition 4.2 there exist Pfister spaces g = ⟨⟨ v_{1}, v_{2}, v_{3} ⟩⟩ and h = ⟨⟨ w_{1}, w_{2}, w_{3}, w_{4}, w_{5} ⟩⟩ over R such that f_{3}(G) = g_{K} and f_{5}(G) = h_{K}. Here v_{1}, v_{2}, v_{3}, w_{1}, …, w_{5} are units in R. Since f_{3}(G) is a subform of f_{5}(G) Theorem 1.3 shows that there are units v_{3}, v_{4} in R such that h = ⟨⟨ v_{1}, v_{2}, v_{3}, v_{4}, v_{5} ⟩⟩. We are done.

□

We are now in position to complete the proof of Theorem 1.1. Take the cocycle λ from Lemma 4.1 with the parameters u = v_{1}, u_{1} = v_{2}, …, u_{4} = v_{5} and the twisted group scheme H = ^{λ}G_{0} over R. Let τ be the nontrivial automorphism of S = R(\sqrt{u}) over R. Let

{ H_{α_{1}}, …, H_{α_{4}}, X_{α}, α ∈ Σ }

be a Chevalley basis of the Lie algebra of H over S with respect to T_{0}. The twisted action of τ on X_{α_{i}} is given by

τ(X_{α_{i}}) = ct(X_{α_{i}})(ct)^{−1} = c ∏_{j} α̌_{j}(u_{j})(X_{α_{i}}) ∏_{j} α̌_{j}(u_{j}^{−1})c^{−1} = u_{i}X_{−α_{i}},

so that in the terminology of [Ch] the structure constants of H are u_{1}, …, u_{4}. By [Ch, Theorem 6.1 and Theorem 6.6] the f_{3} and f_{5} invariants of H_{K} are ⟨⟨ u, u_{1}, u_{2} ⟩⟩ and ⟨⟨ u, u_{1}, u_{2}, u_{3}, u_{4} ⟩⟩. Thus H_{K} and G have the same f_{3}, f_{5} invariants, hence they are isomorphic. Theorem 1.1 is proven.

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Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Steklov Mathematical Institute at St.Petersburg, Fontanka 27, 191023, Russia

   E-mail address: chernous@math.ualberta.ca
   E-mail address: panin@pdmi.ras.ru