Rationally Isotropic Exceptional Projective

Homogeneous Varieties Are Locally Isotropic

I. Panin, V. Petrov^{1}

Received: September 20, 2014 Revised: May 31, 2015

Abstract. Assume that R is a regular local ring that contains an infinite field and whose field of fractions K has charactertistic ≠ 2. Let X be an exceptional projective homogeneous scheme over R. We prove that in most cases the condition X(K) ≠ ∅ implies X(R) ≠ ∅.

2010 Mathematics Subject Classification: 14M17, 20G35

Keywords and Phrases: projective homogeneous varieties, rational points, exceptional groups

1. Introduction

The main result of the present article extends the main results of [Pa3] and [PP] to the case of exceptional groups. In the latter paper one can find historical remarks which might help the general reader. All the rings in the present paper are commutative and Noetherian. We prove the following theorem.

Theorem 1. Let R be a regular local ring that contains an infinite field and whose field of fractions K has characteristic ≠ 2. Let G be a split simple group of exceptional type (that is, E_{6}, E_{7}, E_{8}, F_{4}, or G_{2}), P be a parabolic subgroup of G, [ξ] be a class from H^{1}(R, G), and X = (G/P)_{ξ} be the corresponding homogeneous space over R. Assume that P ≠ P_{7}, P_{8}, P_{7,8} in case G = E_{8}, P ≠ P_{7} in case G = E_{7}, and P ≠ P_{1} in case G = E_{7}^{ad}. Then the condition X(K) ≠ ∅ implies X(R) ≠ ∅.

The results of the present paper depend on the following yet unpublished results: [FP, Corollary of Theorem 1] and [Pa, Theorem 10.0.30].

1The research is supported by Russian Science Foundation grant 14-11-00456

Documenta Mathematica · Extra Volume Merkurjev (2015) 491–500

2. Purity of some H^{1} functors

Let R be a commutative noetherian domain of finite Krull dimension with a fraction field F. We say that a functor F from the category of commutative R-algebras to the category of sets satisfies purity for R if we have

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

An element a ∈ F(F) is called R-unramified if it belongs to ⋂_{ht 𝔭=1} Im [F(R_{𝔭}) → F(F)]. If 𝔭 is a height one prime ideal in R, the element a is called 𝔭-unramified, if it belongs to Im [F(R_{𝔭}) → F(F)]. If H is an étale group sheaf we write H^{i}(−, H) for H_{ét}^{i}(−, H) below through the text. The following theorem is proven in the characteristic zero case [Pa2, Theorem 4.0.3]. We extend it here to reductive group schemes. Let R be a commutative noetherian ring. Recall that an R-group scheme G is called reductive, if it is affine and smooth as an R-scheme and if, moreover, for each algebraically closed field Ω and for each ring homomorphism R → Ω the scalar extension G_{Ω} is a connected reductive algebraic group over Ω. This definition of a reductive R-group scheme coincides with [SGA, Exp. XIX, Definition 2.7].

Theorem 2. Let R be the local ring of a closed point on a smooth scheme over an infinite field. Let G be a reductive R-group scheme. Let i: Z ↪ G be a closed subgroup scheme of the center Cent(G). It is known that Z is of multiplicative type. Let G′ = G/Z be the factor group, π: G → G′ be the projection. If the functor H^{1}(−, G′) satisfies purity for R, then the functor H^{1}(−, G) satisfies purity for R as well.

It is known that π is surjective and strictly flat. Thus the exact sequence of R-group schemes

(*)  {1} → Z ⟶^{i} G ⟶^{π} G′ → {1}

induces an exact sequence of group sheaves in the fppf-topology.

Lemma 1. Consider the category of R-algebras. The functor

R′ ↦ F(R′) = H_{fppf}^{1}(R′, Z)/ Im (δ_{R′}),

where δ is the connecting homomorphism associated to sequence (*), satisfies purity for R.

Proof. The lemma coincides with [Pa, Theorem 10.0.30]. □

Lemma 2. The map

H_{fppf}^{2}(R, Z) → H_{fppf}^{2}(K, Z)

is injective.

Proof. See [C-TS, Theorem 4.3]. □

Proof of Theorem 2. Reproduce the diagram chase from the proof of [Pa2, Theorem 4.0.3]. For this purpose consider the commutative diagram

\xymatrix{\{1\} \ar[r] & F(K) \ar[r]^{δ_{K}} & H^{1}(K, G) \ar[r]^{π_{K}} & H^{1}(K, G′) \ar[r]^{Δ_{K}} & H_{fppf}^{2}(K, Z) \\ \{1\} \ar[r] & F(R) \ar[r]^{δ} \ar[u] & H^{1}(R, G) \ar[r]^{π} \ar[u] & H^{1}(R, G′) \ar[u] \ar[r]^{Δ} & H_{fppf}^{2}(R, Z) \ar[u]^{α}}

Let [ξ] ∈ H^{1}(K, G) be an R-unramified class and let [ξ̄] = π_{K}([ξ]). Clearly, [ξ̄] ∈ H^{1}(K, G′) is R-unramified. Thus there exists an element [ξ̄′] ∈ H^{1}(R, G′) such that [ξ̄′]_{K} = [ξ̄]. The map α is injective by Lemma 2. One has Δ([ξ̄′]) = 0, since Δ_{K}([ξ̄]) = 0. Thus there exists [ξ′] ∈ H^{1}(R, G) such that π([ξ′]) = [ξ̄′]. Twisting G by ξ′ we may assume that [ξ̄] = ∗, so that [ξ] comes from some a ∈ F(K).

Lemma 3. The above constructed element a ∈ F(K) is R-unramified.

Assume Lemma 3; we use it to complete the proof of Theorem 2. By Lemma 1 the functor F satisfies the purity for regular local rings containing the field k. Thus there exists an element a′ ∈ F(R) with a′_{K} = a. It is clear that [δ(a′)]_{K} = [ξ]. It remains to prove Lemma 3. First we need a small variation of Nisnevich’s theorem.

Lemma 4. Let H be a reductive group scheme over a discrete valuation ring A. Let K be the fraction field of A. Then the map

H^{1}(A, H) → H^{1}(K, H)

is injective.

Proof. Let [ξ_{0}], [ξ_{1}] be classes from H^{1}(A, H). Let H_{0} be a principal homogeneous H-bundle representing the class ξ_{0}. Let H_{0} be the inner form of the group scheme H, corresponding to H_{0}. Let X = Spec(A). For each X-scheme S there is a well-known bijection ϕ_{S}: H^{1}(S, H) → H^{1}(S, H_{0}) of non-pointed sets. That bijection takes the principal homogeneous H-bundle H_{0} ×_{X} S to the trivial principal homogeneous H_{0}-bundle H_{0} ×_{X} S. That bijection is functorial with respect to morphisms of X-schemes. Assume that [ξ_{0}]_{K} = [ξ_{1}]_{K}. Then one has ∗ = ϕ_{K}([ξ_{0}]_{K}) = ϕ_{K}([ξ_{1}]_{K}) ∈ H^{1}(K, H_{0}). The kernel of the map H^{1}(A, H_{0}) → H^{1}(K, H_{0}) is trivial by Nisnevich’s theorem [Ni]. Thus ϕ_{A}([ξ]_{1}) = ∗ = ϕ_{A}([ξ]_{0}) ∈ H^{1}(A, H_{0}). Whence [ξ]_{1} = [ξ]_{0} ∈ H^{1}(A, H). □

Now we go back to the proof of Lemma 3. Consider a height 1 prime ideal 𝔭 in R. Since [ξ] is R-unramified there exists its lift up to an element [ξ̃] in H^{1}(R_{𝔭}, G). The map

H^{1}(R_{𝔭}, G′) → H^{1}(K, G′)

is injective by Lemma 4. But

(π_{𝔭}([ξ̃]))_{K} = π_{K}([ξ]) = ∗,

so π_{𝔭}[ξ̃] = ∗. Therefore there exists a unique class a_{𝔭} ∈ F(R_{𝔭}) such that δ(a_{𝔭}) = [ξ̃] ∈ H^{1}(R_{𝔭}, G). So, δ_{K}(a_{𝔭,K}) = [ξ] ∈ H^{1}(K, G) and finally a = a_{𝔭,K}. Lemma 3 is proven and Theorem 2 is proven as well. □

3. Purity of some H^{1} functors, continued

Theorem 3. Let R be such as in Theorem 1. The functor H^{1}(−, PGL_{n}) satisfies purity for R.

Proof. Let [ξ] ∈ H^{1}(K, PGL_{n}) be an R-unramified element. Let δ: H^{1}(−, PGL_{n}) → H^{2}(−, 𝔾_{m}) be the boundary map corresponding to the short exact sequence of étale group sheaves

1 → 𝔾_{m} → GL_{n} → PGL_{n} → 1.

Let D_{ξ} be a central simple K-algebra of degree n corresponding ξ. If D_{ξ} ≅ M_{l}(D′) for a skew-field D′, then there exists [ξ′] ∈ H^{1}(K, PGL_{n′}) such that D′ = D_{ξ′}. Then δ([ξ′]) = [D′] = [D] = δ(ξ). Replacing ξ by ξ′, we may assume that D := D_{ξ} is a central skew-field over K of degree n and the class [D] is R-unramified. Since the functor H^{2}(−, 𝔾_{m}) satisfies purity for R, there exists an Azumaya R-algebra A and an integer d such that A_{K} = M_{d}(D). There exists a projective left A-module P of finite rank such that each projective left A-module Q of finite rank is isomorphic to the left A-module P^{m} for an appropriative integer m (see [DeM, Cor.2]). In particular, two projective left A-modules of finite rank are isomorphic if they have the same rank as Rmodules. One has an isomorphism A ≅ P^{s} of left A-modules for an integer s. Thus one has R-algebra isomorphisms A ≅ End_{A}(P^{s}) ≅ M_{s}(End_{A}(P)). Set B = End_{A}(P). Observe, that B_{K} = End_{A_{K}} (P_{K}), since P is a finitely generated projective left A-module. The class [P_{K}] is a free generator of the group K_{0}(A_{K}) = K_{0}(M_{d}(D)) ≅ ℤ, since [P] is a free generator of the group K_{0}(A) and K_{0}(A) = K_{0}(A_{K}). The P_{K} is a simple A_{K}-module, since [P_{K}] is a free generator of K_{0}(A_{K}). Thus End_{A_{K}} (P_{K}) = B_{K} is a skew-field. We claim that the K-algebras B_{K} and D are isomorphic. In fact, A_{K} = M_{r}(B_{K}) for an integer r, since P_{K} is a simple A_{K}-module. From the other side A_{K} = M_{d}(D). As D, so B_{K} are skew-fields. Thus r = d and D is isomorphic to B_{K} as K-algebras. We claim further that B is an Azumaya R-algebra. That claim is local with respect to the étale topology on Spec(R). Thus it suffices to check the claim assuming that Spec(R) is strictly henselian local ring. In that case A = M_{l}(R) and P = (R^{l})^{m} as an M_{l}(R)-module. Thus B = End_{A}(P) = M_{m}(R), which proves the claim. Since B_{K} is isomorphic to D, one has m = n. So, B is an Azumaya R-algebra, and the K-algebra B_{K} is isomorphic to D. Let [ζ] ∈ H^{1}(R, PGL_{n}) be class corresponding to B. Then [ζ]_{K} = [ξ], since δ([ζ])_{K} = [B_{K}] = [D] = δ([ξ]) ∈ H^{2}(K, 𝔾_{m}). □

We denote by Sim_{n} the group of similitudes of a split quadratic form of rank n and by Sim_{n}^{+} its connected component. Recall that H^{1}(−, Sim_{n}) classifies similarity classes of nondegenerate quadratic forms of rank n (see [KMRT, (29.15)]).

Theorem 4. Let R be such as in Theorem 1. The functor H^{1}(−, Sim_{n}) satisfies purity for R.

Proof. Let [ξ] ∈ H^{1}(K, Sim_{n}) be an R-unramified element. Let φ be a quadratic form over K whose similarity class represents [ξ]. Diagonalizing φ we may assume that φ = ∑_{i=1}^{n} f_{i} · t_{i}^{2} for certain non-zero elements f_{1}, f_{2}, …, f_{n} ∈ K. For each i write f_{i} in the form f_{i} = \frac{g_{i}}{h_{i}} with g_{i}, h_{i} ∈ R and h_{i} ≠ 0. There are only finitely many height one prime ideals 𝔮 in R such that there exists 0 ≤ i ≤ n with f_{i} not in R_{𝔮}. Let 𝔮_{1}, 𝔮_{2}, …, 𝔮_{s} be all height one prime ideals in R with that property and let 𝔮_{i} ≠ 𝔮_{j} for i ≠ j. For all other height one prime ideals 𝔭 in R each f_{i} belongs to the group of units R_{𝔭}^{×} of the ring R_{𝔭}. If 𝔭 is a height one prime ideal of R which is not from the list 𝔮_{1}, 𝔮_{2}, …, 𝔮_{s}, then φ = ∑_{i=1}^{n} f_{i} · t_{i}^{2} may be regarded as a quadratic space over R_{𝔭}. We will write _{𝔭}φ for that quadratic space over R_{𝔭}. Clearly, one has (_{𝔭}φ) ⊗_{R_{𝔭}} K = φ as quadratic spaces over K. For each j ∈ {1, 2, …, s} choose and fix a quadratic space _{j}φ over R_{𝔮_{j}} and a non-zero element λ_{j} ∈ K such that the quadratic spaces (_{j}φ) ⊗_{R_{𝔮_{j}}} K and λ_{j} · φ are isomorphic over K. The ring R is factorial since it is regular and local. Thus for each j ∈ {1, 2, …, s} we may choose an element π_{j} ∈ R such that firstly π_{j} generates the only maximal ideal in R_{𝔮_{j}} and secondly π_{j} is an invertible element in R_{𝔫} for each height one prime ideal 𝔫 different from the ideal 𝔮_{j}. Let v_{j}: K^{×} → ℤ be the discrete valuation of K corresponding to the prime ideal 𝔮_{j}. Set λ = ∏_{i=1}^{s} π_{j}^{v_{j}(λ_{j})} and

φ_{new} = λ · φ.

Claim. The quadratic space φ_{new} is R-unramified. In fact, if a height one prime ideal 𝔭 is different from each of 𝔮_{j}’s, then v_{𝔭}(λ) = 0. Thus, λ ∈ R_{𝔭}^{×}. In that case λ · (_{𝔭}φ) is a quadratic space over R_{𝔭} and moreover one have isomorphisms of quadratic spaces (λ · (_{𝔭}φ)) ⊗_{R_{𝔭}} K = λ · φ = φ_{new}. If we take one of 𝔮_{j}’s, then \frac{λ}{λ_{j}} ∈ R_{𝔮_{j}}^{×}. Thus, \frac{λ}{λ_{j}} · (_{j}φ) is a quadratic space over R_{𝔮_{j}}. Moreover, one has

\frac{λ}{λ_{j}} · (_{j}φ) ⊗_{R_{𝔮}} K = \frac{λ}{λ_{j}} · λ_{j} · φ = φ_{new}.

The Claim is proven. By [PP, Corollary 3.1] there exists a quadratic space φ̃ over R such that the quadratic spaces φ̃⊗_{R} K and φ_{new} are isomorphic over K. This shows that the similarity classes of the quadratic spaces φ̃ ⊗_{R} K and φ coincide. The theorem is proven. □

Theorem 5. Let R be such as in Theorem 1. The functor H^{1}(−, Sim_{n}^{+}) satisfies purity for R.

Proof. Consider an element [ξ] ∈ H^{1}(K, Sim_{n}^{+}) such that for any 𝔭 of height 1 [ξ] comes from [ξ_{𝔭}] ∈ H^{1}(R_{𝔭}, Sim_{n}^{+}). Then the image of [ξ] in H^{1}(K, Sim_{n}) by Theorem 4 comes from some [ζ] ∈ H^{1}(R, Sim_{n}). We have a short exact sequence

1 → Sim_{n}^{+} → Sim_{n} → μ_{2} → 1,

and R^{×}/(R^{×})^{2} injects into K^{×}/(K^{×})^{2}. Thus the element [ζ] comes actually from some [ζ′] ∈ H^{1}(R, Sim_{n}^{+}). It remains to show that the map

H^{1}(K, Sim_{n}^{+}) → H^{1}(K, Sim_{n})

is injective, or, by twisting, that the map

H^{1}(K, Sim^{+}(q)) → H^{1}(K, Sim(q))

has trivial kernel. The latter follows from the fact that the map

Sim(q)(K) → μ_{2}(K)

is surjective (indeed, any reflection goes to −1 ∈ μ_{2}(K)). □

4. Proof Theorem 1

Till the end of the proof of Lemma 9 we suppose that R is the local ring of a closed point on a smooth scheme over an infinite field. Let [ξ] be a class from H^{1}(R, G), and X = (G/P)_{ξ} be the corresponding homogeneous space. Denote by L a Levi subgroup of P.

Lemma 5. Consider a parabolic subgroup P_{1} in PGO_{n}^{+}, which is the stabilizer of an isotropic line. A Levi subgroup of P_{1} is isomorphic to Sim_{n−2}^{+}.

Proof. Is is clear from the matrix representation that a Levi subgroup of a parabolic subgroup P_{1} in O_{n}^{+} is isomorphic to O_{n−2}^{+} ×𝔾_{m}. Now the homomorphism

O_{n−2}^{+} ×𝔾_{m} → Sim_{n−2}^{+}

induced by the natural inclusions is surjective in the sense of groups schemes, and its kernel is μ_{2}. The claim follows. □

Recall that a subset Ψ of a root system Φ is called closed if for any α, β ∈ Ψ such that α + β ∈ Φ we have α + β ∈ Ψ.

Lemma 6. Let L modulo its center be isomorphic to PGO_{2m}^{+} (resp., PGO_{2m+1}^{+} or PGO_{2m}^{+} × PGL_{2}). Denote by Φ the root system of G with respect to T, and by Ψ the root system of L with respect to T, where T is a maximal split torus in L. Assume that there is a root λ ∈ Φ such that the smallest closed set of roots Ψ′ containing Ψ and ±λ is a root subsystem of type D_{m+1} (resp. B_{m+1} or D_{m+1} + A_{1}), and Ψ is the standard subsystem of type D_{m} (resp.

B_{m} or D_{m} + A_{1}) therein. Then there is a surjective map L → Sim_{2m}^{+} (resp., L → Sim_{2m+1}^{+} or L → Sim_{2m}^{+} × PGL_{2}) whose kernel is a central closed subgroup scheme in L. In particular, the functor H^{1}(−, L) satisfies purity for R.

Proof. Consider the subgroup H_{Ψ′} of G corresponding to Ψ′ in the sense of [SGA, Exp. XXII, Definition 5.4.2]. Then H_{Ψ′} is split reductive of type D_{m+1} (resp. B_{m+1} or D_{m+1} + A_{1}) by [SGA, Exp. XXII, Proposition 5.10.1], so it maps onto the split adjoint group of the same type. Under this map L maps onto a Levi subgroup of a parabolic subgroup P_{1}, which is isomorphic to Sim_{2m}^{+} (resp. Sim_{2m+1}^{+} or Sim_{2m}^{+} × PGL_{2}) by Lemma 5. The purity claim follows from Theorem 5, Theorem 3 and Theorem 2. □

Lemma 7. For any semi-local R-algebra S the map

H^{1}(S, L) → H^{1}(S, G)

is injective. Moreover, X(S) ≠ ∅ if and only if [ξ]_{S} comes from H^{1}(S, L).

Proof. See [SGA, Exp. XXVI, Cor. 5.10]. □

Lemma 8. Assume that the functor H^{1}(−, L) satisfies purity for R. Then X(K) ≠ ∅ implies X(R) ≠ ∅.

Proof. By Lemma 7 [ξ]_{K} comes from some [ζ] ∈ H^{1}(K, L), which is uniquely determined. Since X is smooth projective, for any prime ideal 𝔭 of height 1 we have X(R_{𝔭}) ≠ ∅. By Lemma 7 ξ_{R_{𝔭}} comes from some [ζ_{𝔭}] ∈ H^{1}(R_{𝔭}, L). Now [ζ_{𝔭}]_{K} = [ζ], and so by the purity assumption there is [ζ′] ∈ H^{1}(R, L) such that [ζ′]_{K} = [ζ]. Set [ξ′] to be the image of ζ′ in H^{1}(R, G). We claim that [ξ′] = [ξ]. Indeed, by the construction [ξ′]_{K} = [ξ]_{K}. It remains to recall that the map H^{1}(R, G) → H^{1}(K, G_{K}) is injective by [FP, Corollary of Theorem 1]. □

Lemma 9. Let Q ≤ P be another parabolic subgroup, Y = (G/Q)_{ξ}. Assume that X(K) ≠ ∅ implies Y (K) ≠ ∅, and Y (K) ≠ ∅ implies Y (R) ≠ ∅. Then X(K) ≠ ∅ implies X(R) ≠ ∅.

Proof. Indeed, there is a map Y → X, so Y (R) ≠ ∅ implies X(R) ≠ ∅. □

Proof of Theorem 1. We first suppose that R is the local ring of a closed point on a smooth scheme over an infinite field. By Lemma 9 we may assume that P_{K} is a minimal parabolic subgroup of (G_{ξ})_{K}. All possible types of such P_{K} are listed in [T, Table II]: the Dynkin diagram with circled vertices erased corresponds to the type of L. We show case by case that H^{1}(−, L) satisfies purity for R, hence we are in the situation of Lemma 8. If P = B is the Borel subgroup, obviously H^{1}(S, L) = {∗} for any semi-local R-algebra S. In the case of index E_{7,4}^{9} (resp. ^{1}E_{6,2}^{16}) L modulo its center is isomorphic to PGL_{2} × PGL_{2} × PGL_{2} (resp. PGL_{3} × PGL_{3}), and we may apply Theorem 2 and Theorem 3. In the all other cases we provide an element λ ∈ X^{∗}(T) such that the assumption of Lemma 6 holds (α̃ stands for the maximal root, enumeration follows [B]). The indices E_{7,1}^{78}, E_{8,1}^{133} and E_{8,2}^{78} are not in the list below since in those cases the L does not belong to one of the type D_{m}, B_{m}, D_{m} × A_{1}. The index E_{7,1}^{66} is not in the list below since in that case we need a weight λ which is not in the root lattice. So, the indices E_{7,1}^{78}, E_{8,1}^{133}, E_{8,2}^{78} and E_{7,1}^{66} are the exceptions in the statement of the Theorem.

\begin{matrix}Index & ^{1}E_{6,2}^{28} & E_{7,1}^{48} & E_{7,2}^{31} & E_{7,3}^{28} & E_{8,1}^{91} & E_{8,2}^{66} & E_{8,4}^{28} & F_{4,1}^{21} \\ λ & α_{1} & −α̃ & α_{1} & α_{1} & −α̃ & α_{8} & α_{1} & −α̃ \end{matrix}

It remains to settle the case P = P_{1} for G = E_{7}^{sc}. Denote by Ẽ_{7} a Levi subgroup of a parabolic subgroup P_{8} in E_{8}. Comparing the exact sequences

H^{1}(R, E_{7}^{sc}) → H^{1}(R, E_{7}^{ad}) → H^{2}(R, μ_{2})

and

H^{1}(R, Ẽ_{7}^{sc}) → H^{1}(R, E_{7}^{ad}) → H^{2}(R, 𝔾_{m})

and one sees that the image of [ξ] in H^{1}(R, E_{7}^{ad}) comes from some [ζ] ∈ H^{1}(R, Ẽ_{7}). Let P̃_{1} denote the corresponding parabolic subgroup in Ẽ_{7}; then we have (E_{7}^{sc}/P_{1})_{ξ} ≃ (Ẽ_{7}/P̃_{1})_{ζ}. We claim that H^{1}(−, L̃) satisfies purity for R, where L̃ is a Levi subgroup of P̃_{1}. Indeed, consider a Levi subgroup G′ of a parabolic subgroup P_{1} inside E_{8}; then G′ has type D_{7} and L̃ is a Levi subgroup of a parabolic subgroup P_{1} in G′. The rest of the proof goes exactly the same way as in Lemma 6. Now suppose that R is a regular local ring containing an infinite field k. We first prove a general lemma. Let k′ be an infinite field, X be a k′-smooth irreducible affine variety, Denote by k′[X] the ring of regular functions on X and by k′(X) the field of rational functions on X. Let 𝔭 be prime ideal in k′[X], and let O_{𝔭} be the corresponding local ring.

Lemma 10. Theorem 1 holds for the local ring O_{𝔭}.

Proof. Choose a maximal ideal 𝔪 ⊂ k′[X] containing 𝔭. One has inclusions of k′-algebras O_{𝔪} ⊂ O_{𝔭} ⊂ k′(X). We already proved Theorem 1 for the ring O_{𝔪}. Thus Theorem 1 holds for the ring O_{𝔭}. □

The rest of the proof of Theorem 1 follows the arguments from [FP, page 5], which we reproduce here. Namely, let 𝔪 be the maximal ideal of R. Let k′ be the algebraic closure of the prime field of R in k. Note that k′ is perfect. It follows from Popescu’s theorem ([P, Sw]) that R is a filtered inductive limit of smooth k′-algebras R_{α}. Modifying the inductive system R_{α} if necessary, we can assume that each R_{α} is integral. There are an index α, a 1-cocycle ξ_{α} ∈ Z^{1}(R_{α}, G), and an element f_{α} ∈ R_{α} such that ξ = φ_{α}(ξ_{α}), f is the image of f_{α} under the homomorphism ϕ_{α} : R_{α} → R, the homogeneous space X_{α} := (G/H)_{ξ_{α}} over R_{α} has a section over (R_{α})_{f_{α}}. If the field k′ is infinite, then set 𝔭 = ϕ_{α}^{−1}(𝔪). The homomorphism ϕ_{α} induces a homomorphism of local rings (R_{α})_{𝔭} → R. By Lemma 10 one has X_{α}(R_{α}) ≠ ∅, whence X(R) ≠ ∅.

If the field k′ is finite, then k contains an element t transcendental over k′. Thus R contains the subfield k′(t) of rational functions in the variable t. So, if R′_{α} := R_{α}⊗_{k′} k′(t), then ϕ_{α} can be decomposed as follows R_{α} ⟶^{i_{α}} R_{α}⊗_{k′} k′(t) = R′_{α} ⟶^{ψ_{α}} R. Let ξ′ = i_{α}(ξ_{α}), f′_{α} = f_{α} ⊗ 1 ∈ R′_{α}, then the homogeneous space X′_{α} := (G/H)_{ξ′_{α}} over R′_{α} has a section over (R′_{α})_{f′_{α}}. Let 𝔮 = ψ_{α}^{−1}(𝔪). The ring R′_{α} is a k′(t)-smooth algebra over the infinite field k′(t), and the homogeneous space X′_{α} := (G/H)_{ξ′_{α}} over R′_{α} has a section over (R′_{α})_{f′_{α}}. By Lemma 10 one has X′_{α}((R′_{α})_{𝔮}) ≠ ∅. The homomorphism ψ_{α} can be factored as R′_{α} → (R′_{α})_{𝔮} → R. Thus X(R) ≠ ∅. □

The authors heartily thank Anastasia Stavrova for discussion on the earlier version of this work.

References

[B] N. Bourbaki, Groupes et algèbres de Lie. Chapitres 4, 5 et 6, Masson, Paris, 1981.

[C-TS] J.-L. Colliot-Thélǹe, J.-J. Sansuc, Principal homogeneous spaces un-

der flasque tori: applications, J. Algebra 106 (1987), 148–205.

[SGA] M. Demazure, A. Grothendieck, Schémas en groupes, Lecture Notes in Mathematics, Vol. 151–153, Springer-Verlag, Berlin-Heidelberg-New York, 1970.

[DeM] F.R. DeMeyer, Projective modules over central separable algebras, Canad. J. Math. 21 (1969), 39–43.

[Gr2] A. Grothendieck. Le group de Brauer II, in Dix exposés sur la cohomologique de schémes, Amsterdam,North-Holland, 1968.

[Gr3] A. Grothendieck. Le group de Brauer III: Exemples et compl’ements, in Dix exposés sur la cohomologique de schémes, Amsterdam,North-Holland, 1968.

[FP] R. Fedorov, I. Panin, A proof of Grothendieck–Serre conjecture on principal bundles over a semilocal regular ring containing an infinite field, Preprint, April 2013, http://www.arxiv.org/abs/1211.2678v2.

[KMRT] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The Book of Involutions, AMS Colloquium Pub. 44, Providence, RI, 1998.

[Ni] Y. Nisnevich, Rationally Trivial Principal Homogeneous Spaces and Arithmetic of Reductive Group Schemes Over Dedekind Rings, C. R. Acad. Sci. Paris, Série I, 299, no. 1, 5–8 (1984).

[Pa] I. Panin, On Grothendieck—Serre’s conjecture concerning principal G-bundles over reductive group schemes:II, Preprint (2013), http://www.math.org/0905.1423v3.

[Pa2] I. Panin, Purity conjecture for reductive groups, in Russian, Vestnik SPbGU ser. I, no. 1 (2010), 51–56.

[Pa3] I. Panin, Rationally isotropic quadratic spaces are locally isotropic, Invent. math. 176 (2009), 397-403.

[PP] I. Panin, K. Pimenov, Rationally Isotropic Quadratic Spaces Are Locally Isotropic: II, Documenta Mathematica, Vol. Extra Volume: 5. Andrei A. Suslin’s Sixtieth Birthday, P. 515-523, 2010.

[P] D. Popescu. General Néron desingularization and approximation, Nagoya Math. Journal, 104 (1986), 85–115.

[Sw] R.G. Swan. Néron-Popescu desingularization, Algebra and Geometry (Taipei, 1995), Lect. Algebra Geom. Vol. 2, Internat. Press, Cambridge, MA, 1998, pp. 135–192.

[T] J. Tits, Classification of algebraic semisimple groups, Algebraic groups and discontinuous subgroups, Proc. Sympos. Pure Math., 9, Amer. Math. Soc., Providence RI, 1966, 33–62.

I. Panin, V. Petrov

St. Petersburg Department of the V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences 27 Fontanka St. Petersburg Russia paniniv@gmail.com victorapetrov@gmail.com