Rationally trivial quadratic spaces are locally trivial:III

I.Panin^{∗}, K.Pimenov ^{†}

25.12.2012

Abstract

It is proved the following. Let R be a regular semi-local domain containing a field such that all the residue fields are infinite. Let K be the fraction field of R. If a quadratic space (R^{n}, q : R^{n} → R) over R is isotropic over K, then there is a unimodular vector v ∈ R^{n} such that q(v) = 0. If char(R) = 2, then in the case of even n we assume that q is a non-singular space in the sense of [Kn] and in the case of odd n > 2 we assume that q is a semi-regular in the sense of [Kn].

1 Introduction

Let k be an infinite field,possibly char(k) = 2, and let X be a k-smooth irreducible affine scheme,let x_{1}, x_{2}, …, x_{s} ∈ X be closed points. Let P be a free k[X]-module of rank n > 0. If n is odd, then let (P, q : P → k[X]) be a semi-regular quadratic module over k[X] in the sense of [Kn, Ch.IV, §3]. If n is even, then let (P, q : P → k[X]) be a non-singular quadratic space in the sense of [Kn, Ch.I, (5.3.5))]. (In both cases it is equivalent of saying that the X-scheme Q := {q = 0} ⊂ P_{X}^{n−1} is smooth over X).

Let p : Q → X be the projection. For a nonzero element g ∈ k[X] let Q_{g} = p^{−1}(X_{g}). Let U = Spec(𝒪_{X,{x_{1},x_{2},...,x_{s}}}). Set _{U}Q = U ×_{X} Q. For a k-scheme D equipped with k-morphisms U ← D and D → X_{g} set _{D}Q = _{U}Q ×_{U} D and Q_{D,g} = D ×_{X_{g}} Q_{g}.

1.0.1 Proposition. If n > 1, then there exists a finite surjective étale k-morphism U ← D of odd degree, a morphism D → X_{g} and an isomorphism of the D-schemes _{D}Q ⟵^{Φ̄} Q_{D,g}.

Given this Proposition we may prove the following Theorem

1.0.2 Theorem (Main). Assume that g ∈ k[X] is a non-zero element such that there is a section s : X_{g} → Q of the projection Q_{g} → X_{g}. Then there is a section s_{U} : U → _{U}Q of the projection _{U}Q → U.

∗The research of the first author is partially supported by RFBR-grant 10-01-00551-a

†The research of the second author is partially supported by RFBR grant 12-01-33057 ”Motivic homotopic cohomology theories on algebraic varieties”

Proof of Main Theorem. We will give a proof of the Theorem only in the local case and left to the reader the semi-local case. So, s = 1 and we will write x for x_{1} and 𝒪_{X,x} for 𝒪_{X,{x_{1}}}. If g ∈ k[X] − m_{x}, then there is nothing to prove. Now let g ∈ m_{x} then by Proposition 1.0.1 there is a a finite surjective étale k-morphism U ← D of odd degree, a morphism D → X_{g} and an isomorphism of the D-schemes _{D}Q ⟵^{Φ̄} Q_{D,g}.

The section s defines a section s_{D} = (id, s) : D → Q_{D,g} of the projection Q_{D,g} → D. Further Φ̄∘s_{D} : D → _{D}Q is a section of the projection_{D}Q → D. Finally, if p_{1} : _{D}Q → _{U}Q is the projection, then p_{1} ∘ Φ̄ ∘ s_{D} : D → _{U}Q is a U-morphism of U-schemes. Recall that U ← D is a a finite surjective étale k-morphism of odd degree and U is local with an infinite residue field. Whence by a variant of Springer’s theorem proven in [PR] there is a section s_{U} : U → _{U}Q of the projection _{U}Q → U. (If char(k)=2 the proof a variant of Springer’s theorem given in [PR] works well with a very mild modification). The Theorem is proven.

The Main Theorem has the following corollaries

1.0.3 Corollary (Main1). Let 𝒪_{X,{x_{1},x_{2},...,x_{s}}} be the semi-local ring as above and let k(X) be the rational function field on X. Let P be a free 𝒪_{X,{x_{1},x_{2},...,x_{s}}}-module of rank n > 1 and q : P → 𝒪_{X,{x_{1},x_{2},...,x_{s}}} be a form over 𝒪_{X,{x_{1},x_{2},...,x_{s}}} as above, that is the 𝒪_{X,{x_{1},x_{2},...,x_{s}}}- scheme Q := {q = 0} ⊂ P_{𝒪_{X,x}}^{n−1} is smooth over 𝒪_{X,x}. If the equation q = 0 has a non-trivial solution over k(X), then it has a unimodular solution over 𝒪_{X,{x_{1},x_{2},...,x_{s}}}.

1.0.4 Corollary (Main2). Let R be a semi-local regular domain containing a field and R is such that all the residue fields are infinite. Let K be the fraction field of R. Let P be a free R-module of rank n > 1 and q : P → R be a quadratic form over R such that the R-scheme Q := {q = 0} ⊂ P_{R}^{n−1} is smooth over R. If the equation q = 0 has a non-trivial solution over K, then it has a unimodular solution over R.

1.0.5 Corollary (Main3). Let R be a semi-local regular domain containing a field and R is such that all the residue fields are infinite. Let K be the fraction field of R. Let P be a free R-module of even rank n > 0 and q : P → R be a quadratic form over R such that the R-scheme Q := {q = 0} ⊂ P_{R}^{n−1} is smooth over R. Let u ∈ R^{×} be a unit. If u is represented by q over K, then u is represented by q already over R.

If 1/2 ∈ R, then the same holds for a quadratic space of an arbitrary rank.

Proof of Proposition 1.0.1. The following Lemma is a corollary from Lemma 2.2.1 and Proposition 3.1.7. from [Kn]

1.0.6 Lemma. For n > 1 there exists an affine open subset X^{0} containing x and a Galois étale cover X̃^{0} ⟶^{π} X^{0} such that the k[X^{0}]-module P ⊗_{k[X]} k[X^{0}] coincides with k[X^{0}]^{n} and π^{∗}(q) is proportional to the quadratic space ⊥_{i=1}^{m}T_{i}T_{i+m} in the case n = 2m and is proportional to the semi-regular quadratic module ⊥_{i=1}^{m}T_{i}T_{i+m} ⊥ T_{n}^{2} in the case n = 2m + 1.

By this Lemma we may and will assume that P = k[X]^{n} and that we are given with a Galois étale cover π : X̃ ⟶^{π} X such that the quadratic space π^{∗}(q) is proportional to a split quadratic space. Let Γ be the Galois group of X̃ over X. Let Ũ = π^{−1}(U) ⊂ X̃.

Let \overline{U × X} := (Ũ × X̃)/Δ(Γ). Clearly, U × X = (Ũ × X̃)/(Γ × Γ). Let ρ : \overline{U × X} → U × X be the obvious map.

Let p_{2} : U × X → X be projection to X and p_{1} : U × X → U be the projection to U. The quadratic spaces p_{1}^{∗}(q) and p_{2}^{∗}(q) over U × X are not proportional in general. However the following Proposition holds (see Appendix, Lemma 2.0.8)

1.0.7 Proposition. The quadratic spaces ρ^{∗}(p_{2}^{∗}(q)) and ρ^{∗}(p_{1}^{∗}(q)) are proportional.

Further by [PSV, Prop. 3.3, Prop. 3.4] and [PaSV] we may find an open X′ in X containing x and an open affine S ⊂ P^{d−1} (d=dim(X)) and a smooth morphism f′ : X′ → S making X′ into a smooth relative curve over S with the geometrically irreducible fibres. Moreover we may find f′ such that f′|_{X′∩Z} : Z′ = X′ ∩ Z → S is finite, where Z is the vanishing locus of g ∈ k[X]. Moreover f′ can be written as pr_{S} ∘ Π′ = f′, where Π′ : X′ → A^{1} × S is a finite surjective morphism. Set X̃′ = π^{−1}(X′). Replacing notation we write X for X′, X̃ for X̃′, Z for Z′, f : X → S for f′ : X′ → S, Π : X → A^{1} × S for Π′ : X′ → A^{1} × S.

Let \overline{U ×_{S} X} := (Ũ ×_{S} X̃)/Δ(Γ). Clearly, U ×_{S} X = (Ũ ×_{S} X̃)/(Γ × Γ). Let

ρ_{S} : \overline{U ×_{S} X} → U ×_{S} X

be the obvious map.

Let p_{X} : U ×_{S} X → X be projection to X and p_{U} : U ×_{S} X → U be the projection to U. By Proposition 1.0.7 the quadratic spaces ρ_{S}^{∗}(p_{X}^{∗}(q)) and ρ_{S}^{∗}(p_{U}^{∗}(q)) are ...

Now the pull-back of Π be means of the morphism U ↪ X → S defines a finite surjective morphism Θ : U ×_{S} X → A^{1} × U. So, Θ ∘ ρ_{S} : \overline{U ×_{S} X} → A^{1} × U is a finite surjective morphism of U-schemes. The U-scheme \overline{U ×_{S} X} is smooth over U since U ×_{S} X is smooth over U and ρ_{S} is étale. The subscheme Δ(Ũ)/Δ(Γ) ⊂ \overline{U ×_{S} X} projects isomorphically onto U. So, we are given with a section Δ̃ of the morphism

\overline{U ×_{S} X} ⟶^{ρ_{S}} U ×_{S} X ⟶^{p_{U}} U.

The recollection from the latter paragraph shows that we are under the hypotheses of Lemma 3.0.9 from Appendix B for the relative U-curve 𝒳 := \overline{U ×_{S} X} and its closed subset 𝒵 := ρ_{S}^{−1}(U ×_{S} Z). (If to be more accurate, then one should take the connected component 𝒳^{c} of 𝒳 containing Δ̃(U) and the closed subset 𝒵 ∩ 𝒳^{c} of 𝒳^{c} ).

By Lemma 3.0.9 there exists an open subscheme 𝒳^{0} ↪ 𝒳 and a finite surjective morphism α : 𝒳^{0} → A^{1} × U such that α is étale over 0 × U and 1 × U and α^{−1}(0 × U) = Δ̃(U) ∐ D_{0}. Moreover if we define D_{1} as α^{−1}(1 × U), then D_{1} ∩ Z = ∅ and D_{0} ∩ Z = ∅. One has [D_{1} : U] = [D_{0} : U] + 1. Thus either [D_{1} : U] is odd or [D_{0} : U] is odd.

Assume [D_{1} : U] is odd. Then the morphism 1 × U ⟵^{α|_{D_{1}}} D_{1}, the morphism D_{1} ⟶^{p_{X}∘ρ_{S}} X − Z and the isomorphism Φ̄ := Φ|_{D_{1}} satisfy the conclusion of the Proposition 1.0.1 (here Φ is from the Proposition 1.0.7). The Proposition is proven.

2 Appendix A: Equating Lemma

Let k be a field, X be a k-smooth affine scheme, G be a reductive k-group, 𝒢/X be a principle G-bundle over X. Let π : X̃ → X be a finite étale Galois cover of X with a Galois group Γ and let s : X̃ → 𝒢 be an X-scheme morphism (in other words 𝒢 splits over X̃). Let \overline{X × X} := (X̃ × X̃)/Δ(Γ). Clearly, X × X = (X̃ × X̃)/(Γ × Γ). Let π : \overline{X × X} → X × X be the obvious map. Observe that the map X̃ × X̃ → \overline{X × X} is an étale Galois cover with the Galois group Γ.

Let q_{i} : X̃ × X̃ → X̃ be projection to the i-th factor and let p_{i} : X × X → X be projection to the i-th factor. The principal G bundles 𝒢_{1} := p_{1}^{∗}(𝒢) and 𝒢_{2} := p_{2}^{∗}(𝒢) over X × X are not isomorphic in general. However the following Proposition holds

2.0.8 Lemma. The principal G-bundles π^{∗}(𝒢_{1}) and π^{∗}(𝒢_{2}) are isomorphic and moreover there is such an isomorphism Φ : π^{∗}(𝒢_{1}) → π^{∗}(𝒢_{2}) that the restriction of Φ to the subscheme X = Δ(X̃)/(Γ) ⊂ \overline{X × X} is the identity isomorphism.

Proof. The morphism s : X̃ → 𝒢 gives rise to a 1-cocycle a : Γ → G(X̃) defined as follows: given γ ∈ Γ consider the composition s ∘ γ and set a_{γ} ∈ G(X̃) to be a unique element with a_{γ} · s = s ∘ γ in G(X̃).

It’s straight forward to check that the 1-cocycle corresponding to the principal G bundle π^{∗}(𝒢_{1}) and the morphism X̃ × X̃ ⟶^{q_{1}} X̃ ⟶^{s} 𝒢 coincides with the one

Γ ⟶^{a} G(X̃) ⟶^{q_{1}^{∗}} G(X̃ × X̃).

Similarly the 1-cocycle corresponding to the principal G bundle π^{∗}(𝒢_{2}) and the morphism X̃ × X̃ ⟶^{q_{2}} X̃ ⟶^{s} 𝒢 coincides with the one

Γ ⟶^{a} G(X̃) ⟶^{q_{2}^{∗}} G(X̃ × X̃).

Let b ∈ G(X̃ × X̃) be an element defined by the equality b · (s ∘ q_{2}) = s ∘ q_{1}. To prove that the principal G bundles π^{∗}(𝒢_{1}) and π^{∗}(𝒢_{2}) are isomorphic it suffices to check that for every γ ∈ Γ the following relation holds in G(X̃ × X̃)

γb · q_{2}^{∗}(a)(γ) · b^{−1} = q_{1}^{∗}(a)(γ),  (1)

where q_{i}^{∗}(a)(γ) := q_{i}^{∗} ∘ a for i = 1, 2.

To prove the relation (1) it suffices to check the following one in 𝒢(X̃ × X̃)

(^{γ}b · q_{2}^{∗}(a)(γ) · b^{−1}) · (s ∘ q_{1}) = q_{1}^{∗}(a)(γ) · (s ∘ q_{1}).  (2)

One has the following chain of relations

(^{γ}b · q_{2}^{∗}(a)(γ) · b^{−1}) · (s ∘ q_{1}) = ( ^{γ}b · q_{2}^{∗}(a)(γ)) · (s ∘ q_{2}) = ^{γ}b · ^{γ}(s ∘ q_{2}) =
= ^{γ}(s ∘ q_{1}) = s ∘ q_{1} ∘ (γ × γ)

The first one follows from the definition of the element b, the second one follows from the commutativity of the diagram

G × 𝒢 ⟶^{ν} 𝒢
(a(γ),s)↑  ↑^{s}
X̃ ⟶^{γ}  X̃
q_{2}↑  ↑^{q_{2}}
X̃ × X̃ ⟶^{γ×γ} X̃ × X̃,

the third one follows from the commutativity of the diagram

G × 𝒢 ⟶^{ν} 𝒢
(b,s∘q_{2})↑  ↑^{s}
X̃ × X̃ ⟶^{q_{1}} X̃
γ×γ↑
X̃ × X̃.

Thus (^{γ}b · q_{2}^{∗}(a)(γ) · b^{−1}) · (s ∘ q_{1}) = s ∘ q_{1} ∘ (γ × γ). The right hand side of the relation (2) is equal to s ∘ q_{1} ∘ (γ × γ) as well, as follows from the commutativity of the diagram

G × 𝒢 ⟶^{ν} 𝒢
(a(γ),s)↑  ↑^{s}
X̃ ⟶^{γ}  X̃
q_{1}↑  ↑^{q_{1}}
X̃ × X̃ ⟶^{γ×γ} X̃ × X̃.

So, the relation (2) holds. Whence the relation (1) holds. Whence the principal G bundles π^{∗}(𝒢_{1}) and π^{∗}(𝒢_{2}) are isomorphic.

The composite X̃ ⟶^{Δ} X̃ × X̃ ⟶^{q_{2}} X̃ ⟶^{s} 𝒢 equals s and equals the composite s ∘ q_{1} ∘ Δ. Whence Δ^{∗}(b) = 1 ∈ G(X̃). This shows that the restriction to X = Δ(X)/Δ(Γ) of the isomorphism π^{∗}(𝒢_{1}) and π^{∗}(𝒢_{2}) corresponding to the element b is the identity isomorphism. The Lemma is proved.

3 Appendix B: a variant of geometric lemma

Let k be an infinite field, Y be a k-smooth algebraic variety, y ∈ Y be a point, 𝒪 = 𝒪_{Y,y} be the local ring, U = Spec(𝒪). Let 𝒳/U be a U-smooth relative curve with geometrically connected fibres equipped with a finite surjective morphism π : 𝒳 → A^{1}×U and equipped with a section Δ : U → 𝒳 of the projection p : 𝒳 → U. Let 𝒵 ⊂ 𝒳 be a closed subset finite over U. The following Lemma is a variant of Lemma 5.1 from [OP].

3.0.9 Lemma. There exists an open subscheme 𝒳^{0} ↪ 𝒳 and a finite surjective morphism α : 𝒳^{0} → A^{1} ×U such that α is étale over 0×U and 1×U and α^{−1}(0×U) = Δ(U) ∐ D_{0}. Moreover if we define D_{1} as α^{−1}(1 × U), then D_{1} ∩ 𝒵 = ∅ and D_{0} ∩ 𝒵 = ∅.

Proof. Let 𝒳̄ be the normalization of the scheme P^{1} × U in the function field k(𝒳) of 𝒳. Let π̄ : 𝒳̄ → P^{1} × U be the morphism. Let 𝒳_{∞} = π^{−1}(∞ × U) be the set theoretic preimage of ∞ × U. Let p̄: 𝒳̄ → U be the structure map. Let u ∈ U be the closed point and X̄_{u} = 𝒳̄ ×_{U} u.

Let , L′ = π̄^{∗}(𝒪_{P^{1}×U}(1)), L′′ = 𝒪X̄(Δ(U)). Let D_{∞} = (π^{∗})(∞×U) be the pull-back of the Cartier divisor ∞ × U ⊂ P^{1} × U. Choose and fix a closed embedding i : 𝒳̄ ↪ P^{n} × U of U-schemes. Set L = i^{∗}(𝒪_{P^{n}×U}(1)).

The sheaf L is very ample. Thus the sheaf L′′⊗L is very amply as well. So, there exists a closed embedding i′′ : 𝒳̄ ↪ P^{n′′} × U of U-schemes such that L′′ ⊗ L = (i′′)^{∗}(𝒪_{P^{n′′}×U}(1)). Using Bertini theorem choose a hyperplane H′′ ⊂ P^{n′′} × U such that (a′′) H′′ ∩ Δ(U) = ∅, H′′ ∩ 𝒵 = ∅, H′′ ∩ D_{∞} = ∅. Define a Cartier divisor D′′ on 𝒳̄ as the the closed subscheme H′′ ∩ 𝒳̄ of 𝒳̄. Regard D′′_{1} := D′′ ∐ D_{∞} as a Cartier divisor on 𝒳̄. Clearly, one has 𝒪𝒳̄(D′′_{1}) = L′′ ⊗L⊗L′.

The sheaf L is very ample. Thus the sheaf L′ ⊗L is very ample as well. So, there exists a closed embedding i′ : 𝒳̄ ↪ P^{n′} × U of U-schemes such that L′ ⊗ L = (i′)^{∗}(𝒪_{P^{n′}×U}(1)). Using Bertini theorem choose a hyperplane H′ ⊂ P^{n′} × U such that (a′) H′ ∩ Δ(U) = ∅, H′ ∩ 𝒵 = ∅, H′ ∩ D′′_{1} = ∅; (b′) the scheme theoretic intersection H′ ∩ X̄_{u} is a k(u)-smooth scheme. Define a Cartier divisor D′ on 𝒳̄ as the closed subscheme D′ = H′ ∩ 𝒳̄ of 𝒳̄. Regard D′_{1} := D′ ∐ Δ(U) as a Cartier divisor on 𝒳̄. Clearly, one has 𝒪𝒳̄(D′_{1}) = L′⊗L⊗L′′.

Observe that D′ is an essentially k-smooth scheme finite and étale over U. Let s′ and s′′ be global sections of L′ ⊗ L ⊗ L′′ such that the vanishing locus of s′ is the Cartier divisor D′_{1} and the vanishing locus of s′′ is the Cartier divisor D′′_{1}. Clearly D′_{1} ∩ D′′_{1} = ∅. Thus f = [s′ : s′′] : 𝒳̄ → P^{1} is a regular morphism of U-schemes. Set

ᾱ = (f, p̄) : 𝒳̄ → P^{1} × U.

Clearly, ᾱ is a finite surjective morphism. Set 𝒳^{0} = ᾱ^{−1}(A^{1} × U) and

α = ᾱ|_{𝒳^{0}} : 𝒳^{0} → A^{1} × U.

Clearly, α is a finite surjective morphism and 𝒳^{0} is an open subscheme of 𝒳. Since α is a finite surjective morphism and 𝒳^{0}, A^{1} × U are regular schemes the morphism α is flat by a theorem of Grothendieck. Since D′_{1} is finite étale over U the morphism α is étale over 0 × U. So, we may choose a point 1 ∈ P^{1} such that the α is étale over 1 × U and (α)^{−1}(1 × U) ∩ 𝒵 = ∅. If we set D_{0} = D′_{1}, then α^{−1}(0 × U) = Δ(U) ∐ D_{0} and D_{0} ∩ 𝒵 = ∅. The Lemma is proven.

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