On the norm principle for quadratic forms

M.Ojanguren, I.Panin, K.Zainoulline

September 6, 2004

Abstract We prove a version of Knebusch’s Norm Principle for finite étale extensions of semi-local Noetherian domains with infinite residue fields of characteristic different from 2. As an application we prove Grothendieck’s conjecture on principal homogeneous spaces for the spinor group of a quadratic space.

Keywords: quadratic space, spinor group, local ring

MSC: 20G35, 13H05

1 Introduction

Let E/F be a finite field extension and q : V → F a quadratic space over F. Let D_{q}(F) ⊂ F^{∗} (resp. D_{q}(E)) be the subgroup generated by the set of non-zero elements of the field F (resp. E) represented by the form q (resp. q_{E} = q ⊗_{F} E). The well-known Knebusch’s Norm Principle for quadratic forms over fields [5], [4, VII.5.1] says that N_{F}^{E}(D_{q}(E)) ⊂ D_{q}(F), where N_{F}^{E} : E^{∗} → F^{∗} is the norm map. The goal of the present article is to show that the Knebusch’s Norm Principle holds for finite étale extensions of semi-local Noetherian domains with infinite residue fields of characteristic different from 2 (see Theorem 3.2). Previously, the Norm Principle for quadratic spaces over semi-local rings was proved for F of characteristic 0 in [9]. As an application we prove Grothendieck’s conjecture on principal homogeneous spaces for the spinor group of a quadratic space (see Theorem 4.1).

This article is organized as follows. Section 2 is devoted to some preliminary results. In Section 3 we prove the Norm Principle for quadratic spaces over local rings. Finally, we prove Grothendieck’s conjecture (Sect. 4).

Acknowledgments The second author thanks very much for the support the RTN-Network HPRN-CT-2002-00287, the grant of the years 2001 to 2003 of the “Support Fund of National Science” at the Russian Academy of Science, the RFFI-grant 03-01-00633a and the Swiss National Science Foundation. The last author is also grateful to SFB478, Münster and AvH Foundation for hospitality and support.

2 Preliminary Lemmas

In the present section we state and prove few auxiliary results which are the main tools in the proof of Theorem 3.2.

2.1. Let F be an infinite field of characteristic different from 2. Let E be a finite étale F -algebra of degree n. Let (V, q) be a quadratic space over F of rank m. Let (V_{E}, q_{E}) be the base change of (V, q) via the extension E/F, i.e., V_{E} = V ⊗_{F} E and q_{E} = q ⊗_{F} E. Sometimes we identify the vector space V_{E} with the set 𝔸_{E}^{m}(E) of E-points of the affine space 𝔸_{E}^{m}.

2.2. Since E is étale over F, there exists an element α ∈ E^{∗} such that the powers 1, α, …, α^{n−1} form a basis of the F -vector space E. Such an element is called primitive. In other words, E can be written as the quotient F [t]/(f_{α}(t)) of the polynomial ring F [t] modulo the ideal generated by a monic separable polynomial f_{α}(t) of degree n. In this case α is identified with the image of t by means of the quotient map F [t] → F [t]/(f_{α}(t)). The polynomial f_{α}(t) is called a minimal polynomial of α.

2.3. Observe that the subset of primitive elements of E is big enough in the following sense: Let β = b_{0} + b_{1}α + ⋯ + b_{n−1}α^{n−1} with b_{0}, …, b_{n−1} ∈ F be an element of E. Then, for every integer i with 0 ≤ i ≤ n − 1,

β^{i} = b_{0}^{(i)} + b_{1}^{(i)}α + ⋯ + b_{n−1}^{(i)}α^{n−1}, with b_{j}^{(i)} ∈ F.

Clearly, each b_{j}^{(i)} is a polynomial in b_{0}, …, b_{n−1}. The condition that β be primitive is the non-vanishing of the determinant of (b_{j}^{(i)}) and can, thus, be expressed by the non-vanishing of a polynomial d(b_{0}, …, b_{n−1}). We define an open subset P of the Weil restriction R_{E/F} (𝔾_{m}_{,E}) of the multiplicative group 𝔾_{m}_{,E} by

P = {(b_{0}, …, b_{n−1}) | d(b_{0}, …, b_{n−1}) ≠ 0} .

By definition, the set P (F) of the F -points of P is the subset of primitive invertible elements of E.

2.4 Lemma. Let X be an F -rational variety. Then any non-empty open subset U of X has a rational point.

Proof. The proof follows from the fact that F is infinite.

From this point onwards we will use the following notations and terminology.

2.5. Assume there is given an element α ∈ P (E) and a vector v ∈ V_{E} such that αq(v) = 1. Denote by f_{α}(t) ∈ F [t] the minimal polynomial of α.

2.6. Let Q be the quadric over E given by

Q = {ω ∈ 𝔸_{E}^{m} | αq(ω) = 1}.

Since the element v is an E-point of Q, the variety Q is E-rational. Set Y = R_{E/F} (Q) to be the Weil restriction of Q (see [2]). The variety Y is a closed subvariety of dimension (m−1)n of the affine space R_{E/F} (𝔸_{E}^{m}) = 𝔸_{F}^{nm} and we have the bijection Y (F) = Q(E) between the sets of F -points of Y and E-points of Q. Since Q is rational over E, Y is rational over F.

2.7. In order to define the next variety U we identify R_{E/F} (𝔸_{E}^{m}) with the affine space M_{n,m} of n × m-matrices over F by choosing {1, α, …, α^{n−1}} as a basis of the vector space E over F. Thus, a vector ω = (ω_{0}, …, ω_{m−1}) ∈ R_{E/F} (𝔸_{E}^{m})(F) corresponds to the matrix (ω_{i,j})_{i,j=0}^{n−1,m−1}, where the entries ω_{i,j} ∈ F are defined by ω_{j} = ∑_{i=0}^{n−1} ω_{i,j}α^{i}. For any ω = (ω_{=}, …, ω_{n−1}) we define ω(t) = (ω_{0}(t), …, ω_{m−1}(t)), where ω_{j}(t) = ∑_{i=0}^{n−1} ω_{i,j}t^{i}. Note that each ω_{j}(t) is of degree at most n − 1. We define U to be the open subset of the affine space R_{E/F} (𝔸_{E}^{m}) defined by

U = {ω ∈ M_{n,m} | Φ_{ω}(t) = tq(ω(t)) − 1 is separable of degree 2n − 1}.

Clearly, the coefficients of the polynomial Φ_{ω}(t) ∈ F [t] depend on the choice of the isomorphism R_{E/F} (𝔸_{E}^{m}) ≅ M_{n,m}, i.e., they depend on the choice of α.

2.8 Lemma. The open subset U ∩ Y of Y contains an F -point.

Proof. Since Y is F -rational and U ∩ Y is open in Y, by Lemma 2.4 it is enough to show that U ∩ Y contains a point ρ over the algebraic closure F̄ of the field F. Let g(t) be a polynomial over F such that 1. deg g(t) = n − 1

2. g(t) is coprime with the polynomial f_{α}(t)

3. −g(0)f_{α}(0) = 1

4. g(t) is separable

Choose a hyperbolic plane ℍ in the quadratic space (VF̄, qF̄) over F̄. Choose a basis {e_{1}, e_{2}} of ℍ such that q(e_{1}) = q(e_{2}) = 0 and q(e_{1} + e_{2}) = 1. Let g_{1}(t), g_{2}(t) be polynomials over F̄ of degree n − 1 such that

tg_{1}(t)g_{2}(t) = g(t)f_{α}(t) + 1.

Set ρ(t) = g_{1}(t)e_{1} + g_{2}(t)e_{2} to be a vector of polynomials over F̄ which can be identified with an F̄-point of R_{E/F} (𝔸_{E}^{m}) following 2.7. Then we have

tq(ρ(t)) = t(q(e_{1})g_{1}(t)^{2} + q(e_{1} + e_{2})g_{1}(t)g_{2}(t) + q(e_{2})g_{2}(t)^{2}) =
= tg_{1}(t)g_{2}(t) = g(t)f_{α}(t) + 1

and

• αq(ρ(α)) = g(α)f_{α}(α) + 1 = 1

• Φ_{ρ}(t) = tq(ρ(t)) − 1 = g(t)f_{α}(t) is separable of degree 2n − 1.

Hence ρ ∈ (U ∩ Y)(F̄) is the desired point.

2.9. Define one more open subset of Y. For that consider a closed subset Z ⊂ 𝔸_{E}^{m} defined by

Z = {ω ∈ 𝔸_{E}^{m} | ⟨v, ω⟩ − 1 ∈ {0}_{E}},

where ⟨ , ⟩ : V_{E} × V_{E} → E is the bilinear form associated with the quadratic form αq_{E} and {0}_{E} the image of the zero section Spec E → 𝔸_{E}^{m} of 𝔸_{E}^{m}. Set W = R_{E/F} (Q ∖ Z). Passing to the algebraic closure F̄ we see that W is a non-empty open subset of Y = R_{E/F} (Q). The set W (F) of F -points of W consists of all w ∈ Q(E) satisfying the condition ⟨v, ω⟩ − 1 ∈ E^{∗}.

2.10 Lemma. There exists ω′ ∈ V_{E} such that

(i) αq(ω′) = 1 (ii) ⟨v, ω′⟩ − 1 ∈ E^{∗}

(iii) the polynomial Φ_{ω′}(t) is separable of degree 2n − 1.

Proof. By Lemma 2.8 the set U ∩ Y is non-empty open in Y. The set W defined in 2.9 is also non-empty open in Y. Since the variety Y is irreducible, the set W ∩ U ∩ Y is non-empty open in Y. Since Y is F -rational, the set W ∩ U ∩ Y contains an F -point. Recall that Y (F) = Q(E) ⊂ V_{E}. Let ω′ ∈ (W ∩ U ∩ Y)(F) ⊂ V_{E}. We claim that ω′ satisfies (i) to (iii). In fact, property (i) holds because ω′ ∈ Q(E), property (ii) holds because ω′ ∈ W (F) = (Q ∖ Z)(E) and property (iii) holds because ω′ ∈ U(F).

3 The Norm Principle

3.1. Let R be a semi-local Noetherian domain with infinite residue fields of characteristic different from 2 (in this case \frac{1}{2} ∈ R). Let S/R be a finite étale R-algebra (not necessarily a domain). Let (V, q) be a quadratic space of rank m over R. Let (V_{S}, q_{S}) be the base change of (V, q) via the extension S/R. Let D_{q}(R) (resp. D_{q}(S)) be the group generated by the invertible elements of R (resp. S) represented by the form q.

The goal of the present section is to prove the following

3.2 Theorem. There is an inclusion of the subgroups of R^{∗}

N_{R}^{S}(D_{q}(S)) ⊂ D_{q}(R),

where N_{R}^{S} : S^{∗} → R^{∗} is the norm map for the finite étale extension S/R.

3.3. For simplicity we will consider only the case of local R. By a variety over R we will mean a reduced separated scheme of finite type over Spec R. From this point onwards, by “bar” we mean the reduction modulo the maximal ideal 𝔪 of R. So that \overline{S} = S/𝔪S. We will write F for \overline{R} and E for \overline{S}. So E/F is a finite étale algebra.

To prove Theorem 3.2 we need the following auxiliary results.

3.4 Lemma. Let (S^{m}, ϕ) be a quadratic space over S and let ⟨ , ⟩ : S^{m}×S^{m} → S be the associated bilinear form. Let v ∈ S^{m} be such that ϕ(v) = 1. Let ω′ ∈ E^{m} be such that \overline{ϕ}(ω′) = ¯1 and ⟨\overline{v}, ω′⟩ − ¯1 ∈ E^{∗} (a unit). Then there exists ω ∈ S^{m} satisfying the conditions (i) ϕ(ω) = 1

(ii) \overline{ω} = ω′ in E^{m}.

Proof. If ω̃ is a lift of ω′ we have ϕ(ω̃) = 1 + h and ⟨v, ω̃⟩ − 1 = u with h ∈ 𝔪 and u ∈ R^{∗}. Putting

ω = \frac{λv + ω̃}{λ + 1}

we find

ϕ(ω) = \frac{λ^{2} + 2λ(1 + u) + 1 + h}{(λ + 1)^{2}} .

Thus, for λ = −h/2u we have ϕ(ω) = 1 and since h ∈ 𝔪, ω is a lift of ω′.

3.5. Recall that a polynomial f(t) ∈ R[t] is said to be separable if the quotient ring R[t]/(f(t)) is a finite étale extension of R. Since R is local, a polynomial f is separable iff its reduction \overline{f} modulo the maximal ideal 𝔪 is a separable polynomial over \overline{R}.

3.6. Similar to 2.2, since, S being étale over R, there exists an element α ∈ S^{∗} such that the powers 1, α, …, α^{n−1} form a basis of the free R-module S. As in the case of fields such an element is called primitive. In other words, S can be written as a quotient R[t]/(f_{α}(t)) of the polynomial ring R[t], where f_{α}(t) is a monic separable polynomial called the minimal polynomial for α. In this case α is identified with the image of t by means of the quotient map R[t] → R[t]/(f_{α}(t)).

3.7. As in 2.3 consider a primitive element α of the extension S/R. An element β ∈ S^{∗} can be written as β = b_{0} + b_{1}α + ⋯ + b_{n−1}α^{n−1} in S^{∗} with b_{0}, …, b_{n−1} ∈ R. Then, for every integer i with 0 ≤ i ≤ n − 1,

β^{i} = b_{0}^{(i)} + b_{1}^{(i)}α + ⋯ + b_{n−1}^{(i)}α^{n−1} with b_{j}^{(i)} ∈ R.

Clearly, each b_{j}^{(i)} is a polynomial in b_{0}, …, b_{n−1}. The condition that β be primitive is the non-vanishing of the determinant of (b_{j}^{(i)}) and can thus be expressed by the non-vanishing of a polynomial d(b_{0}, …, b_{n−1}). We define an open subset P of R_{S/R}(𝔾_{m}_{,S}) by

P = {(b_{0}, …, b_{n−1}) | d(b_{0}, …, b_{n−1}) ∈ R^{∗}} .

Observe that P (R) is the set of primitive invertible elements of S.

3.8. Following 2.7, for a given primitive element α and any vector ω ∈ V_{S} we define the polynomial Φ_{ω}(t) ∈ R[t] as follows. We identify the free S-module V_{S} with the set of matrices M_{n,m}(R) using {1, α, …, α^{n−1}} as a basis of S over R. Thus, a vector ω = (ω_{0}, …, ω_{m−1}) ∈ V_{S} corresponds to the matrix (ω_{i,j})_{i,j=0}^{n−1,m−1}, where the entries ω_{i,j} ∈ R are defined by ω_{j} = ∑_{i=0}^{n−1} ω_{i,j}α^{i}. We set

Φ_{ω}(t) = tq(ω(t)) − 1 ∈ R[t],

where ω(t) = (ω_{0}(t), …, ω_{m−1}(t)) is the vector of m polynomials ω_{j}(t) = ∑_{i=0}^{n−1} ω_{i,j}t^{i} of degree at most n−1. Clearly, the coefficients of the polynomial Φ_{ω}(t) depend on the choice of α.

3.9 Lemma. Let α ∈ S be a primitive invertible element of the finite étale extension S/R. Consider the quadratic space (V_{S}, αq_{S}) over S. Assume αq(v) = 1 for a vector v ∈ V_{S}. Then there exists an ω ∈ V_{S} such that

(i) αq(ω) = 1;

(ii) the polynomial Φ_{ω}(t) (defined in 3.8) is separable of degree 2n − 1.

Proof. According to our notation we write V_{F} for V/𝔪V and V_{E} for V_{S}/𝔪V_{S}. The element \overline{α} ∈ E is a primitive element of E/F. It satisfies the relation \overline{α}\overline{q}(\overline{v}) = ¯1 for the vector \overline{v} ∈ V_{E}. By Lemma 2.10 applied to \overline{α} and \overline{v} there exists ω′ ∈ V_{E} satisfying the conditions \overline{α}\overline{q}(ω′) = ¯1 and ⟨\overline{v}, ω′⟩ − ¯1 ∈ E^{∗}, where ⟨ , ⟩ : V_{E} × V_{E} → E is the bilinear form associated with \overline{α}\overline{q}.

Now apply Lemma 3.4 to the quadratic space (V_{S}, αq_{S}) and the vectors v ∈ V_{S}, ω′ ∈ V_{E}. We find a vector ω ∈ V_{S} such that

(i) αq(ω) = 1 in S;

(ii) \overline{ω} = ω′ in V_{E}.

Property (ii) implies that the reduction modulo 𝔪 of the polynomial Φ_{ω}(t) ∈ R[t] coincides with the polynomial Φ_{ω′}(t) ∈ F [t]. The polynomial Φ_{ω′}(t) is separable of degree 2n−1 by property (iii) of 2.10. Hence, Φ_{ω}(t) is separable of degree 2n − 1 (see 3.5).

3.10 Lemma. Let (V, q) be a quadratic space over R. Then the group of squares (R^{∗})^{2} is contained in D_{q}(R). In particular, a ∈ D_{q}(R) if and only if a^{−1} ∈ D_{q}(R).

Proof. For any b ∈ S^{∗} and q(u) ∈ R^{∗} we have b^{2} = q(ub)/q(u) ∈ D_{q}(R).

3.11 Proposition. Let α ∈ S be a primitive invertible element of the finite étale extension S/R. In particular, the ring S can be written as S = R[t]/(f_{α}(t)), where f_{α}(t) is the minimal polynomial of α. Assume αq(v) = 1 for a vector v ∈ V_{S}. Then N_{R}^{S}(q(v)) ∈ D_{q}(R).

Proof. By Lemma 3.9 there exists ω ∈ V_{S} such that

(i) αq(ω) = 1;

(ii) the polynomial Φ_{ω}(t) is separable of degree 2n − 1.

We have Φ_{ω}(α) = αq(ω) − 1 = 0. This implies Φ_{ω}(t) = c · h(t)f_{α}(t), where c ∈ R^{∗} and h(t) is some monic polynomial of degree n − 1. The polynomial h(t) is separable, since Φ_{ω}(t) is separable. Clearly, we have the relation

1 + c · h(t)f_{α}(t) = tq(ω(t)).  (1)

The proof proceeds by the induction on degree of the extension S/R. The case n = 1 is obvious. Assume that the proposition holds for all finite étale extensions of degree strictly less than n.

Consider the finite étale extension T = R[t]/(h(t)) over R, where h(t) is the polynomial appearing in (1). Observe that the degree of T/R is n − 1. Let β be the image of t under the quotient map R[t] → R[t]/(h(t)). Observe that β is a primitive element of the R-algebra T with the minimal polynomial h(t).

Consider the reduction modulo the ideal (h(t)) of the relation (1). We get 1 = βq(u) in T for some u ∈ V_{T}. By Lemma 3.10 we get β ∈ D_{q}(T). Substituting t = 0 in (1) we get f_{α}(0) = −1/(c·h(0)). Together with the fact that for the extension S = R[t]/(f_{α}(t)) over R, N_{R}^{S}(α) = (−1)^{deg(S/R)}f_{α}(0), this implies the following chain of relations in R:

N_{R}^{S}(q(v)) = 1/N_{R}^{S}(α) = (−1)^{n}/f_{α}(0) = c · (−1)^{n−1}h(0) = c · N_{R}^{T}(β).

Since c is the leading coefficient of the polynomial tq(ω(t)) of degree 2n − 1, it is represented by the form q. Namely, c = q(ω_{n−1,0}, …, ω_{n−1,m−1}), where ω_{n−1,j} ∈ R are the leading coefficients of polynomials ω_{j}(t). By the induction hypothesis the norm N_{R}^{T}(β) lies in D_{q}(R). Hence N_{R}^{S}(q(v)) ∈ D_{q}(R). This completes the proof of Proposition 3.11 3.12 Lemma. Consider the multiplicative group 𝔾_{m}_{,S} over the semi-local scheme Spec S. Let W be an open subset of 𝔾_{m}_{,S} such that for each closed point x ∈ Spec S the fiber W_{x} over x is non-empty. Then there exists b ∈ S^{∗} such that b^{2} ∈ W (S).

Proof. For each closed point x of Spec S there is an element a_{x} in the residue field of x such that a_{x}^{2} ∈ W_{x}. Since S is semi-local there is an element b ∈ S^{∗} such that b_{x} = a_{x} in the residue field of x, for each x. Clearly, b^{2} ∈ W (S).

Proof of Theorem 3.2. Let a = q(u) ∈ S^{∗} for certain u ∈ V_{S}. Consider the non-empty open subset P of R_{S/R}(𝔾_{m}_{,S}) defined in 3.7. Clearly, each closed fiber of aP over Spec S is non-empty. By Lemma 3.12 there exists an element b ∈ S^{∗} such that b^{2} ∈ aP (S). It means that α = a^{−1}b^{2} is in P (S), i.e., primitive. Replacing a by α and u by v = u · b^{−1}, we get αq(v) = 1. Then, by Lemma 3.10 and Proposition 3.11, we get the desired inclusion N_{R}^{S}(a) = N_{R}^{S}(q(v)) · N_{R}^{S}(b)^{2} ∈ D_{q}(R).

3.13. As before let R be a semi-local Noetherian domain with infinite residue fields of characteristic different from 2. Let S/R be a finite étale R-algebra of degree n. Let (V, q) be a quadratic space over R of rank m. By D_{q}^{0}(S) (resp. D_{q}^{1}(S)) we denote the set of all even (resp. odd) products of invertible elements of S represented by q_{S}, i.e.,

D_{q}^{i}(S) = {∏_{j=0}^{l} q(v_{j}) | v_{j} ∈ S^{m}, q(v_{j}) ∈ S^{∗}, l ≡ i mod 2}, i = 0, 1.

Observe that D_{q}^{0}(S) is a subgroup of the group D_{q}(S) and (S^{∗})^{2} ⊂ D_{q}^{0}(S). Clearly, if c ∈ D_{q}^{i}(S), i = 0, 1, and b ∈ D_{q}^{0}(S), then cb ∈ D_{q}^{i}(S). The following result is an obvious consequence of the proof of Theorem 3.2.

3.14 Theorem. Let N_{R}^{S} : S^{∗} → R^{∗} be the norm map. Then

N_{R}^{S}(D_{q}^{0}(S)) ⊂ D_{q}^{0}(R),

Proof. For a positive integer n we set D_{q}^{n}(S) = D_{q}^{0}(S) if n is even and D_{q}^{n}(S) = D_{q}^{1}(S) if n is odd. By the proof of Proposition 3.11 and Theorem 3.2 it follows that N_{R}^{S}(a) ∈ D_{q}^{n}(R), where a is an element represented by q_{S} and S/R is an extension of degree n.

4 Grothendieck’s conjecture for the spinor group

Let R be a local domain with residue field of characteristic different from 2 and q be a quadratic space over R. Following [6, IV.6] we define the spinor group (scheme) Spin_{q} to be Spin_{q}(R) = {x ∈ SΓ_{q}(R) | xσ(x) = 1}, where σ is the canonical involution, SΓ_{q}(R) = {c ∈ C_{0}(V, q)^{∗} | cV c^{−1} ⊂ V } is the special Clifford group and C_{0}(V, q) is the even part of the Clifford algebra of the respective quadratic space (V, q) over R. The present section is devoted to the proof of the following result:

4.1 Theorem. Let R be a local regular ring containing an infinite field of characteristic different from 2. Let K be its quotient field. Let q be a quadratic space over R. Then the induced map on the sets of principal homogeneous spaces

H_{et}^{1}(R, Spin_{q}) → H_{et}^{1}(K, Spin_{q})

has trivial kernel, where Spin_{q} is the spinor group for the quadratic space q.

4.2. Observe that the theorem is a particular case of Grothendieck’s conjecture on principal homogeneous spaces [3], which states that, for a smooth reductive group scheme G over R, the induced map H_{et}^{1}(R, G) → H_{et}^{1}(K, G) has trivial kernel.

Proof. The proof is based on the results of [7], [10] and [11].

Assume R is a local regular ring containing a field of characteristic different from 2. Let K be its quotient field. We have the following commutative diagram (see [6, IV.8.2.7]):

\xymatrix{SO_{q}(R) \ar[r]^{SN} \ar[d] & R^{∗}/(R^{∗})^{2} \ar[r] \ar[d] & H_{et}^{1}(R, Spin_{q}) \ar[r] \ar[d] & H_{et}^{1}(R, SO_{q}) \ar[d] \\ SO_{q}(K) \ar[r]^{SN} & K^{∗}/(K^{∗})^{2} \ar[r] & H_{et}^{1}(K, Spin_{q}) \ar[r] & H_{et}^{1}(K, SO_{q}),}

where SN : SO_{q}(R) → H_{et}^{1}(R, μ_{2}) = R^{∗}/(R^{∗})^{2} is the spinor norm. The main result of [7] says that the vertical arrow on the right hand side has trivial kernel (see also [10, 3.4]). Thus, in order to show that the middle one has trivial kernel, it is enough to check that the induced map on the cokernels coker(SN)(R) → coker(SN)(K) is injective. First, we prove the geometric case. To do that we use the following slight modification of the main result of section 2 of [10].

4.3 Proposition. Let R be a local regular ring of geometric type over an infinite field. Let F be a presheaf from the category of affine schemes over R to abelian groups. Assume F satisfies the axioms C, E of [10] and the weak versions of the axioms TE, TA, TB of [10] (see 4.4). Then the canonical map F (R) → F (K) is injective.

4.4. The weak versions of the axioms TE, TA and TB state

• For a finite étale R-algebra T (instead of finitely generated projective considered in [10]) there is given a transfer map T r_{R}^{T} : F (T) → F (R).

• This transfer map is additive in the sense of TA of [10].

• For a finitely generated projective R[t]-algebra S such that the algebras S/(t) and S/(t − 1) are finite étale over R there is the commutative diagram of the axiom TB of [10]

\xymatrix{F (S) \ar[r] \ar[d] & F (S/(t)) \ar[d]^{T r} \\ F (S/(t − 1)) \ar[r]^{T r} & F (R)}

The proof of 4.3 follows immediately after one replaces the Geometric Presentation Lemma [8, 10.1] used in section 1.1 of [10] by its stronger (étale) version

4.5 Lemma (Étale Geometric Presentation Lemma). Let R be a local essentially smooth algebra over an infinite field k, m its maximal ideal and S an essentially smooth k-algebra which is an integral domain and finite over the polynomial algebra R[t′]. Suppose that e : S → R is an R-augmentation and let I = ker e. Assume that S/mS is smooth over the residue field R/m at the maximal ideal e^{−1}(m)/mS. Then, given a regular function f ∈ S such that S/(f) is finite over R, we can find a t ∈ I such that

• S is finite over R[t];

• There is an ideal J comaximal with I and such that I ∩ J = (t);

• (f) and J are comaximal; (f) and (t − 1) are comaximal;

• S/(t) is étale over R; S/(t − 1) is étale over R.

Proof. See [11, 6.1]

Consider the presheaf of abelian groups F : T ↦ coker(SN)(T). According to 4.3 to prove the mentioned injectivity we have to show that the functor F satisfies axioms C, E and the weak versions of axioms TE, TA, TB of [10] (see 4.4). Axioms C, E, TA and TB hold by the same arguments as in sections 3.2 and 3.4 of [10].

Consider, for instance, the proof of axiom E. First, following the proof of E.(a) and E.(b) of [10, 3.2] for a given quadratic space q over R we construct the R-algebra S̃, two quadratic spaces q_{1} = q ⊗_{A} S̃ and q_{2} = q ⊗_{R} S̃ over S̃ and the isomorphism Ψ between them such that the restrictions q_{1}|_{R} and q_{2}|_{R} coincide with q and the restriction Ψ|_{R} is the identity. Then following the proof of E.(c) the isomorphism Ψ induces a commutative diagram

\xymatrix{SO_{q_{1}}(T) \ar[r]^{Ψ} \ar[d]^{SN} & SO_{q_{2}}(T) \ar[d]^{SN} \\ T^{∗}/(T^{∗})^{2} \ar[r]^{Ψ} & T^{∗}/(T^{∗})^{2}}

for any S̃-algebra T. Taking the cokernels of the vertical arrows we obtain the desired functor transformation Φ : F_{1}(T) → F_{2}(T) of E.(c).

Hence, in order to prove the injectivity, it remains to produce a welldefined transfer map T r_{R}^{S} : F (S) → F (R) for any finite étale extension S/R of a local regular ring of geometric type over an infinite field.

To produce such a map it suffices to take the norm map N_{R}^{S} : S^{∗} → R^{∗} and to check the inclusion

N_{R}^{S}(SN(SO_{q}(S))) ⊂ SN(SO_{q}(R)).

Since in the semi-local case SN(SO_{q}(S)) = D_{q}^{0}(S) and SN(SO_{q}(R)) = D_{q}^{0}(R) (see [6, IV.6], [1, III.3.21]) it remains to check the inclusion

N_{R}^{S}(D_{q}^{0}(S)) ⊂ D_{q}^{0}(R).

This inclusion holds by Theorem 3.14. Hence, the norm map N_{R}^{S} : S^{∗} → R^{∗} induces the desired transfer map T r_{R}^{S} : F (S) → F (R). This completes the proof of Theorem 4.1 in the geometric case.

Finally, to extend our result to the case of a local regular ring R containing an infinite field of characteristic different from 2 we use Popescu’s approximation theorem [11, 7.5]. We refer to the item 1 of section 5 of [10] for the precise arguments.

References

[1] R. Baeza: Quadratic forms over semi-local rings, Lecture Notes in Mathematics, Vol. 655, Springer-Verlag, 1978.

[2] M. Demazure and P. Gabriel: Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, North-Holland, 1970.

[3] A. Grothendieck, Le groupe de Brauer. II. Théorie cohomologique. (French) Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam; Masson, Paris, 1968, pp.67-87.

[4] T.Y. Lam: The algebraic theory of quadratic forms, W.A. Benjamin, 1973.

[5] M. Knebusch, Ein Satz über die Werte von quadratischen Formen über Körpern, Invent. Math. 12 (1971), 300–303.

[6] M.-A. Knus: Quadratic and Hermitian Forms over Rings, Grund. Math. Wiss. 294, Springer-Verlag, 1991.

[7] M. Ojanguren, Quadratic forms over regular rings, J. Indian Math. Soc. 44: 109–116 (1980).

[8] M. Ojanguren, I. Panin, Rationally trivial hermitian spaces are locally trivial, Math. Z. 237: 181–198, 2001.

[9] K. Zainoulline: On Knebusch’s Norm Principle for quadratic forms over semi-local rings, Preprints of Max-Planck-Institute für Mathematik 40, 2003.

[10] K. Zainoulline, On Grothendieck’s conjecture on principal homogeneous spaces for some classical algebraic groups, St. Petersburg Math. J. 12(1): 117–143, 2001.

[11] K. Zainoulline, The Purity Problem for Functors with Transfers, K-Theory 22(4): 303–333, 2001.

Manuel Ojanguren Ecole Polytechnique Fédérale de Lausanne Institut de Géométrie, Algèbre et Topologie Bâtiment BCH 1015 Lausanne Switzerland e-mail: manuel.ojanguren@epfl.ch

Ivan Panin St. Petersburg Department of V.A. Steklov Institute of Mathematics 27, Fontanka St. Petersburg 191023 Russia e-mail: panin@pdmi.ras.ru

Kirill Zainoulline Fakultät für Mathematik Universität Bielefeld Postfach 100131 33501 Bielefeld Germany e-mail: kirill@mathematik.uni-bielefeld.de