A purity theorem for linear algebraic groups

Ivan Panin ^{∗}

01.03.2005

Abstract Given a characteristic zero field k and a dominant morphism of affine algebraic k-groups μ : G → C one can form a functor from k-algebras to abelian groups R ↦ ℱ(R) := C(R)/μ(G(R)). Assuming that C is commutative we prove that this functor satisfies a purity theorem for any regular local k-algebra. Few examples are considered in the very end of the preprint.

1 Introduction

Let ℱ be a covariant functor from the category of commutative rings to the category of abelian groups. For any domain R consider the sequence

ℱ(R) → ℱ(K) → ⨁_{𝔭}ℱ(K)/ℱ(R_{𝔭})

where p runs over the height 1 primes of R. We say that ℱ satisfies purity for R if this sequence — which is clearly a complex — is exact. The purity for R is equivalent to the following property

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

Let k be a characteristic zero field and k̄ be its algebraic closure. By a linear algebraic k-group we mean a reduced affine group k-scheme. In particular, a linear algebraic k-group is always k-smooth. The main results of this paper is the following purity theorem

Theorem 1.0.1 (A). Let

μ : G → C

be a dominant morphism of linear algebraic k-groups, with C commutative. Let A be a local ring of a smooth algebraic variety X over k. The functor

ℱ : R ↦ C(R)/μ(G(R))

satisfies purity for A.

∗The author thanks very much for the support the the Ellentuck Fund, the RTN-Network HPRN-CT-2002-00287, the RFFI-grant 03-01-00633a, the Swiss National Science Foundation and INTAS-03-51-3251.

Theorem 1.0.2 (B). Let R be a regular local ring containing the field k. Let

μ : G → C

be a dominant morphism of linear algebraic k-groups, with C commutative. The functor

ℱ : A ↦ C(A)/μ(G(A))

satisfies purity for R. If K is the fraction field of R this statement can be restated in an explicit way as follows: given an element c ∈ C(K) suppose that for each height 1 prime ideal 𝔭 in A there exist a_{𝔭} ∈ C(R_{𝔭}), g_{𝔭} ∈ G(K) with a = a_{𝔭} · μ(g_{𝔭}) ∈ C(K). Then there exist g_{𝔪} ∈ G(K), a_{𝔪} ∈ C(R) such that

a = a_{𝔪} · μ(g_{𝔪}) ∈ C(K).

After the pioneering articles [C-T/P/S] and [R] on purity theorems for algebraic groups, various versions of purity theorems were proved in [C-T/O], [PS], [Z], [Pa]. The most general result in the so called constant case was given in [Z, Exm.3.3]. This result follows now from our Theorem (A) by taking G to be a k-rational reductive group, C = 𝔾_{m,k} and μ : G → 𝔾_{m,k} a dominant k-group morphism. The papers [PS], [Z], [Pa] contain results for the nonconstant case. However they only consider specific examples of algebraic scheme morphisms μ : G → C.

It seems plausible to expect purity theorem in the following context. Let R be a regular local ring. Let μ : G → T be a smooth dominant morphism of R-group schemes with an R-smooth reductive group scheme G and an R-torus T. Let ℱ be the covariant functor from the category of commutative rings to the category of abelian groups given by S ↦ T (S)/μ(G(S)). Then ℱ should satisfies purity for R provided that R contains an infinite field. It might even happen that ℱ satisfies purity for any regular local ring R.

Another functor for which one should expect purity is the following one. Let R be a regular local ring, G and G′ smooth reductive R-group schemes, π : G → G′ a smooth dominant R-group scheme morphism. Assume Z = ker(π) is finite étale of multiplicative type over R. The boundary operator δ_{π,R} : G′(R) → H1_{ét}(R, Z) makes sense and is a group homomorphism [Se, Ch.II, §5.6, Cor.2]. For an R-algebra S set

ℱ(S) = H_{ét}^{1}(S, Z)/Im(δ_{π,S}).  (1)

It seems plausible that the functor ℱ satisfies purity for R provided that R contains an infinite field. It may even happen that, for any regular local ring R, ℱ satisfies purity. Theorem 2.0.4 states that this is the case if R contains an infinite field k of characteristic zero and G, G′ and π are already defined over k.

Note that we use transfers for the functor R ↦ C(R), but we do not use at all the norm principle for the map μ : G → C.

The preprint is organized as follows. In Section 2 we construct norm maps following a method from [SV, Sect.6]. In Section 3 we prove the main geometric Lemmas 1.2.3 and 1.2.5. In Section 4 we discuss unramified elements. A key point here is Lemma 1.3.5. In Section 5 we discuss specialization maps. A key point here is Corollary 1.4.6. In Section 6 we prove Theorem (A). In Section 7 we prove Theorem (B). In Section 8 we consider the functor (1) and prove in 2.0.4 a purity theorem for it. Finally in Section 9 we collect several examples.

Acknowledgments The author thanks very much M.Ojanguren for useful discussions on the subject of the present article.

1.1 Norms

Let k ⊂ K ⊂ L be field extensions and assume that L is finite separable over K. Let K^{sep} be a separable closure of K and

σ_{i} : K → K^{sep}, 1 ≤ i ≤ n

the different embeddings of K into L. As in §1, let C be a commutative algebraic group scheme defined over k. We can define a norm map

𝒩_{L/K} : C(L) → C(K)

by

𝒩_{L/K}(α) = ∏_{i} C(σ_{i})(α) ∈ C(K^{sep})^{𝒢(K)} = C(K) .

Following Suslin and Voevodsky [SV, Sect.6] we generalize this construction to finite flat ring extensions. Let p : X → Y be a finite flat morphism of affine schemes. Suppose that its rank is constant, equal to d. Denote by S^{d}(X/Y) the d-th symmetric power of X over Y.

Lemma 1.1.1. There is a canonical section

𝒩_{X/Y} : Y → S^{d}(X/Y)

which satisfies the following three properties:

(1) Base change: for any map f : Y′ → Y of affine schemes, putting X′ = X ×_{Y} Y′ we have a commutative diagram

\xymatrix{Y′ \ar[r]^{𝒩_{X′/Y′}} \ar[d]_{f} & S^{d}(X′/Y′) \ar[d]^{S^{d}(Id_{X}×f)} \\ Y \ar[r]^{𝒩_{X/Y}} & S^{d}(X/Y)}

(2) Additivity: If f_{1} : X_{1} → Y and f_{2} : X_{2} → Y are finite flat morphisms of degree d_{1} and d_{2} respectively, then, putting X = X_{1} ∐ X_{2}, f = f_{1} ∐ f_{2} and d = d_{1} + d_{2}, we have a commutative diagram

\xymatrix{S^{d_{1}}(X_{1}/Y) × S^{d_{2}}(X_{2}/Y) \ar[rr]^{σ} &  & S^{d}(X/Y) \\  & Y \ar[ul]^{𝒩_{X_{1}/Y} ×𝒩_{X_{2}/Y}} \ar[ur]_{𝒩_{X/Y}}}

where σ is the canonical imbedding.

(3) Normalization: If X = Y and p is the identity, then 𝒩_{X/Y} is the identity.

Proof. We construct a map 𝒩_{X/Y} and check that it has the desired properties. Let B = k[X] and A = k[Y], so that B is a locally free A-module of finite rank d. Let B^{⊗d} = B ⊗_{A} B ⊗_{A} ⋯ ⊗_{A} B be the d-fold tensor product of B over A. The permutation group 𝔖_{d} acts on B^{⊗d} by permuting the factors. Let S^{d}(B) ⊆ B^{⊗d} be the A-algebra of all the 𝔖_{d}-invariant elements of B^{⊗d}. We consider B^{⊗d} as an S^{d}(B)-module through the inclusion S^{d}(B) ⊆ B^{⊗d} of A-algebras. Let I be the kernel of the canonical homomorphism B^{⊗d} → ⋀^{d}(B) mapping b_{1} ⊗ ⋯ ⊗ b_{d} to b_{1} ∧ ⋯ ∧ b_{d}. It is well-known (and easily checked locally on A) that I is generated by all the elements x ∈ B^{⊗d} such that τ(x) = x for some transposition τ. If s is in S^{d}(B), then τ(sx) = τ(s)τ(x) = sx, hence sx is in S^{d}(B) too. In other words, I is an S^{d}(B)-submodule of B^{⊗d}. The induced S^{d}(B)-module structure on ⋀^{d}(B) defines an A-algebra homomorphism

φ : S^{d}(B) → End_{A}(⋀^{d}(B)) .

Since B is locally free of rank d over A, ⋀^{d}(B) is an invertible A-module and we can canonically identify End_{A}(⋀^{d}(B) with A. Thus we have a map

φ : S^{d}(B) → A

and we define

T r_{X/Y} : Y → S^{d}(X)

as the morphism of Y -schemes induced by φ. The verification of properties (1), (2) and (3) is straightforward.

Let now C be a commutative k-group scheme and f : X → C any morphism. We define the norm N_{X/Y} (f) of f as the composite map

Y ⟶^{T r_{X/Y}} S^{d}(X) ⟶^{S^{d}(f)} S^{d}(C) ⟶^{×} C  (2)

Here we write ” × ” for the group law on C. The norm maps N_{X/Y} satisfy the following conditions (1) Base change: for any map f : Y′ → Y of affine schemes, putting X′ = X ×_{Y} Y′ we

have a commutative diagram

C(X) ⟶^{(id×f)^{∗}} C(X′)
N_{X/Y} ↓  ↓^{N_{X′/Y′}}
C(Y) ⟶^{f^{∗}} C(Y′)

(2) multiplicativity: if X = X_{1} ⨿ X_{2} then the diagram commutes

C(X) ⟶^{(id×f)^{∗}} C(X_{1}) × C(X_{2})
N_{X/Y} ↓  ↓^{N_{X_{1}/Y} N_{X_{2}/Y}}
C(Y) ⟶^{id}  C(Y)

(3) normalization: if X = Y and the map X → Y is the identity then N_{X/Y} = id_{C(X)}.

1.2 Geometric lemmas

In this Section few geometric lemmas will be proved. Let k be a field of characteristic zero. The main result of this Section are the lemmas 1.2.3 and 1.2.5. Lemma 1.2.3 is a refinement of [OP, Lemma 2] and Lemma 1.2.5 is a refinement of Quillen’s trick.

Lemma 1.2.1. Let F be a field of characteristic zero and let S be an F -smooth algebra which is a domain of dimension 1. Let 𝔪_{0} be a maximal ideal with S/𝔪_{0} = F. Let 𝔪_{1}, 𝔪_{2}, …, 𝔪_{n} be different maximal ideals of S (it might be that 𝔪_{0} = 𝔪_{i} for an index i). Then there exists an element β ∈ S such that S is finite over F [β] and

(1) the ideals 𝔫_{i} := 𝔪_{i} ∩ F [t] (i = 1, 2, …, n) are all different from each other and different from 𝔫_{0} := 𝔪_{0} ∩ F [t] provided that 𝔪_{0} is different from 𝔪_{i}’s;

(2) the extension S/F [t] is étale at each 𝔪_{i}’s and at 𝔪_{0};

(3) F [β]/𝔫_{i} = S/𝔪_{i} for each i = 1, 2, …, n;

(4) 𝔫_{0} = βF [β].

Proof. Consider a closed imbedding Spec(S) ↪ A_{F}^{n} and use a generic linear projection to A1_{F}.

Lemma 1.2.2. Under the hypotheses of Lemma 1.2.1 let 0 ≠ f ∈ S be an element that does not belong to a maximal ideal different of 𝔪_{0}, 𝔪_{1}, …, 𝔪_{n}. Let N(f) = N_{S/F [β]} be the norm of f. Then for an element β ∈ S satisfying (1) to (4) of Lemma 1.2.1 one has

(a) N(f) = fg for an element g ∈ S; (b) fS + gS = S;

(c) the map F [β]/(N(f)) → S/(f) is an isomorphism.

Proof. It is straightforward.

Lemma 1.2.3. Let A be a local essentially smooth k-algebra with a maximal ideal 𝔪. Let A[t] ⊂ R be an A[t]-algebra which is essentially smooth as a k-algebra and is finite over the polynomial algebra A[t]. Assume R/𝔪R is a domain and assume there exists an element h ∈ R \ 𝔪R such that R_{h} is A-smooth. Let ϵ : R → A be an A-augmentation and I = ker(ϵ). Given an f ∈ R with 0 ≠ ϵ(f) ∈ A one can find an s ∈ R satisfying the following conditions

(1) R is finite over A[s];

(2) R/sR = R/I × R/J for certain ideal J;

(3) J + fR = R;

(4) (s − 1)R + fR = R;

(5) if N(f) = N_{R/A[s]}(f) then N(f) = fg ∈ R for certain g ∈ R (R/A[t] is a projective module since R and A[t] are regular and R is finite over A[t]);

(6) fR + gR = R;

(7) the composition map A[s]/(N(f)) → R/(N(f)) → R/(f) is an isomorphism.

Proof. Replacing t by t−ϵ(t) we may assume that ϵ(t) = 0. Since R is finite over A[t] and R/fR is finite over A we conclude that A[t]/(N(f)) is finite over A, hence A/(tN(f)) is finite over R too. So setting u = tN(f) we get an integral extension A[t]/A[u]. Consider the characteristic polynomial of the operator R ⟶^{tf} R as an A[u]-module operator. This polynomial vanishes on tf and its free coefficient is N(tf) up to a sing. Thus N(tf) = tfg for an element g ∈ R. Now replacing t by u we may assume that R/A[t] is still integral, ϵ(t) = 0 and t ∈ fR. In fact, u = N(ft) = ftg and whence ϵ(u) = ϵ(t)ϵ(fg) = 0 and u ∈ fR.

We use below “bar” to denote the reduction modulo the ideal 𝔪R. Let F = Ā = A/𝔪. By the assumption of the lemma the F -algebra R̄ is an F -smooth domain of dimension 1. Let 𝔪_{1}, 𝔪_{2}, …, 𝔪_{n} be different maximal ideals of R̄ dividing f̄ and let 𝔪_{0} = ker(ϵ̄). Let β ∈ R̄ be such that the extension R̄/F [β] satisfies the conditions (1) to (4) of Lemma 1.2.1.

Let α ∈ R be a lift of β, that is ᾱ = β in R̄. Replacing α by α − ϵ(α) we may assume that ϵ(α) = 0 and still ᾱ = β.

Let α^{n} + p_{1}(t)α^{n−1} + ⋯ + p_{n}(t) = 0 be an integral dependence equation for α. Let N be an integer large than the max{2, deg(p_{i}(t))}, where i = 1, 2, …, n. Then for any r ∈ k^{×} the element s = α − rt^{N} is such that t is integral over A[s]. Thus for any r ∈ k^{×} the ring R is integral over A[s].

On the other hand the element s̄ = ᾱ − rt̄^{N} still satisfies the conditions (1) to (4) of Lemma 1.2.1 because for any i = 1, 2, …, n one has t̄∈ 𝔪_{i}.

We claim that the element s ∈ R is the required element (for almost all r ∈ k^{×}). In fact, for almost all r ∈ k^{×} the element s satisfies the conditions (1) to (4) of Lemma

[OP, Lemma 2]. It remains to show that the conditions (5) to (7) are satisfied for all r ∈ k^{×}.

To prove (5) consider the characteristic polynomial of the operator R ⟶^{f} R as an A[s]- module operator. This polynomial vanishes on f and its free coefficient is ±N(f) (the norm of f). Thus f^{n} − a_{1}f^{n−1} + ⋯ ± N(f) = 0 and N(f) = fg for an element g ∈ R.

To prove (6) one has to check that this g is a unit modulo the ideal fR. It suffices to check that ḡ ∈ R̄ is a unit modulo the ideal f̄R̄. The field F = A/𝔪, the F -algebra S = R̄, its maximal ideals 𝔪_{0}, 𝔪_{1}, …, 𝔪_{n} and the element satisfy the hypotheses of Lemma 1.2.2 with β replaced by s̄. Now ḡ is a unit modulo the ideal f̄R̄ by the item (b) of Lemma 1.2.2.

To prove (7) note that R/fR and A[s]/(N(f)) are finite A-modules. So it remains to check that the map φ : A[s]/(N(f)) → R/fR is an isomorphism modulo the maximal ideal 𝔪. For that it suffices to verify that the map φ̄ : F [s̄]/(N(f̄)) → R̄/f̄R̄ is an isomorphism where N(f̄) := N_{R̄/F [s̄]}. Now φ̄ is an isomorphism by the item (c) of Lemma 1.2.2. The lemma follows.

Corollary 1.2.4. Under the hypotheses of Lemma 1.2.3 let K be the quotient field of A, R_{K} = R ⊗_{A} K and ϵ_{K} = ϵ ⊗_{A} id : R_{K} → K. Consider the inclusion K[s] ⊂ R_{K}. Then the norm N(f) ∈ K[s] does not vanish at the points 1 and 0 of the affine line A1_{K}.

Proof. The condition (4) of 1.2.3 implies that N(f) does not vanish at the point 1. Since ϵ_{K}(f) ≠ 0 ∈ K the conditions (2) and (3) imply that N(f) does not vanish at 0 either.

Lemma 1.2.5. Let X be a geometrically irreducible k-smooth affine variety of dimension d. Let x ∈ X be a closed point, 0 ≠ f ∈ k[X] and Z be the vanishing locus of f. Then there exists a commutative diagram

\xymatrix{X \ar[rr]^{ϱ} \ar[dr]_{q} &  & A^{d} \ar[dl]^{pr} \\  & A^{d−1}}  (3)

with a linear projection pr such that

(1) there exists a principle open set X_{h} ⊂ X containing the fibre q^{−1}q(x) such that the map q : X_{h} → A^{d−1} is smooth;

(2) the fibre q^{−1}q(x) is geometrically irreducible; (3) the map ϱ is finite;

(4) the map q|_{Z} : Z → A^{d−1} is finite;

(5) k(x)=k(q(x)).

Proof. Bertini plus generic projection. Consider a closed imbedding X ↪ A_{k}^{n}, the closure X̄ of X in the projective space P_{k}^{n} and the complement X_{∞} of X in X̄. Let Z̄ be the closure of Z in P_{k}^{n} and Z_{∞} be the complement of Z in Z̄. Since dimX = d one has dimX_{∞} = d − 1. Set P_{k}^{n−1} = P_{k}^{n} − A_{k}^{n}. Now extend the scalars to an algebraic closure k̄ of k. Let x_{1}, x_{2}, …, x_{n} ∈ X(k̄) be the points of x⊗_{k} k̄. Let Π ⊂ Pn−1 ; k̄ be a linear subspace of dimension n − d which is general in the sense that

(i) Π ∩ X_{∞, k̄} consists of finitely many points;

(ii) Π ∩ Z_{∞, k̄} = ∅;

(iii) if < Π, x_{i} > is the space generated by Π and by an x_{i} then the < Π, x_{i} > ∩Xk̄ is k̄-smooth and irreducible for each i = 1, 2, …, n;

(iv) < Π, x_{i} >≠< Π, x_{j} > for i ≠ j. The requirement (iii) can be satisfied by a Bertini type theorem for a hyperplane section crossing a given rational point because we are in characteristic zero.

Since the set of linear spaces Π satisfying conditions (i) to (iii) is an open set of a Grassmannian we may choose a Π which is already defined over the ground field k. We may then choose a hyperplaneplane π ⊂ Π of dimension n − d − 1, defined over k and such that

π ∩ X_{∞, k̄} ≠ ∅.  (4)

Now consider the linear projection P_{k}^{n} \ π → P_{d}^{n}. Its restriction to X̄ defines a finite morphism ρ̄: X̄ → P_{k}^{d}. And its restriction to X defines a finite morphism ρ : X → A_{k}^{d}, because it is a base change of ρ̄ by means of the open imbedding A_{k}^{d} ↪ P_{k}^{d}.

The linear projection with the center Π defines a morphism q : X → A_{k}^{n−1} which clearly fits in the commutative diagram (3) with a linear projection pr : A_{k}^{d} → A_{k}^{d−1}. We claim that these ρ, q and pr are the required morphisms.

To prove this recall that ϱ is finite. The morphism q|_{Z} : Z → A_{k}^{d−1} is finite too, because Π ∩ Z_{∞} = ∅. Condition (5) is satisfied because p ⊗_{k} k̄ separates the closed points x_{i}’s of the scheme x ⊗_{k} k̄. The fibre of the morphism p ⊗_{k} k̄ over the point x_{i} coincides with the intersection < Π, x_{i} > ∩Xk̄ for each i = 1, 2, …, n. After the scalar extension k to k̄ the scheme theoretic fibre of q over q(x) becomes a disjoint union of the fibres of q ⊗_{k} k̄ over the points (q ⊗_{k} k̄)(x_{i}). Since k(q(x)) = k(x) and for each i the intersection < Π, x_{i} > ∩Xk̄ is irreducible we conclude that the fibre q^{−1}q(x) is irreducible too, and thus condition (2) is verified. Since for each i the intersection < Π, x_{i} > ∩Xk̄ is smooth we conclude that the scheme theoretic fibre of q over q(x) is smooth too. Thus condition (1) is satisfied. The lemma is proved.

1.3 Unramified elements

Let k be the characteristic zero field. We work in this section with the category of commutative k-algebras. Let ℱ be a covariant functor from the category of commutative k-algebras to the category of abelian groups. Let K be a field containing k and R ⊂ K be a k-subalgebra whose field of fractions is K. We define the subgroup of R-unramified elements of K as

ℱ_{nr,R}(K) = ⋂_{𝔭∈Spec(R)^{(1)}} Im[ℱ(R_{𝔭}) → ℱ(K)],

where Spec(R)^{(1)} is the set of hight 1 prime ideals in R. Obviously the image of ℱ(R) in ℱ(K) is contained in ℱ_{nr,R}(K). In most cases ℱ(R_{𝔭}) injects into ℱ(K) and ℱ_{nr,R}(K) is simply the intersection of all ℱ(R_{p}).

A basic functor we are interested in is the following one. Let μ : G → C be the morphism of linear algebraic k-groups from Theorem 1.0.1. For a commutative k-algebra R set

ℱ(R) = C(R)/μ(G(R)).  (5)

For an element α ∈ C(R) we will write ᾱ for its image in ℱ(R). In this section we will write ℱ for the functor (5), the only exception being Lemma 1.3.6. We will repeatedly use the following result

Theorem 1.3.1 ([C-T/O]). Let H be a linear algebraic group over the field k. Let R be a discrete valuation ring with a fraction field K. Then the map of pointed sets

H_{ét}^{1}(R, H) → H_{ét}^{1}(K, H)

induced by the inclusion R ⊂ K has the trivial kernel.

Corollary 1.3.2 ([Ni]). Let R be a discrete valuation ring with fraction field K. The map ℱ(R) → ℱ(K) is injective.

Proof. Let H be the kernel of μ. Then the boundary map ∂ : C(R) → H_{ét}^{1}(R, H) fits in a commutative diagram

C(R)/μ(G(R)) ⟶ C(K)/μ(G(K))
↓  ↓
H_{ét}^{1}(R, H) ⟶ H_{ét}^{1}(K, H).

in which the vertical arrows have trivial kernels. The bottom arrow has trivial kernel by Theorem 1.3.1. Thus the top arrow has trivial kernel too.

Lemma 1.3.3. Let μ : G → C be the morphism of linear algebraic groups. Let H = ker(μ). Then for any field K the boundary map ∂ : C(K)/μ(G(K)) → H_{ét}^{1}(K, H) is injective.

Proof. For a K-rational point t ∈ C set H_{t} = μ^{−1}(t). The action by left multiplication of H on H_{t} makes H_{t} into a left principal homogeneous H-space and moreover ∂(t) ∈ H_{ét}^{1}(K, H) coincides with the isomorphism class of H_{t}. Now suppose that s, t ∈ C(K) are such that ∂(s) = ∂(t). This means that H_{t} and H_{s} are isomorphic as principal homogeneous H-spaces. We must check that for certain g ∈ G(K) one has t = sg.

Let K^{sep} be a separable closure of K. Let ψ : H_{s} → H_{t} be an isomorphism of left H-spaces. For any r ∈ H_{s}(K^{sep}) and h ∈ H_{s}(K^{sep}) one has

(hr)^{−1}ψ(hr) = r^{−1}h^{−1}hψ(r) = r^{−1}ψ(r).

Thus for any σ ∈ Gal(K^{sep}/K) and any r ∈ H_{s}(K^{sep}) one has

r^{−1}ψ(r) = (r^{σ})^{−1}ψ(r^{σ}) = (r^{−1}ψ(r))^{σ}

which means that the point u = r^{−1}ψ(r) is a Gal(K^{sep}/K)-invariant point of G(K^{sep}). So u ∈ G(K). The following relation shows that the ψ coincides with the right multiplication by u. In fact, for any r ∈ H_{s}(K^{sep}) one has ψ(r) = rr^{−1}ψ(r) = ru. Since ψ is the right multiplication by u one has t = sμ(u), which proves the lemma.

Let K be a field and x : K^{∗} → ℤ be a discrete valuation vanishing on k. Let A_{x} be the valuation ring of x. Let Â_{x} and K̂_{x} be the completions of A_{x} and K with respect to x. Let i : K ↪ K̂_{x} be the inclusion. By Lemma 1.3.2 the map ℱ(Â_{x}) → ℱ(K̂_{x}) is injective. We will identify ℱ(Â_{x}) with its image under this map. Set

ℱ_{x}(K) = i_{∗}^{−1}(ℱ(Â_{x})).

The inclusion A_{x} ↪ K induces a map ℱ(A_{x}) → ℱ(K) which is injective by Lemma 1.3.2. So both groups ℱ(A) and ℱ_{x}(K) are subgroups of ℱ(K). The following lemma shows that ℱ_{x}(K) coincides with the subgroup of ℱ(K) consisting of all elements unramified at x.

Lemma 1.3.4. ℱ(A_{x}) = ℱ_{x}(K).

Proof. We only have to check the inclusion ℱ_{x}(K) ⊆ ℱ(A_{x}). Let a_{x} ∈ ℱ_{x}(K) be an element. It determines the elements a ∈ ℱ(K) and â ∈ ℱ(Â_{x}) which coincide when regarded as elements of ℱ(K̂_{x}). We denote this common element in ℱ(K̂_{x}) by â_{x}. Let H = ker(μ) and let ∂ : C(−) → H_{ét}^{1}(−, H) be the boundary map.

Let ξ = ∂(a) ∈ H_{ét}^{1}(K, H), ξ̂ = ∂(â) ∈ H_{ét}^{1}(Â_{x}, H) and ξ̂_{x} = ∂(â_{x}) ∈ H_{ét}^{1}(K̂_{x}, H). Clearly, ξ̂ and ξ both coincide with ξ̂_{x} when regarded as elements of H_{ét}^{1}(K̂_{x}, H). Thus one can glue ξ and ξ̂ to get a ξ_{x} ∈ H_{ét}^{1}(A_{x}, H) which maps to ξ under the map induced by the inclusion A_{x} ↪ K and maps to ξ̂ under the map induced by the inclusion A_{x} ↪ Â_{x}.

We now show that ξ_{x} has the form ∂(a′_{x}) for a certain a′_{x} ∈ ℱ(A_{x}). In fact, observe that the image ζ of ξ in H_{ét}^{1}(K, G) is trivial. By Theorem 1.3.1 the map

H_{ét}^{1}(A_{x}, G) → H_{ét}^{1}(K, G)

has trivial kernel. Therefore the image ζ_{x} of ξ_{x} in H_{ét}^{1}(A_{x}, G) is trivial. Thus there exists an element a′_{x} ∈ ℱ(A_{x}) with ∂(a′_{x}) = ξ_{x} ∈ H_{ét}^{1}(A_{x}, H).

We now prove that a′_{x} coincides with a_{x} in ℱ_{x}(K). Since ℱ(A_{x}) and ℱ_{x}(K) are both subgroups of ℱ(K), it suffices to show that a′_{x} coincides with the element a in ℱ(K). By Lemma 1.3.3 the map

ℱ(K) ⟶^{∂} H_{ét}^{1}(K, H)

is \underline{injective}. Thus it suffices to check that ∂(a′_{x}) = ∂(a) in H_{ét}^{1}(K, H). This is indeed the case because ∂(a′_{x}) = ξ_{x} and ∂(a) = ξ, and ξ_{x} coincides with ξ when regarded over K. We have proved that a′_{x} ∈ ℱ(A_{x}) coincides with a_{x} in ℱ_{x}(K). Thus the inclusion ℱ_{x}(K) ⊆ ℱ(A_{x}) is proved, whence the lemma.

Lemma 1.3.5. Let A ⊂ B be a finite extension of Dedekind k-algebras. Let 0 ≠ f ∈ B be such that B/fB is reduced.

Suppose N_{B/A}(f) = fg ∈ B for a certain g ∈ B coprime with f. Suppose the composite map A/N(f)A → B/N(f)B → B/fB is an isomorphism. Let F and E be the quotient fields of A and B respectively. Let β ∈ C(B_{f}) be such that β̄ ∈ ℱ(E) is B-unramified. Then, for α = N_{E/F} (β), the class ᾱ ∈ ℱ(F) is A-unramified.

Proof. The only primes at which ᾱ could be ramified are those which divide N(f). Let 𝔭 be one of them. Check that ᾱ is unramified at 𝔭.

To do this we consider all primes 𝔮_{1}, 𝔮_{2}, …, 𝔮_{n} in B lying over 𝔭. Let 𝔮_{1} be the unique prime dividing f and lying over 𝔭. Then

B̂_{𝔭} = B̂_{𝔮_{1}} × ∏_{i≠1} B̂_{𝔮_{i}}

with B̂_{𝔮_{1}} = Â_{𝔭}. If F, E are the fields of fractions of A and B then

E ⊗ F̂_{𝔭} = Ê_{𝔮_{1}} × ⋯ × Ê_{𝔮_{n}}

and Ê_{𝔮_{1}} = F̂_{𝔭}. We will write Ê_{i} for Ê_{𝔮_{i}} and B̂_{i} for B̂_{𝔮_{i}}. Let β ⊗1 = (β_{1}, …, β_{n}) ∈ C(Ê_{1})× ⋯ × C(Ê_{n}). Clearly for i ≥ 2 β_{i} ∈ C(B̂_{i}) and β_{1} = μ(γ_{1})β′_{1} with β′_{1} ∈ C(B̂_{1}) = C(Â_{𝔭}) and γ_{1} ∈ G(Ê_{1}) = G(F̂_{𝔭}). Now α ⊗ 1 ∈ C(F̂_{𝔭}) coincides with the product

β_{1}N_{L̂_{2}/K̂_{𝔭}}(β_{2}) ⋯ N_{L̂_{n}/K̂_{𝔭}}(β_{n}) = μ(γ_{1})[β′_{1}N_{L̂_{2}/K̂_{𝔭}}(β_{2}) ⋯ N_{L̂_{n}/K̂_{𝔭}}(β_{n})].

Thus \overline{α ⊗ 1} = β̄′_{1}\overline{N_{L̂_{2}/K̂_{𝔭}}(β_{2})} ⋯ \overline{N_{L̂_{n}/K̂_{𝔭}}(β_{n})} ∈ ℱ(Â_{𝔭}). Let i : F ↪ F̂_{𝔭} be the inclusion and i_{∗} : ℱ(F) → ℱ(F̂_{𝔭}) be the induced map. Clearly i_{∗}(ᾱ) = \overline{α ⊗ 1} in ℱ(F̂_{𝔭}). Now Lemma 1.3.4 shows that the element ᾱ ∈ ℱ(F) belongs to ℱ(A_{𝔭}). Hence ᾱ is A-unramified.

Lemma 1.3.6 (Unramifiedness Lemma). Let ℱ be a covariant functor from the category of commutative k-algebras to the category of abelian groups. Let S and R be noetherian domains with quotient fields K and L respectively. Let S ⟶^{i} R be an injective flat homomorphism of finite type and let j : K → L be the induced inclusion of the quotient fields. Then for each localization R′ ⊃ R of R the map

j_{∗} : ℱ(K) → ℱ(L)

takes S-unramified elements to R′-unramified elements.

Proof. Let v ∈ K^{∗} and let 𝔯 be height 1 primes of R′. Then 𝔮 = R ∩ 𝔯 is a height 1 prime of R. Let 𝔭 = S ∩ 𝔮. Since the S-algebra R is flat of finite type one has ht(𝔮) ≥ ht(𝔭). Thus ht(𝔭) is 1 or 0. The commutative diagram

ℱ(K) ⟶ ℱ(L)
↑  ↑
ℱ(S_{𝔭}) ⟶ ℱ(R′_{𝔯})

shows that the class j_{∗}(v) is in the image of ℱ(R′_{𝔯}) and hence the class j_{∗}(v) ∈ ℱ(L) is R′-unramified.

1.4 Specialization maps

In this Section we consider the functor R ↦ ℱ(R) defined by (5). For a regular ring R containing k and each height one prime 𝔭 in R we construct specialization maps s_{𝔭} : ℱ_{nr,R}(K) → ℱ(k(𝔭)), where K is the quotient field of R and k(𝔭) is the residue field of R at the prime 𝔭.

Definition 1.4.1. Let Ev_{𝔭} : C(R_{𝔭}) → C(k(𝔭)) and ev_{𝔭} : ℱ(R_{𝔭}) → ℱ(k(𝔭)) be the maps induced by R_{𝔭} → k(𝔭). Define a homomorphism s_{𝔭} : ℱ_{nr,R}(K) → ℱ(k(𝔭)) by s_{𝔭}(α) = ev_{𝔭}(α̃) where α̃ is a lift of α in ℱ(R_{𝔭}). Corollary 1.3.2 shows that the map s_{𝔭} is welldefined. It is called the specialization map. The map ev_{𝔭} is called the evaluation map at the prime 𝔭.

Obviously for α ∈ C(R_{𝔭}) one has s_{𝔭}(ᾱ) = \overline{Ev_{𝔭}(α)} ∈ ℱ(k(𝔭)).

Lemma 1.4.2 ([H]). Let H be a linear algebraic group over the field k. Let R be a k-algebra which is a Dedekind domain with quotient field K. If ξ ∈ H_{ét}^{1}(K, H) is an R-unramified element for the functor H_{ét}^{1}(−, H) then ξ can be lifted to an element of H_{ét}^{1}(R, H).

Proof. Patching.

Theorem 1.4.3. Let K be a field of characteristic zero and let H be a linear algebraic group over K. Then the canonical map H_{ét}^{1}(K, H) → H_{ét}^{1}(K[t], H) is bijective.

Proof. Let i : {0} ↪ A1_{K} be the origin of the affine line. Let i^{∗} : H_{ét}^{1}(K[t], H) → H_{ét}^{1}(K, H) be the pull-back map. To prove the theorem we need the following Claim 1.4.4. The following two statements are equivalent.

(1) For any linear algebraic group H over K the map i^{∗} is injective.

(2) For any linear algebraic group H over K the map i^{∗} has the trivial kernel.

Clearly the first statement implies the second one. Now suppose the second statement holds and prove the first one. For that consider ξ, ξ′ ∈ H_{ét}^{1}(K[t], H) and set ξ_{0} = i^{∗}(ξ), ξ′_{0} = i^{∗}(ξ′). Let H_{0} be the inner form of the group H with respect to the H-torsor ℋ_{0}. For any K-scheme S there is a well-known bijection ϕ_{S} : H_{ét}^{1}(S, H) → H_{ét}^{1}(S, H_{0}) between non-pointed sets. It takes the H-torsor ℋ_{0} ×_{K} S to the trivial H_{0}-torsor H_{0} ×_{K} S, where ℋ_{0} is an H-torsor over K representing the class ξ_{0}. These bijections respect K-scheme morphisms. Suppose ξ_{0} = ξ′_{0}, then

i^{∗}(ϕ_{A1}(ξ)) = ϕ_{K}(ξ_{0}) = ∗ = ϕ_{K}(ξ′_{0}) = i^{∗}(ϕ_{A1}(ξ′)) ∈ H_{ét}^{1}(K, H_{0}).

So ϕ_{A1}(ξ) = ∗ = ϕ_{A1}(ξ′) ∈ H_{ét}^{1}(K[t], H_{0}). Whence ξ = ξ′ and the injectivity follows. With this Claim in hand prove the theorem as follows.

The surjectivity of the map i^{∗} is obvious. To prove its injectivity just use the Claim and mimic the proof of [C-T/O, Thm.2.1, Property P2]. Let p : A1_{K} → Spec(K) be the structural morphism. Since p ∘ i = id and i^{∗} is bijective p^{∗} is bijective.

We need the following theorem.

Theorem 1.4.5 (Homotopy invariance). Let k be the field and μ : G → C be the algebraic group morphism from Theorem 1.0.1. Let k ⊂ K be a field extension and K(t) be the rational function field in one variable. Let R ↦ ℱ(R) be the functor defined in 5. Then one has

ℱ(K) = ℱ_{nr,K[t]}(K(t)).

Proof. Injectivity is clear, because the composition

ℱ(K) → ℱ_{un}(K(t)) ⟶^{s_{0}} ℱ(K)

coincides with the identity (here s_{0} is the specialization map at the point zero defined in 4.6).

It remains to check the surjectivity. Let a ∈ ℱ_{un}(K(t)) and let H = ker(μ). Then by Lemma 1.3.2 the element ∂(a) ∈ H_{et}^{1}(K(t), H) is a class which for every x ∈ 𝔸_{K}1 belongs to the image of H_{et}^{1}(𝒪_{x}, H). Thus by Lemma 1.4.2, ∂(a) can be represented by an element ξ ∈ H_{et}^{1}(K[t], H), where K[t] is the polynomial ring. By Theorem 1.4.3, the map

H_{et}^{1}(K, H) → H_{et}^{1}(K[t], H)

is an isomorphism. Then ξ = ρ(ξ_{0}) for an element ξ_{0} ∈ H_{et}^{1}(K, H). Consider the diagram

\xymatrix{ & a \ar@{|->}[r] & ξ \ar@{|->}[r] & ∗ \\ 1 \ar[r] & ℱ(K(t)) \ar[r]^{∂} & H_{et}^{1}(K(t), H) \ar[r] & H_{et}^{1}(K(t), G) \\ 1 \ar[r] & ℱ(K) \ar[r]^{∂} \ar[u]_{ϵ} & H_{et}^{1}(K, H) \ar[r] \ar[u]_{ρ} & H_{et}^{1}(K, G) \ar[u]_{η} \\  & a_{0} \ar@{|->}[r] & ξ_{0} ,}

in which all the maps are canonical and all the vertical arrows have trivial kernels. Since ξ goes to the trivial element in H_{et}^{1}(K(t), G), one concludes that ξ_{0} goes to the trivial element in H_{et}^{1}(K, G). Thus there exists an element a_{0} ∈ ℱ(K) such that ∂(a_{0}) = ξ_{0}. The map ℱ(K(t)) → H_{et}^{1}(K(t), H) is injective by Lemma 1.3.3. Thus ϵ(a_{0}) = a.

Corollary 1.4.6. Let R ↦ ℱ(R) be the functor defined in (5). Let

s_{0}, s_{1} : ℱ_{un}(K(t)) → ℱ(K)

be the specialization maps at zero and at one (at the primes (t) and (t-1)). Then s_{0} = s_{1}.

Proof. It is an obvious consequence of Theorem 1.4.5.

1.5 Proof of Theorem (A)

Proof. By assumption there exists a smooth d-dimensional k-algebra S = k[t_{1}, t_{2}, …, t_{n}] and a prime ideal 𝔭 in R such that A = S_{𝔭}. We first reduce the proof to the case in which 𝔭 is maximal. To do this, choose a maximal ideal m containing p. Since k is of characteristic zero it is infinite. By a standard general position argument we can find d algebraically independent elements X_{1}, …, X_{d} such that S is finite over k[X_{1}, …, X_{d}] and étale at m. After a linear change of coordinates we may assume that S/p is finite over B = k[X_{1}, …, X_{m}], where m is the dimension of S/p. Clearly S is smooth over B at m and thus, for some h ∈ S \ m, the localization S_{h} is smooth over B. Let 𝒮 be the set of nonzero elements of B, k′ = 𝒮^{−1}B the field of fractions of B and S′ = 𝒮^{−1}S_{h}. The prime ideal p′ = 𝒮^{−1}p_{h} is maximal in S′, the k′-algebra S′ is smooth and S = S′_{p′}.

From now on and till the end of this proof we assume that A = 𝒪_{X,x} is the local ring of a closed point x of a smooth d-dimensional irreducible affine variety X over k. Replacing k by its algebraic closure in k[X] we may assume furthermore that this X is a k-smooth geometrically irreducible affine variety over k.

Let ξ ∈ C(K) be such that the ξ̄ ∈ ℱ(K) is A-unramified. We may assume that ξ ∈ C(k[X]_{f}) for an appropriative non-zero f ∈ k[X] with a reduced k-algebra k[X]/(f). Shrinking X we may assume that ξ̄ ∈ ℱ(k[X]_{f}) is k[X]-unramified. By Lemma 1.2.5 we can find a commutative diagram

\xymatrix{X \ar[rr]^{ϱ} \ar[dr]_{q} &  & A^{d} \ar[dl]^{pr} \\  & A^{d−1}}  (6)

with a linear projection pr and a function h ∈ k[X] such that the conditions (1) to (5) of are satisfied. Let U = Spec(A) the point x ∈ X is the closed point of U. Consider the Cartesian square of schemes

\xymatrix{ & 𝒳 \ar[r]^{p_{X}} \ar[d]_{p} & X \ar[d]^{q} \\ U \ar[ur]^{δ} \ar[r]^{id} & U \ar[r]^{r} & A^{d−1}}

where r = q|_{U}, 𝒳 = U ×_{A^{d−1}} X, p is the first projection and δ is the diagonal map. Applying base change by means of r to the diagram (6) we get a commutative triangle

\xymatrix{𝒳 \ar[rr]^{λ} \ar[dr]_{p} &  & U × A1 \ar[dl]^{pr} \\  & U}  (7)

with a finite surjective λ. Set f = p_{X}^{∗}(f) and h = p_{X}^{∗}(h), where h is from the condition (1) of Lemma 1.2.5.

Now check that the field k, the k-algebra A with the maximal ideal 𝔪 := 𝔪_{x}, the ring R of regular functions on the affine scheme 𝒳, the inclusion λ^{∗} : A[t] ↪ R induced by λ, the function h, the A-augmentation ϵ := Δ^{∗} : R → A and the function f satisfy the assumption of Lemma 1.2.3.

As a local ring of a k-smooth variety A is essentially k-smooth. Since r is essentially smooth and X is k-smooth, 𝒳 is essentially k-smooth and so is R. Since ϱ is finite R is finite over A[t] too. Since k(q(x)) = k(x) and the fibre of q over q(x) is geometrically irreducible so is the fibre of p over x ∈ U. This implies that R/𝔪 is a domain. By base change, the open subscheme 𝒳_{h} contains the closed fibre of p and the restriction of p to 𝒳_{h} is smooth. The assumptions are satisfied.

Now we can find an s ∈ R satisfying the conditions (1) to (7) of Lemma 1.2.3. The function s defines a finite morphism π : 𝒳 → U × A1. Since 𝒳 and 𝒴 := U × A1 are regular schemes and π is finite surjective it is finite and flat by a theorem of Grothendieck [E, Cor.18.17]. So for any map f : 𝒴′ → 𝒴 of affine schemes, putting 𝒳′ = 𝒳 ×_{𝒴} 𝒴′ we have the norm map given by (2)

C(𝒳′) → C(𝒴′).

Set 𝒟 = Spec(R/J) and 𝒟_{1} = Spec(R/(s − 1)R). Clearly 𝒟_{1} is the scheme theoretic pre-image of U × {1} and the disjoint union 𝒟_{0} := 𝒟 ∐ δ(U) is the scheme theoretic pre-image of U × {0} under the morphism π. In particular 𝒟 is finite flat over U. We will write Δ for δ(U). Recall that ξ ∈ C(k[X]_{f}) and set ζ = p_{X}^{∗}(ξ) ∈ C(R_{f}). Since the function f ∈ R is co-prime to the ideals J and (s−1)R one can form the following element

ξ′ = N_{𝒟_{1}/U} (ζ|_{𝒟_{1}} )N_{𝒟/U} (ζ|_{𝒟} )^{−1} ∈ C(U) := C(A).  (8)

Claim 1.5.1 (Main). One has ξ̄′_{K} = ξ̄_{K} in ℱ(K), where K is the fraction field of both k[X] and k[U] = A.

To complete the proof, it remains to prove the claim. To do this it is convenient to fix some notations. For an A-module M set M_{K} = M ⊗_{A} K, for an 𝒴-scheme 𝒵 set 𝒵_{K} = Spec(K) ×_{𝒴} 𝒵, for a 𝒴-morphism φ : 𝒵 → 𝒲 set φ_{K} = Spec(K) ×_{𝒴} φ. Clearly one has A_{K} = K, U_{K} = Spec(K), 𝒳_{K} = Spec(R_{K}) and (A1 × U_{K})_{K} is just the affine line A1_{K}. Its closed subschemes U_{K} × {1} and U_{K} × {0} coincide with the points 1 and 0 of A1_{K}. The morphism p_{K} : 𝒳_{K} → Spec(K) is smooth by the condition (1) of Lemma 1.2.3. The morphism π_{K} is finite flat and fits in the commutative triangle

\xymatrix{𝒳_{K} \ar[rr]^{π_{K}} \ar[dr]_{p_{K}} &  & A1_{K} \ar[dl] \\  & Spec(K)}  (9)

One has 𝒟_{K} = Spec(R_{K}/J_{K}), 𝒟_{1,K} = Spec(R_{K}/(s − 1)R_{K}). Clearly 𝒟_{1,K} is the scheme theoretic pre-image of the point {1} ∈ A1_{K} and the disjoint union 𝒟_{0,K} := 𝒟_{K} ∐ Δ_{K} is the scheme theoretic pre-image of the point {0} ∈ A1_{K} under the morphism π_{K}. Let

ζ_{s} := N_{K(𝒳_{K})/K(A1)} (ζ) ∈ C(K(A1)) = C(K(s)).

To prove Claim 1.5.1 we need the following one:

Claim 1.5.2. The class ζ̄_{s} ∈ ℱ(K(s)) is K[s]-unramified.

Assuming this claim we complete the proof of Claim 1.5.1. By Claim 1.5.2 we can apply the specialization maps to ζ̄_{s}. By Corollary 1.4.6 the specializations at 0 and 1 of the element ζ̄_{s} coincide, that is

s_{1}(ζ̄_{s}) = s_{0}(ζ̄_{s}) ∈ ℱ(K).  (10)

By Corollary 1.2.4 the function N_{R_{K}/K[s]}(f) ∈ K[s] does not vanishes as at 1, so at 0 and by the condition (5) of Lemma 1.2.3 one has ζ_{s} ∈ C(K[s]_{N(f)}). Using the relation between specialization and evaluation maps described in Definition 1.4.1 one has a chain of relations

\overline{Ev_{1}(ζ_{s})} = s_{1}(ζ̄_{s}) = s_{0}(ζ̄_{s}) = \overline{Ev_{0}(ζ_{s})} ∈ ℱ(K).  (11)

Base change and the multiplicativity properties of the norm map (2) imply relations

Ev_{1}(ζ_{s}) = N_{𝒟_{1,K}/U_{K}} (ζ|_{𝒟_{1,K}} )  (12)

and

Ev_{0}(ζ_{s}) = N_{𝒟_{K} ∐ Δ_{K}/U_{K}} (ζ|_{𝒟_{K} ∐ Δ_{K}} ) = N_{𝒟_{K}/U_{K}} (ζ|_{𝒟_{K}} )N_{Δ_{K}/U_{K}}(ζ|_{Δ_{K}})  (13)

So we have a chain of relations in ℱ(K)

\overline{N_{𝒟_{1,K}/U_{K}} (ζ|_{𝒟_{1,K}} )} = \overline{Ev_{1}(ζ_{s})} = \overline{Ev_{0}(ζ_{s})} = \overline{N_{𝒟_{K}/U_{K}} (ζ|_{𝒟_{K}} )} · \overline{N_{Δ_{K}/U_{K}}(ζ|_{Δ_{K}})}

By the normalization property of the norm map (2) one has N_{Δ_{K}/U_{K}}(ζ|_{Δ_{K}}) = δ_{K}^{∗}(ζ). Since δ_{K}^{∗}(ζ) = ξ_{K} ∈ C(K) we have a chain of relations in ℱ(K)

ξ̄_{K} = \overline{δ_{K}^{∗}(ζ)} = \overline{N_{Δ_{K}/U_{K}}(ζ|_{Δ_{K}})} = \overline{N_{𝒟_{1,K}/U_{K}} (ζ|_{𝒟_{1,K}} )} · (\overline{N_{𝒟_{K}/U_{K}} (ζ|_{𝒟_{K}} )})^{−1}

By the base change property of the norm map (2) one has

ξ′_{K} = N_{𝒟_{1,K}/U_{K}} (ζ|_{𝒟_{1,K}} )N_{𝒟_{K}/U_{K}} (ζ|_{𝒟_{K}} )^{−1} ∈ C(U_{K}) := C(K).  (14)

So ξ̄′_{K} = ξ̄_{K}. Claim 1.5.1 follows.

It remains to prove Claim 1.5.2. Recall that the class ξ̄ ∈ ℱ(k[X]_{f}) is k[X]-unramified. Since q|_{X_{h}} : X_{h} → A^{d−1} is flat (it is smooth) so is r : U ↪ X_{h} → A^{d−1}. Since the p_{X} : 𝒳 → X is the base change of r it coincides with a composition map 𝒳 ↪ X_{h} ×_{A^{d−1}} X ⟶^{q×id} X where q × id is a flat morphism of finite type. By Lemma 1.3.6 the class ζ̄ ∈ ℱ(R_{f}) is R-unramified. Thus the same class ζ̄ when regarded in ℱ(K[𝒳_{K}]_{f}) is K[𝒳_{K}]-unramified.

Now check that the inclusion of K-algebras K[s] ⊂ R_{K} = K[𝒳_{K}] and the function f ∈ R satisfy the hypotheses of Lemma 1.3.5. We first check that the K-algebra R_{K}/fR_{K} is reduced. Recall that the ring k[X]/(f) is reduced. As we mentioned above, the morphism p_{X} : 𝒳 → X is essentially smooth, hence the ring R/(f) is reduced too and thus its localization R_{K}/fR_{K} is reduced. Since the extension K[s] ⊂ R_{K} = K[𝒳_{K}] and the functions f ∈ R satisfy the conditions (5) to (7) of Lemma 1.2.3 they also satisfy the hypotheses of Lemma 1.3.5. Thus by Lemma 1.3.5 the class ζ̄_{s} is K[s]-unramified. This implies Claim 1.5.2. The proof of the theorem is completed.

1.6 Proof of Theorem (B)

Proof. To prove Theorem (B) we now recall a celebrated result of Dorin Popescu (see [P] or, for a self-contained proof, [Sw]).

Let k be a field and R a local k-algebra. We say that R is geometrically regular if k′ ⊗_{k} R is regular for any finite extension k′ of k. A ring homomorphism A → R is called geometrically regular if it is flat and for each prime ideal 𝔮 of R lying over 𝔭, R_{𝔮}/𝔭R_{𝔮} = k(𝔭) ⊗_{A} R_{𝔮} is geometrically regular over k(𝔭) = A_{𝔭}/𝔭_{𝔭}.

Observe that any regular local ring containing a field k is geometrically regular over the prime field of k.

Theorem 1.6.1 (Popescu’s theorem). A homomorphism A → R of noetherian rings is geometrically regular if and only if R is a filtered direct limit of smooth A-algebras.

Proof of Theorem B. Let R be a regular local ring containing the field k. Since k is of characteristic zero one can apply Popescu’s theorem. So R can be presented as a filtered direct limit of smooth k-algebras A_{α}, We first observe that we may replace the direct system of the A_{α}’s by a system of essentially smooth local k-algebras. In fact, if 𝔪 is the maximal ideal of R, we can replace each A_{α} by (A_{α})_{𝔭_{α}}, where 𝔭_{α} = 𝔪 ∩ A_{α}. Note that in this case the canonical morphisms φ_{α} : A_{α} → R are local and every A_{α} is a regular local ring, in particular a factorial ring.

Let now L be the field of fractions of R and, for each α, let K_{α} be the field of fractions of A_{α}. For each index α let 𝔞_{α} be the kernel of the map φ_{α} : A_{α} → R and B_{α} = (A_{α})_{𝔞_{α}}. Clearly, for each α, K_{α} is the field of fractions of B_{α}. The composition map A_{α} → R → L factors through B_{α} and hence it also factors through the residue field k_{α} of B_{α}. Since R is a filtering direct limit of the A_{α}’s we see that L is a filtering direct limit of the B_{α}’s.

Let ξ ∈ C(L) be such that the class ξ̄ ∈ ℱ(L) is R-unramified. We need the following two lemmas.

Lemma 1.6.2. Let B = 𝒪_{X,x} be the local ring of a k-smooth variety at a point x. Let k(x) be its residue field and K be its field of fractions. Let η, ρ ∈ ℱ(B) be such that η_{K} = ρ_{K} ∈ ℱ(K). Then η(x) = ρ(x) ∈ ℱ(k(x)).

Lemma 1.6.3. There exists an index α and an element ξ_{α} ∈ C(B_{α}) such that the class ξ̄_{α} ∈ ℱ(K_{α}) is A_{α}-unramified.

Assuming these two claims we complete the proof as follows. Consider a commutative diagram

\xymatrix{A_{α} \ar[d] \ar[rr]^{φ_{α}} \ar[d] &  & R \ar[d] \\ B_{α} \ar[d] \ar[r] & k_{α} \ar[r] & L \\ K_{α}.}

By Lemma 1.6.3 the class ξ̄_{α} ∈ ℱ(K_{α}) is A_{α}-unramified. Hence by Theorem A there exists an element η ∈ C(A_{α}) such that ξ̄_{α} = η̄ ∈ ℱ(K_{α}). By Lemma 1.6.2 the elements ξ̄_{α} and η̄ have the same image in ℱ(k_{α}). Hence ξ̄ ∈ ℱ(L) coincides with the image of the element φ_{α}(η̄) in ℱ(L). It remains to prove the two Lemmas.

Proof of Lemma 1.6.2. Induction on dim(B). The case of dimension 1 follows fromTheorem 1.3.1. To prove the general case choose an f ∈ B such that η = ρ ∈ ℱ(B_{f}). Let π be a regular parameter in B without common factors with f and let B̄ = B/πB. Then for the image \overline{η − ρ} of η − ρ in ℱ(B̄) we have (\overline{η − ρ})_{f} = \overline{(η − ρ)_{f}} = 0 ∈ ℱ(B̄f̄). By the inductive hypotheses one has \overline{η − ρ} = 0 ∈ ℱ(B̄). Thus η(x) = ρ(x) ∈ ℱ(k(x)).

Proof of Lemma 1.6.3. Choose an f ∈ R such that ξ is defined over R_{f}. Then ξ is ramified at most at those hight one primes 𝔭_{1}, …, 𝔭_{r} which contains f. Since the class ξ̄ ∈ ℱ(L) is R-unramified there exists, for any 𝔭_{i}, an element σ_{i} ∈ G(L) and an element ξ_{i} ∈ C(R_{𝔭_{i}}) such that ξ = μ(σ_{i})ξ_{i} ∈ C(L). We may assume that ξ_{i} is defined over R_{h_{i}} for some h_{i} ∈ R − 𝔭_{i} and that σ_{i} is defined over R_{g_{i}} for some g_{i} ∈ R.

We can find an index α such that A_{α} contains lifts f_{α}, h_{1,α}, …, h_{r,α}, g_{1α}, …, g_{r,α} and moreover

(1) C(A_{α,f_{α}}) contains a lift ξ_{α} of ξ,

(2) C(A_{α,h_{i,α}}) contains a lift of ξ_{i,α} of ξ_{i},

(3) G(A_{α,g_{i,α}}) contains a lift of σ_{i,α} of σ_{i}.

Since none of the f_{α}, h_{1,α}, …, h_{r,α}, g_{1α}, …, g_{r,α} vanishes in R, the elements ξ_{α}, ξ_{1,α}, …, ξ_{ir,α} and σ_{1,α}, …, σ_{r,α} may be regarded as elements of C(B_{α}) and G(B_{α}) respectively.

We know that ξ_{i,α}μ(σ_{i,α}) and ξ_{α} map to the same element in C(L). Hence replacing α by a larger index, we may assume that ξ_{α} = ξ_{i,α}μ(σ_{i,α}) ∈ C(B_{α}). We claim that the class ξ̄_{α} ∈ ℱ(K_{α}) is A_{α}-unramified. To prove this note that the only primes at which ξ̄_{α} could be ramified are those which divide f_{α}. Let 𝔮_{α} be one of them. Check that ξ̄_{α} is unramified at 𝔮_{α}. Let q_{α} ∈ A_{α} be a prime element such that q_{α}A_{α} = 𝔮_{α}. Then q_{α}r_{α} = f_{α} for an element r_{α}. Thus qr = f ∈ R for the images of q_{α} and r_{α} in R. Since the homomorphism φ_{α} : A_{α} → R is local, q ∈ 𝔪_{R}. The relation qr = f shows that q ∈ 𝔭_{i} for some index i. Thus q_{α} ∈ φ_{α}^{−1}(𝔭_{i}) and 𝔮_{α} ⊂ φ_{α}^{−1}(𝔭_{i}). On the other hand h_{i,α} ∈ A_{α} − φ_{α}^{−1}(𝔭_{i}), because h_{i} ∈ R − 𝔭_{i}. Thus h_{i,α} ∈ A_{α} − 𝔮_{α}. Now the relation ξ_{α} = ξ_{i,α}μ(σ_{i,α}) ∈ C(B_{α}) with ξ_{i,α} ∈ C(A_{α,h_{i,α}}) shows that ξ̄_{α} is unramified at 𝔮_{α}. Thus ξ̄_{α} is unramified at each hight one prime in A_{α} containing f_{α}. Since ξ_{α} ∈ C(A_{α,f_{α}}) we conclude that ξ̄_{α} is A_{α}-unramified. The lemma follows. The theorem is proved.

2 Applications

In this Section we prove a purity theorem for reductive groups. Let G be a reductive group over the characteristic zero field k and Z ⟶^{i} G a closed central subgroup of G. Let G′ = G/Z be the factor group, π : G → G′ be the projection. For any k-algebra R consider the boundary operator δ_{π,R} : G′(R) → H_{ét}^{1}(R, Z). It is a group homomorphism [Se, Ch.II, §5.6, Cor.2]. Set

ℱ(R) = H_{ét}^{1}(R, Z)/Im(δ_{π,R}).

Theorem 2.0.4. Let R be a regular local ring containing the field k. The functor ℱ satisfies purity for R. If K is the fraction field of R this statement can be restated in an explicit way as follows:

given an element ξ ∈ H_{ét}^{1}(K, Z) suppose that for each height 1 prime ideal 𝔭 in R there exist ξ_{𝔭} ∈ H_{ét}^{1}(R_{𝔭}, Z), g_{𝔭} ∈ G′(K) with ξ = ξ_{𝔭} + δ_{π}(g_{𝔭}) ∈ H_{ét}^{1}(K, Z). Then there exist ξ_{𝔪} ∈ H_{ét}^{1}(R, Z), g_{𝔪} ∈ G′(K), such that

ξ = ξ_{𝔪} + δ_{π}(g_{𝔪}) ∈ H_{ét}^{1}(K, Z).

Proof. Since G is reductive the group Z is of multiplicative type. So we can find a commutative separable k algebra l and a closed embedding Z ↪ R_{l/k}(𝔾_{m, l}) into the permutation torus T^{+} = R_{l/k}(𝔾_{m, l}). Let G^{+} = (G × T^{+})/Z and T = T^{+}/Z, where Z is embedded in G × T^{+} diagonally. Clearly G^{+}/G = T. Consider a commutative diagram

\xymatrix{ & \{1\} & \{1\} \\  & G′ \ar[r]^{id} \ar[u] & G′ \ar[u] \\ \{1\} \ar[r] & G \ar[r]^{j^{+}} \ar[u]^{π} & G^{+} \ar[r]^{μ^{+}} \ar[u]^{π^{+}} & T \ar[r] & \{1\} \\ \{1\} \ar[r] & Z \ar[r]^{j} \ar[u]^{i} & T^{+} \ar[r]^{μ} \ar[u]^{i^{+}} & T \ar[u]_{id} \ar[r] & \{1\} \\  & \{1\} \ar[u] & \{1\} \ar[u]}

with exact rows and columns. For a local k-algebra A one has H_{ét}^{1}(A, T^{+}) = {∗} by Hilbert 90 and this diagram gives rise to a commutative diagram of pointed sets

\xymatrix{ &  & H_{ét}^{1}(A, G′) \ar[r]^{id} & H_{ét}^{1}(A, G′) \\ G^{+}(A) \ar[r]^{μ_{A}^{+}} & T (A) \ar[r]^{δ_{A}^{+}} & H_{ét}^{1}(A, G) \ar[r]^{j_{∗}^{+}} \ar[u]^{π_{∗}} & H_{ét}^{1}(A, G^{+}) \ar[u]^{π_{∗}^{+}} \\ T^{+}(A) \ar[r]^{μ_{A}} \ar[u]^{i_{∗}^{+}} & T (A) \ar[r]^{δ_{A}} \ar[u]^{id} & H_{ét}^{1}(A, Z) \ar[r]^{μ} \ar[u]^{i_{∗}} & \{∗\} \ar[u]^{i_{∗}^{+}} \\  &  & G′(A) \ar[u]^{δ_{π}}}

with exact rows and columns. It follows that π_{∗}^{+} has trivial kernel and one has a chain of group isomorphisms

H_{ét}^{1}(A, Z)/Im(δ_{π,A}) = ker(π_{∗}) = ker(j_{∗}^{+}) = T (A)/μ^{+}(G^{+}(A)).

Clearly these isomorphisms respect k-homomorphisms of local k-algebras. The functor A ↦ T (A)/μ^{+}(G^{+}(A)) satisfies purity for the regular local k-algebra R by Theorem (B). Hence the functor A ↦ H_{ét}^{1}(A, Z)/Im(δ_{π,A}) satisfies purity for R.

3 Examples

We follow here the notation of The Book of Involutions [KMRT]. The field k is a characteristic zero field. The functors (15) to (29) satisfy purity for regular local rings containing k as follows either from Theorem 2.0.4 or from Theorem (B).

(1) Let G be a simple algebraic group over the field k, Z a central subgroup, G′ = G/Z, π : G → G′ the canonical morphism. For any k-algebra A let δ_{π,R} : G′(R) → H_{ét}^{1}(R, Z) be the boundary operator. One has a functor

R ↦ H_{ét}^{1}(R, Z)/Im(δ_{π,R}).  (15)

(2) Let (A, σ) be a finite separable k-algebra with an orthogonal involution. Let π : Spin(A, σ) → PGO^{+}(A, σ) be the canonical morphism of the spinor k-group scheme to the projective orthogonal k-group scheme. Let Z = ker(π). For a k-algebra R let δ_{R} : PGO^{+}(A, σ)(R) → H_{ét}^{1}(R, Z) be the boundary operator. One has a functor

R ↦ H_{ét}^{1}(R, Z)/Im(δ_{R}).  (16)

In (2a) and (2b) below we describe this functor somewhat more explicitly following [KMRT].

(2a) Let C(A, σ) be the Clifford algebra. Its center l is an étale quadratic k-algebra. Assume that deg(A) is divisible by 4. Let Ω(A, σ) be the extended Clifford group [KMRT, Definition given just below (13.19)]. Let \underline{σ} be the canonical involution of C(A, σ) as it is described in [KMRT, just above (8.11)]. Then \underline{σ} is either orthogonal or symplectic by [KMRT, Prop.8.12]. Let \underline{μ} : Ω(A, σ) → R_{l/k}(𝔾_{m,l}) be the multiplier map defined in [KMRT, just above (13.25)] by \underline{μ}(ω) = \underline{σ}(ω)·ω. Set R_{l} = R⊗_{k} l. For a field or a local ring R one has H_{ét}^{1}(R, Z)/Im(δ_{R}) = R_{l}^{×}/\underline{μ}(Ω(A, σ)(R)) by [KMRT, the diagram in (13.32)]. Consider the functor

R ↦ R_{l}^{×}/\underline{μ}(Ω(A, σ)(R)).  (17)

It coincides with the functor R ↦ H_{ét}^{1}(R, Z)/Im(δ_{R}) on local rings containing k.

(2b) Now let deg(A) = 2m with odd m. Let τ : l → l be the involution of l/k. The kernel of the morphism R_{l/k}(𝔾_{m,l}) ⟶^{id−τ} R_{l/k}(𝔾_{m,l}) coincides with 𝔾_{m,k}. Thus id−τ induces a k-group scheme morphism which we denote \overline{id − τ} : R_{l/k}(𝔾_{m,l})/𝔾_{m,k} ↪ R_{l/k}(𝔾_{m,l}). Let \underline{μ} : Ω(A, σ) → R_{l/k}(𝔾_{m,l}) be the multiplier map defined in [KMRT, just above (13.25)] by \underline{μ}(ω) = \underline{σ}(ω) · ω. Let κ : Ω(A, σ) → R_{l/k}(𝔾_{m,l})/𝔾_{m,k} be the k-group scheme morphism described in [KMRT, Prop.13.21]. The composition \overline{(id − τ)} ∘ κ lands in R_{l/k}(𝔾_{m,l}). Let U ⊂ 𝔾_{m,k} × R_{l/k}(𝔾_{m,l}) be a closed k-subgroup consisting of all (α, z) such that α^{4} = N_{l/k}(z).

Set μ_{∗} = (\underline{μ}, [\overline{(id − τ)} ∘ κ] · \underline{μ}2) : Ω(A, σ) → 𝔾_{m,k} × R_{l/k}(𝔾_{m,l}). This k-group scheme morphism lands in U. So we get a k-group scheme morphism μ_{∗} : Ω(A, σ) → U. On the level of k-rational points it coincides with the one described in [KMRT, just above (13.35)]. For a field or a local ring one has

H_{ét}^{1}(R, Z)/Im(δ_{R}) = U(R)/[{(N_{l/k}(α), α^{4})|α ∈ R_{l}^{×}} · μ_{∗}(Ω(A, σ)(R))].

Consider the the functor

R ↦ U(R)/[{(N_{l/k}(α), α^{4})|α ∈ R_{l}^{×}} · μ_{∗}(Ω(A, σ)(R))].  (18)

It coincides with the functor R ↦ H_{ét}^{1}(R, Z)/Im(δ_{R}) on local rings containing k.

(3) Let Γ(A, σ) be the Clifford group k-scheme of (A, σ). Let Sn : Γ(A, σ) → 𝔾_{m,k} be the spinor norm map. It is dominant. Consider the functor

R ↦ R^{×}/Sn(Γ(A, σ)(R)).  (19)

Purity for this functor was originally proved in [Z, Thm.3.1]. In fact, Γ(A, σ) is k-rational.

(4) We follow here the Book of Involutions [KMRT, §23]. Let A be a separable finite dimensional k-algebra with center l and k-involution σ such that k coincides with all σ-invariant elements of l, that is k = l^{σ}. Consider the k-group schemes of similitudes of (A, σ):

Sim(A, σ)(R) = {a ∈ A_{R}^{×} |a · σ_{R}(a) ∈ l_{K}^{×}}.

We have a k-group scheme morphism μ : Sim(A, σ) → 𝔾_{m,k}, a ↦ a · σ(a). It gives an exact sequence of algebraic k-groups

{1} → Iso(A, σ) → Sim(A, σ) → 𝔾_{m,k} → {1}.

One has a the functor

R ↦ R^{×}/μ(Sim(A, σ)(R)).  (20)

Purity for this functor was originally proved in [Pa, Thm.1.2]. Various particular cases are obtained considering unitary, symplectic and orthogonal involutions.

(4a) In the case of an orthogonal involution σ the connected component GO^{+}(A, σ) [KMRT, (12.24)] of the similitude k-group scheme GO(A, σ) := Sim(A, σ) has the index two in GO(A, σ). The restriction of μ to GO^{+}(A, σ) is still a dominant morphism to 𝔾_{m,k}. One has a functor

R ↦ R^{×}/μ(GO^{+}(A, σ)(R))  (21)

It seems that its purity does not follow from [Pa, Thm.1.2]. In fact we do not know whether the norm principle holds for μ : GO^{+}(A, σ) → 𝔾_{m,k} or not.

(5) Let A be a central simple algebra (csa) over k and Nrd : GL_{1,A} → 𝔾_{m,k} the reduced norm morphism. One has a functor

R ↦ R^{×}/Nrd(GL_{1,A}(R)).  (22)

Purity for this functor was originally proved in [C-T/O, Thm.5.2].

(6) Let (A, σ) be a finite separable k-algebra with a unitary involution such that its center l is a quadratic extension of k. Let U(A, σ) be the unitary k-group scheme. Let U_{l}(1) be an algebraic tori given by N_{l/k} = 1. One has a functor

R ↦ U_{l}(1)(R)/Nrd(U_{A,σ}(R)) = {α ∈ R_{l}^{×}|N_{l/k}(α) = 1}/Nrd(U_{A,σ}(R))  (23)

where Nrd is the reduced norm map. Purity for this functor was originally proved in [Z, Thm.3.3].

(7) With the notation of example (5) choose an integer d and consider the k-group scheme morphism μ : GL_{1,A} × 𝔾_{m,k} → 𝔾_{m,k} given by (α, a) ↦ Nrd(α) · a^{d}. One has a functor

R ↦ R^{×}/μ[GL_{1,A} × 𝔾_{m,k})(R)] = R^{×}/Nrd(A_{R}^{×}) · R^{×}^{d}  (24)

Purity for this functor was originally proved in [Z, Thm.3.2].

(8) With the notation of example (5) choose an integer d and consider the functor

R ↦ U_{l}(1)(R)/[Nrd(U_{A,σ}(R)) · {α ∈ R_{l}^{×}|N_{l/k}(α) = 1}^{d}]  (25)

Purity for this functor was originally proved in [Z, Thm.3.2].

(9) Let G_{1},G_{2},C be affine k-group schemes. Assume that C is commutative and let μ_{1} : G_{1} → C, μ_{2} : G_{2} → C be k-group scheme morphisms and μ : G_{1} × G_{2} → C be given by μ(g_{1}, g_{2}) = μ_{1}(g_{1}) · μ_{2}(g_{2}). One has a functor

R ↦ C(R)/μ[(G_{1} × G_{2})(R)] = C(R)/[μ_{1}(G_{1}(R)) · μ_{2}(G_{2}(R))]  (26)

In this style one could get a lot of curious examples of functors, one of which is given here.

(10) Let (A_{1}, σ_{1}) be a finite separable k-algebra with an orthogonal involution. Let A_{2} be a csa over k. Let μ_{1} be the multiplier map for (A_{1}, σ_{1}) and Nrd_{2} be the reduced norm for A_{2}. One has a functor

R ↦ R^{×}/[μ_{1}(GO^{+}(A_{1}, σ_{1})(R)) · Nrd_{2}(A_{2,R}^{×}) · R^{×}^{d}].  (27)

(10) Let A be a csa of degree 3 over k, Nrd the reduced norm and Trd be the reduced trace. Consider the cubic form on the 27-dimensional k-vector space A × A × A given by N := Nrd(x) + Nrd(y) + Nrd(z) − Trd(xyz). Let Iso(A, N) be the k-group scheme of isometries of N and Sim(N) be the k-group scheme of similitudes of N. It is known that Iso(N) is a normal algebraic subgroup in Sim(A, N) and the factor group coincides with 𝔾_{m,k}. So we have a canonical k-group morphism (the multiplier) μ : Sim(N) → 𝔾_{m,k}. Now one has a functor

R ↦ R^{×}/μ(Sim(N)(R)).  (28)

Note that the connected component of Iso(N) is a simply connected algebraic kgroup of the type E_{6}.

(11) Let (A, σ) be a csa of degree 8 over k with a symplectic involution. Let V ⊂ A be the subspace of all skew-symmetric elements. It is of dimension 28. Let Pfd be the reduced Pfaffian on V and Trd be the reduced trace on A. Consider the degree 4 form on the space V × V given by F := Pfr(x) + Pfr(y) − 1/4Trd((xy)2) − 1/16Trd(xy)^{2}. Consider the symplectic form on V × V given by ϕ((x_{1}, y_{1}), (x_{2}, y_{2})) = Trd(x_{1}y_{2} − x_{2}y_{1}). Let Iso(F) (resp. Iso(ϕ)) be the k-group scheme of isometries of the pair F (resp. of ϕ)). Let Sim(F) (resp. Sim(ϕ)) be the k-group scheme of similitudes of F (resp. of ϕ). Set G = Iso(F) ∩ Iso(ϕ) and G^{+} = Sim(F) ∩ Sim(ϕ). It is known that G is a normal algebraic subgroup in G^{+} and the factor group coincides with 𝔾_{m,k}. So we have a canonical k-group morphism μ : G^{+} → 𝔾_{m,k}. Now one has a functor

R ↦ R^{×}/μ(G^{+}(R)).  (29)

Note that G is a simply-connected group of the type E_{7}.

References

[C-T/O] Colliot-Thélène, J.-L.; Ojanguren, M. Espaces Principaux Homogènes Localement Triviaux, Publ. Math. IHES 75, no. 2, 97–122 (1992).

[C-T/P/S] Colliot-Thélène, J.-L.; Parimala, R.; Sridaran, R. Un théorème de pureté local, C. R. Acad. Sc. Paris 309, ser.I, 857–862 (1989).

[E] Eisenbud, D. Commutative Algebra, Graduate Texts in Mathematics 150, Springer (1994).

[KMRT] Knus, M.A.; Merkurjev, A.S.; Rost, M.; Tignol, J.-P., The Book of Involutions, Colloquium Publication, no.44, AMS, 1998.

[H] Harder, G. Halbeinfache Gruppenschemata über Dedekindringen, Inventiones Math. 4, 165–191 (1967).

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

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

[OP] Ojanguren, M; Panin, I. A purity theorem for the Witt group.

Ann. Sci. Ecole Norm. Sup. (4) 32, no. 1, 71–86 (1999).

[PS] Panin, I.; Suslin, A. On a conjecture of Grothendieck concerning Azumaya algebras, St.Petersburg Math. J., 9, no.4, 851–858 (1998).

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

[Pa] Panin, I. Purity for similarity factors, www.math.uni-bielefeld.de/LAG/man/119

[R1] Raghunathan, M.S. Principal bundles admitting a rational section. (English) Invent. Math. 116, No.1–3, 409–423 (1994).

[R2] Raghunathan, M.S. Erratum: Principal bundles admitting a rational section. (English) Invent. Math. 121, No.1, 223 (1995).

[RR] Raghunathan, M.S.; Ramanathan, A. Principal bundles on the affine line. (English) Proc. Indian Acad. Sci., Math. Sci. 93, 137–145 (1984).

[R] Rost, M. Durch Normengruppen definierte birationale Invarianten. C. R. Acad. Sc. Paris 10, ser.I, 189–192 (1990).

[Q] Quillen, D. Higher algebraic K-theory I, in Algebraic K-Theory I.

Lect. Notes in Math. 341, Springer-Verlag, 1973.

[Se] Serre, J-P. Cohomologie Galoisienne. Springer-Verlag (1964).

[SV] Suslin, A.; Voevodsky, V. Durch Normengruppen definierte birationale Invarianten. C. R. Acad. Sc. Paris 10, ser.I, 189–192 (1990).

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

[Z] Zainoulline, K. The Purity Problem for Functors with Transfers, K-theory, 22 (2001), no.4, 303-333.