On Grothendieck—Serre’s conjecture concerning principal G-bundles over reductive group schemes:II

Ivan Panin ^{∗}

08.05.2009

Abstract

Let R be a regular semi-local ring containing an infinite perfect subfield and let K be its field of fractions. Let G be a reductive R-group scheme satifying a mild ”isotropy condition”. Then each principal G-bundle P which becomes trivial over K is trivial itself. If R is of geometric type, then it suffices to assume that R is of geometric type over an infinite field. Two main Theorems of Panin’s, Stavrova’s and Vavilov’s [PSV] state the same results for semi-simple simply connected R-group schemes. Our proof is heavily based on those two Theorems of [PSV, Thm.1.1], on the main result of [C-T/S] and on two purity Theorems proven in the present preprint.

Those purity result look as follows. Given an R-torus C and a smooth R-group scheme morphism μ : G → C one can form a functor from R-algebras to abelian groups S ↦ ℱ(S) := C(S)/μ(G(S)). We prove that this functor satisfies a purity theorem for R. If R is of geometric type, then it suffices to assume that R is of geometric type over an infinite field.

Examples to mentioned purity results are considered in the very end of the preprint.

1 Introduction

Recall that an R-group scheme G is called reductive (respectively, semi-simple; respectively, simple), if it is affine and smooth as an R-scheme and if, moreover, for each ring homomorphism s : R → Ω(s) to an algebraically closed field Ω(s), its scalar extension G_{Ω(s)} is a reductive (respectively, semi-simple; respectively, simple) algebraic group over Ω(s). The class of reductive group schemes contains the class of semi-simple group schemes which in turn contains the class of simple group schemes. This notion of a simple R-group scheme coincides with the notion of a simple semi-simple R-group scheme from Demazure—Grothendieck [D-G, Exp. XIX, Defn. 2.7 and Exp. XXIV, 5.3]. Throughout the paper R denotes an integral domain and G denotes a semi-simple R-group scheme, unless explicitly stated otherwise.

∗The author acknowledges support of the joint DFG - RFBR-project 09-01-91333-NNIO-a

A well-known conjecture due to J.-P. Serre and A. Grothendieck [Se, Remarque, p.31], [Gr1, Remarque 3, p.26-27], and [Gr2, Remarque 1.11.a] asserts that given a regular local ring R and its field of fractions K and given a reductive group scheme G over R the map

H_{ét}^{1}(R, G) → H_{ét}^{1}(K, G),

induced by the inclusion of R into K, has trivial kernel.

Let R be a semi-local ring and G be a reductive group scheme over R. Recall an ”isotropy condition ” for G.

(I) Each R-simple component of the derived group G_{der} of the group G is isotropic over R.

The hypotheses (I) means more precicely the following : consider the derived group G_{der} of the group G and the simply-connected cover G_{der}^{sc} of G_{red}; that group G_{der}^{sc} is a product over R (in a unique way) of R-indecomposable groups, which are required to be isotropic.

Theorem 1.0.1. Let R be regular semi-local domain containing an infinite perfect field and let K be its field of fractions. Let G be a reductive R-group scheme satisfying the condition (I). Then the map

H_{ét}^{1}(R, G) → H_{ét}^{1}(K, G),

induced by the inclusion R into K, has trivial kernel.

In other words, under the above assumptions on R and G each principal G-bundle P over R which has a K-rational point is itself trivial.

• One can ask whether two elements ξ, ζ ∈ H_{ét}^{1}(R, G) which are equal over K are equal over R. Theorem 1.0.1 does not imply that in general. However it does imply that provided that at least one of the two group schemes G(ξ) or G(ζ) is isotropic.

• Clearly, this Theorem extends as the geometric case of the main result of J.-L. Colliot-Thélène and J.-J. Sansuc [C-T/S], so main results of I.Panin, A.Stavrova and N.Vavilov [PSV, Thm.1.1, Thm. 1.2]. However our proof of Theorem 1.0.1 is based heavily on those results and on two purity results (Theorems 1.0.3 and 1.0.4) proven in the present preprint.

• The case of arbitrary reductive group scheme over a discrete valuation ring is completely solved by Y.Nisnevich in [Ni].

• The case when G is an arbitrary tori over a regular local ring is done by J.-L. Colliot-Thélène and J.-J. Sansuc in [C-T/S].

• For simple group schemes of classical series this result follows from more general results established by the first author, A. Suslin, M. Ojanguren and K. Zainoulline [PS], [OP], [Z], [OPZ], [Pa]. In fact, unlike our Theorem 1.0.1, no isotropy hypotheses was imposed there. However our result is new, say for simple adjoint groups of exceptional type (and for many others).

• The case of arbitrary simple adjoint group schemes of type E_{6} and E_{7} is done by the first author, V.Petrov and A.Stavrova in [PaPS].

• There exists a folklore result, concerning type G_{2}. It gives affirmative answer in this case, also independent of isotropy hypotheses, see the paper by V. Chernousov and the first author [ChP].

• The case when the group scheme G comes from the ground field k is completely solved by J.-L. Colliot-Thélène, M. Ojanguren, M. S. Raghunatan and O. Gabber in [C-T/O], when k is perfect, in [R1]; O. Gabber announced a proof for a general ground field k.

A geometric counterpart of Theorem 1.0.1 is the following result

Theorem 1.0.2. Let k be an infinite field. Let 𝒪 be the semi-local ring of finitely many points on a smooth irreducible k-variety X and let K be its field of fractions of 𝒪. Let G be a reductive 𝒪-group scheme satisfying the condition (I) above. Then the map

H_{ét}^{1}(𝒪, G) → H_{ét}^{1}(K, G),

induced by the inclusion 𝒪 into K, has trivial kernel.

In other words, under the above assumptions on R and G each principal G-bundle P over R which has a K-rational point is itself trivial.

To state our purity Theorems recall certain notions. Let ℱ be a covariant functor from the category of commutative R-algebras 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 and K is the field of fractions 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)].

Theorem 1.0.3 (A). Let 𝒪 be a semi-local ring of finitely many points on a k-smooth scheme X with an infinite field k. Let

μ : G → C

be a smooth 𝒪-morphism of reductive algebraic 𝒪-group schemes, with a torus C. The functor

ℱ : S ↦ C(S)/μ(G(S))

defined on the category of 𝒪-algebras satisfies purity for 𝒪.

Theorem 1.0.4 (B). Let R be a regular local ring containing an infinite perfect field k. Let

μ : G → C

be a smooth R-group scheme morphism to a tori C. The functor

ℱ : S ↦ C(S)/μ(G(S))

defined on the category of R-algebras satisfies purity for R.

If K is the field of fractions of R, then 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 of reductive R-group schemes with 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.

Another functor for which one should expect purity is the following one. Let R be a regular local ring, G and G′ reductive R-group schemes, π : G → G′ a smooth central R-group scheme morphism. Assume Z = ker(π) is finite étale of multiplicative type over R. The boundary operator δ_{π,R} : G′(R) → H_{ét}^{1}(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. Theorem 12.0.34 states that this is the case if R contains an infinite perfect field k.

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

The preprint is organized as follows. In Section 2 we construct norm maps following a method from [SuVo, Sect.6]. In Section 3 we prove the main geometric Lemma 3.0.8. In Section 4 we discuss unramified elements. A key point here is Lemma 4.0.13. In Section 5 we discuss specialization maps. A key point here is Corollary 5.0.20. In Section 7 certain Artin’s results are generalized. In Section 8 a convenient technical tool is introduced. In Section 10 Theorem (A) is proved. In Section 11 Theorem (B) is proved. In Section 12 we consider the functor (1) and prove a purity Theorem 12.0.34 for that functor. In Section 13 Theorems 1.0.1 and 1.0.2 are proved. Finally in Section 14 we collect several examples illustrating our Purity Theorems.

Acknowledgments The author thanks very much M.Ojanguren for useful discussions concerning purity results obtained in the present article. The author thanks very much N.Vavilov and A.Stavrova. Long discussions with them over our joint work [PSV] resulted unexpectedly to me with the present preprint.

2 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 [SuVo, 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 2.0.5. 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)}.

3 Geometric lemmas

The aim of this Section is to prove an analog of three Lemmas from [?] making them characteristic free. Lemma 3.0.8 is a refinement of [OP, Lemma 2].

Lemma 3.0.6. Let k be an infinite field and let S be an k-smooth equi-dimensional kalgebra of dimension 1. Let f ∈ S be a non-zero divisor

Let 𝔪_{0} be a maximal ideal with S/𝔪_{0} = k. 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 a non-zero divisor s̄ ∈ S such that S is finite over k[s̄] and

(1) the ideals 𝔫_{i} := 𝔪_{i} ∩ k[s̄] (i = 1, 2, …, n) are all different from each other and different from 𝔫_{0} := 𝔪_{0} ∩ k[s̄] provided that 𝔪_{0} is different from 𝔪_{i}’s;

(2) the extension S/k[s̄] is étale at each 𝔪_{i}’s and at 𝔪_{0};

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

(4) 𝔫_{0} = s̄k[s̄].

Proof. For each index i let x_{i} be the point on Spec(S) corresponding to the ideal 𝔪_{i}. Consider a closed imbedding Spec(S) ↪ A_{k}^{n} and find a generic k-defined linear projection p : A_{k}^{n} → A_{k}^{1} to such that for each index i one has

(1) for each i ≥ 0, j ≥ 0 one has p(x_{i}) ≠ p(x_{j}) provided that x_{i} ≠ x_{j};

(2) for each index i ≥ 0 the map p|_{Spec(S)} : Spec(S) → A^{1} is étale at the point x_{i}; (3) the separable degree extension k(x_{i})/k(p(x_{i})) is one.

With this in hand one gets for each the equality k(p(x_{i})) = k(x_{i}). In fact the extension k(x_{i})/k(p(x_{i})) is separable by the item (2). By the item (3) we conclude that k(p(x_{i})) = k(x_{i}). The Lemma follows.

Lemma 3.0.7. Under the hypotheses of Lemma 3.0.6 let f ∈ S be a non-zero divisor which does not belong to a maximal ideal different of 𝔪_{0}, 𝔪_{1}, …, 𝔪_{n}. Let N(f) = N_{S/k[s]}(f) be the norm of f. Then for an element s̄ ∈ S satisfying (1) to (4) of Lemma 3.0.6 one has

(a) N(f) = fg for an element g ∈ S;

(b) fS + gS = S;

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

Proof. It is straightforward.

Lemma 3.0.8. Let R be a semi-local essentially smooth k-algebra with maximal ideals 𝔭_{i}’s, where i runs from 1 to r. Let R[t] ⊂ A be an R[t]-algebra which is smooth as an R-algebra and is finite over the polynomial algebra R[t]. Assume that for each index i the R/𝔭_{i}-algebra A_{i} = A/𝔭_{i}A is equi-dimensional of dimension one. Let ϵ : A → R be an R-augmentation and I = ker(ϵ). Given an f ∈ A with 0 ≠ ϵ(f) ∈ A and such that the R-module A/fA is finite, one can find an u ∈ A satisfying the following conditions

(1) A is finite projective as an R[u]-module;

(2) A/uA = A/I × A/J for certain ideal J;

(3) J + fA = A;

(4) (u − 1)A + fA = A;

(5) if N(f) = N_{A/R[u]}(f), then N(f) = fg ∈ A for certain g ∈ A

(6) fA + gA = A;

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

Proof. Replacing t by t − ϵ(t) we may assume that ϵ(t) = 0. Since A is finite over R[t] we conclude that it is a finite projective R[t]-module by a theorem of Grothendieck [E, Cor.17.18]. Since A is finite over R[t] and A/fA is finite over R we conclude that R[t]/(N(f)) is finite over R, hence R/(tN(f)) is finite over R too. So setting v = tN(f) we get an integral extension A over R[v] and one has equalities

v = tN_{A/R[t]}(f) = (ft)g = tfg.

We claim that A/R[v] is integral, ϵ(v) = 0 and u ∈ fA. In fact, u = tN_{A/R[t]}(f) = t(fg) and whence ϵ(v) = ϵ(t)ϵ(fg) = 0. Finally, and v ∈ fA.

We use below “bar” and lower script i to denote the reduction modulo the ideal 𝔭_{i}A. Let l_{i} = R̄_{i} = R/𝔭_{i}. By the assumption of the lemma the l_{i}-algebra Ā_{i} is an l_{i}-smooth equi-dimensional of dimension 1. Let 𝔪_{1}^{(i)}, 𝔪_{2}^{(i)}, …, 𝔪_{n}^{(i)} be different maximal ideals of Ā_{i} dividing f̄_{i} and let 𝔪_{0}^{(i)} = ker(ϵ̄_{i}). Let s̄_{i} ∈ Ā_{i} be such that the extension Ā_{i}/l_{i}[s̄_{i}] satisfies the conditions (1) to (4) of Lemma 3.0.6.

Let s ∈ A be a common lift of s̄_{i}’s , that is for each index i one has \overline{s}_{i} = s̄_{i} in Ā_{i}. Replacing s by s − ϵ(s) we may assume that ϵ(s) = 0 and still for each index i one has \overline{s}_{i} = s̄_{i}.

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

On the other hand for each index i the element ū_{i} = s̄_{i} − rv̄_{i}^{N} still satisfies the conditions (1) to (4) of Lemma 3.0.6 because for each index j = 0, 1, 2, …, n one has v̄_{i} ∈ 𝔪_{j}^{(i)}.

We claim that the element u ∈ R is the required element (for almost all r ∈ k^{×}).

In fact, for almost all r ∈ k^{×} the element u satisfies the conditions (1) to (4) of Lemma [OP, Lemma 5.2]. It remains to show that the conditions (5) to (7) are satisfied for all r ∈ k^{×}.

Since A is finite over R[u] we conclude that it is a finite projective R[u]-module by a theorem of Grothendieck [E, Cor.18.17] To prove (5) consider the characteristic polynomial of the operator A ⟶^{f} A as an R[u]-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 fA. It suffices to check that for each index i the element ḡ_{i} ∈ Ā_{i} is a unit modulo the ideal f̄_{i}Ā_{i}. The field l_{i} = R/𝔭_{i}, the l_{i}-algebra S_{i} = Ā_{i}, its maximal ideals 𝔪_{0}^{(i)}, 𝔪_{1}^{(i)}, …, 𝔪_{n}^{(i)} and the element ū_{i} satisfy the hypotheses of Lemma 3.0.7 with u replaced by ū_{i}. Now ḡ_{i} is a unit modulo the ideal f̄_{i}R̄_{j} by the item (b) of Lemma 3.0.7.

To prove (7) note that R[u]/(N_{A/k[X]}(f)) are A/fA finite A-modules. So it remains to check that the map φ : R[u]/(N_{A/k[X]}(f)) → A/fA is an isomorphism modulo each maximal ideal 𝔪_{i}^{(j)}. For that it suffices to verify that the map φ̄_{i} : l_{i}[ū_{i}]/(N(f̄_{i})) → Ā_{i}/f̄_{i}Ā_{i} is an isomorphism for each index i, where N(f̄_{i}) := N_{Ā_{i}/l_{i}[ū]}(f̄_{i}). Now φ̄_{i} is an isomorphism by the item (c) of Lemma 3.0.7. The lemma follows.

Corollary 3.0.9. Under the hypotheses of Lemma 3.0.8 let K be the field of fractions of R, A_{K} = K ⊗_{R} A and ϵ_{K} = id_{K} ⊗ ϵ : A_{K} → K. Consider the inclusion K[u] ⊂ A_{K}. Then the norm N(f) = N_{A_{K}/K[u]}(f) ∈ K[u] does not vanish at the points 1 and 0 of the affine line A_{K}^{1}.

Proof. The condition (4) of 3.0.8 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.

4 Unramified elements

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

ℱ_{nr,S}(K) = ⋂_{𝔭∈Spec(S)^{(1)}} Im[ℱ(S_{𝔭}) → ℱ(K)],  (3)

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

A basic functor we are interested in is the following one. Let μ : G → C be the morphism of reductive R-group schemes from Theorem 1.0.3. For a commutative Ralgebra S set

ℱ(S) = C(S)/μ(G(S)).  (4)

For an element α ∈ C(S) we will write ᾱ for its image in ℱ(S). In this section we will write ℱ for the functor (7), the only exception being Lemma 4.0.14. We will repeatedly use the following result

Theorem 4.0.10 ([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 a Theorem from [Ni]. Thus the top arrow has trivial kernel too.

Lemma 4.0.11. Let μ : G → C be the morphism of our reductive group schemes. Let H = ker(μ). Then for an R-algebra K, where K is a field, 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 the field of fractions of R and 𝒦 be a field containing K 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 4.0.10 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 4.0.10. 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 4.0.12. 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 [Ni] 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 4.0.11 the map

ℱ(𝒦) ⟶^{∂} H_{ét}^{1}(𝒦, H)  (5)

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 4.0.13. Let A ⊂ B be a finite extension of Dedekind K-algebras. Let 0 ≠ f ∈ B be such that B/fB is finite over K and 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 field of fractions 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 4.0.12 shows that the element ᾱ ∈ ℱ(F) belongs to ℱ(A_{𝔭}). Hence ᾱ is A-unramified.

Lemma 4.0.14 (Unramifiedness Lemma). Let ℱ be a covariant functor from the category of commutative R-algebras to the category of abelian groups. Let S′ and R′ be two Ralgebras which are noetherian domains with fields of fractions K′ and L′ respectively. Let S′ ⟶^{i} R′ be an injective flat R-algebra homomorphism of finite type and let j : K′ → L′ be the induced inclusion of the field of fractions. 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 ∈ ℱ(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.

5 Specialization maps

Let K be an infinite field. Let

μ : G → C  (6)

be a smooth K-morphism of reductive algebraic K-group schemes, with a torus C. We work in this section with the category of commutative K-algebras and with the functor

ℱ : S ↦ C(S)/μ(G(S))  (7)

defined on the category of A-algebras. So, we assume in this Section that each ring from this Section is equipped with a distinguished K-algebra structure and each ring homomorphism from this Section respects that structures.

For a regular ring R with the fraction field 𝒦 and each height one prime 𝔭 in R we construct specialization maps s_{𝔭} : ℱ_{nr,R}(𝒦) → ℱ((𝔭)), where 𝒦 is the field of fractions of R and k(𝔭) is the residue field of R at the prime 𝔭.

Definition 5.0.15. Let Ev_{𝔭} : C(R_{𝔭}) → C(k(𝔭)) and ev_{𝔭} : ℱ(R_{𝔭}) → ℱ(k(𝔭)) be the maps induced by the canonical ring homomorphism R_{𝔭} → k(𝔭). Define a homomorphism s_{𝔭} : ℱ_{nr,R}(𝒦) → ℱ((𝔭)) by s_{𝔭}(α) = ev_{𝔭}(α̃), where α̃ is a lift of α in ℱ(R_{𝔭}). Theorem 4.0.10 shows that the map s_{𝔭} is well-defined. 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 5.0.16 ([H]). Let H be a smooth linear algebraic group over the field K. Let R be a K-algebra which is a Dedekind domain with field of fractions 𝒦. If ξ ∈ H_{ét}^{1}(𝒦, H) is an R-unramified element for the functor H_{ét}^{1}(−, H) (see (3) for the Definition), then ξ can be lifted to an element of H_{ét}^{1}(R, H).

Proof. Patching.

Theorem 5.0.17. The canonical map H_{ét}^{1}(K, H) → H_{ét}^{1}(K[t], H) is bijective.

Proof. Let i : {0} ↪ A_{K}^{1} be the origin of the affine line. Let H_{ét}^{1}(K[t], H) ⟶^{i^{∗}} H_{ét}^{1}(K, H) be the pull-back map. To prove the theorem we need the following

Claim 5.0.18. 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 5.0.19 (Homotopy invariance). Let K be the field and μ : G → C be the algebraic group morphism (6). Let K(t) be the rational function field in one variable. Let R ↦ ℱ(R) be the functor defined by the formulae (7). Then one has

F (K) = F_{nr,K[t]}(K(t)).

Proof. The injectivity is clear, since the composition

ℱ(K) → ℱ_{nr,K[t]}(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_{nr}(K(t)) and let H = ker(μ). Then by Lemma 4.0.10 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 5.0.16, ∂(a) can be represented by an element ξ ∈ H_{et}^{1}(K[t], H), where K[t] is the polynomial ring. By Theorem 5.0.17, 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 4.0.11. Thus ϵ(a_{0}) = a.

Corollary 5.0.20. Let S ↦ ℱ(S) be the functor defined in (7). Let

s_{0}, s_{1} : ℱ_{nr,K[t]}(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 5.0.19.

6 Equating Groups

The aim of this Section is to sketch a proof of Theorem ??. The following Proposition is a straightforward analog of [OP, Prop.7.1]

Proposition 6.0.21. Let S be a regular semi-local irreducible scheme. Let μ_{1} : G_{1} → C_{1} and μ_{2} : G_{2} → C_{2} be two smooth S-group scheme morphisms with tori C_{1} and C_{2}. Let T ⊂ S be a non-empty closed sub-scheme of S, and φ : G_{1}|_{T} → G_{2}|_{T}, ψ : C_{1}|_{T} → C_{2}|_{T} be S-group scheme isomorphisms such that (μ_{2}|_{T}) ∘ φ = ψ ∘ (μ_{1}|_{T}). Then there exists a finite étale morphism S̃ ⟶^{π} S together with its section δ : T → S̃ over T and S̃-group scheme isomorphisms Φ : π^{∗}G_{1} → π^{∗}G_{2} and Ψ : π^{∗}C_{1} → π^{∗}C_{2} such that (i) δ^{∗}(Φ) = φ,

(ii) δ^{∗}(Ψ) = ψ,

(iii) π^{∗}(μ_{2}) ∘ Φ = Ψ ∘ π^{∗}(μ_{1}) : π^{∗}(G_{1}) → π^{∗}(C_{2}).

7 Elementary fibrations

In this Section we extend a result of Artin from [A] concerning existence of nice neighborhoods. The following notion is due to Artin [A, Exp.XI,Defn.3.1]

Definition 7.0.22. An elementary fibration is a morphism of schemes p : X → S which can be included in a commutative diagram

\xymatrix{X \ar[dr]_{p} \ar[r]^{j} & \overline{X} \ar[d]_{\overline{p}} & Y \ar[l]_{i} \ar[dl]_{q} \\  & S}  (8)

satisfying the following conditions

(i) j is an open immersion dense at each fibre of \overline{p}, and X = \overline{X} − Y;

(ii) \overline{p} is a smooth projective such that all fibres are geometrically irreducible of dimension one;

(iii) q is a finite étale morphism such that each fibre is non-empty.

The following result is an extension of Artin’s result [A, Exp.XI,Thm.2.1]

Theorem 7.0.23. Let k be an infinite field, V ⊂ P_{k}^{n} a locally closed subscheme of pure dimension r. Let V′ ⊂ V be an open subscheme consisting of all points x ∈ V such that V is k-smooth at x. Let p_{1}, p_{2}, …, p_{m} ∈ P_{k}^{n} be a family of closed points with p_{i} ≠ p_{j} for i ≠ j. For a family H_{1}(d), H_{2}(d), …, H_{s}(d) (s ≤ r) of hyperplanes of degree d containing all points p_{i} (i = 1, 2, …, m) set Y = H_{1}(d) ∩ H_{2}(d) ∩ ⋯ ∩ H_{s}(d).

There exists an integer d depending of the family p_{1}, p_{2}, …, p_{m} such that if the family H_{1}(d), H_{2}(d), …, H_{s}(d) with s ≤ r is general enough, then Y crosses V transversally at each point of Y ∩ V′.

If V is irreducible (geometrically irreducible) and s < r then for the same integer d and for a general enough family H_{1}(d), H_{2}(d), …, H_{s}(d) the intersection Y ∩ V is additionally irreducible (geometrically irreducible).

Using this Bertini type Theorem one can prove the following result, which is a slight extension of Artin’s result [A, Exp.XI,Prop.3.3 ] Proposition 7.0.24. Let k be an infinite field, X/k be a smooth geometrically irreducible variety, x_{1}, x_{2}, …, x_{n} ∈ X be closed points. Then there exists a Zarisky open neighborhood X^{0} of the family {x_{1}, x_{2}, …, x_{n}} and an elementary fibration p : X^{0} → S, where S is an open subscheme of the projective space P^{dimX−1}. Additionally, if Z is a closed codimension one subvariety in X, then one can choose X^{0} and p such that p|_{Z ⋂ X^{0}} : Z ⋂ X^{0} → S is finite surjective.

Proposition 7.0.25. Let p : X → S be an elementary fibration. If S is a regular semilocal scheme, then there exists a commutative diagram of S-schemes such that the left hand side square is Cartesian.

\xymatrix{X \ar[r]^{j} \ar[d]_{π} & \overline{X} \ar[d]^{\overline{π}} & Y \ar[l]_{i} \ar[d] \\ A^{1} × S \ar[r]^{in} & P^{1} × S & \{∞\} × S \ar[l]_{i}}  (9)

where j and i are from Definition 7.0.22 and pr_{S} ∘ π = p, where pr_{S} is the projection A^{1} × S → S.

In particular, π : X → A^{1} × S is a finite surjective morphism of S-schemes, where X (resp. A^{1} × S ) is regarded as an S-scheme via the morphism p (resp. via the projection pr_{S}).

We omit for now the proof of this proposition.

8 Nice triples

We study in the present Section certain packages of geometric data and morphisms of that packages. The concept of ”nice triples” is very closed to the one of ”standard triples ” [Vo, Defn.4.1] and is inspired by the latter one. Let k be an infinite field, X/k be a smooth geometrically irreducible variety, x_{1}, x_{2}, …, x_{n} ∈ X be closed points. Let Spec(𝒪_{X,{x_{1},x_{2},...,x_{n}}}) be the respecting semi-local ring.

Definition 8.0.26. Let U := Spec(𝒪_{X,{x_{1},x_{2},...,x_{n}}}). A nice triple over U consists of the following family of data:

(i) a smooth morphism q_{U} : 𝒳 → U,

(iii) an element f ∈ Γ(𝒳, 𝒪_{𝒳})

(ii) a section Δ of the morphism q_{U}.

These data must satisfy the following conditions:

(a) each component of each fibre of the morphism q_{U} has dimension one,

(b) the Γ(𝒳, 𝒪_{𝒳})/f · Γ(𝒳, 𝒪_{𝒳}) is a finite Γ(U, 𝒪_{U}) = O_{X,{x_{1},x_{2},...,x_{n}}}-module, (c) there exists a finite surjective U-morphism Π : 𝒳 → A^{1} × U,

(d) Δ^{∗}(f) ≠ 0 ∈ Γ(U, 𝒪_{U}).

A morphism of two nice triples (𝒳′, f′, Δ′) → (𝒳, f, Δ) is an étale morphism of Uschemes θ : 𝒳′ → 𝒳 such that

(1) q′_{U} = q_{U} ∘ θ,

(2) f′ = θ^{∗}(f) · g′ for an element g′ ∈ Γ(𝒳′, 𝒪_{𝒳′} ) (in particular, Γ(𝒳′, 𝒪_{𝒳′} )/θ^{∗}(f) · Γ(𝒳′, 𝒪_{𝒳′} ) is a finite O_{X,{x_{1},x_{2},...,x_{n}}}-module ),

(3) Δ = θ ∘ Δ′.

(Stress that there are no conditions concerning an interaction of Π′ and Π ).

Theorem 8.0.27. Let U be as in Definition 8.0.26. Let (𝒳, f, Δ) be a nice triple over U. Let G_{𝒳} be a reductive 𝒳-group scheme and G_{U} := Δ^{∗}(G_{𝒳}) and G_{const} be the pull-back of G_{U} to 𝒳. Let C_{𝒳} be an 𝒳-tori and C_{U} := Δ^{∗}(C_{𝒳}) and C_{const} be the pull-back of C_{U} to 𝒳. Then there exist a morphism θ : (𝒳′, f′, Δ′) → (𝒳, f, Δ) of nice triples and isomorphisms

Φ : θ^{∗}(G_{const}) → θ^{∗}(G_{𝒳}), Ψ : θ^{∗}(C_{const}) → θ^{∗}(C_{𝒳})

of 𝒳′-group schemes such that (Δ′)^{∗}(Φ) = id_{G_{U}}, (Δ′)^{∗}(Φ) = id_{G_{U}} and

θ^{∗}(μ_{2}) ∘ Φ = Ψ ∘ θ^{∗}(μ_{1})  (10)

This Theorem is a consequence of Proposition 6.0.21. The proof of this theorem is omitted for now.

9 A basic nice triple

With Propositions 7.0.24 and 7.0.25 at our disposal we may form a basic nice triple, namely the triple (14). This is the main aim of the present Section. Namely, fix a smooth irreducible affine k-scheme X, a finite family of points x_{1}, x_{2}, …, x_{n} on X, and a non-zero function f ∈ k[X]. We always assume that the set {x_{1}, x_{2}, …, x_{n}} is contained in the vanishing locus of the function f.

Replacing k with its algebraic closure in k[X], we may assume that X is a geometrically irreducible k-variety. By Proposition 7.0.24 there exist a Zariski open neighborhood X^{0} of the family {x_{1}, x_{2}, …, x_{n}} and an elementary fibration p : X^{0} → S, where S is an open sub-scheme of the projective space P^{dimX−1} such that

p|_{{f=0}∩X^{0}} : {f = 0} ∩ X^{0} → S

is finite surjective. Let s_{i} = p(x_{i}) ∈ S, for each 1 ≤ i ≤ n. Shrinking S, we may assume that S is affine and still contains the family {s_{1}, s_{2}, …, s_{n}}. Clearly, in this case p^{−1}(S) ⊆ X^{0} contains the family {x_{1}, x_{2}, …, x_{n}}. Now we replace X by p^{−1}(S) and f by its restriction to this new X.

In this way we get an elementary fibration p : X → S such that

{x_{1}, …, x_{n}} ⊂ {f = 0} ⊂ X,

S is an open affine sub-scheme in the projective space P^{dimX−1}, and the restriction of p|_{{f=0}} : {f = 0} → S to the vanishing locus of f is a finite surjective morphism. In other words, k[X]/(f) is finite as a k[S]-module.

As an open affine sub-scheme of the projective space P^{dimX−1} the scheme S is regular. By Proposition 7.0.25 one can shrink S such that S is still affine, contains the family {s_{1}, s_{2}, …, s_{n}} and there exists a finite surjective morphism

π : X → A^{1} × S

such that p = pr_{S}∘π. Clearly, in this case p^{−1}(S) ⊆ X contains the family {x_{1}, x_{2}, …, x_{n}}. Now we replace X by p^{−1}(S) and f by its restriction to this new X.

In this way we get an elementary fibration p : X → S such that

{x_{1}, …, x_{n}} ⊂ {f = 0} ⊂ X,

S is an open affine sub-scheme in the projective space P^{dimX−1}, and the restriction of p|_{{f=0}} : {f = 0} → S to the vanishing locus of f is a finite surjective morphism. And eventually there exists a finite surjective morphism

π : X → A^{1} × S

such that p = pr_{S} ∘ π.

Now set U := Spec(𝒪_{X,{x_{1},x_{2},...,x_{n}}}), and denote by can : U ↪ X the canonical inclusion of schemes, and p_{U} = p ∘ can : U → S. Further, we consider the fibre product 𝒳 := U ×_{S} X. Then the canonical projections q_{U} : 𝒳 → U and q_{X} : 𝒳 → X and the diagonal morphism Δ : U → 𝒳 can be included in the following diagram

\xymatrix{𝒳 \ar[d]_{q_{U}} \ar[r]^{q_{X}} & X \\ U \ar[ur]_{can} \ar[u]_{Δ}}  (11)

where

q_{X} ∘ Δ = can  (12)

and

q_{U} ∘ Δ = id_{U}.  (13)

Note that q_{U} is a smooth morphism with geometrically irreducible fibres of dimension one. Indeed, observe that q_{U} is a base change via p_{U} of the morphism p which has the desired properties. Taking the base change via p_{U} of the finite surjective morphism π : X → A^{1} × S we get a finite surjective morphism

Π : 𝒳 → A^{1} × U

such that q_{U} = pr_{U} ∘ Π. Set f := q_{X}^{∗}(f). The 𝒪_{X,{x_{1},x_{2},...,x_{n}}}-module Γ(𝒳, 𝒪_{𝒳})/f · Γ(𝒳, 𝒪_{𝒳}) is finite, since the k[S]-module k[X]/f · k[X] is finite.

Note that the data

(q_{U} : 𝒳 → U, f, Δ)  (14)

form an example of a nice triple, as defined in Definition 8.0.26.

Claim 9.0.28. Note that the schemes Δ(U) and {f = 0} are both semi-local and that the set of closed points of Δ(U) is contained in the set of closed points of {f = 0}.

This holds since the set {x_{1}, x_{2}, …, x_{n}} is contained in the vanishing locus of the function f.

10 Proof of Theorem (A)

Proof. We begin with the following data. Fix a smooth irreducible affine k-scheme X, a finite family of points x_{1}, x_{2}, …, x_{r} on X, and set 𝒪 := 𝒪_{X,{x_{1},x_{2},...,x_{r}}} and U := Spec(𝒪). Replacing k with its algebraic closure in Γ(X, 𝒪_{X}) we may and will assume that X is k-smooth and geometrically irreducible. Further, consider the reductive U-group scheme G, the U-tori C and the smooth U-group scheme morphism

μ : G → C.

Let K be the fraction field of 𝒪. Let ξ ∈ C(K) be such that the element ξ̄ ∈ ℱ(K) is 𝒪-unramified (see (3)).

Shrinking X if necessary, we may secure the following properties:

(i) The points x_{1}, x_{2}, …, x_{r} are still in X;

(ii) The group schemes G, C and the group morphism μ are defined over X, G is reductive over X, C is a tori over X and μ is a smooth X-group morphism. We will often denote this X-group scheme G by G_{X} and write G_{U} for the original U-group scheme G; We will often denote this X-group scheme C by C_{X} and write C_{U} for the original U-group scheme C;

(iii) The element ξ is defined over X_{f}, that is ξ ∈ C(k[X]_{f}) for a non-zero function f ∈ k[X] such that f vanishes at each x_{i}’s and the k-algebra k[X]/(f) is reduced;

(iv) we may assume further that ξ̄ ∈ ℱ(k[X]_{f}) is k[X]-unramified.

So, we are given now with a geometically irreducible k-smooth scheme X, a finite family points x_{1}, x_{2}, …, x_{r} on X and a non-zero function f ∈ k[X] vanishing at each x_{i}’s. Beginning with these data the nice triple (14)

(q_{U} : 𝒳 → U, f, Δ)

was constructed in Section 9.

Set G_{𝒳} = q_{X}^{∗}(G_{X}) and G_{U} = Δ^{∗}(G_{𝒳}) = can^{∗}(G_{X}). Let G_{const} be the pull-back of G_{U} to 𝒳.

Set C_{𝒳} = q_{X}^{∗}(C_{X}) and C_{U} := Δ^{∗}(C_{𝒳}) = can^{∗}(C_{X}). Let C_{const} be the pull-back of C_{U} to 𝒳.

By Theorem 8.0.27 there exist a morphism θ : (𝒳′, f′, Δ′) → (𝒳, f, Δ) of nice triples and isomorphisms

Φ : θ^{∗}(G_{const}) → θ^{∗}(G_{𝒳}), Ψ : θ^{∗}(C_{const}) → θ^{∗}(C_{𝒳})

of 𝒳′-group schemes such that (Δ′)^{∗}(Φ) = id_{G_{U}}, (Δ′)^{∗}(Φ) = id_{G_{U}} and

θ^{∗}(μ_{2}) ∘ Φ = Ψ ∘ θ^{∗}(μ_{1}) : θ^{∗}(G_{const}) → θ^{∗}(C_{𝒳})  (15)

The equality (15) implies that for each U-scheme 𝒴′ and each U-scheme morphism t : 𝒴′ → 𝒳′ the group isomorphism Ψ(𝒴′) : θ^{∗}(C_{const})(𝒴′) → θ^{∗}(C_{𝒳})(𝒴′) induces a group isomorphism

C_{U}(𝒴′)/μ(G_{U}(𝒴′)) = ℱ_{const}(𝒴′) ≅ ℱ_{𝒳} (𝒴′) = C(𝒴′)/(μ(G(𝒴′)))  (16)

Group isomorphisms Ψ(𝒴′) form a natural functor transformation of functors defined on the category of U-schemes and U-scheme morphisms. The same holds concerning isomorphisms (16).

Taking the open inclusion 𝒳′_{f′} ↪ 𝒳′ and the element ζ := (q′_{X})^{∗}(ξ) ∈ θ^{∗}(C_{𝒳})(𝒳′_{f′}) and applying the equality (15) we get an element ζ_{U} ∈ θ^{∗}(C_{const})(𝒳′_{f′}).

Let U = Spec(𝒪_{X,{x_{1},x_{2},...,x_{r}}}) be as in Definition 8.0.26. Write R for 𝒪_{X,{x_{1},x_{2},...,x_{n}}}. It is a local essentially smooth k-algebra with maximal ideals 𝔪_{i}’s where i runs from 1 to r. Let (𝒳′, f′, Δ′) be the above mentioned nice triple over U. Show that it gives rise to certain data subjecting the hypotheses of Lemma 3.0.8.

Let A = Γ(𝒳′, 𝒪_{𝒳′} ). It is an R-algebra via the ring homomorphism (q′_{U})^{∗} : R → Γ(𝒳, 𝒪_{𝒳}). Moreover this R-algebra is smooth. The triple (𝒳′, f′, Δ′) is a nice triple. Thus there exists a finite surjective U-morphism Π : 𝒳′ → A_{U}^{1}. It induces the respecting Ralgebras inclusion R[t] ↪ Γ(𝒳′, 𝒪_{𝒳′} ) = A such that the A is finitely generated as an R[t]- module. For each index i the R/𝔪_{i}-algebra A/𝔪_{i}A is equi-dimensional of dimension one since (𝒳′, f′, Δ′) is a nice triple. Let ϵ = (Δ′)^{∗} : A → R be an R-algebra homomorphism induced by the section Δ′ of the morphism q′_{U}. Clearly, that ϵ is an augmentation. Further, ϵ(f′) ≠ 0 ∈ R since (𝒳′, f′, Δ′) is a nice triple. Set I = ker(ϵ). The R-module A/f′A is finite since (𝒳′, f′, Δ′) is a nice triple. So, we are under the hypotheses of Lemma 3.0.8. Thus we may use the conclusion of that Lemma.

So, there exists an element u ∈ A subjecting (1) to (7) of Lemma 3.0.8. The function u defines a finite morphism π : 𝒳′ → U × A^{1}. Since 𝒳′ and 𝒴′ := U × A^{1} 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 t : 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 = (q′_{X})^{∗}(ξ) ∈ C(A_{f′}). Since the function f′ ∈ A 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(R).  (17)

Claim 10.0.29 (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} = K ⊗_{R} M, for an U-scheme 𝒵 set 𝒵_{K} = Spec(K) ×_{U} 𝒵, for a U-morphism φ : 𝒵 → 𝒲 set φ_{K} = Spec(K) ×_{U} φ. Clearly one has R_{K} = K, U_{K} = Spec(K), 𝒳′_{K} = Spec(A_{K}) and (U × A^{1})_{K} is just the affine line A_{K}^{1}. Its closed subschemes U_{K} × {1} and U_{K} × {0} coincide with the points 1 and 0 of A_{K}^{1}. The morphism (q′_{U})_{K} : 𝒳′_{K} → U_{K} = Spec(K) is smooth. In fact, the morphism θ is étale and q_{U} : 𝒳 → U is smooth. The morphism π_{K} is finite flat and fits in the commutative triangle

\xymatrix{𝒳′_{K} \ar[rr]^{π_{K}} \ar[dr]_{(q′_{U})_{K}} &  & A_{K}^{1} \ar[dl]^{pr_{K}} \\  & Spec(K)}  (18)

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} ∈ A_{K}^{1} and the disjoint union 𝒟_{0,K} := 𝒟_{K} ∐ Δ_{K} is the scheme theoretic pre-image of the point {0} ∈ A_{K}^{1} under the morphism π_{K}. Let

ζ_{u} := N_{K(𝒳′ _{K})/K(A^{1})}(ζ) ∈ C(K(A^{1})) = C(K(u)).  (19)

To prove Claim 10.0.29 we need the following one:

Claim 10.0.30. The class ζ̄_{u} ∈ ℱ(K(s)) is K[s]-unramified.

Assuming this claim we complete the proof of Claim 10.0.29. By Claim 10.0.30 we can apply the specialization maps to ζ̄_{u}. By Corollary 5.0.20 the specializations at 0 and 1 of the element ζ̄_{u} coincide, that is

s_{1}(ζ̄_{u}) = s_{0}(ζ̄_{u}) ∈ ℱ(K).  (20)

By Corollary 3.0.9 the function N_{A_{K}/K[u]}(f) ∈ K[s] does not vanishes as at 1, so at 0 and by the condition (5) of Lemma 3.0.8 one has ζ_{u} ∈ C(K[u]_{N(f)}). Using the relation between specialization and evaluation maps described in Definition 5.0.15 one has a chain of relations

\overline{Ev_{1}(ζ_{u})} = s_{1}(ζ̄_{u}) = s_{0}(ζ̄_{u}) = \overline{Ev_{0}(ζ_{u})} ∈ ℱ(K).  (21)

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

Ev_{1}(ζ_{u}) = N_{𝒟_{1,K}/U_{K}} (ζ|_{𝒟_{1,K}} )  (22)

and

Ev_{0}(ζ_{u}) = 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}})  (23)

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_{u})} = \overline{Ev_{0}(ζ_{u})} = \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).  (24)

So ξ̄′_{K} = ξ̄_{K}. Claim 10.0.29 follows.

It remains to prove Claim 10.0.30. Recall that the class ξ̄ ∈ ℱ(k[X]_{f}) is k[X]- unramified. Since the morphism p : X → S is smooth so is the morphism r : U ↪ X → S. Since the p_{X} : 𝒳 → X is the base change of r it coincides with a composition map 𝒳 ↪ X ×_{S} X ⟶^{id×pr_{X}} X where id × pr_{X} is a smooth morphism. Whence id × pr_{X} is flat and of finite type. By Lemma 4.0.14 the class ζ̄ ∈ ℱ(A_{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[u] ⊂ A_{K} = K[𝒳′_{K}] and the function f′ ∈ A satisfy the hypotheses of Lemma 4.0.13. We first check that the K-algebra A_{K}/f′A_{K} is reduced. Recall that the ring k[X]/(f) is reduced. As we mentioned above, the morphism q′_{𝒳′} : 𝒳′ → X is essentially smooth, hence the ring A/(f′) is reduced too and thus its localization A_{K}/f′A_{K} is reduced. Since the extension K[u] ⊂ A_{K} = K[𝒳′_{K}] and the functions f′ ∈ A satisfy the conditions (5) to (7) of Lemma 3.0.8 they also satisfy the hypotheses of Lemma 4.0.13 for the functor ℱ_{const} from equality (16). Thus by Lemma 4.0.13 the class ζ̄_{u} is K[u]-unramified. This implies Claim 10.0.30. The proof of the Theorem A is completed.

11 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 semi- local ring containing a perfect field k is geometrically regular over k.

Theorem 11.0.31 (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 semi-local ring containing an infinite perfect field k. Since k is perfect one can apply Popescu’s theorem. So, R can be presented as a filtered direct limit of smooth k-algebras A_{α} over the infinite field k. We first observe that we may replace the direct system of the A_{α}’s by a system of essentially smooth semi-local k-algebras. In fact, if 𝔪_{1}, 𝔪_{2}, …, 𝔪_{r} are all maximal ideal of R and S = R − (∪ 𝔪_{i}) we can replace each A_{α} by (A_{α})_{S_{α}}, where S_{α} = S ∩ A_{α}. Note that in this case the canonical morphisms φ_{α} : A_{α} → R are local (sends a maximal ideal to a maximal one) and every A_{α} is a regular semi-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 11.0.32. Let B be a local ring which is a domain and let K be its field of fractions. Let 𝔪 be a maximal ideal of B and B̄ = B/𝔪. For an element θ ∈ ℱ(B) write θ̄ for its image in ℱ(B̄) and θ_{K} for its image in ℱ(K). Let η, ρ ∈ ℱ(B) be such that η_{K} = ρ_{K} ∈ ℱ(K). Then η̄ = ρ̄ ∈ ℱ(B̄).

Lemma 11.0.33. There exists an index α and an element ξ_{α} ∈ C(B_{α}) such that φ(ξ_{α}) = ξ and 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 11.0.33 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 11.0.32 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 11.0.32. Induction on dim(B). The case of dimension 1 follows from Theorem 4.0.10 applied to the local ring B. To prove the general case choose an f ∈ B such that η = ρ ∈ ℱ(B_{f}). Let π ∈ B be such that π is regular parameter in B, having no common factors with f. Let \overline{B} = B/πB. Then for the image \overline{η − ρ} of η − ρ in ℱ(\overline{B}) we have (\overline{η − ρ})_{f} = \overline{(η − ρ)_{f}} = 0 ∈ ℱ(\overline{B}\overline{f}). By the inductive hypotheses one has \overline{η − ρ} = 0 ∈ ℱ(B̄). Thus η̄ = ρ̄ ∈ ℱ(B̄).

Proof of Lemma 11.0.33.

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.

12 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 12.0.34. 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.

13 Proof of Theorems 1.0.1 and 1.0.2

Proof of Theorem 1.0.1. Consider a short exact sequence of reductive R-group schemes

{1} → G_{der} ⟶^{i} G ⟶^{μ} C → {1},  (25)

where G_{der} is the derived R-group scheme of G and C be a tori over R. Prove that

ker[H_{ét}^{1}(R, G) → H_{ét}^{1}(K, G)] = ∗  (26)

provided that one has

ker[H_{ét}^{1}(R, G_{der}) → H_{ét}^{1}(K, G_{der})] = ∗  (27)

This sequence of R-group schemes gives a short exact sequence of the corresponding sheaves in the étale topology on the big étale site. That sequence of sheaves gives rise to a commutative diagram with exact arrows of pointed sets

\xymatrix{\{1\} \ar[r] & C(R)/μ(G(R)) \ar[r]^{∂} \ar[d]_{α} & H_{ét}^{1}(R, G_{der}) \ar[r]^{i_{∗}} \ar[d]^{β} & H_{ét}^{1}(R, G) \ar[r]^{μ} \ar[d]_{γ} & H_{ét}^{1}(R, C) \ar[d]^{δ} \\ \{1\} \ar[r] & C(R_{𝔭})/μ(G(R_{𝔭})) \ar[r]^{∂} \ar[d]_{α_{𝔭}} & H_{ét}^{1}(R_{𝔭}, G_{der}) \ar[r]^{i_{∗}} \ar[d]^{β_{𝔭}} & H_{ét}^{1}(R_{𝔭}, G) \ar[r]^{μ} \ar[d]_{γ_{𝔭}} & H_{ét}^{1}(R_{𝔭}, C) \ar[d]^{δ_{𝔭}} \\ \{1\} \ar[r] & C(K)/μ(G(K)) \ar[r]^{∂} & H_{ét}^{1}(K, G_{der}) \ar[r]^{i_{∗}} & H_{ét}^{1}(K, G) \ar[r]^{μ} & H_{ét}^{1}(K, C)}

(28)

Set α_{K} = α_{𝔭} ∘ α, β_{K} = β_{𝔭} ∘ β, γ_{K} = γ_{𝔭} ∘ γ, δ_{K} = δ_{𝔭} ∘ δ. By a theorem of Nisnevich [Ni] one has

ker(α_{𝔭}) = ker(β_{𝔭}) = ker(γ_{𝔭}) = ker(δ_{𝔭}) = ∗  (29)

Let ξ ∈ ker(γ_{K}), then μ(ξ) ∈ ker(δ_{K}). By [C-T/S] ker(δ_{K}) = ∗, whence μ(ξ) = ∗ and ξ = i_{∗}(ζ) for an ζ ∈ H_{ét}^{1}(R, G_{der}). By a Theorem of Nisnevich [Ni, ] ker(γ_{𝔭}) = ∗. Since γ_{K}(ξ) = ∗ we see that γ(ξ) = ∗. Whence i_{∗}(β(ζ)) = ∗ and β(ζ) = ∂(ϵ_{𝔭}) for an ϵ_{𝔭} ∈ C(R_{𝔭})/μ(G(R_{𝔭})). A diagram chaise shows that there exists a unique element ϵ_{K} ∈ C(K)/μ(G(K)) for each hight one prime ideal 𝔭 of R one has α_{𝔭}(ϵ_{𝔭}) = ϵ_{K} ∈ C(K)/μ(G(K)). By Purity Theorem 1.0.4 there exists an element ϵ ∈ C(R)/μ(G(R)) such that α_{K}(ϵ) = ϵ_{K}. The element ϵ_{K} has the property that ∂(ϵ_{K}) = β_{K}(ζ). Whence ∂(ϵ) = ζ and ξ = i(∂(ϵ)) = ∗. The proof equality (26) is reduced to the proof of the equality (31).

To prove (31) recall that G_{der} is a semi-simple R-group scheme. Let G_{red}^{sc} be the corresponding simply-connected semi-simple R-group scheme and let π : G_{red}^{sc} → G be the corresponding R-group scheme morphism. Let Z′ = ker(π). It is known that Z′ is contained in the center of G_{red}^{sc} and Z′ is a finite group scheme of multiplicative type. Consider a short exact sequence of R-group schemes

{1} → Z′ → G_{der}^{cs} ⟶^{π} G_{der} → {1},  (30)

By the main result of [PSV, Thm.1.1] one has

ker[H_{ét}^{1}(R, G_{der}^{sc}) → H_{ét}^{1}(K, G_{der}^{sc})] = ∗  (31)

The short exact sequence (30) give rises to an exact sequence of pointed sets

H_{ét}^{1}(R, Z′)/∂(G_{red}) → H_{ét}^{1}(R, G_{der}^{sc}) → H_{ét}^{1}(R, G_{der}) → H_{ét}^{2}(R, Z′)  (32)

and furthermore to a commutative diagram like (28) with exact arrows. Now a diagram chaise similar to the one above in this proof completes the proof of Theorem 1.0.4.

Proof of Theorem 1.0.2. Repeat literally the proof of Theorem 1.0.1.

14 Examples

We follow here the notation of The Book of Involutions [KMRT]. The field k is a characteristic zero field. The functors (33) to (47) satisfy purity for regular local rings containing k as follows either from Theorem 12.0.34 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}).  (33)

(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}).  (34)

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)).  (35)

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))].  (36)

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)).  (37)

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)).  (38)

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))  (39)

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)).  (40)

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))  (41)

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}  (42)

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}]  (43)

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))]  (44)

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}].  (45)

(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)).  (46)

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)).  (47)

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

References

[A] Artin, M. Comparaison avec la cohomologie classique: cas d’un préschéma lisse. Lect. Notes Math., vol. 305, Exp XI.

[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).

[C-T/S] Colliot-Thélène, J.-L.; Sansuc, J.-J. Principal homogeneous spaces under flasque tori: Applications, Journal of Algebra, 106 (1987), 148–205.

[ChP] V.Chernousov, I.Panin Purity of G_{2}-torsors. C. R. Math. Acad. Sci. Paris 345 (2007), no. 6, 307–312.

[D-G] Demazure, Grothendieck. Structure des schémas en groupes réductifs. Lect. Notes Math., vol 153.

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

[Gr1] Grothendieck, A. Torsion homologique et section rationnalles, in Anneaux de Chow et applications, Séminaire Chevalley, 2-e année, Secrétariat mathématique, Paris, 1958.

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

[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).

[OPZ] Ojanguren, M; Panin, I.; Zainoulline, K. On the norm principle for quadratic forms. J. Ramanujan Math. Soc. 19, No.4 (2004) 1–12.

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

[PSV] Panin, I.; Stavrova A,; Vavilov, N. On Grothendieck—Serre’s conjecture concerning principal G-bundles over reductive group schemes I, Preprint (2009), http://www.math.uiuc.edu/K-theory/

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

[Pa] Panin, I. Purity for multipliers, in Proceedings of the Silver Jubilee Conference University of Hyderabad (Ed. R.Tandon) Algebra and Number Theory, pp.66–89

[PaPS] Panin, I.; Petrov, V.; Stavrova, A. On Grothendieck–Serre’s for simple adjoint group schemes of types E_{6} and E_{7}. Preprint (2009), http://www.math.uiuc.edu/K-theory/

[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. Espaces fibrés algébriques, in Anneaux de Chow et applications, Séminaire Chevalley, 2-e année, Secrétariat mathématique, Paris, 1958.

[SuVo] Suslin, A.; Voevodsky, V. Singular homology of abstract algebraic varieties. Invent. Math. 123 (1996), no. 1, 61–94.

[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.

[Vo] Voevodsky, V. Cohomological theory of presheaves with transfers. In Cycles, Transfers, and Motivic Homology Theories. Ann. Math. Studies, 2000, Princeton University Pres.