Proof of Grothendieck–Serre conjecture on principal bundles over regular local rings containing a finite field

Ivan Panin^{∗}

September 10, 2015

Abstract

Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. In other words, if K is the fraction field of R, then the map of non-abelian cohomology pointed sets

H_{ét}^{1}(R, G) → H_{ét}^{1}(K, G),

induced by the inclusion of R into K, has a trivial kernel. If the regular local ring R contains an infinite field this result is proved in [FP]. Thus the conjecture holds for regular local rings containing a field.

1 Main results

Let R be a commutative unital ring. Recall that an R-group scheme G is called reductive, if it is affine and smooth as an R-scheme and if, moreover, for each algebraically closed field Ω and for each ring homomorphism R → Ω the scalar extension G_{Ω} is a connected reductive algebraic group over Ω. This definition of a reductive R-group scheme coincides with [SGA3, Exp. XIX, Definition 2.7]. A well-known conjecture due to J.-P. Serre and A. Grothendieck (see [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 a trivial kernel. The following theorem, which is the main result of the present paper, asserts that this conjecture holds, provided that R contains a finite field. If R contains an infinite field, then the conjecture is proved in [FP].

∗The author acknowledges support of the RNF-grant 14-11-00456.

Theorem 1.1. Let R be a regular semi-local domain containing a finite field, and let K be its field of fractions. Let G be a reductive group scheme over R. Then the map

H_{ét}^{1}(R, G) → H_{ét}^{1}(K, G),

induced by the inclusion of R into K, has a trivial kernel. In other words, under the above assumptions on R and G, each principal G-bundle over R having a K-rational point is trivial.

Theorem 1.1 has the following

Corollary 1.2. Under the hypothesis of Theorem 1.1, the map

H_{ét}^{1}(R, G) → H_{ét}^{1}(K, G),

induced by the inclusion of R into K, is injective. Equivalently, if 𝒢_{1} and 𝒢_{2} are two principal bundles isomorphic over SpecK, then they are isomorphic.

Proof. Let 𝒢_{1} and 𝒢_{2} be two principal G-bundles isomorphic over Spec K. Let Iso(𝒢_{1}, 𝒢_{2}) be the scheme of isomorphisms. This scheme is a principal Aut𝒢_{2}-bundle. By Theorem 1.1 it is trivial, and we see that 𝒢_{1} ≅ 𝒢_{2}.

Note that, while Theorem 1.1 was previously known for reductive group schemes G coming from the ground field (an unpublished result due to O.Gabber), in many cases the corollary is a new result even for such group schemes.

For a scheme U we denote by 𝔸_{U}^{1} the affine line over U and by ℙ_{U}^{1} the projective line over U. Let T be a U-scheme. By a principal G-bundle over T we understand a principal G ×_{U} T -bundle. We refer to [SGA3, Exp. XXIV, Sect. 5.3] for the definitions of a simple simply-connected group scheme over a scheme and a semi-simple simply-connected group scheme over a scheme.

In Section 2 we deduce Theorem 1.1 from the following three results.

Theorem 1.3. Let k be a field. Let 𝒪 be the semi-local ring of finitely many closed points on a k-smooth irreducible affine k-variety X and let K be its field of fractions. Let G be a simple simply connected group scheme over 𝒪. Let G be a reductive group scheme over 𝒪. Let 𝒢 be a principal G-bundle over 𝒪 which is trivial over K. Then there exists a principal G-bundle 𝒢_{t} over 𝒪[t] and a monic polynomial h(t) ∈ 𝒪[t] such that

(i) the G-bundle 𝒢_{t} is trivial over 𝒪[t]_{h},

(ii) the evaluation of 𝒢_{t} at t = 0 coincides with the original G-bundle 𝒢.

If the field k is infinite this result is proved in [PSV, Thm.1.2].

Theorem 1.4. Let k be a field. Let 𝒪 be the semi-local ring of finitely many closed points on a k-smooth irreducible affine k-variety X. Set U = Spec 𝒪. Let G be a simple simply-connected group scheme over U. Let ℰ_{t} be a principal G-bundle over the affine line 𝔸_{U}^{1} = Spec 𝒪[t], and let h(t) ∈ 𝒪[t] be a monic polynomial. Denote by (𝔸_{U}^{1})_{h} the open subscheme in 𝔸_{U}^{1} given by h(t) ≠ 0 and assume that the restriction of ℰ_{t} to (𝔸_{U}^{1})_{h} is a trivial principal G-bundle. Then for each section s : U → 𝔸_{U}^{1} of the projection 𝔸_{U}^{1} → U the G-bundle s^{∗}ℰ_{t} over U is trivial.

If the field k is infinite this result is proved in [FP, Thm.2].

Theorem 1.5. Let k be a field. Assume that for any irreducible k-smooth affine variety X and any finite family of its closed points x_{1}, x_{2}, …, x_{n} and the semi-local k-algebra 𝒪 := 𝒪_{X,x_{1},x_{2},...,x_{n}} and all semi-simple simply connected reductive 𝒪-group schemes H the pointed set map

H_{ét}^{1}(𝒪, H) → H_{ét}^{1}(k(X), H),

induced by the inclusion of 𝒪 into its fraction field k(X), has trivial kernel.

Then for any regular semi-local domain 𝒪 of the form 𝒪_{X,x_{1},x_{2},...,x_{n}} above and any reductive 𝒪-group scheme G the pointed set map

H_{ét}^{1}(𝒪, G) → H_{ét}^{1}(K, G),

induced by the inclusion of 𝒪 into its fraction field K, has trivial kernel.

Theorem 1.4 is an easy consequence of Theorem 1.6 proven in section 3 and of Proposition 4.2 proven in section 4.

Theorem 1.6. Let k be a finite field. Let R, U, and G be as in Theorem 1.4. Let Z ⊂ ℙ_{U}^{1} be a closed subscheme finite over U. Let Y ⊂ ℙ_{U}^{1} be a closed subscheme finite and étale over U and such that

(i) G_{Y} := G ×_{U} Y is quasi-split, (ii) Y ∩ Z = ∅ and Y ∩ {∞} × U = ∅ = Z ∩ {∞} × U, (iii) for any closed point u ∈ U one has P ic(ℙ_{u}^{1} − Y_{u}) = 0, where Y_{u} := ℙ_{u}^{1} ∩ Y. Let 𝒢 be a principal G-bundle over ℙ_{U}^{1} such that its restriction to ℙ_{U}^{1} − Z is trivial. Then the restriction of 𝒢 to ℙ_{U}^{1} − Y is also trivial. In particular, the principal G-bundle 𝒢 is trivial locally for the Zarisky topology.

If the field k is infinite then a stronger result is proved in [FP, Thm.3]. Proof of Theorem 1.3 is given in Section 5. Proof of Theorem 1.5 is given in Section 8.

The article is organized as follows. In Section 2 Theorem 1.1 is reduced to Theorems 1.4 and 1.3. In Section 4 Theorem 1.4 is proved. In Section 5 two major technical theorems (Theorems 5.1 and 5.2) are stated and Theorem 1.3 is reduced to those two theorems. In Section 6 a purity theorem is stated and it is reduced to Theorem 5.2. In Section 7 one more purity theorem is stated and it is reduced to Theorem 5.2. In Section 8 Theorem 1.5 is reduced to Theorem 5.2. The strategy of the proof of Theorems 5.1 and 5.2 is described in Section 9. Theorems 5.1 and 5.2 are proved in Section 15 after preliminary work which is done in Sections 10 to 14. Finally, in Sections 16 to 18 few technical results are proved. We refer to those results reducing Theorems 6.1 and 1.5 to Theorem 5.2.

The history of the topic is described in [FP].

The author thanks J.-P.Serre, A.Suslin, A.Merkurjev, V.Chernousov, J.-L.Colliot-The’l‘ene, Ph.Gille, R.Parimala, A.Stavrova, B.Totaro, C.Voisin, R.Fedorov and many other mathematicians for their interest to the topic of the present article. I would like to thank A.Stavrova for drawing my attention to B.Poonen’s works on Bertini theorems for varieties over finite fields.

2 Reducing Theorem 1.1 to Theorems 1.3, 1.4, 1.5.

If the regular semi-local domain contains an infinite field, then such a reduction is given in details in [FP, Section 3]. This is why we give here just a short sketch of the reduction. So, suppose that Theorems 1.3, 1.4 and 1.5 are true.

Theorems 1.4 and 1.3 yield Theorem 1.1 in the simple simply-connected case, when the ring R is the semi-local ring of finitely many closed points on a k-smooth variety (the field k is finite). Indeed, let k be a field and let 𝒪, G and 𝒢 be as in Theorem 1.3. Assume additionally that the group scheme G is simple and simply-connected. Take 𝒢_{t} from the conclusion of Theorem 1.3 for the bundle ℰ_{t} from Theorem 1.4. Then take h(t) from the conclusion of Theorem 1.3 for the bundle ℰ_{t} from Theorem1.4. Finally take the inclusion i_{0} : U ↪ U × 𝔸^{1} for the section s from the Theorem 1.4. Theorem 1.4 with this choice of ℰ_{t}, h(t) and s yields triviality of the G bundle 𝒢 from Theorem 1.3.

Now standard arguments (see for instance [PSV, Thm.11.1]) show that Theorem 1.1 holds for the ring R as above in this section and for arbitrary semi-simple and simplyconnected group scheme G over R. (Note that to use those arguments it is necessary to work with semi-local rings).

Now Theorem 1.5 yields Theorem 1.1 for the ring R as above in this section and for arbitrary reductive group scheme G over R.

This latter case implies easily Theorem 1.1 for arbitrary reductive group scheme over a ring R, where R is the semi-local ring of finitely many arbitrary points on a k-smooth irreducible affine k-variety (see [FP, Lemma 3.3]). Auguments using Popescu’s Theorem completes the proof of Theorem 1.1 (see [FP, Proof of Theorem 1]). Those arguments runs now easier since Theorem 1.1 is established already for semi-local rings of finitely many arbitrary points on a k-smooth variety (with a finite field k).

3 Proof of Theorem 1.6

Proof. Let Y′ be a semi-local scheme. We will call a simple Y′-group scheme quasisplit if its restriction to each connected component of Y′ contains a Borel subgroup scheme. Let Y be a semi-local scheme. We will call a simple Y -group scheme isotropic, if its restriction to each connected component of Y contains a proper parabolic subgroup scheme.

Since G_{Y} := G ×_{U} Y is quasi-split it is isotropic. For any of the closed point u ∈ U the residue field k(u) is a finite field. Hence the reductive k(u)-group G_{u} = G ×_{U} u is quasi-split. Thus it is isotropic. The finite étale U-scheme Y does not satisfy (in general) the condition that for any of the closed point u ∈ U there is a k(u)-rational point as required by the assuptions of [FP, Thm.3]. However for any of the closed point u ∈ U one has P ic(ℙ_{u}^{1} − Y_{u}) = 0, where Y_{u} := ℙ_{u}^{1} ∩ Y. The G_{u}-bundle 𝒢_{u} is trivial over A_{u}^{1} − Z_{u}. Thus, by [Gil1, Corollary 3.10(a)] it is trivial locally for Zariski topology on ℙ_{u}^{1}. And again by [Gil1, Corollary 3.10(a)], it is trivial over ℙ_{u}^{1} − Y_{u}. Now one can repeat literally the proof of [FP, Thm.3] and conclude Theorem 1.6.

4 Proof of Theorem 1.4.

Theorem 1.4 in the case of an infinite field k is proved in [FP, Thm.2]. So, we prove the remaining case, when the field k is finite.

Let k, U and G be as in Theorem 1.4 and let the field k be finite. Let u_{1}, …, u_{n} be all the closed points of U. Let k(u_{i}) be the residue field of u_{i}. Consider the reduced closed subscheme u of U, whose points are u_{1}, …, u_{n}. Thus

u ≅ ∐_{i} Spec k(u_{i}).

Set G_{u} = G×_{U} u. By G_{u_{i}} we denote the fiber of G over u_{i}; it is a simple simply-connected algebraic group over k(u_{i}). The following lemma is a simple version of Lemma 13.4.

Lemma 4.1. Let S = Spec(A) be a regular semi-local scheme such that the residue field at any of its closed point is finite. Let T be a closed subscheme of S. Let W be a closed subscheme of the projective space ℙ_{S}^{d}. Assume that over T there exists a section δ : T → W of the projection W → S. Suppose further that W is S-smooth and equidimensional over S of relative dimension r. Then there exists a closed subscheme S̃ of W which is finite étale over S and contains δ(T).

Proof. Since S is semilocal, after a linear change of coordinates we may assume that δ maps T into the closed subscheme of P_{T}^{d} defined by X_{1} = ⋯ = X_{d} = 0. For each closed fibre P_{s}^{d} of P_{S}^{d} using repeatedly [Poo, Thm.1.2], we can choose a family of homogeneous polynomials H_{1}(s), …, H_{r}(s) (in general of increasing degrees) such that the subscheme Y (s) of P_{S}^{d}(s) defined by the equations

H_{1}(s) = 0 , …, H_{r}(s) = 0

intersects W (s) transversally and contains the point [1 : 0 : ⋯: 0]. By the chinese remainders’ theorem there exists a common lift H_{i} ∈ A[X_{0}, …, X_{d}] of all polynomials H_{i}(s), s ∈ Max(A). We may choose this common lift H_{i} such that H_{i}(1, 0, …, 0) = 0. Let V be the closed subscheme of P_{S}^{d} defined by

H_{1} = 0 , …, H_{r} = 0 .

We claim that the subscheme S̃ = V ∩ X has the required properties. Note first that X ∩ V is finite over S. In fact, X ∩ V is projective over S and every closed fibre (hence every fibre) is finite. Since the closed fibres of X ∩ V are finite étale over the closed points of S, to show that X ∩ V is finite étale over S it only remains to show that it is flat over S. Noting that X ∩ V is defined in every closed fibre by a regular sequence of equations and localizing at each closed point of S, we see that flatness follows from [OP2, Lemma 7.3].

Proposition 4.2. Let Z ⊂ 𝔸_{U}^{1} be a closed subscheme finite over U. There is a closed subscheme Y ⊂ 𝔸_{U}^{1} which is étale and finite over U and such that

(i) G_{Y} := G ×_{U} Y is quasi-split, (ii) Y ∩ Z = ∅, (iii) for any closed point u ∈ U one has P ic(ℙ_{u}^{1} − Y_{u}) = 0, where Y_{u} := ℙ_{u}^{1} ∩ Y. (Note that Y and Z are closed in ℙ_{U}^{1} since they are finite over U).

Proof. For every u_{i} in u choose a Borel subgroup B_{u_{i}} in G_{u_{i}}. The latter is possible since the fields k(u_{i}) are finite. Let ℬ be the U-scheme of Borel subgroup schemes of G. It is a smooth projective U-scheme (see [SGA3, Cor. 3.5, Exp. XXVI]). The subgroup B_{u_{i}} in G_{u_{i}} is a k(u_{i})-rational point b_{i} in the fibre of ℬ over the point u_{i}. Now apply Lemma 4.1 to the scheme U for S, the scheme u for T, the scheme ℬ for W and to a section δ : u → ℬ, which takes the point u_{i} to the point b_{i} ∈ ℬ. Since the scheme ℬ is U-smooth and is equidimensional over S we are under the assumption of Lemma 4.1. Hence there is a closed subscheme Y′ of ℬ such that Y′ is étale over U and all the b_{i}’s are in Y

To continue the proof of the Proposition we need the following

Lemma 4.3. Let U be as in the Proposition. Let Z ⊂ 𝔸_{U}^{1} be a closed subscheme finite over U. Let Y′ → U be a finite étale morphism such that for any closed point u_{i} in U the fibre Y′_{u_{i}} of Y′ over u_{i} contains a k(u_{i})-rational point. Then there are finite field extensions k_{1} and k_{2} of the finite field k such that

(i) the degrees [k_{1} : k] and [k_{2} : k] are coprime, (ii) k(u_{i}) ⊗_{k} k_{r} is a field for r = 1 and r = 2, (iii) the degrees [k_{1} : k] and [k_{2} : k] are strictly greater than any of the degrees [k(z) : k] , where z runs over all closed points of Z, (iv) there is a closed embedding of U-schemes Y′′ = ((Y′ ⊗_{k} k_{1}) ∐(Y′ ⊗_{k} k_{2})) ⟶^{i} 𝔸_{U}^{1},

(v) for Y = i(Y′′) one has Y ∩ Z = ∅, (vi) for any closed point u_{i} in U one has P ic(ℙ_{u_{i}}^{1} − Y_{u_{i}}) = 0.

To prove this Lemma note that it’s easy to find field extensions k_{1} and k_{2} subjecting (i) to (iii). To satisfy (iv) it suffices to require that for any closed point u_{i} in U and for r = 1 and r = 2 the number of closed points in Y′_{u_{i}} ⊗_{k} k_{r} is the same as the number of closed points in Y′_{u_{i}}, and to require that for any integer n > 0 and any closed point u_{i} in U the number of points y ∈ Y′′_{u_{i}} with [k(y) : k(u_{i})] = n is not more than the number of points x ∈ 𝔸_{u_{i}}^{1} with [k(x) : k(u_{i})] = n. Clearly, these requirements can be satisfied, which proves the item (iv).

The condition (v) holds for any closed U-embedding i : Y′′ ↪ 𝔸_{U}^{1} from item (iv), since the property (iii). The condition (vi) holds since the property (i).

Now complete the proof of Proposition 4.2. Take the U-scheme Y′ ⊂ B as in the beginning of the proof. This U-scheme Y′ satisfies the assumption of Lemma 4.3. Take the closed subscheme Y of 𝔸_{U}^{1} as in the item (v) of the Lemma. For this Y the conditions (ii) and (iii) of the Proposition are obviously satisfied. The condition (i) is satisfied too, since already it is satisfied for the U-scheme Y′. The Proposition follows.

Proof of Theorem 1.4. Set Z := {h = 0} ∪ s(U) ⊂ 𝔸_{U}^{1}. Clearly, Z is finite over U. Since the principal G-bundle ℰ_{t} is trivial over (𝔸_{U}^{1})_{h} it is trivial over 𝔸_{U}^{1} −Z. Note that {h = 0} is closed in ℙ_{U}^{1} and finite over U because h is monic. Further, s(U) is also closed in ℙ_{U}^{1} and finite over U because it is a zero set of a degree one monic polynomial. Thus Z ⊂ ℙ_{U}^{1} is closed and finite over U.

Since the principal G-bundle ℰ_{t} is trivial over (𝔸_{U}^{1})_{h}, and G-bundles can be glued in Zariski topology, there exists a principal G-bundle 𝒢 over ℙ_{U}^{1} such that

(i) its restriction to 𝔸_{U}^{1} coincides with ℰ_{t};

(ii) its restriction to ℙ_{U}^{1} − Z is trivial.

Now choose Y in 𝔸_{U}^{1} as in Proposition 4.2. Clearly, Y is finite étale over U and closed in ℙ_{U}^{1}. Moreover, Y ∩ {∞} × U = ∅ = Z ∩ {∞} × U and Y ∩ Z = ∅. Applying Theorem 1.6 with this choice of Y and Z, we see that the restriction of 𝒢 to ℙ_{U}^{1} − Y is a trivial G-bundle. Since s(U) is in 𝔸_{U}^{1} − Y and 𝒢|_{𝔸_{U}^{1}} coincides with ℰ_{t}, we conclude that s^{∗}ℰ_{t} is a trivial principal G-bundle over U.

5 Reducing Theorem 1.3 to Theorems 5.1 and 5.2

Theorem 5.1 is a purely geometric one. Theorem 5.2 contains additionally group scheme data in its condition and contains additional equating results concerning those group scheme data in its conclusion. Reducing Theorem 1.5 to Theorem 5.2 is in Section 8. Let k be a field.

Theorem 5.1. Let X be an affine k-smooth irreducible k-variety, and let x_{1}, x_{2}, …, x_{n} be closed points in X. Let U = Spec(𝒪_{X,{x_{1},x_{2},...,x_{n}}}). Given a non-zero function f ∈ k[X], vanishing at each the point x_{i} there is a diagram of the form

\xymatrix{A^{1} × U \ar[dr]_{_{pr_{U}}} & 𝒳 \ar[d] \ar[l]_{σ} \ar[d]_{q_{U}} \ar[r]^{q_{X}} & X \\  & U \ar[ur]_{can} \ar[u]_{Δ}}  (1)

with an irreducible scheme 𝒳, a smooth morphism q_{U}, a finite surjective morphism σ and an essentially smooth morphism q_{X}, and a function f′ ∈ q_{X}^{∗}(f )k[𝒳], which enjoys the following properties:

(a) if 𝒵′ is a closed subscheme of 𝒳 defined by the principal ideal (f′), the morphism σ|_{𝒵′} : 𝒵′ → A^{1} × U is a closed embedding and the morphism q_{U}|_{𝒵′} : 𝒵′ → U is finite;

(a’) q_{U} ∘ Δ = id_{U} and q_{X} ∘ Δ = can and σ ∘ Δ = i_{0};

(b) σ is étale in a neighborhood of 𝒵′ ∪ Δ(U);

(c) σ^{−1}(σ(𝒵′)) = 𝒵′ ∐ 𝒵′′ scheme theoretically and 𝒵′′ ∩ Δ(U) = ∅;

(d) 𝒟_{0} := σ^{−1}({0} × U) = Δ(U) ∐ 𝒟′_{0} scheme theoretically and 𝒟′_{0} ∩ 𝒵′ = ∅; (e) for 𝒟_{1} := σ^{−1}({1} × U) one has 𝒟_{1} ∩ 𝒵′ = ∅.

(f) there is a monic polinomial h ∈ 𝒪[t] such that (h) = Ker[𝒪[t] ⟶^{¯ ∘σ^{∗}} Γ(𝒳, 𝒪_{𝒳})/(f′)].

Theorem 5.2. Let X, {x_{1}, x_{2}, …, x_{n}} ⊂ X, U = Spec(𝒪_{X,{x_{1},x_{2},...,x_{n}}}), and f ∈ k[X] be as in Theorem 5.1. Let G be a reductive X-group scheme and G_{U} := can^{∗}(G). Let C be an X-tori and C_{U} := can^{∗}(C). Let μ : G → C be an X-group scheme morphism smooth as a scheme morphism. Let μ_{U} = can^{∗}(μ) : G_{U} → C_{U}.

Then there exists a diagram of the form (1) with an irreducible scheme 𝒳, a smooth morphism q_{U}, a finite surjective morphism σ and an essentially smooth morphism q_{X}, and a function f′ ∈ q_{X}^{∗}(f )k[𝒳], which enjoys the conditions (a) to (f ) from Theorem 5.1, and additionally there are 𝒳-group scheme isomorphisms

Φ : q_{U}^{∗}(G_{U}) → q_{X}^{∗}(G) and Ψ : q_{U}^{∗}(C_{U}) → q_{X}^{∗}(C)

such that Δ^{∗}(Φ) = id_{G_{U}}, Δ^{∗}(Ψ) = id_{C_{U}} and q_{X}^{∗}(μ) ∘ Φ = Ψ ∘ q_{U}^{∗}(μ_{U}).

The proof of these two theorems we postpone till Section 15. Right now we formulate its first consequence (see Corollary 5.3). To do that note that using the items (b) and (c) of Theorem 5.1 one can find an element g ∈ I(𝒵′′) such that

(1) (f′) + (g) = Γ(𝒳, 𝒪_{𝒳}),

(2) Ker((Δ)^{∗}) + (g) = Γ(𝒳, 𝒪_{𝒳}),

(3) σ_{g} = σ|_{𝒳_{g}} : 𝒳_{g} → A_{U}^{1} is étale.

Corollary 5.3 (Corollary of Theorem 5.1). The function f′ from Theorem 5.1, the polinomial h from the item (f ) of that Theorem, the morphism σ : 𝒳 → A_{U}^{1} and the function g ∈ Γ(𝒳, 𝒪_{𝒳}) defined just above enjoy the following properties:

(i) the morphism σ_{g} = σ|_{𝒳_{g}} : 𝒳_{g} → A^{1} × U is étale,

(ii) data (𝒪[t], σ_{g}^{∗} : 𝒪[t] → Γ(𝒳, 𝒪_{𝒳})_{g}, h) satisfies the hypotheses of [C-T/O, Prop.2.6], i.e. Γ(𝒳, 𝒪_{𝒳})_{g} is a finitely generated as the 𝒪[t]-algebra, the element (σ_{g})^{∗}(h) is not a zero-divisor in Γ(𝒳, 𝒪_{𝒳})_{g} and 𝒪[t]/(h) = Γ(𝒳, 𝒪_{𝒳})_{g}/hΓ(𝒳, 𝒪_{𝒳})_{g} ,

(iii) (Δ(U) ∪ 𝒵) ⊂ 𝒳_{g} and σ_{g} ∘ Δ = i_{0} : U → A^{1} × U,

(iv) 𝒳_{gh} ⊆ 𝒳_{gf′} ⊆ 𝒳_{f′} ⊆ 𝒳_{q_{X}^{∗}(f)} ,

(v) 𝒪[t]/(h) = Γ(𝒳, 𝒪_{𝒳})/(f′) and hΓ(𝒳, 𝒪_{𝒳}) = (f′)∩I(𝒵′′) and (f′)+I(𝒵′′) = Γ(𝒳, 𝒪_{𝒳}).

The same properties holds for the function f′ from Theorem 5.2, the polinomial h from the item (f ) of that Theorem, the morphism σ : 𝒳 → A_{U}^{1} and the function g ∈ Γ(𝒳, 𝒪_{𝒳}) defined just above.

Before proving this Corollary let us make one remark Remark 5.4. The item (ii) of this corollary shows that the cartesian square

\xymatrix{𝒳_{gh} \ar[r]^{inc} \ar[d]_{σ_{gh}} & 𝒳_{g} \ar[d]^{σ_{g}} \\ (A^{1} × U)_{h} \ar[r]^{inc} & A^{1} × U}  (2)

can be used to glue principal G-bundles for a reductive U-group scheme G. The items (i) and (ii) show that the square (2) is an elementary distinguished square in the category of smooth U-schemes in the sense of [MV, Defn.3.1.3]. The item (iv) guaranties that a principal G-bundle on 𝒳, which is trivial being restricted to 𝒳_{q_{X}^{∗}(f)} is trivial being restricted to 𝒳_{gh}.

Proof of Corollary 5.3. We will use notation from Theorem 5.1. Since 𝒳 is a regular affine irreducible and σ : 𝒳 → A_{U}^{1} is finite surjective the induced 𝒪-algebra homomorphism σ^{∗} : 𝒪[t] → Γ(𝒳, 𝒪_{𝒳}) is a monomorphism. We will regard below the 𝒪-algebra 𝒪[t] as a subalgebra via σ^{∗}.

The assertions (i) and (iii) of the Corollary hold by our choice of g. The assertion (iv) holds, since σ^{∗}(h) is in the principal ideal (f′) (use the properties (a) and ( f) from Theorem 5.1). It remains to prove the assertion (ii). The morphism σ is finite. Hence the 𝒪[t]-algebra Γ(𝒳, 𝒪_{𝒳})_{g} is finitely generated. The scheme 𝒳 is regular and irreducible. Thus, the ring Γ(𝒳, 𝒪_{𝒳}) is a domain. The homomorphism σ^{∗} is injective. Hence, the element h is not zero and is not a zero divisor in Γ(𝒳, 𝒪_{𝒳})_{g}.

It remains to check that 𝒪[t]/(h) = Γ(𝒳, 𝒪_{𝒳})_{g}/hΓ(𝒳, 𝒪_{𝒳})_{g}. Firstly, by the choice of h and by the item (a) of Theorem 5.1 one has 𝒪[t]/(h) = Γ(𝒳, 𝒪_{𝒳})/(f′). Secondly, by the property (1) of the element g one has Γ(𝒳, 𝒪_{𝒳})/(f′) = Γ(𝒳, 𝒪_{𝒳})_{g}/f′Γ(𝒳, 𝒪_{𝒳})_{g}. Finally, by the items (c) and (a) of Theorem 5.1 one has

Γ(𝒳, 𝒪_{𝒳})/(f′) × Γ(𝒳, 𝒪_{𝒳})/I(𝒵′′) = Γ(𝒳, 𝒪_{𝒳})/(h).  (3)

Localizing both sides of (3) in g one gets an equality

Γ(𝒳, 𝒪_{𝒳})_{g}/f′Γ(𝒳, 𝒪_{𝒳})_{g} = Γ(𝒳, 𝒪_{𝒳})_{g}/hΓ(𝒳, 𝒪_{𝒳})_{g},

hence 𝒪[t]/(h) = Γ(𝒳, 𝒪_{𝒳})/(f′) = Γ(𝒳, 𝒪_{𝒳})_{g}/f′Γ(𝒳, 𝒪_{𝒳})_{g} = Γ(𝒳, 𝒪_{𝒳})_{g}/hΓ(𝒳, 𝒪_{𝒳})_{g}. Whence the Corollary.

Reducing Theorem 1.3 to Theorem 5.2. The k-algebra 𝒪 is of the form 𝒪_{X,{x_{1},x_{2},...,x_{n}}}, where X is a k-smooth irreducible affine variety. We may and will assume in this proof that the reductive group scheme G and the principal G-bundle 𝒢 are both defined over the variety X. Futhermore we may and will assume that there is given a non-zero function f ∈ k[X] such that the G-bundle 𝒢 is trivial on X_{f} and the function f vanishes at each point x_{i} in {x_{1}, x_{2}, …, x_{n}}. By Theorem 5.2 there is a diagram of the form (1) enjoying the properties (a) to (f) from Theorem 5.1. Moreover there is a 𝒳-group scheme isomorphisms Φ : q_{U}^{∗}(G_{U}) → q_{X}^{∗}(G) such that Δ^{∗}(Φ) = id_{G_{U}}. Let f′, h, σ and g be as in Corollary 5.3.

Given a G-bundle 𝒢 over X, which is trivial on X_{f} take its pull-back q_{X}^{∗}(𝒢) to 𝒳. Using the isomorphism Φ we may and will regard the q_{X}^{∗}(G)-bundle q_{X}^{∗}(𝒢) as a q_{U}^{∗}(G_{U})-bundle, t.e. as a G_{U}-bundle. We will denote that G_{U}-bundle by _{U} q_{X}^{∗}(𝒢).

The q_{X}^{∗}(G)-bundle q_{X}^{∗}(𝒢) is trivial on 𝒳_{q_{X}^{∗}(f)}. Thus the G_{U}-bundle _{U} q_{X}^{∗}(𝒢) is trivial on 𝒳_{q_{X}^{∗}(f)}. By the property (iv) of the Corollary 5.3 it is also trivial on 𝒳_{gh}.

Take a trivial G_{U}-bundle over (A_{U}^{1})_{h} and glue it with the G_{U}-bundle _{U} q_{X}^{∗}(𝒢)|_{𝒳_{g}} patching over 𝒳_{gh} (it can be done due to Remark 5.4). We get a G_{U}-bundle 𝒢_{t} over A_{U}^{1} which has particularly the following properties:

(a) the restriction of 𝒢_{t} to (A_{U}^{1})_{h} is trivial (by the construction);

(b) the G_{U}-bundle σ_{g}^{∗}(𝒢_{t}) is isomorphic to the G_{U}-bundle _{U} q_{X}^{∗}(𝒢)|_{𝒳_{g}}. It remains to check that the restriction of the G_{U}-bundle 𝒢_{t} to 0 × U is isomorphic to the G_{U}-bundle can^{∗}(𝒢). To do that note that the equalities q_{U} ∘ Δ = id_{U} and q_{X} ∘ Δ = can yield the equalities

G_{U} = Δ^{∗}(q_{X}^{∗}(G)) and can^{∗}(𝒢) = Δ^{∗}(q_{X}^{∗}(𝒢)).

There are two interesting G_{U}-bundles over U. Namely, the G_{U}-bundle can^{∗}(𝒢) = Δ^{∗}(q_{X}^{∗}(𝒢)) and the G_{U}-bundle Δ^{∗}(_{U} q_{X}^{∗}(𝒢)). They coincide since Δ^{∗}(Φ) = id_{G_{U}}. Thus

can^{∗}(𝒢) = Δ^{∗}(_{U} q_{X}^{∗}(𝒢)) ≅ Δ^{∗}(σ_{g}^{∗}(𝒢_{t})) = 𝒢_{t}|_{0×U}.

The middle isomorphism is well-defined by the property (iii) of Corollary 5.3. The latter equality holds also by the property (iii) of Corollary 5.3. Whence the Theorem 1.3.

Remark 5.5. Here is the motivic view point on the above arguments (in the constant case). The distinguished elementary square (2) defines a motivic space isomorphism 𝒳_{g}/𝒳_{gh} ⟵^{σ} A_{U}^{1}/(A^{1}×U)_{h} (just a Nisnevich sheaf isomorphism), hence there is a composite morphism of motivic spaces of the form

φ : A_{U}^{1}/(A^{1} × U)_{h} ⟶^{σ^{−1}} 𝒳_{g}/𝒳_{gh} → 𝒳_{g}/𝒳_{q_{X}^{∗}(f)} ⟶^{q} X/X_{f}.

Let i_{0} : 0 × U → A_{U}^{1}/(A_{U}^{1})_{h} be the natural morphism. By the properties (a’) and (d) from Theorem 5.1 the morphism φ ∘ i_{0} equals to the one

U ⟶^{can} X ⟶^{p} X/X_{f},

where p : X → X/X_{f} is the canonical morphisms.

Now assume that G_{0} is a reductive group scheme over the field k. A G_{0}-bundle over X, trivialized on X_{f}, is ”classified” by a morphism ρ : X/X_{f} → (BG_{0})_{et} in an appropriate category. Thus the morphism ρ ∘ φ ”classifies” a G_{0}-bundle 𝒢_{t} over A_{U}^{1} trivialized on (A_{U}^{1})_{h}. The equality φ ∘ i_{0} = p ∘ can shows that the G_{0}-bundles 𝒢_{t}|_{0×U} and can^{∗}(𝒢) are isomorphic. This ”proves” Theorem 1.3 in the constant case.

6 A purity theorem

Let k be a field.

Theorem 6.1. Let 𝒪 be the semi-local ring of finitely many closed points on a k-smooth irreducible affine k-variety X. Let K = k(X). Let

μ : G → C

be a smooth 𝒪-morphism of reductive 𝒪-group schemes, with a torus C. Suppose additionally that the kernel of μ is a reductive 𝒪-group scheme. Then the following sequence

\{1\} → C(𝒪)/μ(G(𝒪)) → C(K)/μ(G(K)) \underrightarrow{∑ res_{𝔭}} ⨁_{𝔭} C(K)/[C(𝒪_{𝔭}) · μ(G(K))] → \{1\}

(4)

is exact in the middle term, where 𝔭 runs over the height 1 primes of 𝒪 and res_{𝔭} is the natural map (the projection to the factor group).

Remark 6.2. After the proof of Theorem 1.1 will be completed it will have the following corollary: the sequence (4) is exact. In fact, the exactness at the lefthand side term is a direct consequence Theorem 1.1. The exactness at the righthand side term is proved in [C-T/S].

Definition 6.3. Let a ∈ C(k(X)). Its class ā ∈ C̄(k(X)) := C(K)/μ(G(K)) is called unramified at a height 1 prime ideal 𝔭 of k[X], if the element ā is in the image of the group C̄(𝒪_{𝔭}). Let S ⊂ k[X] be a multiplicative system The class ā ∈ C̄(k(X)) is called k[X]_{S} -unramified, if it is unramified at any codimension one prime ideal of k[X]_{S}.

The following lemma is obvious.

Lemma 6.4. Let φ : Y → X be a smooth morphism of smooth irreducible affine kvarieties. This morphism induces an obvious map φ̄^{∗} : C̄(k(X)) → C̄(k(Y)), which takes X-unramified elements to Y -unramified elements. If S ⊂ k[Y] be a multiplicative system, then the homomorphism φ̄^{∗} takes X-unramified elements to k[Y]_{S} -unramified elements.

Reducing Theorem 6.1 to Theorem 5.2. Assume firstly that μ is ”constant”, i.e there are a reductive group G_{0}, a torus C_{0} over the field k and an algebraic k-group morphism μ_{0} and U-group schemes isomorphisms

Φ : G_{0,U} = G_{0} ×_{Spec(k)} U → G and Ψ : C_{0,U} = C_{0} ×_{Spec(k)} U → C

such that Ψ ∘ μ_{0,U} = μ ∘ Φ.

Let a_{K} ∈ C(k(X)) be such that its class in C̄(K) is 𝒪-unramified. Then there is a non-zero function f ∈ k[X] such that the element a_{K} is defined over X_{f}, that is there is given an element a ∈ C(k[X_{f}]) for a non-zero function f ∈ k[X] such that the image of a in C(K) coincides with the element a_{K}. Shrinking X we may assume further that f vanishes at each x_{i}’s and the k-algebra k[X]/(f) is reduced. Shrinking X once again we may and will assume also that ā ∈ C̄(k[X]_{f}) is k[X]-unramified. By Theorem 5.1 there is a diagram of the form (1) together with with the scheme 𝒳, the morphisms q_{U}, σ and q_{X}, and the function f′ ∈ q_{X}^{∗}(f )k[𝒳], which enjoys the properties (a) to (f) from that Theorem. From now on and till the end of this proof we will use the notation from Theorem 5.1.

The morphism σ from that theorem is finite surjective and the schemes A_{U}^{1} and 𝒳 are regular. Thus by a theorem of Grothendieck [E] the morphism σ is flat and finite. Thus any base change of σ is finite and flat. Set α := q_{f}^{∗}(a) ∈ C(𝒳_{f}) where q_{f} : 𝒳_{f} → X_{f} is the restriction of f to 𝒳 and set

a_{U} := N_{𝒟_{1}/U}(α|_{𝒟_{1}}) · N_{𝒟′_{0}/U}(α|_{𝒟′_{0}})^{−1} ∈ C(U).  (5)

Claim 6.5. Let η_{U} : Spec(k(X)) → Spec(𝒪) = U and η : Spec(k(X)) → X_{f} be the generic points of U and X_{f} respectively. Then η_{U}^{∗}(ā_{U}) = η^{∗}(ā) ∈ C̄(k(X)).

Since η^{∗}(a) = a_{K}, this Claim completes the proof of Theorem 6.1 in the constant case. To prove the Claim consider the scheme 𝒳 and its closed and open subschemes as U-schemes via the morphism q_{U}. Set K = k(X). Taking the base change of 𝒳, A_{U}^{1} and σ via the morphism η_{U} : Spec(K) → U we get a morphism of the K-shemes A_{K}^{1} ⟵^{σ_{K}} 𝒳_{K}. Recall that the class ā ∈ C̄(X_{f}) is X-unramified. By Lemma 6.4 the class ᾱ ∈ C̄(𝒳_{f}) is 𝒳-unramified. Hence its image ᾱ_{K} in C̄(K(𝒳_{K})) is X_{K}-unramified too. The item (v) of Corollary 5.3 and Lemma 17.4 show that for the element β_{t} := N_{K(𝒳_{K})/K(A_{K}^{1})}(α_{K}) ∈ C(K(t)) the class β̄_{t} ∈ C̄(K(t)) is A_{K}^{1}-unramified. By Theorem 18.4 the class β̄_{t} is constant, t.e. it comes from the field K. By Corollary 18.5 its specializations at the K-points 0 and 1 of the affine line A_{K}^{1} coincide: s_{0}(β̄_{t}) = s_{1}(β̄_{t}) ∈ C̄(K). The properties (d),(c) and (e) and the equality q_{X} ∘ Δ = can from Theorem 5.1 show that 𝒟_{1,K}, 𝒟′_{0,K}, Δ(Spec(K)) ⊂ (𝒳_{f})_{K}. Thus there is a Zariski open neighborhood V of the Kpoints 0 and 1 in A_{K}^{1} such that W := (σ_{K})^{−1}(V) ⊂ (𝒳_{f})_{K}. Hence for β_{V} := N_{W/V} (α|_{W}), one has the equality β_{V} = β_{t} in C(K(t)). Thus one has equalities

\overline{β(1)} = s_{1}(β̄_{t}) = s_{0}(β̄_{t}) = \overline{β(0)}

(see the remark at the end of Definition 18.1). By the properties (i′), (ii′) and (iii′) of the norm maps (see Section 16) one has equalities

N_{𝒟_{1,K}/K}(α|_{𝒟_{1,K}}) = β(1) and β(0) = N_{𝒟_{0,K}/K}(α|_{𝒟_{0,K}}) = N_{𝒟′_{0,K}/K}(α|_{𝒟′_{0,K}}) · Δ_{K}^{∗}(α_{K})

By the base change property of the norm maps one has the equality

η_{U}^{∗}(a_{U}) = N_{𝒟_{1,K}/K}(α|_{𝒟_{1,K}}) · [N_{𝒟′_{0,K}/K}(α|_{𝒟′_{0,K}})]^{−1}

Hence Δ_{K}^{∗}(ᾱ_{K}) = η_{U}^{∗}(ā_{U}) in C̄(k(X)). Finally, the composite map Spec(K) ⟶^{Δ_{K}} (𝒳_{f})_{K} → 𝒳_{f} ⟶^{q_{f}} X_{f} coincides with the canonical map η : Spec(K) → X_{f}. Hence Δ_{K}^{∗}(ᾱ_{K}) = η^{∗}(ā), which proves the Claim. Whence the Theorem 6.1 in the constant case.

In the general case there are two functors on the category of 𝒳-schemes. Namely, C̄ and _{U} C̄. If r : 𝒴 → 𝒳 is a scheme morphism, then C̄(𝒴) := C(𝒴)/(μ(G(𝒴))) and U C̄(𝒴) := _{U}C(𝒴)/(μ(_{U}G(𝒴))). Here 𝒴 is regarded as an X-scheme via the morphism q_{X} ∘ r and is regarded as an U-scheme via the morphism q_{U} ∘ r. The 𝒳-group scheme isomorphisms Φ and Ψ from Theorem 5.2 induce a group isomorphism

Ψ̄_{𝒴} : _{U} C̄(𝒴) → C̄(𝒴),

which respect to 𝒳-schemes morphisms. Moreover, if the scheme U is regarded as an 𝒳-scheme via the morphism Δ, then the isomorphism Ψ̄_{𝒴} is the identity. And similarly for any U-scheme g : W → U regarded as an 𝒳-scheme via the morphism Δ ∘ g the the isomorphism Ψ̄_{W} is the identity.

Set α := q_{f}^{∗}(a) ∈ C(𝒳_{f}) where q_{f} : 𝒳_{f} → X_{f} is above in this proof. Let _{U} α ∈ _{U}C(𝒳) be a unique element such that Ψ̄_{𝒳}(_{U} α) = α. Set

U a := N_{𝒟_{1}/U}((_{U} α)|_{𝒟_{1}}) · N_{𝒟′_{0}/U}((_{U} α)|_{𝒟′_{0}})^{−1} ∈ _{U} C(U) and a_{U} := Ψ_{U}(_{U} a) ∈ C(U) (6)

We left to the reader to proof the following Claim

Claim 6.6. Let η_{U} : Spec(k(X)) → Spec(𝒪) = U and η : Spec(k(X)) → X_{f} be as above in this proof. Then

η_{U}^{∗}(ā_{U}) = η^{∗}(ā) ∈ C̄(k(X))

Since η^{∗}(a) = a_{K}, this Claim completes the reduction of Theorem 6.1 to Theorem 5.2.

7 One more purity theorem

In this section we reduce another purity theorem for reductive group schemes to Theorem 5.2. Let k, 𝒪 and K be as in Theorem 6.1. Let G be a semi-simple 𝒪-group scheme. Let i : Z ↪ G be a closed subgroup scheme of the center Cent(G). It is known that Z is of multiplicative type. Let G′ = G/Z be the factor group, π : G → G′ be the projection. It is known that π is finite surjective and strictly flat. Thus the sequence of 𝒪-group schemes

{1} → Z ⟶^{i} G ⟶^{π} G′ → {1}  (7)

induces an exact sequence of group sheaves in fppt-topology. Thus for every 𝒪-algebra R the sequence (7) gives rise to a boundary operator

δ_{π,R} : G′(R) → H_{fppt}^{1}(R, Z)  (8)

One can check that it is a group homomorphism (compare [Se, Ch.II, §5.6, Cor.2]). Set

ℱ(R) = H_{fttp}^{1}(R, Z)/Im(δ_{π,R}).  (9)

Clearly we get a functor on the category of 𝒪-algebras.

Theorem 7.1. Let 𝒪 be the semi-local ring of finitely many closed points on a k-smooth irreducible affine k-variety X. Let G be a semi-simple 𝒪-group scheme. Let i : Z ↪ G be a closed subgroup scheme of the center Cent(G). Let ℱ be the functor on the category 𝒪-algebras given by (9). Then the sequence

ℱ(𝒪) → ℱ(K) \underrightarrow{∑ can_{𝔭}} ⨁_{𝔭} ℱ(K)/Im[ℱ(𝒪_{𝔭}) → ℱ(K)]  (10)

is exact, where 𝔭 runs over the height 1 primes of 𝒪 and can_{𝔭} is the natural map (the projection to the factor group).

Reducing Theorem 7.1 to Theorem 5.2. The group Z is of multiplicative type. So we can find a finite étale 𝒪-algebra A and a closed embedding Z ↪ R_{A/𝒪}(𝔾_{m, A}) into the permutation torus T^{+} = R_{A/𝒪}(𝔾_{m, A}). 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. By the known fact (see Lemma 8.1) and Hilbert 90 for the semi-local 𝒪-algebra A one has H_{fppt}^{1}(𝒪, T^{+}) = H_{ét}^{1}(𝒪, T^{+}) = H_{ét}^{1}(A, 𝔾_{m,A}) = {∗}. So, the latter diagram gives rise to a commutative diagram of pointed sets

\xymatrix{ &  & H_{fppt}^{1}(𝒪, G′) \ar[r]^{id} & H_{fppt}^{1}(𝒪, G′) \\ G^{+}(𝒪) \ar[r]^{μ_{𝒪}^{+}} & T (𝒪) \ar[r]^{δ_{𝒪}^{+}} & H_{fppt}^{1}(𝒪, G) \ar[r]^{j_{∗}^{+}} \ar[u]^{π_{∗}} & H_{fppt}^{1}(𝒪, G^{+}) \ar[u]^{π_{∗}^{+}} \\ T^{+}(𝒪) \ar[r]^{μ_{𝒪}} \ar[u]^{i_{∗}^{+}} & T (𝒪) \ar[r]^{δ_{𝒪}} \ar[u]^{id} & H_{fppt}^{1}(𝒪, Z) \ar[r]^{μ} \ar[u]^{i_{∗}} & \{∗\} \ar[u]^{i_{∗}^{+}} \\  &  & G′(𝒪) \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_{fppt}^{1}(𝒪, Z)/Im(δ_{π,𝒪}) = ker(π_{∗}) = ker(j_{∗}^{+}) = T (𝒪)/μ^{+}(G^{+}(𝒪)).

Clearly these isomorphisms respect 𝒪-homomorphisms of semi-local 𝒪-algebras.

The morphism μ^{+} : G^{+} → T is a smooth 𝒪-morphism of reductive 𝒪-group schemes, with the torus T. The kernel ker(μ^{+}) is equal to G and G is a reductive 𝒪-group scheme. Now by Theorem 6.1 the sequence (4) is exact. Thus the sequence (10) is exact too.

8 Reducing Theorem 1.5 to Theorem 5.2

Reducing the semi-simple case of Theorem 1.5 to Theorem 5.2. Let 𝒪 and G be the same as in Theorem 1.5 and assume additionally that G is semi-simple. We need to prove that

ker[H_{ét}^{1}(𝒪, G) → H_{ét}^{1}(K, G)] = ∗.  (11)

Let G^{sc} be the corresponding simply-connected semi-simple 𝒪-group scheme and let π : G^{sc} → G be the corresponding 𝒪-group scheme morphism. Let Z = ker(π). It is known that Z is contained in the center Cent(G^{sc}) of G^{sc} and Z is a finite group scheme of multiplicative type. It is known that G = G^{sc}/Z and π is finite surjective and strictly flat. Thus the sequence of 𝒪-group schemes

{1} → Z ⟶^{i} G^{sc} ⟶^{π} G → {1},  (12)

gives rise to an exact sequence of pointed sets

H_{fppt}^{1}(𝒪, Z)/∂(G(𝒪)) → H_{fppt}^{1}(𝒪, G^{sc}) → H_{fppt}^{1}(𝒪, G) → H_{fppt}^{2}(𝒪, Z)  (13)

(here H_{fppt}^{1}(𝒪, Z)/∂(G(𝒪)) is the factor-group; compare ([Se, Ch.I,Sect.5, Prop.39 and Cor.1 of Prop.40,Cor.2 of Prop.40])). By the Theorem 7.1 the functor

ℱ(R) = H_{fppt}^{1}(R, Z)/Im(δ_{π,R}) = H_{fppt}^{1}(R, Z)/∂(G(R)).

satisfies purity for the ring 𝒪. The following result is known (see [Gr3, Thm.11.7])

Lemma 8.1. Let R be a noetherian ring. Then for a reductive R-group scheme H and for n = 0, 1 the canonical map H_{ét}^{n}(R, H) → H_{fppt}^{n}(R, H) is a bijection of pointed sets. For a R-tori T and for each integer n ≥ 0 the canonical map H_{ét}^{n}(R, T) → H_{fppt}^{n}(R, T) is an isomorphism.

Lemma 8.2. For the ring 𝒪 above the map H_{fppt}^{2}(𝒪, Z) → H_{fppt}^{2}(K, Z) is injective.

Proof. See [C-T/S, Thm.4.3].

Continue the proof of the equality (11). We have the exact sequence of pointed sets

H_{fppt}^{1}(𝒪, Z)/∂(G(𝒪)) → H_{fppt}^{1}(𝒪, G^{sc}) → H_{fppt}^{1}(𝒪, G) → H_{fppt}^{2}(𝒪, Z)  (14)

and furthermore a commutative diagram with exact arrows

\xymatrix{H_{fppt}^{1}(𝒪, Z)/∂(G(𝒪)) \ar[r]^{i_{∗}} \ar[d]_{α} & H_{fppt}^{1}(𝒪, G^{sc}) \ar[r]^{π_{∗}} \ar[d]^{β} & H_{fppt}^{1}(𝒪, G) \ar[r]^{∂} \ar[d]_{γ} & H_{fppt}^{2}(𝒪, Z) \ar[d]^{δ} \\ H_{fppt}^{1}(𝒪_{𝔭}, Z)/∂(G(𝒪_{𝔭})) \ar[r]^{i′_{∗}} \ar[d]_{α_{𝔭}} & H_{fppt}^{1}(𝒪_{𝔭}, G^{sc}) \ar[r]^{π_{∗}} \ar[d]^{β_{𝔭}} & H_{fppt}^{1}(𝒪_{𝔭}, G) \ar[r]^{∂} \ar[d]_{γ_{𝔭}} & H_{fppt}^{2}(𝒪_{𝔭}, Z) \ar[d]^{δ_{𝔭}} \\ H_{fppt}^{1}(K, Z)/∂(G(K)) \ar[r]^{i′′_{∗}} & H_{fppt}^{1}(K, G^{sc}) \ar[r]^{π_{∗}} & H_{fppt}^{1}(K, G) \ar[r]^{∂} & H_{fppt}^{1}(K, Z)}

(15)

Here 𝔭 ⊂ 𝒪 is a hight one prime ideal in 𝒪. The maps i_{∗}, i′_{∗} and i′′_{∗} are injective (compare ([Se, Ch.I,Sect.5, Prop.39 and Cor.1 of Prop.40])). 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 [Ni2] and Lemma 8.1 one has

ker(β_{𝔭}) = ker(γ_{𝔭}) = ∗ .  (16)

Thus ker(α_{𝔭}) = ∗. By the assumptions of Theorem 1.5 and by Lemma 8.1 one has

ker[H_{fppt}^{1}(𝒪, G^{sc}) → H_{fppt}^{1}(K, G^{sc})] = ∗ .  (17)

By Lemma 8.2 the map δ_{K} is injective. As mentioned right above Lemma 8.1 the functor ℱ(R) = H_{fppt}^{1}(R, Z)/∂(G(R)) satisfies purity for the ring 𝒪. Now we are ready to make a diagram chaise.

Let ξ ∈ ker(γ_{K}), then ∂(ξ) ∈ ker(δ_{K}). By Lemma 8.2 one has ker(δ_{K}) = ∗, whence ∂(ξ) = ∗ and ξ = π_{∗}(ζ) for an ζ ∈ H_{ét}^{1}(𝒪, G^{sc}). Since γ_{K}(ξ) = ∗ and ker(γ_{𝔭}) = ∗ we see that γ(ξ) = ∗. Thus π_{∗}(β(ζ)) = ∗ and β(ζ) = i′_{∗}(ϵ_{𝔭}) for an ϵ_{𝔭} ∈ H_{fppt}^{1}(𝒪_{𝔭}, Z)/∂(G(𝒪_{𝔭})). A diagram chaise shows that there exists a unique element ϵ_{K} ∈ H_{fppt}^{1}(K, Z)/∂(G(K)) such that for each hight one prime ideal 𝔭 of 𝒪 one has

α_{𝔭}(ϵ_{𝔭}) = ϵ_{K} ∈ H_{fppt}^{1}(K, Z)/∂(G(K)).

By purity Theorem 7.1 there exists an element ϵ ∈ H_{fppt}^{1}(𝒪, Z)/∂(G(𝒪)) such that α_{K}(ϵ) = ϵ_{K}. The element ϵ_{K} has the property that i′′_{∗}(ϵ_{K}) = β_{K}(ζ). Whence β_{K}(i_{∗}(ϵ)) = β_{K}(ζ). The map β_{K} : H_{ét}^{1}(𝒪, G^{sc}) → H_{ét}^{1}(K, G^{sc}) is injective since by the hypotheses of Theorem 1.5 such a map has trivial kernel for all semi-simple simply-connected reductive 𝒪-group schemes. Whence i_{∗}(ϵ) = ζ and ξ = i_{∗}(π_{∗}(ϵ)) = ∗. The semi-simple case of Theorem 1.5 is reduced to Theorem 5.2.

Claim 8.3. Under the hypotheses of Theorem 1.5 for all semi-simple reductive 𝒪-group scheme G the map H_{ét}^{1}(𝒪, G) → H_{ét}^{1}(K, G) is injective.

In fact, let ξ, ζ ∈ H_{ét}^{1}(𝒪, G) be two elements such that its images ξ_{K}, ζ_{K} in H_{ét}^{1}(K, G) are equal. Let _{ξ}G, _{ζ}G be the corresponding principal G-bundles over 𝒪 and G(ζ) be the inner form of the 𝒪-group scheme G corresponding to ζ. The 𝒪-scheme \underline{Iso}(_{ξ}G, _{ζ}G) is a principal G(ζ)-bundle over 𝒪, which is trivial over K. Since G(ζ) is semi-simple reductive over 𝒪, the 𝒪-scheme \underline{Iso}(_{ξ}G, _{ζ}G) has an 𝒪-point. Whence the Claim.

Reducing Theorem 1.5 to Theorem 5.2. Let 𝒪 and G be the same as in Theorem 1.5. Consider a short sequence of reductive 𝒪-group schemes

{1} → G_{der} ⟶^{i} G ⟶^{μ} C → {1},  (18)

where G_{der} is the derived 𝒪-group scheme of G and C = corad(G) be a tori over 𝒪 and μ = f_{0} (see [D-G, Exp.XXII, Thm.6.2.1]). By that Theorem the morphism μ is smooth and its kernel is the reductive 𝒪-group scheme G_{der}. Moreover G_{der} is a semi-simple 𝒪-group scheme. By Claim 8.3 the map

H_{ét}^{1}(𝒪, G_{der}) → H_{ét}^{1}(K, G_{der})  (19)

is injective. We need to prove that

ker[H_{ét}^{1}(𝒪, G) → H_{ét}^{1}(K, G)] = ∗.  (20)

The sequence (18) of 𝒪-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(𝒪)/μ(G(𝒪)) \ar[r]^{∂} \ar[d]_{α} & H_{ét}^{1}(𝒪, G_{der}) \ar[r]^{i_{∗}} \ar[d]^{β} & H_{ét}^{1}(𝒪, G) \ar[r]^{μ} \ar[d]_{γ} & H_{ét}^{1}(𝒪, C) \ar[d]^{δ} \\ \{1\} \ar[r] & C(𝒪_{𝔭})/μ(G(𝒪_{𝔭})) \ar[r]^{∂} \ar[d]_{α_{𝔭}} & H_{ét}^{1}(𝒪_{𝔭}, G_{der}) \ar[r]^{i_{∗}} \ar[d]^{β_{𝔭}} & H_{ét}^{1}(𝒪_{𝔭}, G) \ar[r]^{μ} \ar[d]_{γ_{𝔭}} & H_{ét}^{1}(𝒪_{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)}

(21)

Here 𝔭 ⊂ 𝒪 is a hight one prime ideal in 𝒪. 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 [Ni2] one has

ker(α_{𝔭}) = ker(β_{𝔭}) = ker(γ_{𝔭}) = ∗  (22)

Let ξ ∈ ker(γ_{K}), then μ(ξ) ∈ ker(δ_{K}). By [C-T/S] one has ker(δ_{K}) = ∗, whence μ(ξ) = ∗ and ξ = i_{∗}(ζ) for an ζ ∈ H_{ét}^{1}(𝒪, G_{der}). Since γ_{K}(ξ) = ∗ and ker(γ_{𝔭}) = ∗ we see that γ(ξ) = ∗. Whence i_{∗}(β(ζ)) = ∗ and β(ζ) = ∂(ϵ_{𝔭}) for an ϵ_{𝔭} ∈ C(𝒪_{𝔭})/μ(G(𝒪_{𝔭})). A diagram chaise AND Lemma 17.2 show that there exists a unique element ϵ_{K} ∈ C(K)/μ(G(K)) such that for each hight one prime ideal 𝔭 of 𝒪 one has α_{𝔭}(ϵ_{𝔭}) = ϵ_{K} ∈ C(K)/μ(G(K)).

By purity Theorem 6.1 there exists an element ϵ ∈ C(𝒪)/μ(G(𝒪)) such that α_{K}(ϵ) = ϵ_{K}. The element ϵ_{K} has the property that ∂(ϵ_{K}) = β_{K}(ζ). Whence β_{K}(∂(ϵ)) = β_{K}(ζ). The map β_{K} is injective as indicated in the beginning of the proof. Whence ∂(ϵ) = ζ and ξ = i_{∗}(∂(ϵ)) = ∗. The reduction of Theorem 1.5 to Theorem 5.2 is completed.

9 The strategy of the proof of Theorems 5.1 and 5.2

A toy task: given a finite field k, a k-smooth irreducible affine curve Y, a k-rational point y_{0} ∈ Y, a ”bad” locus B ⊂ Y finite over Spec(k) and containing the point y_{0}, find a diagram of the form A_{k}^{1} ⟵^{τ} Y′ ⟶^{θ} Y with an irreducible affine curve Y′ and a k-rational point y′_{0} ∈ Y′ and a closed subset B′ of Y′ (a new ”bad” locus) such that the morphism θ is étale, θ(y′_{0}) = y_{0}, θ^{−1}(B) ⊂ B′ and B′ has the following properies hold:

(1) τ is finite surjective,

(2) τ|_{B′} : B′ → A_{k}^{1} is a closed embedding,

(3) τ is étale at each the point of B′,

(4) τ^{−1}(0) = {y′_{0}} ⊔ D (scheme-wise) and D ∩ B′ = ∅,

(5) τ^{−1}(1) ∩ B′ = ∅. The following properties of the subset B′ are useful: (1′) the point y′_{0} ∈ B′ is the only k-rational point in B′; (2′) for any integer d ≥ 0 the amount of the degree d points on the locus B′ is less or equal to the amount of points of degree d on A_{k}^{1}. The toy task can be solved in two steps: (a) find appropriate an θ, y′ and B′; (b) find the required τ (once the conditions (1′), (2′) are full-filled it’s easy to find the required τ).

If G is a reductive Y -group scheme and G_{0} = G|_{y_{0}} and G_{const} = q^{∗}(G_{0}), then one can modify the toy task adding to the original one more conditions on Y′ and y′_{0}:

(6) there is a Y′-group scheme isomorphism Φ : θ^{∗}(G_{const}) → θ^{∗}(G) such that Φ|_{y′_{0}} = id. To solve the modified task one can firstly find an étale morphism θ′′ : Y′′ → Y

(with an irreducible Y′′) and a k-rational point y′′_{0} and a Y′′-group scheme isomorphism Φ′′ : (θ′′)^{∗}(G_{const}) → (θ′′)^{∗}(G) such that θ′′(y′′_{0}) = y and Φ′′|_{y′′_{0}} = id. Once θ′′, y′′_{0} and Φ′′ subjecting these conditions are found the modified task is reduced to the original one for a new data. Namely, for the data Y′′, y′′_{0} and B′′ := (θ′′)^{−1}(B).

Theorems 5.1 and 5.2 are proved in Section 15. One of the main idea is this: given a non-zero function f ∈ X shrink X and to make it a smooth relative curve over a smooth affine base S equipped with a finite surjective S-morphism to the affine line A^{1} × S and such that the locus {f = 0} is finite over S. Then the fibre product U ×_{S} X is a smooth relative curve over U, equipped with the section Δ, with a finite surjective U-morphism to the affine line A^{1} × U and such that the locus p_{X}^{−1}({f = 0}) is finite over U. It turns out that with this relative U-curve one can work exactly as with the k-curve Y in the toy task. This idea is inspired by one of the main idea from [Vo]. Here we refined that idea following [PSV].

10 Elementary fibrations

In this Section we extend a result of M. Artin from [A] concerning existence of nice neighborhoods. The following notion is introduced by Artin in [A, Exp. XI, Déf. 3.1].

Definition 10.1. 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}  (23)

of morphisms 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 smooth projective all of whose fibres are geometrically irreducible of dimension one;

(iii) q is finite étale all of whose fibres are non-empty.

Remark 10.2. Clearly, an elementary fibration is an almost elementary fibration in the sense of [PSV, Defn.2.1].

Using repeatedly [Poo, Thm.1.3] and [ChPoo, Thm.1.1] and modifying Artin’s arguments [A, Exp. XI,Prop. 3.3], one can prove the following result, which is a slight extension of Artin’s result [A, Exp. XI,Prop. 3.3].

Proposition 10.3. Let k be a finite field, X be a smooth geometrically irreducible affine variety over k, x_{1}, x_{2}, …, x_{n} ∈ X be a family of closed points. Then there exists 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}.

If, moreover, Z is a closed co-dimension one subvariety in X, then one can choose X^{0} and p in such a way that p|_{Z ⋂ X^{0}} : Z ⋂ X^{0} → S is finite surjective.

The following result is proved in [PSV, Prop.2.4].

Proposition 10.4. Let p : X → S be an elementary fibration. If S is a regular semi-local irreducible scheme, then there exists a commutative diagram of S-schemes

\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}}  (24)

such that the left hand side square is Cartesian. Here j and i are the same as in Definition 10.1, while 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 and A^{1} × S are regarded as S-schemes via the morphism p and the projection pr_{S}, respectively.

11 Nice triples

In the present section we recall and study certain collections of geometric data and their morphisms. The concept of a nice triple was introduced in [PSV, Defn. 3.1] and is very similar to that of a standard triple introduced by Voevodsky [Vo, Defn. 4.1], and was in fact inspired by the latter notion. Let k be a field, let X be a k-smooth irreducible affine k-variety, and let x_{1}, x_{2}, …, x_{n} ∈ X be a family of closed points. Further, let 𝒪 = 𝒪_{X,{x_{1},x_{2},...,x_{n}}} be the corresponding geometric semi-local ring.

Definition 11.1. Let U := Spec(𝒪_{X,{x_{1},x_{2},...,x_{n}}}). A nice triple over U consists of the following data:

(i) a smooth morphism q_{U} : 𝒳 → U, where 𝒳 is an irreducible scheme,

(ii) an element f ∈ Γ(𝒳, 𝒪_{𝒳}),

(iii) a section Δ of the morphism q_{U},

subject to the following conditions:

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

(b) the module Γ(𝒳, 𝒪_{𝒳})/f · Γ(𝒳, 𝒪_{𝒳}) is finite as a Γ(U, 𝒪_{U}) = 𝒪-module,

(c) there exists a finite surjective U-morphism Π : 𝒳 → A^{1} × U,

(d) Δ^{∗}(f) ≠ 0 ∈ Γ(U, 𝒪_{U}).

Definition 11.2. A morphism between two nice triples over U

(q′ : 𝒳′ → U, f′, Δ′) → (q : 𝒳 → U, f, Δ)

is an étale morphism of U-schemes θ : 𝒳′ → 𝒳 such that

(1) q′_{U} = q_{U} ∘ θ,

(2) f′ = θ^{∗}(f) · h′ for an element h′ ∈ Γ(𝒳′, 𝒪_{𝒳′}),

(3) Δ = θ ∘ Δ′.

Two observations are in order here.

• Item (2) implies in particular that Γ(𝒳′, 𝒪_{𝒳′})/θ^{∗}(f) · Γ(𝒳′, 𝒪_{𝒳′}) is a finite 𝒪-module.

• It should be emphasized that no conditions are imposed on the interrelation of Π′ and Π.

After substituting k by its algebraic closure k̃ in k[X], we can assume that X is a k̃-smooth geometrically irreducible affine k̃-scheme. To simplify the notation, we will continue to denote this new k̃ by k. Let U be as in Definition 11.1 and can : U ↪ X be the canonical inclusion of schemes.

Definition 11.3. A nice triple (q_{U} : 𝒳 → U, Δ, f) over U is called special if the set of closed points of Δ(U) is contained in the set of closed points of {f = 0}.

Remark 11.4. Clearly the following holds: let (𝒳, f, Δ) be a special nice triple over U and let θ : (𝒳′, f′, Δ′) → (𝒳, f, Δ) be a morphism between nice triples over U. Then the triple (𝒳′, f′, Δ′) is a special nice triple over U.

Proposition 11.5. One can shrink X such that x_{1}, x_{2}, …, x_{n} are still in X and X is affine, and then to construct a special nice triple (q_{U} : 𝒳 → U, Δ, f) over U and an essentially smooth morphism q_{X} : 𝒳 → X such that q_{X} ∘ Δ = can, f = q_{X}^{∗}(f ).

Proof of Proposition 11.5. If the field k is infinite, then this proposition is proved in [PSV, Prop. 6.1]. So, we may and will assume that k is finite. To prove the proposition repeat literally the proof of [PSV, Prop. 6.1]. One has to replace the references to [PSV, Prop. 2.3] and [PSV, Prop.2.4] with the reference to Propositions 10.3 and 10.4 respectively.

Let us state two crucial results which are used in Section 15 to prove Theorems 5.1 and 5.2. Their proofs are given in Sections 12 and 14 respectively. If U as in Definition 11.1 then for any U-scheme V and any closed point u ∈ U set V_{u} = u ×_{U} V. For a finite set A denote ♯A the cardinality of A.

Definition 11.6. Let (𝒳, f, Δ) be a special nice triple over U. We say that the triple (𝒳, f, Δ) satisfies conditions 1_{U}^{∗} and 2_{U}^{∗} if either the field k is infinite or (if k is finite) the following holds

(1^{∗}) for any closed point u ∈ U, any integer d ≥ 1 one has

♯{z ∈ 𝒵_{u}|deg[k(z) : k(u)] = d} ≤ ♯{x ∈ A_{u}^{1}|deg[k(z) : k(u)] = d}

(2^{∗}) for the vanishing locus 𝒵 of f and for any closed point u ∈ U the point Δ(u) ∈ 𝒵_{u} is the only k(u)-rational point of 𝒵_{u}.

Theorem 11.7. Let U be as in Definition 11.1. Let (q′_{U} : 𝒳′ → U, f′, Δ′) be a special nice triple over U subjecting to the conditions (1^{∗}) and (2^{∗}) from Definition 11.6. Then there exists a finite surjective morphism

A^{1} × U ⟵^{σ} 𝒳′

of U-schemes which enjoys the following properties:

(a) the morphism A^{1} × U ⟵^{σ|_{𝒵′}} 𝒵′ is a closed embedding;

(b) σ is étale in a neighborhood of 𝒵′ ∪ Δ′(U);

(c) σ^{−1}(σ(𝒵′)) = 𝒵′ ∐ 𝒵′′ scheme theoretically and 𝒵′′ ∩ Δ′(U) = ∅;

(d) σ^{−1}({0} × U) = Δ′(U) ∐ 𝒟 scheme theoretically and 𝒟 ∩ 𝒵′ = ∅; (e) for 𝒟_{1} := σ^{−1}({1} × U) one has 𝒟_{1} ∩ 𝒵′ = ∅.

(f) there is a monic polinomial h ∈ 𝒪[t] such that (h) = Ker[𝒪[t] ⟶^{¯ ∘σ^{∗}} Γ(𝒳′, 𝒪_{𝒳′})/(f′)].

Theorem 11.8. Let U be as in Definition 11.1. Let (𝒳, f, Δ) be a special nice triple over U. Let G_{𝒳} be a reductive 𝒳-group scheme and G_{U} := Δ^{∗}(G_{𝒳}) and G_{const} := q_{U}^{∗}(G_{U}). Let C_{𝒳} be an 𝒳-tori and C_{U} := Δ^{∗}(C_{𝒳}) and C_{const} := q_{U}^{∗}(C_{U}). Let μ_{𝒳} : G_{𝒳} → C_{𝒳} be an 𝒳-group scheme morphism smooth as a scheme morphism. Let μ_{U} = Δ^{∗}(μ_{𝒳}) and μ_{const} : G_{const} → C_{const} be the the pull-back of μ_{U} to 𝒳. Then there exist a morphism θ : (q′ : 𝒳′ → U, f′, Δ′) → (q : 𝒳 → U, f, Δ) between nice triples over U such that the triple (𝒳′, f′, Δ′) is a special nice triple over U subjecting to the conditions (1^{∗}) and (2^{∗}) from Definition 11.6. And additionally there are 𝒳-group schemes isomorphisms

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

such that (Δ′)^{∗}(Φ) = id_{G_{U}}, (Δ′)^{∗}(Φ) = id_{G_{U}} and

θ^{∗}(μ_{𝒳}) ∘ Φ = Ψ ∘ θ^{∗}(μ_{const}).  (25)

Forgetting about group-schemes in the hypotheses and in the conclusion of Theorem 11.8 one gets the following result (we will not use this result in the present paper).

Theorem 11.9. Let U be as in Definition 11.1. Let (𝒳, f, Δ) be a special nice triple over U. Then there exists a morphism θ : (𝒳′, f′, Δ′) → (𝒳, f, Δ) of nice triples over U such that (𝒳′, f′, Δ′) is a special nice triple satisfying the conditions (1^{∗}) and (2^{∗}) from Definition 11.6.

12 Proof of Theorem 11.7

Proof of Theorem 11.7. For any closed point u ∈ U and any U-scheme V let V_{u} = u×_{U} V be the scheme fibre of the scheme V over the point u.

Step (i). For any closed point u ∈ U and any point z ∈ 𝒵′_{u} there is a closed embedding z^{(2)} ↪ A_{u}^{1}, where z^{(2)} := Spec(Γ(𝒳′_{u}, 𝒪_{𝒳′_{u}})/𝔪_{z}^{2}) for the maximal ideal 𝔪_{z} ⊂ Γ(𝒳′_{u}, 𝒪_{𝒳′_{u}}) of the point z regarded as a point of 𝒳′. This holds, since the k(u)-scheme 𝒳′_{u} is equidimensional of dimension one, affine and k(u)-smooth.

Step (ii). For any closed point u ∈ U there is a closed embedding i_{u} : ∐_{z∈𝒵′_{u}} z^{(2)} ↪ A_{u}^{1} of the k(u)-schemes. To see this apply Step (i) and use that the triple (𝒳, f, Δ) satisfies the condition 1_{U}^{∗} from Definition 11.6.

Step(iii) is to introduce some notation. Since (𝒳′, f′, Δ′) is a nice triple over U there is a finite surjective morphism 𝒳′ ⟶^{Π} A^{1} × U of the U-schemes. Take the composite 𝒳′ ⟶^{Π} A^{1} × U ↪ P^{1} × U morphism and denote by 𝒳̄′ the normalization of P^{1} × U in the fraction field k(𝒳′) of the ring Γ(𝒳′, 𝒪_{𝒳′}). The normalization of A^{1} × U in k(𝒳′) coincides with the scheme 𝒳′, since 𝒳′ is a regular scheme. So, we have a Cartesian diagram of U-schemes

\xymatrix{𝒳′ \ar[r]^{inc} \ar[d]_{Π} & 𝒳̄′ \ar[d]^{Π̄} \\ A^{1} × U \ar[r]^{inc} & P^{1} × U,}  (26)

in which the horizontal arrows are open embedding.

Let 𝒳′_{∞} be the Cartie-divisor (Π̄)^{−1}(∞ × U) in 𝒳̄′. Let ℒ := 𝒪𝒳̄′(𝒳′_{∞}) be the corresponding invertible sheaf and let s_{0} ∈ Γ(𝒳̄′, ℒ) be its section vanishing exactly on 𝒳′_{∞}. One has a Cartesian square of U-schemes

\xymatrix{𝒳′_{∞,u} \ar[r]^{j_{∞}} \ar[d]_{in_{u}} & 𝒳′_{∞} \ar[d]^{in} \\ 𝒳̄′_{u} \ar[r]^{j} & 𝒳̄′,}  (27)

which shows that the closed embedding in_{u} : 𝒳′_{∞,u} ↪ 𝒳̄′_{u} is a Cartie-divisor on 𝒳̄′_{u}. Set ℒ_{u} = j^{∗}(ℒ) and s_{0,u} = s_{0}|𝒳̄′_{u} ∈ Γ(𝒳̄′_{u}, ℒ_{u}).

Step (iv). There exists an integer n > 0 and a section s_{1,u} ∈ Γ(𝒳̄′_{u}, ℒ_{u}^{⊗n}) which has no zeros on 𝒳′_{∞,u} and such that the morphism

[s_{0,u}^{n} : s_{1,u}] : 𝒳̄′_{u} → P_{u}^{1}

has the following two properties

(a) the morphism σ_{u} = s_{1,u}/s_{0,u}^{n} : 𝒳′_{u} → A_{u}^{1} is finite surjective,

(b) σ_{u}|_{∐_{z∈𝒵′_{u}} z^{(2)}} = i_{u} : ∐_{z∈𝒵′_{u}} z^{(2)} ↪ A_{u}^{1}, where i_{u} is from the step (ii); in particular, σ_{u} is étale at every point z ∈ 𝒵′_{u}.

Step (v). There exists a section s_{1} ∈ Γ(𝒳̄′, ℒ^{⊗n}) such that for any closed point u ∈ U one has s_{1}|_{𝒳′_{u}} = s_{1,u}.

Step (vi). If s_{1} is such as in the step (v), then the morphism

σ = (s_{1}/s_{0}^{n}, pr_{U}) : 𝒳′ → A^{1} × U

is finite and surjective.

We are ready now to check step by step all the statements of the Theorem. The assertion (b). Since the schemes 𝒳′ and A^{1} × U are regular and the morphism σ

is finite and surjective, the morphism σ is flat by a theorem of Grothendieck.

So, to check that σ is étale at a closed point z ∈ 𝒵′ it suffices to check that for the point u = q′_{U}(z) the morphism σ_{u} : 𝒳′_{u} → A_{u}^{1} is étale at the point z. The latter does hold by the step (iv), item (b). Whence σ is étale at all the closed points of 𝒵′. By the hypotheses of the Theorem the set of closed points of Δ′(U) is contained in the set of the closed points of 𝒵′. Whence σ is étale also at all the closed points of Δ′(U). The schemes Δ′(U) and 𝒵′ are both semi-local. Thus, σ is étale in a neighborhood of 𝒵′ ∪ Δ′(U).

The assertion (a). For any closed point u ∈ U and any point z ∈ 𝒵′_{u} the k(u)-algebra homomorphism σ_{u}^{∗} : k(u)[t] → k(u)[𝒳′_{u}] is étale at the maximal ideal 𝔪_{z} of the k(u)-algebra k(u)[𝒳′_{u}] and the composite map k(u)[t] ⟶^{σ_{u}^{∗}} k(u)[𝒳′_{u}] → k(u)[𝒳′_{u}]/𝔪_{z} is an epimorphism. Thus, for any integer r > 0 the k(u)-algebra homomorphism k(u)[t] → k(u)[𝒳′_{u}]/𝔪_{z}^{r} is an epimorphism. The local ring 𝒪_{𝒵′_{u},z} of the scheme 𝒵′_{u} at the point z is of the form k(u)[𝒳′_{u}]/𝔪_{z}^{s} for an integer s. Thus, the k(u)-algebra homomorphism

k(u)[t] ⟶^{σ_{u}^{∗}} k(u)[𝒳′_{u}] → 𝒪_{𝒵′_{u},z}

is surjective. Since σ_{u}|_{∐_{z∈𝒵′_{u}} z^{(2)}} = i_{u} and i_{u} is a closed embedding one concludes that the k(u)-algebra homomorphism

k(u)[t] → ∏_{z/u} O_{𝒵′_{u},z} = Γ(𝒵′_{u}, 𝒪_{𝒵′_{u}})

is surjective. Let u = ∐ Spec(k(u)) regarded as the closed sub-scheme of U, where u runs over all closed points of U. Then, for the scheme 𝒵′_{u} = u ×_{U} 𝒵′ the k[u]-algebra homomorphism

k[u][t] → Γ(𝒵′_{u}, 𝒪_{𝒵′_{u}})  (28)

is surjective.

Since (𝒳′, f′, Δ′) is a nice triple over U, the 𝒪-module Γ(𝒵′, 𝒪_{𝒵′}) is finite. Thus, the k[u]-module Γ(𝒵′_{u}, 𝒪_{𝒵′_{u}}) is finite. Now the surjectivity of the k[u]-algebra homomorphism (28) and the Nakayama lemma show that the 𝒪-algebra homomorphism 𝒪[t] → Γ(𝒵′, 𝒪_{𝒵′}) is surjective. Thus, σ|_{𝒵′} : 𝒵′ → A^{1} × U is a closed embedding.

The assertion (e). The morphism Δ′ is a section of the structure morphism q′_{U} : 𝒳′ → U and the morphism σ is a morphism of the U-schemes. Hence the composite morphism t_{0} := σ ∘ Δ′ is a section of the projection pr_{U} : A^{1} × U → U. This section is defined by an element a ∈ 𝒪. There is another section t_{1} of the projection pr_{U} defined by the element 1 − a ∈ 𝒪. Making an affine change of coordinates on A_{U}^{1} we may and will assume that t_{0}(U) = {0} × U and t_{1}(U) = {1} × U. If the field k is infinite we can choose a non-zero λ ∈ k such that for t_{1}^{new} := λt_{1} one has: 𝒟_{1}^{new} ∩ 𝒵′ = ∅ and t_{0}^{new} := λt_{0} = t_{0}.

Since 𝒟_{1} and 𝒵′ are semi-local, to prove the assertion (e) it suffices to check that 𝒟_{1} and 𝒵′ have no common closed points. In the infinite field case this is checked just above. It remains to check the finite field case. Let z ∈ 𝒟_{1} ∩ 𝒵′ be a common closed point. Then σ(z) ∈ {1} × U. Let u = q′_{U}(z). We already know that σ|_{𝒵′} is a closed embedding. Thus deg[z : u] = deg[σ(z) : u] = 1. The triple (𝒳, f, Δ) satisfies the condition 2_{U}^{∗} from Definition 11.6. Thus, z = Δ′(u). In this case σ(z) ∈ {0} × U. But σ(z) ∈ {1} × U. This is a contradiction. Whence 𝒟_{1} ∩ 𝒵′ = ∅.

The assertion (c). The finite morphism σ is étale in a neighborhood of 𝒵′ by the item (b) of the Theorem. By the item (a) of the Theorem σ|_{𝒵′} is a closed embedding. Thus, the morphism σ^{−1}(σ(𝒵′)) → σ(𝒵′) of affine schemes is finite and there is an affine open sub-scheme V of the scheme σ^{−1}(σ(𝒵′)) such that the morphism V → σ(𝒵′) is étale. Since σ|_{𝒵′} is a closed embedding there is a unique section s of the morphism σ^{−1}(σ(𝒵′)) → σ(𝒵′) with the image 𝒵′ and this image is contained in V. By [OP1, Lemma 5.3] the scheme σ^{−1}(σ(𝒵′)) has the form σ^{−1}(σ(𝒵′)) = 𝒵′ ∐ 𝒵′′. By a similar reasoning the scheme σ^{−1}({0} × U) has the form Δ′(U) ∐ 𝒟. The triple (𝒳, f, Δ) is a special nice triple. Thus all the closed points of Δ′(U) are closed points of 𝒵′ and 𝒵′ ∩ 𝒵′′ = ∅. Thus, Δ′(U) ∩ 𝒵′′ = ∅.

The assertion (d). It remains to show that 𝒟 ∩ 𝒵′ = ∅. Recall that σ is étale in a neighborhood of 𝒵′∪Δ′(U). It’s easy to check that σ|_{𝒵′∪Δ′(U)} is a closed embedding. Thus arguing as above one gets a disjoint union decomposition

σ^{−1}(σ(𝒵′ ∪ Δ′(U))) = (𝒵′ ∪ Δ′(U)) ⊔ E

for a closed subscheme E in 𝒳′. It’s checked already that Δ′(U) ∐ 𝒟 = ∅. Thus, 𝒟 ⊂ E. Hence 𝒟 ∩ 𝒵′ = ∅ .

The assertion (f). Recall that 𝒳′ is affine irreducible and regular. So, the principal ideal (f′) has the form 𝔭_{1}^{r_{1}}𝔭_{2}^{r_{2}} ⋯ 𝔭_{n}^{r_{n}}, where 𝔭_{i}’s are hight one prime ideals in Γ(𝒳′, 𝒪_{𝒳′}). Let 𝒵′_{i} be the closed subscheme in 𝒳′ defined by the ideal 𝔭_{i}. Let 𝔮_{i} = 𝒪[t] ∩ 𝔭_{i}. The morphism σ|_{𝒵′} : 𝒵′ → A^{1} × U is a closed embedding by the item (a) of Theorem 5.1. This yields that σ|_{𝒵′_{i}} : 𝒵′_{i} → A^{1} × U is a closed embedding too. Thus 𝔭_{i} is a hight one prime ideal in 𝒪[t]. So, it is a principal prime ideal. Since 𝒵′ is finite over U the scheme 𝒵′_{i} is finite over U too. Hence the principal prime ideal 𝔭_{i} is of the form (h_{i}) for a unique monic polinomial h_{i} ∈ 𝒪[t].

Set h = h_{1}^{r_{1}}h_{2}^{r_{2}} … h_{n}^{r_{n}}. Clearly, h ∈ Ker[𝒪[t] ⟶^{¯ ∘σ^{∗}} Γ(𝒳′, 𝒪_{𝒳′})/(f′)]. Since the map

𝒪[t] ⟶^{¯ ∘σ^{∗}} Γ(𝒳′, 𝒪_{𝒳′})/(f′) is surjective, to prove the assertion (f) it suffices to check that the surjective 𝒪-module homomorphism

¯ ∘ σ^{∗} : 𝒪[t]/(h) → Γ(𝒳′, 𝒪_{𝒳′})/(f′)

is an isomorphism. Both sides are finitely generated projective 𝒪-modules. It remains to check that both sides have the same rank as the 𝒪-modules. For that it suffices to know that 𝒪[t]/(h_{i}) and Γ(𝒳′, 𝒪_{𝒳′})/𝔭_{i} are of the same rank as the 𝒪-modules. This is the case since they are isomorphic 𝒪-modules. Indeed, the composite map

𝒪[t] ⟶^{¯ ∘σ^{∗}} Γ(𝒳′, 𝒪_{𝒳′})/(f′) → Γ(𝒳′, 𝒪_{𝒳′})/𝔭_{i}

is an 𝒪-algebra epimorphism and the kernel of this epimorphism is the ideal 𝔮_{i} = (h_{i}). Whence the assertion (f) and whence the Theorem.

13 Equating group schemes

Theorem 13.1 is proved in the present Section.

Theorem 13.1. 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}. Assume as well that G_{1} and G_{2} are reductive S-group schemes which are forms of each other. Assume that C_{1} and C_{2} are forms of each other. Let T ⊂ S be a connected 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})

(iv) the scheme S̃ is irreducible.

We begin with the following result which extends [PSV, Prop.5.1].

Theorem 13.2. Let S be a regular semi-local irreducible scheme. Assume that G_{1} and G_{2} are reductive S-group schemes which are forms of each other. Let T ⊂ S be a connected non-empty closed sub-scheme of S, and φ : G_{1}|_{T} → G_{2}|_{T} be S-group scheme isomorphism. 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} such that

(i) δ^{∗}(Φ) = φ,

(ii) the scheme S̃ is irreducible.

Proposition 13.3. Theorem 13.2 holds in the case when the group schemes G_{1} and G_{2} are semi-simple.

Proof of Proposition 13.3. The proof literally repeats the proof of [PSV, Prop.5.1] except exactly one reference, which is the reference to [OP2, Lemma 7.2]. That reference one has to replace with the reference to the following

Lemma 13.4. Let S = Spec(R) be a regular semi-local scheme. Let T be a closed subscheme of S. Let X̄ be a closed subscheme of ℙ_{S}^{d} = Proj(S[X_{0}, …, X_{d}]) and X = X̄ ∩A_{S}^{d}, where 𝔸_{S}^{d} is the affine space defined by X_{0} ≠ 0. Let X_{∞} = X̄ \ X be the intersection of X̄ with the hyperplane at infinity X_{0} = 0. Assume that over T there exists a section δ : T → X of the canonical projection X → S. Assume further that

(1) X is smooth and equidimensional over S, of relative dimension r;

(2) For every closed point s ∈ S the closed fibres of X_{∞} and X satisfy

dim(X_{∞}(s)) < dim(X(s)) = r .

Then there exists a closed subscheme S̃ of X which is finite étale over S and contains δ(T).

If the residue field at any of closed point of the scheme S is infinite, then this lemma is just [OP2, Lemma 7.2]. We give a proof of the lemma in the case when the residue field at any of closed point of the scheme S is finite. The proof of the lemma in this case is given below and repeats literally the proof of [OP2, Lemma 7.2]. The only difference is that we refer below to a Poonen’s article [Poo] on Bertini theorems over finite fields rather than to Artin’s result. We left to the reader the general case.

Since S is semilocal, after a linear change of coordinates we may assume that δ maps T into the closed subscheme of P_{T}^{d} defined by X_{1} = ⋯ = X_{d} = 0. For each closed fibre P_{s}^{d} of P_{S}^{d} using repeatedly [Poo, Thm.1.2], we can choose a family of homogeneous polynomials H_{1}(s), …, H_{r}(s) (in general of increasing degrees) such that the subscheme Y (s) of P_{S}^{d}(s) defined by the equations

H_{1}(s) = 0 , …, H_{r}(s) = 0

intersects X(s) transversally, contains the point [1 : 0 : ⋯: 0] and avoids X_{∞}(s). By the chinese remainders’ theorem there exists a common lift H_{i} ∈ R[X_{0}, …, X_{d}] of all polynomials H_{i}(s), s ∈ Max(R). We may choose this common lift H_{i} such that H_{i}(1, 0, …, 0) = 0. Let Y be the closed subscheme of P_{S}^{d} defined by

H_{1} = 0 , …, H_{r} = 0 .

We claim that the subscheme S̃ = Y ∩ X has the required properties. Note first that X ∩ Y is finite over S. In fact, X ∩ Y = X̄ ∩ Y, which is projective over S and such that every closed fibre (hence every fibre) is finite. Since the closed fibres of X ∩ Y are finite étale over the closed points of S, to show that X ∩ Y is finite étale over S it only remains to show that it is flat over S. Noting that X ∩ Y is defined in every closed fibre by a regular sequence of equations and localizing at each closed point of S, we see that flatness follows from [OP2, Lemma 7.3].

Proposition 13.5. Theorem 13.2 holds in the case when the groups G_{1} and G_{2} are tori and, more generally, in the case when the groups G_{1} and G_{2} are of multiplicative type.

We left a proof of this latter proposition and the next one to the reader.

Proposition 13.6. Let T and S be the same as in Theorem 13.2. Let M_{1} and M_{2} be two S-group schemes of multiplicative type. Let α_{1}, α_{2} : M_{1} ⇉ M_{2} be two S-group scheme morphisms such that α_{1}|_{T} = α_{2}|_{T}. Then α_{1} = α_{2}.

Proof of Theorem 13.2. Let Rad(G_{r}) ⊂ G_{r} be the radical of G_{r} and let der(G_{r}) ⊂ G_{r} be the derived subgroup of G_{r} (r = 1, 2) (see [D-G, Exp.XXII, 4.3]). By the very definition the radical is a tori. The S-group scheme der(G_{r}) is semi-simple (r = 1, 2). Set Z_{r} := Rad(G_{r}) ∩ der(G_{r}). The above embeddings induce natural S-group morphisms

Π_{r} : Rad(G_{r}) ×_{S} der(G_{r}) → G_{r}

with Z_{r} as the kernel (r = 1, 2). By [D-G, Exp.XXII,Prop.6.2.4] Π_{r} is a central isogeny. Particularly, Π_{r} is a faithfully flat finite morphism by [D-G, Exp.XXII,Defn.4.2.9]. Let i_{r} : Z_{r} ↪ Rad(G_{r}) ×_{S} der(G_{r}) be the closed embedding.

The T -group scheme isomorphism φ : G_{1}|_{T} → G_{2}|T induces certain T -group scheme isomorphisms φ_{der} : der(G_{1}|_{T}) → der(G_{2}|_{T}), φ_{rad} : rad(G_{1}|_{T}) → rad(G_{2}|_{T}) and φ_{Z} : Z_{1}|_{T} → Z_{2}|_{T} such that

(Π_{2})|_{T} ∘ (φ_{der} × φ_{rad}) = φ ∘ (Π_{1})|_{T} and i_{2,T} ∘ φ_{Z} = (φ_{rad} × φ_{der}) ∘ i_{1,T} .

By Propositions 13.3 and 13.5 there exist a finite étale morphism π : S̃ → S (with an irreducible scheme S̃) and its section δ : T → S̃ over T and S̃-group scheme isomorphisms

Φ_{der} : der(G_{1,S̃}) → der(G_{2,S̃}) , Φ_{Rad} : Rad(G_{1,S̃}) → Rad(G_{2,S̃}) and Φ_{Z} : Z_{1,S̃} → Z_{2,S̃}

such that δ^{∗}(Φ_{der}) = φ_{der}, δ^{∗}(Φ_{rad}) = φ_{rad} and δ^{∗}(Φ_{Z}) = φ_{Z}.

Since Z_{r} is contained in the center of der(G_{r}) and is of multiplicative type Proposition 13.6 yields the equality

i_{2,S̃} ∘ Φ_{Z} = (Φ_{Rad} × Φ_{der}) ∘ i_{1,S̃} : Z_{1,S̃} → Rad(G_{2,S̃}) ×S̃ der(G_{2,S̃}).

Thus (Φ_{Rad} × Φ_{der}) induces an S̃-group scheme isomorphism

Φ : G_{1,S̃} → G_{2,S̃}

such that Π_{2,S̃} ∘ (Φ_{Rad} × Φ_{der}) = Φ ∘ Π_{1,S̃}. The latter equality yields the following one (Π_{2})|_{T} ∘ δ^{∗}(Φ_{Rad} × Φ_{der}) = δ^{∗}(Φ) ∘ (Π_{1})|_{T}, which in turn yields the equality

(Π_{2})|_{T} ∘ (φ_{rad} × φ_{der}) = δ^{∗}(Φ) ∘ (Π_{1})|_{T} .

Comparing it with the equality (Π_{2})|_{T} ∘ (φ_{rad} × φ_{der}) = φ ∘ (Π_{1})|_{T} and using the fact that (Π_{1})|_{T} is strictly flat we conclude the equality δ^{∗}(Φ) = φ.

Proof of Theorem 13.1. By Theorem 13.2 there exists a finite étale morphism S̃ ⟶^{π} S together with its section δ : T → S̃ over T and S̃-group scheme isomorphisms

Φ : G_{1,S̃} → G_{2,S̃} and Ψ : C_{1,S̃} → C_{2,S̃}

such that δ^{∗}(Φ) = φ, δ^{∗}(Ψ) = ψ. It remains to show that μ_{2,S̃} ∘ Φ = Ψ ∘ μ_{1,S̃}. To prove this equality recall that μ_{r} can be naturally presented as a composition

G_{r} ⟶^{can_{r}} Corad(G_{r}) ⟶^{μ̄_{r}} C_{r}.

Since can_{2,S̃} ∘ Φ = Corad(Φ) ∘ can_{1,S̃} it remains to check that μ̄_{2,S̃} ∘ Corad(Φ) = Ψ ∘ μ̄_{1,S̃}. The latter equality follows from Proposition 13.6 and the equality (μ̄_{2}|_{T}) ∘ Corad(φ) = ψ∘(μ̄_{1}|_{T}), which holds since (μ_{2}|_{T})∘φ = ψ∘(μ_{1}|_{T}) by the very assumption of the Theorem.

14 Proof of Theorem 11.8

Let k be a field. Let U be as in Definition 11.1. Let S′ be an irreducible regular semi-local scheme over k and p : S′ → U be a k-morphism. Let i : T′ ↪ S′ be a closed sub-scheme of S′ such that the restriction p|_{T′} : T′ → U is an isomorphism. We will assume below that dim(T′) < dim(S′), where dim is the Krull dimension. For any closed point u ∈ U and any U-scheme V let V_{u} = u ×_{U} V be the fibre of the scheme V over the point u. For a finite set A denote ♯A the cardinality of A.

Lemma 14.1. If the field k is finite and all the closed points of S′ have finite residue fields. Then there exists a finite étale morphism ρ : S′′ → S′ (with an irreducible scheme S′′) and a section δ′ : T′ → S′′ of ρ over T′ such that the following holds

(1) for any closed point u ∈ U let u′ ∈ T′ be a unique point such that p(u′) = u, then the point δ′(u′) ∈ S′′_{u} is the only k(u)-rational point of S′′_{u},

(2) for any closed point u ∈ U and any integer d ≥ 1 one has

♯{z ∈ S′′_{u}|[k(z) : k(u)] = d} ≤ ♯{x ∈ A_{u}^{1}|[k(x) : k(u)] = d}

If the field k is infinite, then set S′′ = S′, ρ = id, and δ′ = i.

Proof of Theorem 11.8. We can start by almost literally repeating arguments from the proof of [OP1, Lemma 8.1], which involve the following purely geometric lemma [OP1, Lemma 8.2].

For reader’s convenience below we state that Lemma adapting notation to the ones of Section 11. Namely, let U be as in Definition 11.1 and let (𝒳, f, Δ) be a nice triple over U. Further, let G_{𝒳} be a simple simply-connected 𝒳-group scheme, G_{U} := Δ^{∗}(G_{𝒳}), and let G_{const} be the pull-back of G_{U} to 𝒳. Finally, by the definition of a nice triple there exists a finite surjective morphism Π : 𝒳 → A^{1} × U of U-schemes.

Lemma 14.2. Let 𝒴 be a closed nonempty sub-scheme of 𝒳, finite over U. Let 𝒱 be an open subset of 𝒳 containing Π^{−1}(Π(𝒴)). There exists an open set 𝒲 ⊆ 𝒱 still containing q_{U}^{−1}(q_{U}(𝒴)) and endowed with a finite surjective morphism Π^{∗} : 𝒲 → A^{1} × U (in general ≠ Π).

Let Π : 𝒳 → A^{1}×U be the above finite surjective U-morphism. The following diagram summarises the situation:

\xymatrix{ & 𝒵 \ar[d] \\ 𝒳 − 𝒵 \ar@{^{(}->}[r] & 𝒳 \ar[d]^{q_{U}} \ar[r]^{Π} & A^{1} × U \\  & U \ar[u]^{Δ}}

Here 𝒵 is the closed sub-scheme defined by the equation f = 0. By assumption, 𝒵 is finite over U. Let 𝒴 = Π^{−1}(Π(𝒵∪Δ(U))). Since 𝒵 and Δ(U) are both finite over U and since Π is a finite morphism of U-schemes, 𝒴 is also finite over U. Denote by y_{1}, …, y_{m} its closed points and let S = Spec(𝒪_{𝒳,y_{1},...,y_{m}}). Set T = Δ(U) ⊆ S. Further, let G_{U} = Δ^{∗}(G_{𝒳}) be as in the hypotheses of Theorem 11.8 and let G_{const} be the pull-back of G_{U} to 𝒳. Finally, let φ : G_{const}|_{T} → G_{𝒳}|_{T} be the canonical isomorphism. Recall that by assumption 𝒳 is U-smooth and irreducible, and thus S is regular and irreducible.

By Theorem 13.1 there exists a finite étale covering θ_{0} : S′ → S, a section δ : T → S′ of θ_{0} over T and isomorphisms Φ_{0} : θ_{0}^{∗}(G_{const,S}) → θ_{0}^{∗}(G_{𝒳}|_{S}) and Ψ_{0} : θ_{0}^{∗}(C_{const,S}) → θ_{0}^{∗}(C_{𝒳}|_{S}) such that δ^{∗}(Φ_{0}) = φ, δ^{∗}(Ψ_{0}) = ψ and

θ_{0}^{∗}(μ_{𝒳}|_{S}) ∘ Φ_{0} = Ψ_{0} ∘ θ_{0}^{∗}(μ_{const,S}) : θ_{0}^{∗}(G_{const,S}) → θ_{0}^{∗}(C_{𝒳}|_{S})

and the scheme S′ is irreducible. Replacing S′ with a connected component of S′ which contains T′ := δ(T) = δ(Δ(U)) we may and will assume that S′ is irreducible.

Let p = q_{U} ∘ θ_{0} : S′ → U. Let S′′, ρ : S′′ → S′, and δ′ : T′ → S′′ be as in Lemma 14.1. Recall that ρ : S′′ → S′ a finite étale morphism (with an irreducible scheme S′′) and δ′ ∘ ρ = i : T′ ↪ S′. Set δ′′ = δ′ ∘ δ : T → S′′ and θ′′_{0} = θ_{0} ∘ ρ : S′′ → S. We are also given the S′′-group scheme isomorphisms

ρ^{∗}(Φ_{0}) : (θ′′_{0})^{∗}(G_{const,S}) → (θ′′_{0})^{∗}(G_{𝒳}|_{S}) and ρ^{∗}(Ψ_{0}) : (θ′′_{0})^{∗}(C_{const,S}) → (θ′′_{0})^{∗}(C_{𝒳}|_{S})

such that (δ′′)^{∗}(ρ^{∗}(Φ_{0})) = φ, (δ′′)^{∗}(ρ^{∗}(Ψ_{0})) = ψ and

(θ′′)_{0}^{∗}(μ_{𝒳}|_{S}) ∘ ρ^{∗}(Φ_{0}) = ρ^{∗}(Ψ_{0}) ∘ (θ′′)_{0}^{∗}(μ_{const,S}) : (θ′′)_{0}^{∗}(G_{const,S}) → (θ′′)_{0}^{∗}(C_{𝒳}|_{S})

We can extend these data to a neighborhood 𝒱 of {y_{1}, …, y_{n}} and get the diagram

\xymatrix{ & S′′ \ar[d]^{θ′′_{0}} \ar@{^{(}->}[r] & 𝒱′′ \ar[d]_{θ} \\ T \ar@{^{(}->}[r] \ar[ur]^{δ′′} & S \ar@{^{(}->}[r] & 𝒱 \ar@{^{(}->}[r] & 𝒳}  (29)

where θ : 𝒱′′ → 𝒱 finite étale, and isomorphisms Φ : θ^{∗}(G_{const,𝒱}) → θ^{∗}(G_{𝒳}|_{𝒱}) and Ψ : θ^{∗}(C_{const,𝒱}) → θ^{∗}(C_{𝒳}|_{𝒱}) such that one has the equality

θ^{∗}(μ_{𝒳}|_{𝒱}) ∘ Φ = Ψ ∘ θ^{∗}(μ_{const,𝒱}) : θ^{∗}(G_{const,𝒱}) → θ^{∗}(C_{𝒳}|_{𝒱})  (30)

Since T isomorphically projects onto U, it is still closed viewed as a sub-scheme of 𝒱. Note that since 𝒴 is semi-local and 𝒱 contains all of its closed points, 𝒱 contains Π^{−1}(Π(𝒴)) = 𝒴. By Lemma 14.2 there exists an open subset 𝒲 ⊆ 𝒱 containing 𝒴 and endowed with a finite surjective U-morphism Π^{∗} : 𝒲 → A^{1} × U.

Let 𝒳′ = θ^{−1}(𝒲), f′ = θ^{∗}(f), q′_{U} = q_{U} ∘ θ, and let Δ′ : U → 𝒳′ be the section of q′_{U} obtained as the composition of δ′′ with Δ. We claim that the triple (𝒳′, f′, Δ′) is a nice triple over U. Let us verify this. Firstly, the structure morphism q′_{U} : 𝒳′ → U coincides with the composition

𝒳′ ⟶^{θ} 𝒲 ↪ 𝒳 ⟶^{q_{U}} U.

Thus, it is smooth. The element f′ belongs to the ring Γ(𝒳′, 𝒪_{𝒳′}), the morphism Δ′ is a section of q′_{U}. Each component of each fibre of the morphism q_{U} has dimension one, the morphism 𝒳′ ⟶^{θ} 𝒲 ↪ 𝒳 is étale. Thus, each component of each fibre of the morphism q′_{U} is also of dimension one. Since {f = 0} ⊂ 𝒲 and θ : 𝒳′ → 𝒲 is finite, {f′ = 0} is finite over {f = 0} and hence also over U. In other words, the 𝒪-module Γ(𝒳′, 𝒪_{𝒳′})/f′ · Γ(𝒳′, 𝒪_{𝒳′}) is finite. The morphism θ : 𝒳′ → 𝒲 is finite and surjective. We have constructed above in Lemma 14.2 the finite surjective morphism Π^{∗} : 𝒲 → A^{1} × U. It follows that Π^{∗} ∘ θ : 𝒳′ → A^{1} × U is finite and surjective.

Clearly, the étale morphism θ : 𝒳′ → 𝒳 is a morphism between the nice triples

θ : (𝒳′, f′, Δ′) → (𝒳, f, Δ) with h′ = 1

Denote the restriction of Φ and of Ψ to 𝒳′ simply by Φ and by Ψ respectively. Thus, Φ is an 𝒳′-group scheme isomorphism θ^{∗}(G_{const}) → θ^{∗}(G_{𝒳}) and Ψ is an 𝒳′-group scheme isomorphism θ^{∗}(C_{const}) → θ^{∗}(C_{𝒳}). The equalities (Δ′)^{∗}Φ = id_{G_{U}} and (Δ′)^{∗}Ψ = id_{C_{U}} holds by the construction of the isomorphisms Φ and Ψ. The equality (30) yields the following one

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

By Remark 11.4 the triple (𝒳′, f′, Δ′) is a special nice triple over U since the one (𝒳, f, Δ) is a special nice triple over U.

It remains to check that (𝒳′, f′, Δ′) is a special nice triple satisfying the conditions (1^{∗}) and (2^{∗}) from Definition 11.6. To do this recall that all the closed points of the sub-scheme {f = 0} ⊂ 𝒳 are in S. The morphism θ is finite and θ^{−1}(S) = S′′. Thus all the closed points of the sub-scheme {f′ = 0} ⊂ 𝒳′ are in S′′. By the above choice of T′ = δ(Δ(U)) ⊂ S′ the scheme T′ projects isomorphically to U. Thus in the case when k is finite the properties (1) and (2) of the U-scheme S′′ show that the conditions (1^{∗}) and (2^{∗}) are full filled for the closed sub-scheme 𝒵′ of 𝒳′ defined by {f′ = 0}. If the field k is infinite, then by Definition 11.6 any special nice triple satisfies the conditions (1^{∗}) and (2^{∗}). Theorem 11.8 follows.

15 Proof of Theorems 5.1 and 5.2

Clearly, Theorem 5.2 yields Theorem 5.1.

Proof of Theorem 5.2. By Proposition 11.5 one can shrink X such that x_{1}, x_{2}, …, x_{n} are still in X and X is affine, and then to construct a nice triple (q_{U} : 𝒳 → U, Δ, f) over U and an essentially smooth morphism q_{X} : 𝒳 → X such that q_{X} ∘ Δ = can, f = q_{X}^{∗}(f) and the set of closed points of Δ(U) is contained in the set of closed points of {f = 0}.

Set G_{𝒳} = q_{X}^{∗}(G), then Δ^{∗}(G_{𝒳}) = can^{∗}(G). Thus the U-group scheme G_{U} from Theorem 11.8 and the U-group scheme G_{U} from Theorem 5.2 are the same. By Theorem 11.8 there exists a morphism θ : (𝒳′, f′, Δ′) → (𝒳, f, Δ) such that the triple (𝒳′, f′, Δ′) is a special nice triple over U subjecting to the conditions (1^{∗}) and (2^{∗}) from Definition 11.6. And additionally there are isomorphisms

Φ : (q_{U} ∘ θ)^{∗}(G_{U}) = θ^{∗}(G_{const}) → θ^{∗}(G_{𝒳}) = (q_{X} ∘ θ)^{∗}(G)

and

Ψ : (q_{U} ∘ θ)^{∗}(C_{U}) = θ^{∗}(C_{const}) → θ^{∗}(C_{𝒳}) = (q_{X} ∘ θ)^{∗}(G)

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

θ^{∗}(μ_{𝒳}) ∘ Φ = Ψ ∘ θ^{∗}(μ_{const}).  (31)

The triple (𝒳′, f′, Δ′) is a special nice triple over U subjecting to the conditions (1^{∗}) and (2^{∗}) from Definition 11.6. Thus by Theorem 11.7 there is a finite surjective morphism A^{1} × U ⟵^{σ} 𝒳′ of the U-schemes satisfying the conditions (a) to (f) from that Theorem. Hence one has a diagram of the form

\xymatrix{A^{1} × U \ar[dr]_{_{pr_{U}}} & 𝒳′ \ar[d] \ar[l]_{σ} \ar[d]_{q_{U} ∘θ} \ar[r]^{q_{X}∘θ} & X \\  & U \ar[ur]_{can} \ar[u]_{Δ′}}  (32)

with the irreducible scheme 𝒳′, the smooth morphism q_{U} ∘θ, the finite surjective morphism σ and the essentially smooth morphism q_{X} ∘θ and with the function f′ ∈ (q_{X} ∘θ)^{∗}(f)k[𝒳′], which after identifying notation enjoy the properties (a) to (f) from Theorem 5.1. The isomorphisms Φ and Ψ are the required ones. Whence the Theorem 5.2.

16 Norms

In the rest of the paper we prove few results which we refer to reducing Theorems 6.1 and 1.5 to Theorem 5.2.

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. Let C be a k-smooth commutative algebraic group scheme defined over k. One 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 16.1. There is a canonical section

𝒩_{X/Y} : Y → S^{d}(X/Y)

which satisfies the following three properties:

(i) 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)}

(ii) 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.

(iii) 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_{A}^{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_{A}^{d}(B)-module through the inclusion S_{A}^{d}(B) ⊆ B^{⊗d} of A-algebras. Let I be the kernel of the canonical homomorphism B^{⊗d} → ⋀_{A}^{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_{A}^{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_{A}^{d}(B)-submodule of B^{⊗d}. The induced S_{A}^{d}(B)-module structure on ⋀_{A}^{d}(B) defines an A-algebra homomorphism

φ : S_{A}^{d}(B) → End_{A}(⋀_{A}^{d}(B)) .

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

φ : S_{A}^{d}(B) → A

and we define

N_{X/Y} : Y → S^{d}(X/Y)

as the morphism of Y -schemes induced by φ. The verification of properties (i), (ii) and (iii) is straightforward.

Let k be a field. Let 𝒪 be the semi-local ring of finitely many closed points on a smooth affine irreducible k-variety X. Let C be an affine smooth commutative 𝒪-group scheme, Let p : X → Y be a finite flat morphism of affine 𝒪-schemes and f : X → C any 𝒪-morphism. We define the norm N_{X/Y} (f) of f as the composite map

Y ⟶^{N_{X/Y}} S^{d}(X/Y) → S_{𝒪}^{d}(X) ⟶^{S_{𝒪}^{d}(f)} S_{𝒪}^{d}(C) ⟶^{×} C  (33)

Here we write ” × ” for the group law on C. The norm maps N_{X/Y} satisfy the following conditions

(i’) 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′)

(ii’) 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)

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

17 Unramified elements

Let k be a field, 𝒪 be the k-algebra from Theorem 6.1 and K be the fraction field of 𝒪. Let μ : G → C be the morphism of reductive 𝒪-group schemes from Theorem 6.1.

We work in this section with the category of commutative Noetherian 𝒪-algebras. For a commutative 𝒪-algebra S set

ℱ(S) = C(S)/μ(G(S)).  (34)

Let S be an 𝒪-algebra which is a domain and let L be its fraction field. Define the subgroup of S-unramified elements of ℱ(L) as

ℱ_{nr,S}(L) = ⋂_{𝔭∈Spec(S)^{(1)}} Im[ℱ(S_{𝔭}) → ℱ(L)],  (35)

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

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 (34). We will repeatedly use the following result due to Nisnevich.

Theorem 17.1 ([Ni2]). Let S be a 𝒪-algebra which is discrete valuation ring with fraction field L. The map ℱ(S) → ℱ(L) is injective.

Proof. Let H be the kernel of μ. Since μ is smooth and C is a tori, the group scheme sequence

1 → H → G → C → 1

gives rise to a short exact sequence of group sheaves in the étale topology. In turn that sequence of sheaves induces a long exact sequence of pointed sets. So, the boundary map ∂ : C(S) → H_{ét}^{1}(S, H) fits in a commutative diagram

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

in which the vertical arrows have trivial kernels. The bottom arrow has trivial kernel by a Theorem from [Ni2], since H is a reductive 𝒪-group scheme. Thus the top arrow has trivial kernel too.

Lemma 17.2. Let μ : G → C be the above morphism of our reductive group schemes. Let H = ker(μ). Then for an 𝒪-algebra L, where L is a field, the boundary map ∂ : C(L)/μ(G(L)) → H_{ét}^{1}(L, H) is injective.

Proof. For a L-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}(L, H) coincides with the isomorphism class of H_{t}. Now suppose that s, t ∈ C(L) are such that ∂(s) = ∂(t). This means that H_{t} and H_{s} are isomorphic as principal homogeneous Hspaces. We must check that for certain g ∈ G(L) one has t = sμ(g).

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

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

Thus for any σ ∈ Gal(L^{sep}/L) and any r ∈ H_{s}(L^{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(L^{sep}/L)-invariant point of G(L^{sep}). So u ∈ G(L). The following relation shows that the ψ coincides with the right multiplication by u. In fact, for any r ∈ H_{s}(L^{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, 𝒪 and K be as above in this Section. Let 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. Clearly, 𝒪 ⊂ A_{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 Theorem 17.1 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 17.1. So both groups ℱ(A_{x}) 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 17.3. 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 [Ni2] 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 17.2 the map

ℱ(𝒦) ⟶^{∂} H_{ét}^{1}(𝒦, H)  (36)

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} coincides with a_{x} in ℱ_{x}(𝒦). Thus the inclusion ℱ_{x}(𝒦) ⊆ ℱ(A_{x}) is proved, whence the lemma.

Let k, 𝒪 and K be as above in this Section.

Lemma 17.4. Let B ⊂ A be a finite extension of K-smooth algebras, which are domains and each has dimension one. Let 0 ≠ f ∈ A and let h ∈ B ∩ fA be such that the induced map B/hB → A/fA is an isomorphism. Suppose hA = fA ∩ J′′ for an ideal J′′ ⊆ A co-prime to the ideal fA.

Let E and F be the field of fractions of B and A respectively. Let α ∈ C(A_{f}) be such that ᾱ ∈ ℱ(F) is A-unramified. Then, for β = N_{F/E}(α), the class β̄ ∈ ℱ(E) is B-unramified.

Proof. The only primes at which ᾱ could be ramified are those which divide hA. 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 A lying over 𝔭. Let 𝔮_{1} be the unique prime dividing f and lying over 𝔭. Then

A ⊗_{B} B̂_{𝔭} = Â_{𝔮_{1}} × ∏_{i≠1} Â_{𝔮_{i}}

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

F ⊗_{B} B̂_{𝔭} = F̂_{𝔮_{1}} × ∏_{i≠1} F̂_{𝔮_{n}}

and F̂_{𝔮_{1}} = Ê_{𝔭}. We will write F̂_{i} for F̂_{𝔮_{i}} and Â_{i} for Â_{𝔮_{i}}. Let

α ⊗ 1 = (α_{1}, …, α_{n}) ∈ C(F̂_{1}) × ⋯ × C(F̂_{n}).

Clearly for i ≥ 2 one has α_{i} ∈ C(Â_{i}) and α_{1} = μ(γ_{1})α′_{1} with α′_{1} ∈ C(Â_{1}) = C(B̂_{𝔭}) and γ_{1} ∈ G(F̂_{1}) = G(Ê_{𝔭}). Now β ⊗ 1 ∈ C(Ê_{𝔭}) coincides with the product

α_{1}N_{F̂_{2}/Ê_{𝔭}}(α_{2}) ⋯ N_{F̂_{n}/Ê_{𝔭}}(α_{n}) = μ(γ_{1})[α′_{1}N_{F̂_{2}/Ê_{𝔭}}(α_{2}) ⋯ N_{F̂_{n}/Ê_{𝔭}}(α_{n})].

Thus \overline{β ⊗ 1} = ᾱ′_{1}\overline{N_{F̂_{2}/Ê_{𝔭}}(α_{2})} ⋯ \overline{N_{F̂_{n}/Ê_{𝔭}}(β_{n})} ∈ ℱ(B̂_{𝔭}). Let i : E ↪ Ê_{𝔭} be the inclusion and i_{∗} : ℱ(E) → ℱ(Ê_{𝔭}) be the induced map. Clearly i_{∗}(β̄) = \overline{β ⊗ 1} in ℱ(Ê_{𝔭}). Now Lemma 17.3 shows that the element β̄ ∈ ℱ(E) belongs to ℱ(B_{𝔭}). Hence β̄ is B-unramified.

18 Specialization maps

Let k be a field, 𝒪 be the k-algebra from Theorem 6.1 and K be the fraction field of 𝒪. Let μ : G → C be the morphism of reductive 𝒪-group schemes from Theorem 6.1. We work in this section with the category of commutative K-algebras and with the functor

ℱ : S ↦ C(S)/μ(G(S))  (37)

defined on the category of K-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. Let S be an K-algebra which is a domain and let L be its fraction field. Define the subgroup of S-unramified elements ℱ_{nr,S}(L) of ℱ(L) by formulae (35).

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

Definition 18.1. Let Ev_{𝔭} : C(S_{𝔭}) → C(K(𝔭)) and ev_{𝔭} : ℱ(S_{𝔭}) → ℱ(K(𝔭)) be the maps induced by the canonical K-algebra homomorphism S_{𝔭} → K(𝔭). Define a homomorphism s_{𝔭} : ℱ_{nr,S}(𝒦) → ℱ(K(𝔭)) by s_{𝔭}(α) = ev_{𝔭}(α̃), where α̃ is a lift of α to ℱ(S_{𝔭}). Theorem 17.1 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(S_{𝔭}) one has s_{𝔭}(ᾱ) = \overline{Ev_{𝔭}(α)} ∈ ℱ(K(𝔭)).

Lemma 18.2 ([?]). Let H′ be a smooth linear algebraic group over the field K. Let S be a K-algebra which is a Dedekind domain with field of fractions 𝒦. If ξ ∈ H_{ét}^{1}(𝒦, H′) is an S-unramified element for the functor H_{ét}^{1}(−, H′) (see (35) for the Definition), then ξ can be lifted to an element of H_{ét}^{1}(S, H′).

Proof. Patching.

Theorem 18.3 ([C-T/O], Prop.2.2). Let G′ = G_{K}, where G is the reductive 𝒪-group scheme from this Section (it is connected and even geometrically connected, since we follow [D-G, Exp. XIX, Defn.2.7]). Then

ker[H_{ét}^{1}(K[t], G′) → H_{ét}^{1}(K(t), G′)] = ∗ .

We need the following theorem.

Theorem 18.4 (Homotopy invariance). Let S ↦ ℱ(S) be the functor defined by the formulae (37) and let ℱ_{nr,K[t]}(K(t)) be defined by the formulae (35). Let K(t) be the rational function field in one variable. 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

μ_{K} = μ ⊗_{𝒪} K : G_{K} = G ⊗_{𝒪} K → C ⊗_{𝒪} K = C_{K}.

Let a ∈ ℱ_{nr,K[t]}(K(t)) and let H_{K} = ker(μ_{K}). Since μ is smooth the K-group H_{K} is smooth. Since G_{K} is reductive it is K-affine. Whence H_{K} is K-affine. Clearly, the element ∂(a) ∈ H_{et}^{1}(K(t), H_{K}) is a class which for every closed point x ∈ A_{K}^{1} belongs to the image of H_{et}^{1}(𝒪_{x}, H_{K}). Thus by Lemma 18.2, ξ := ∂(a) can be represented by an element ξ̃ ∈ H_{et}^{1}(K[t], H_{K}), where K[t] is the polynomial ring. Consider the diagram

\xymatrix{ & ã \ar@{|->}[r] & ξ̃ \ar@{|->}[r] & ζ̃ \\ 1 \ar[r] & ℱ(K[t]) \ar[r]^{∂} \ar[d]_{ϵ} & H_{et}^{1}(K[t], H_{K}) \ar[r] \ar[d]_{ρ} & H_{et}^{1}(K[t], G_{K}) \ar[d]_{η} \\ 1 \ar[r] & ℱ(K(t)) \ar[r]^{∂} & H_{et}^{1}(K(t), H_{K}) \ar[r] & H_{et}^{1}(K(t), G_{K}) \\  & a \ar@{|->}[r] & ξ \ar@{|->}[r] & ∗}

in which all the maps are canonical, the horizontal lines are exact sequences of pointed sets and ker(η) = ∗ by Theorem 18.3. Since ξ goes to the trivial element in H_{et}^{1}(K(t), G_{K}), one concludes that η(ζ̃) = ∗. Whence ζ̃ = ∗ by Theorem 18.3. Thus there exists an element ã ∈ ℱ(K[t]) such that ∂(ã) = ξ̃. The map ℱ(K(t)) → H_{et}^{1}(K(t), H_{K}) is injective by Lemma 17.2. Thus ϵ(ã) = a. The map ℱ(K) → ℱ(K[t]) induced by the inclusion K ↪ K[t] is surjective, since the corresponding map C(K) → C(K[t]) is an isomorphism. Whence there exists an a_{0} ∈ ℱ(K) such that its image in ℱ(K(t)) coincides with the element a.

Corollary 18.5. Let S ↦ ℱ(S) be the functor defined in (34). 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 18.4.

References

[A] Artin, M. Comparaison avec la cohomologie classique: cas d’un préschéma lisse, in Théorie des topos et cohomologie étale des schémas (SGA 4). Tome 3. Lect. Notes Math., vol. 305, Exp. XI, Springer-Verlag, Berlin-New York, 1973.

[Bh] Bhatwadekar, S.M. Analitic isomorphismsms and category of finitely generated modules, Comm. in Algebra, 16 (1988), 1949–1958.

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

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

[SGA3] Demazure, M.; Grothendieck, A. Schémas en groupes, Lect. Notes Math., vol. 151–153, Springer-Verlag, Berlin-Heidelberg-New York, 1970.

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

[E] Eisenbud, D. Commutative algebra with a view toward algebraic geometry. Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995.

[FP] Fedorov, R.; Panin, I. A proof of Grothendieck–Serre conjecture on principal bundles over a semilocal regular ring containing an infinite field, Publications mathematiques de l’IHES, August, 2015 (published on line), DOI 10.1007/s10240-015-0075-z

[Ga] Gabber, O. announced and still unpublished.

[Gil1] Gille, Ph. Torseurs sur la droite affine, Transform. Groups, 7(3):231–245, 2002.

[Gil2] Gille, Ph. Le problème de Kneser-Tits, Astérisque, (326):Exp. No. 983, vii, 39–81 (2010), 2009. Séminaire Bourbaki. Vol. 2007/2008.

[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.

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

[EGAIII] Grothendieck, A. Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Première partie, Publ. Math. IHÉS 11 (1961), 5–167.

[EGAIV] Grothendieck, A. Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Seconde partie, Publ. Math. IHÉS 24 (1965), 5–231.

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

[MV] Morel, F.; Voevodsky V. A^{1}-homotopy theory of schemes, Publ. Math. IHÉS, 90 (1999), 45–143.

[Ni1] Nisnevich, E.A. Affine homogeneous spaces and finite subgroups of arithmetic groups over function fields, Functional Analysis and Its Applications 11 (1977), no. 1, 64–66.

[Ni2] Nisnevich, Y. Rationally Trivial Principal Homogeneous Spaces and Arithmetic of Reductive Group Schemes Over Dedekind Rings, C. R. Acad. Sci. Paris, Série I, 299 (1984), no. 1, 5–8.

[OP1] Ojanguren, M.; Panin, I. A purity theorem for the Witt group, Ann. Sci. Ecole Norm. Sup. (4) 32 (1999), no. 1, 71–86.

[OP2] Ojanguren, M.; Panin, I. Rationally trivial hermitian spaces are locally trivial, Math. Z. 237 (2001), 181–198.

[PSV] Panin, I.; Stavrova, A.; Vavilov, N. On Grothendieck—Serre’s conjecture concerning principal G-bundles over reductive group schemes: I, Compositio Math. 151 (2015), 535–567.

[P2] Panin, I. On Grothendieck—Serre’s conjecture concerning principal Gbundles over reductive group schemes: II, Preprint, April 2013, http://www.math.org/0905.1423v3.

[Pan2] Panin, I. On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: II, Preprint, May 2014.

[Pan3] Panin, I. Proof of Grothendieck–Serre conjecture on principal G-bundles over semi-local regular domains containing a finite field, Preprint, May 2014.

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

[Poo] Poonen, B., Bertini theorems over finite fields, Annals of Mathematics, 160 (2004), 1099 -1127.

[ChPoo] Charles, F., Poonen, B., Bertini irreducibility theorems over finite fields, arXiv:1311.4960v1, 2013.

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

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

[R3] Raghunathan, M.S. Principal bundles on affine space and bundles on the projective line, Math. Ann. 285 (1989), 309–332.

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

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

[SV] 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. 2, Internat. Press, Cambridge, MA, 1998, 135–192.

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