ON GROTHENDIECK’S CONJECTURE

ABOUT PRINCIPAL HOMOGENEOUS SPACES

FOR SOME CLASSICAL ALGEBRAIC GROUPS

Kirill Zainoulline

April 9, 1999

Abstract. In the present paper we investigate the question about the injectivity of the map 𝔉(R) → 𝔉(K) induced by the canonical inclusion of a local regular ring of geometric type R to its field of fractions K for a homotopy invariant functor 𝔉 with transfers satisfying some additional properties. As an application we get the proof of Special Unitary Case of Grothendieck’s conjecture about principal homogeneous spaces and some other interesting examples.

In the present article we consider a regular local ring R with its field of fractions K. One of our main results is the following generalization of the theorem proved in the joint work [PS] of I. Panin and A. Suslin:

Theorem I. Let A be an Azumaya algebra over a local regular ring R containing a field not necessary infinite. Then

(a) the homomorphism R^{∗}/Nrd(A^{∗}) ⟶ K^{∗}/Nrd(A_{K}^{∗}) is injective;

(b) the canonical map H1_{ét}(R, SL_{1,A}) ⟶ H1_{ét}(K, SL_{1,A_{K}}) on the first cohomology groups induced by the canonical inclusion R ↪ K is injective.

Another main result of the present article concerns the unitary variant of this theorem. Namely, we prove:

Theorem II. Let (A, σ) be an Azumaya algebra with unitary involution (see subsection 3.1) over a local regular ring R containing an infinite field of characteristic ≠ 2. Then

(a) the homomorphism U(C)/Nrd(U(A)) ⟶ U(C_{K})/Nrd(U(A_{K})) is injective;

(b) the kernel of the canonical map H1_{ét}(R, SU_{1,A}) ⟶ H1_{ét}(K, SU_{1,A_{K}}) is trivial.

Theorems I and II prove a conjecture of A. Grothendieck [Gr] for Special Linear and for Special Unitary case (for details see section 3).

It turns out that the following two modifications of these theorems hold as well:

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 13H05, 14M17.

Key words and phrases. Azumaya algebra, principal homogeneous space, transfer.

Partly supported by the CNRS, France

Theorem III. Let A be an Azumaya algebra over a local regular ring R containing a field not necessary infinite and d be some natural number. Then the homomorphism

R^{∗}/Nrd(A^{∗})(R^{∗})^{d} ⟶ K^{∗}/Nrd(A_{K}^{∗})(K^{∗})^{d}

is injective.

Theorem IV. Let (A, σ) be an Azumaya algebra with unitary involution over a local regular ring R containing an infinite field of characteristic ≠ 2 and d be some natural number. Then the homomorphism

U(C)/Nrd(U(A))U(C)^{d} ⟶ U(C_{K})/Nrd(U(A_{K}))U(C_{K})^{d}

is injective.

Note that the method used in the proof of [PS] of Theorem I doesn’t give us the way to prove Theorems II, III and IV.

Let R be a local regular ring of geometric type, i.e. R is the local ring of a point of a smooth affine variety A over a field k. Our main tool in the present work is the following result (for the proof see sections 1 and 2):

Theorem. Let 𝔉 be the functor from the category of A-algebras to the category of abelian groups and let 𝔉_{R} be its restriction to the category of R-algebras along the inclusion A ↪ R. If 𝔉 and 𝔉_{R} satisfy all axioms from sections 1 and 2 below then the homomorphism 𝔉(R) → 𝔉(K) induced by the canonical inclusion is injective.

In particular, using the main result of [PO2] and Norm Principle for the Unitary group (section 4) we get the proof of Special Unitary case of Grothendieck’s conjecture.

Finally, using a well-known theorem of D. Popescu [PO1] we generalize our results to the case when R is a local regular ring containing a field (section 5).

Our work was motivated by the paper of V. Voevodsky [Vo] and, mainly, by the paper of I. Panin and M. Ojanguren on one Grothendieck’s conjecture for hermitian spaces [PO2]. The point was to offer a good axiomatization for the method they introduced.

Acknowledgements. I am personally grateful to Ivan Panin for his attention and help. I thank University Franche-Comte Besancon and Eva Bayer-Fluckiger for hospitality and useful discussions on the subject of this article. I am very grateful to Vladimir Chernousov for the close reading of my paper during preparation and for the ideas led to the proof of Norm Principle for the unitary group.

Agreements

All rings are assumed to be commutative with unit. By k we will mean an infinite field.

Let A and S be any rings. By an A-algebra S we will mean the pair (S, i), where i : A → S is a ring homomorphism. Sometimes, we will write just S, instead of the pair (S, i), keeping in mind the homomorphism i. We will denote an A-algebra S by A →^{i} S.

By a morphism f : (S, i) → (S′, i′) between two A-algebras we will mean the commutative diagram:

\begin{matrix}A & ⟶^{i} & S \\ ∥ & & ↓^{f} \\ A & ⟶_{i′} & S′ \end{matrix}

Let 𝔉 be a covariant functor on the category of A-algebras to the category of abelian groups. Let A →^{i} R be an A-algebra. By restriction of 𝔉 to the category of R-algebras along i we will call the functor denoted by 𝔉_{R} and given as follows: 𝔉_{R}(R →^{t} T) = 𝔉(A →^{i} R →^{t} T) on objects and

Mor(𝔉_{R}(R →^{t_{1}} T_{1}), 𝔉_{R}(R →^{t_{2}} T_{2})) = Mor(𝔉(A →^{i} R →^{t_{1}} T_{1}), 𝔉(A →^{i} R →^{t_{2}} T_{2})),

on morphisms for any R-algebra T, T_{1}, T_{2}.

Further in the paper we will use the result of Grothendieck ([Ei], Corollary 18.17) which says that if we have a finite extension A ↪ B of essentially smooth k-algebras then B is finitely generated projective as the A-module.

1. Constant Case

Let A be a smooth k-algebra. Let R = A_{𝔭} be the local ring at a prime ideal 𝔭. By 𝔪_{R} we will denote the corresponding maximal ideal of R. Thus, we have the A-algebra A ↪^{i_{R}} R, where i_{R} is the canonical inclusion.

Let 𝔉 be a covariant functor from the category of k-algebras to the category of abelian groups. By 𝔉_{R} we denote its restriction to the category of R-algebras along k ↪ A ↪^{i_{R}} R. Let functors 𝔉 and 𝔉_{R} satisfy the following list of axioms:

Axiom for the functor 𝔉. C. (continuity) For any A-algebra S essentially smooth over k and for any multiplica-

tive system M in S the canonical map \underrightarrow{lim}_{g∈M} 𝔉(S_{g}) → 𝔉(M^{−1}S) is an isomorphism, where M^{−1}S is the localization of S with respect to M.

Axioms for the functor 𝔉_{R}.

For any R-algebra T finitely generated and projective as the R-module TE. (existence) It is given a homomorphism Tr_{R}^{T} : 𝔉_{R}(T) → 𝔉_{R}(R) called transfer map; TA. (additivity) For every element x ∈ 𝔉_{R}(R × T) with x_{R} = pr_{R}^{∗}(x) ∈ 𝔉_{R}(R) and

x_{T} = pr_{T}^{∗} (x) ∈ 𝔉_{R}(T) the relation Tr_{R}^{R×T} (x) = x_{R} + Tr_{R}^{T}(x_{T}) holds in 𝔉_{R}(R), where pr_{R}^{∗} and pr_{T}^{∗} are induced by projections;

TB. (base changing and homotopy invariance) For any R[t]-algebra S finitely generated projective as the R[t]-module (thus, S/(t) and S/(t − 1) are finitely generated projective as R-modules) the following diagram commutes:

𝔉_{R}(S)  ⟶^{can_{0}^{∗}} 𝔉_{R}(S/(t))
can_{1}^{∗}↓  ↓^{Tr}_{0}
𝔉_{R}(S/(t − 1)) ⟶_{Tr_{1}} 𝔉_{R}(R)

where can_{0}^{∗}, can_{1}^{∗} are induced by the canonical projections and Tr_{0}, Tr_{1} denote the corresponding transfer maps Tr_{R}^{S/(t)} and Tr_{R}^{S/(t−1)}.

Let K be a field of fractions of the ring A. Then our aim is to prove:

Theorem (Constant case). Let 𝔉 be the functor on the category of k-algebras and let 𝔉_{R} be its restriction to the category of R-algebras. If 𝔉 and 𝔉_{R} satisfy axioms C, TE, TA, TB then the homomorphism 𝔉(R) → 𝔉(K) induced by the canonical inclusion is injective.

Remark 1. To get a better feeling for the axioms above observe that TE, TA, TB are the consequences of the following more strong conditions:

For any R-algebra S essentially smooth over k and for any S-algebras T_{1}, T_{2} and T finitely generated projective as the S-modules H. (homotopy invariance) The map 𝔉_{R}(S) → 𝔉_{R}(S[t]) induced by the inclusion is an

isomorphism; TE′. (existence) It is given a homomorphism Tr_{S}^{T} : 𝔉_{R}(T) → 𝔉_{R}(S) called transfer map; TA′. (additivity) Let T = T_{1} × T_{2}. For every x ∈ 𝔉_{R}(T), x_{1} = pr_{1}^{∗}(x) ∈ 𝔉_{R}(T_{1}) and

x_{2} = pr_{2}^{∗}(x) ∈ 𝔉_{R}(T_{2}) the relation Tr_{S}^{T} (x) = Tr_{S}^{T_{1}}(x_{1}) + Tr_{S}^{T_{2}}(x_{2}) holds in 𝔉_{R}(S),

where pr_{i}^{∗} : 𝔉_{R}(T_{1} × T_{2}) → 𝔉_{R}(T_{i}) are induced by projections;

TB′. (base changing) For any extension U/S of an R-algebras with U essentially smooth over k the following diagram commutes:

𝔉_{R}(T) ⟶ 𝔉_{R}(U ⊗_{S} T)
Tr_{S}^{T} ↓  ↓_{Tr}_{U}^{U⊗_{S} T}
𝔉_{R}(S) ⟶ 𝔉_{R}(U)

Indeed, to show that the axiom TB is a consequence of the axioms H and TB′ let look at two commutative diagrams arising when we apply TB′ to the extension S/R[t] and evaluations i_{0} : R[t] →^{t↦0} R, i_{1} : R[t] →^{t↦1} R at 0 and 1 correspondingly:

\begin{matrix}𝔉_{R}(S) & ⟶^{can_{0}^{∗}} & 𝔉_{R}(S/(t)) \\ ^{Tr}_{R[t]}^{S}↓ & & ↓^{Tr}_{0} \\ 𝔉_{R}(R[t]) & ⟶_{i_{0}^{∗}} & 𝔉_{R}(R) \end{matrix}  \begin{matrix}𝔉_{R}(S) & ⟶^{can_{1}^{∗}} & 𝔉_{R}(S/(t − 1)) \\ ^{Tr}_{R[t]}^{S}↓ & & ↓^{Tr}_{1} \\ 𝔉_{R}(R[t]) & ⟶_{i_{1}^{∗}} & 𝔉_{R}(R) \end{matrix}

By axiom H the map 𝔉_{R}(R) → 𝔉_{R}(R[t]) induced by the inclusion is an isomorphism. Since the compositions R ↪ R[t] →^{i_{0}} R and R ↪ R[t] →^{i_{1}} R are identities, evaluations i_{0}^{∗} and i_{1}^{∗} coincide.

Gluing together these two diagrams via the left-down part we get the required one from TB. □

The rest of this section is organized as follows: 1.1 contains the proof of Specialization Lemma, 1.2 is devoted to a version of Quillen’s Trick, and the last subsection 1.3 contains the proof of our theorem.

1.1. Specialization Lemma

Let R be a local regular ring of a smooth k-algebra A and let R →^{i} S be a given R-algebra with an element f ∈ S such that: S1. a) S is finite over R[t] for some specially chosen t ∈ S; b) The quotient S/(f) is

finite over R; S2. There is an augmentation map ε : S → R such that the composition R →^{i} S →^{ε} R

is the identity; S3. a) S is essentially smooth over the field k; b) S/𝔪_{R}S is smooth over the residue

field R/𝔪_{R} at the maximal ideal ε^{−1}(𝔪_{R})/𝔪_{R}S.

Remark 2. To see that S3.(b) is described correctly observe that we have an obvious inclusion 𝔪_{R}S ⊂ ε^{−1}(𝔪_{R}). Since R is a local we conclude that the ideal ε^{−1}(𝔪_{R}) is the unique maximal ideal lying over the kernel of augmentation I = ker(ε). □

Now we are going to prove:

Theorem (Specialization Lemma). Let (R, S, f) be the triple satisfying conditions S1, S2 and S3. Let 𝔊 be a covariant functor from the category of R-algebras to the category of abelian groups satisfying axioms TE, TA and TB.

If the image in 𝔊(S_{f}) of an element α_{S} ∈ 𝔊(S) is trivial then ε^{∗}(α_{S}) = 0 in 𝔊(R).

Proof. To simplify the notation, for any element α_{S} ∈ 𝔊(S), we will denote by α_{S_{f}} its image under the map induced by the inclusion S ↪ S_{f} and by α_{0}, α_{1} and α_{I} – its images under the maps induced by the canonical projections can_{0}^{∗}, can_{1}^{∗} and can_{I}^{∗} for the ideal I correspondingly.

Since S1 and S3 hold for the triple (R, S, f), by Geometric Presentation Lemma ([PO2], Lemma 10.1) for the given f ∈ S and t ∈ S we can find an element t′ ∈ S such that: G1. S is finite over R[t′]; G2. There exists an ideal J coprime with I and with the property I ∩ J = (t′); G3. (f) is coprime with J and with (t′ − 1).

1. The last property G3 means that we can factorize the canonical projections S →^{can_{J}} S/J and S →^{can_{1}} S/(t′ − 1) through the localization S_{f}.

Hence, if α_{S_{f}} = 0 in 𝔊(S_{f}) for some α_{S} ∈ 𝔊(S) then α_{J} = can_{J}^{∗}(α_{S}) = 0 and α_{1} = can_{1}^{∗}(α_{S}) = 0.

2. By the theorem of Grothendieck (see the end of Agreements), since S is essentially smooth over k and S is finite over R[t′] by G1, S is finitely generated projective over R[t′].

Now look at the commutative diagram from TB:

𝔊(S)  ⟶^{can_{0}^{∗}} 𝔊(S/(t′))
can_{1}^{∗}↓  ↓^{Tr}_{0}
𝔊(S/(t′ − 1)) ⟶_{Tr_{1}} 𝔊(R)

By the previous step we get Tr_{0}(α_{0}) = Tr_{1}(α_{1}) = 0.

3. By G2, there is the decomposition S/(t′) ≅ R × S/J via the projections S/(t′) →^{can_{0I}} S/I →^{ε̄} R and S/(t′) →^{can_{0J}} S/J, where ε̄ is an isomorphism taken from the obvious decomposition ε : S →^{can_{I}} S/I →^{ε̄} R.

Since S/(t′) is finitely generated projective over R, the quotient S/J is finitely generated projective over R, too. Therefore, by TE we have well-defined transfer map Tr_{J} : 𝔊(S/J) → 𝔊(R).

By the additivity TA of the transfer and by the second step, we can write:

0 = Tr_{0}(α_{0}) = ε̄^{∗}(can_{0I}^{∗}(α_{0})) + Tr_{J}(can_{0J}^{∗}(α_{0})).

Since can_{0I}^{∗}(α_{0}) = α_{I} and can_{0J}^{∗}(α_{0}) = α_{J}, we can rewrite the last relation as:

0 = ε̄^{∗}(α_{I}) + Tr_{J}(α_{J}).

By the first step α_{J} = 0, thus, one gets ε^{∗}(α_{S}) = ε̄^{∗}(α_{I}) = 0. This completes the proof of Specialization Lemma. □

1.2. A version of Quillen’s Trick

Let (A, R, f) be a triple, where A is a smooth d-dimensional k-algebra, R be a local regular ring at the prime ideal 𝔭 of A and f ∈ 𝔭 be some fixed regular element.

We would like to produce certain extension S of the ring R, such that properties S1, S2, S3 of subsection 1.1 are satisfied.

We need the following lemma of Quillen (see [Qu], Lemma 5.12):

Theorem (Quillen’s Lemma). Let A be a smooth finite type algebra of dimension d over a field k, let f be a regular element of A, and let ℑ be a finite subset of SpecA. Then there exist elements x_{1}, …, x_{d} in A algebraically independent over k and such that if P = k[x_{1}, …, x_{d−1}] ↪^{q} A, then i) A/(f) is finite over P; ii) A is smooth over P at points of ℑ; and iii) the inclusion q factors as q : P ↪ P [x_{d}] →^{q_{1}} A, where q_{1} is finite.

Set ℑ = {𝔭} and apply Quillen’s Lemma to the given pair (A, f).

Look at the canonical tensor product diagram (we keep all notation coming from Quillen’s Lemma):

A ⟶^{i_{S}} A ⊗_{P} R
q↑  ↑^{i}
P ⟶_{r} R

where the map r is the composition r : P ↪^{q} A ↪^{i_{R}} R and i_{S} : a ↦ a ⊗ 1, i : r ↦ 1 ⊗ r.

We are going to show that the triple (R, S, f ⊗ 1) with S = A ⊗_{P} R satisfies properties S1, S2, S3 of subsection 1.1: S1. a) Since A is finite over P [x_{d}] via q_{1}, we get that A⊗_{P} R is finite over P [x_{d}]⊗_{P} R =

R[x_{d}]; b) Obviously, S/(f ⊗ 1) = A/(f) ⊗_{P} R is finite over R. Further we will identify f with f ⊗ 1 via the map i_{S}.

S2. Take the multiplication map a ⊗ r ↦ ar as augmentation ε. S3. a) The fact that A is smooth over the ring P at 𝔭 is equivalent to the smoothness

of the map r : P ↪^{q} A ↪^{i_{R}} A_{𝔭}. Hence, r is smooth. Thus, i_{S} is smooth also and since A is smooth over k, so is S; b) By the property of smooth extensions, since A is smooth over the P at 𝔭, we have that S must be smooth over R at all primes lying above 𝔭S. Since ε^{−1}(𝔪_{R}) lies over 𝔭S, S is smooth over R at ε^{−1}(𝔪_{R}).

1.3. The Proof of Constant Case

Let 𝔉 be the functor on the category of k-algebras and let 𝔉_{R} be its restriction to the category of R-algebras. By the hypothesis of the theorem 𝔉 and 𝔉_{R} satisfy axioms C, TE, TA and TB. We have to show that the map 𝔉(R) → 𝔉(K) induced by the canonical inclusion is injective.

Let α′ ∈ 𝔉(R) be such that its image α′_{K} in 𝔉(K) is trivial.

Since 𝔉 is continuous (C), we may assume that α′ came from an element α in 𝔉(A_{g}), where A_{g} is the localization of A at some g ∈ A \ 𝔭, and the image of α in 𝔉(K) is trivial. Thus, we can write α′ = i_{R}^{∗}(α), where i_{R} : A_{g} ↪ R is the canonical inclusion.

Observe that A_{g} is again smooth k-algebra and R is its local regular ring, therefore, we can replace A by A_{g} and consider the element α as lying in 𝔉(A).

Using C again we get that there exists an element f ∈ 𝔭 such that the image α_{f} in 𝔉(A_{f}) of the element α is trivial.

Hence, our problem has reduced to the following one:

Proposition. For a given α ∈ 𝔉(A) and f ∈ 𝔭 such that the image α_{f} is trivial in 𝔉(A_{f}) the element α′ = i_{R}^{∗}(α) is trivial in 𝔉(R).

Proof. The proof consists of two steps. On the first step for any triple (A, R, f) we build up a 3 × 3 commutative diagram. On the second step by playing with this diagram for the specially chosen triple we will complete the proof.

1. Let R →^{i} S be the extension of the ring R built up by using Quillen’s Trick for any triple (A, R, f) with R = A_{𝔭} and regular f ∈ 𝔭. In particular, there is the commutative diagram:

(∗)  \begin{matrix}A & ⟶^{i_{S}} & S \\ ∥ & & ↓^{ε} \\ A & ⟶_{i_{R}} & R \end{matrix}

where ε : S → R is the augmentation from S2 for the inclusion i : R → S.

Let 𝔉_{R} be the restriction of the functor 𝔉 to the category of R-algebras.

By the commutativity of the diagram

k ⟶ A ⟶^{i_{S}} S
∥  ↑^{i}
k ⟶ A ⟶_{i_{R}} R

we have the identities

𝔉(S) = 𝔉(k ↪ A →^{i_{S}} S) = 𝔉(k ↪ A ↪^{i_{R}} R →^{i} S) = 𝔉_{R}(R →^{i} S) = 𝔉_{R}(S),

𝔉(S_{f}) = 𝔉_{R}(S_{f}) and 𝔉(R) = 𝔉_{R}(R). Moreover, there is a natural identification of the functors 𝔉 and 𝔉_{R} restricted to the category of S-algebras and the following diagram commutes:

(∗∗)  \begin{matrix}𝔉(S) & 𝔉_{R}(S) \\ _{ε^{∗}}↓ & ↓_{ε^{∗}} \\ 𝔉(R) & 𝔉_{R}(R) \end{matrix}

Consider the 3 × 3 commutative diagram

(∗∗∗)  \begin{aligned}α_{f} ∈ & 𝔉(A_{f}) & ⟶ & 𝔉(S_{f}) & 𝔉_{R}(S_{f}) & ∋ α_{S_{f}} \\ ↑ & & ↑ & ↑ \\ α ∈ & 𝔉(A) & ⟶^{i_{S}^{∗}} & 𝔉(S) & 𝔉_{R}(S) & ∋ α_{S} \\ ∥ & & _{ε^{∗}}↓ & ↓_{ε^{∗}} \\ 𝔉(A) & ⟶_{i_{R}^{∗}} & 𝔉(R) & 𝔉_{R}(R) & ∋ α′ \end{aligned}

where the upper squares are induced by the localization, the left-down square is induced by (∗) and the right-down square coincides with (∗∗).

2. Now let α ∈ 𝔉(A) and f be an element from 𝔭 such that α_{f} = 0.

Consider the commutative diagram (∗∗∗) for the given triple (A, R, f) and consider the elements α′ ∈ 𝔉_{R}(R), α_{S} ∈ 𝔉_{R}(S), α_{f} ∈ 𝔉(A_{f}), α_{S_{f}} ∈ 𝔉_{R}(S_{f}), where α′ = i_{R}^{∗}(α), α_{S} = i_{S}^{∗}(α) and α_{f}, α_{S_{f}} are the images of the elements α, α_{S} under the maps induced by the canonical inclusions.

To finish our proof we apply Specialization Lemma (see 1.1) to the right column of our diagram.

By the very construction the triple (R, S, f) satisfies properties S1, S2 and S3 of subsection 1.1. And by the very assumption the functor 𝔉_{R} satisfies axioms TE, TA, TB. So we are under the hypothesis of Specialization Lemma.

Since α_{f} = 0 in 𝔉(A_{f}), we get α_{S_{f}} = 0 in 𝔉_{R}(S_{f}). Using Specialization Lemma we conclude that ε^{∗}(α_{S}) = 0. By the commutativity of the diagram ε^{∗}(α_{S}) coincides with α′, thus, α′ = 0. This completes the proof of Constant Case. □

2. Non-Constant Case

Consider more general situation, namely, let the functors 𝔉 and 𝔉_{R} (see section 1) are defined only on the category of A-algebras and satisfy axioms C, TE, TA, TB. And we still want to prove the injectivity of the homomorphism 𝔉(R) → 𝔉(K). Recall that A is smooth over k, R = A_{𝔭} is the local regular ring and K is its field of fractions.

In this case arguments used in the last part of the previous proof don’t work because our functor is not defined on k-algebras. Moreover, the objects 𝔉(S) and 𝔉_{R}(S) are not isomorphic in general.

To avoid this problem we will assume that the functors 𝔉 and 𝔉_{R} satisfy the additional axiom: E. (extension property) Given an R-algebra R →^{i} S essentially smooth over the field

k, given an A-algebra i_{S} : A → S and an augmentation ε : S → R of i, such that the following diagram commutes:

\begin{matrix}A & ⟶^{i_{S}} & S \\ ∥ & & ↓^{ε} \\ A & ⟶_{i_{R}} & R \end{matrix}

and given a multiplicative system M with respect to a finite set {𝔪_{i}}_{i∈ℑ} of maximal ideals in S with the property ε^{−1}(𝔪_{R}) ⊂ ∪_{i∈ℑ}𝔪_{i}

there exist

a) the localization S_{g} for a certain g ∈ M with a finite etale extension e : S_{g} → S̃;

b) an augmentation ε̃: S̃ → R for the inclusion R →^{i} S_{g} →^{e} S̃ such that the following

diagram commutes:

S_{g} ⟶^{e} S̃
ε↓  ↓ε̃
R  R

c) a natural transformation Φ : 𝔉 → 𝔉_{R} between two functors 𝔉 and 𝔉_{R} restricted to the category of S̃-algebras along A →^{i_{S}} S_{g} →^{e} S̃ and R →^{i} S_{g} →^{e} S̃ correspondingly, such that the morphism Φ(S̃ →^{ε̃} R) : 𝔉(S̃ →^{ε̃} R) ⟶ 𝔉_{R}(S̃ →^{ε̃} R) is the identity.

In particular, the following diagram commutes:

𝔉(S̃) ⟶^{Φ(S̃)} 𝔉_{R}(S̃)
ε̃^{∗}↓  ↓_{}_{ε̃^{∗}}
𝔉(R)  𝔉_{R}(R)

where R is considered as the S̃-algebra via the augmentation ε̃.

Remark 3. Since ε^{−1}(𝔪_{R}) ∩ M = ∅, we can extend our augmentation ε given on S to the augmentation given on the localization S_{g}, for any g ∈ M, i.e., we have the commutative diagram:

S ⟶^{i_{g}} S_{g}
ε↓  ↓^{ε}
R  R

with the canonical inclusion S ↪^{i_{g}} S_{g}. Indeed, since R is local with the unique maximal ideal 𝔪_{R}, the image ε(g) of any g ∈ M is invertible in R. □

Then our main result can be stated as follows:

Theorem (Non-Constant Case). Let 𝔉 be the functor on the category of A-algebras and let 𝔉_{R} be its restriction to the category of R-algebras. If 𝔉 and 𝔉_{R} satisfy axioms C, TE, TA, TB and E then the homomorphism 𝔉(R) → 𝔉(K) induced by the canonical inclusion is injective.

As in the proof of the Constant case (see the beginning of subsection 1.3), by using continuity C, we can reduce the problem about the injectivity of the map 𝔉(R) → 𝔉(K) to the following one:

Proposition. For a given α ∈ 𝔉(A) and f ∈ 𝔭 such that the image α_{f} in 𝔉(A_{f}) of α is trivial the element i_{R}^{∗}(α) is trivial in 𝔉(R).

Proof. As in Constant Case the proof consists of two steps. On the first step for any triple (A, R, f) we build up a 3 × 5 commutative diagram. On the second step by playing with this diagram for the specially chosen triple we complete the proof of the proposition.

In contrast with Constant Case the first step is more complicated and can be subdivided as follows:

First we produce starting data to apply axiom E. In particular we construct an Ralgebra S and a multiplicative system M. Second we construct a lot of elements h ∈ M such that the extension S_{h}/R satisfies properties S1, S2 and S3 of subsection 1.1. Third for the specially chosen h ∈ M we construct an extension S̃_{h}/R satisfying conditions S1, S2, S3 and such that the diagram (∗∗′) below commutes. And finally we build up the 3 × 5 commutative diagram.

1. Let R →^{i} S be the extension built up by using Quillen’s Trick for any triple (A, R, f). In particular, it means that there is a commutative diagram

(∗)  \begin{matrix}A & ⟶^{i_{S}} & S \\ ∥ & & ↓^{ε} \\ A & ⟶_{i_{R}} & R \end{matrix}

where ε is the augmentation from S2, S is finite over R[t] for specially chosen t in S and S/(f) is finite over R by S1.

Let I = ker(ε), J = (f) ∩ I and J′ = (J ∩ R[t])S (we identify R[t] with its embedding in S). Observe, that J′ ⊂ J ⊂ I.

Lemma 1. The quotients S/J and S/J′ are finite over R.

Proof. Since J = (f)∩I, there is the inclusion S/J ↪ S/(f)×S/I induced by projections. It is easy to see now that S/J is finite over R.

Denote J_{t} = J ∩ R[t] the ideal in R[t]. Then S/J′ = S/J_{t}S = S ⊗_{R[t]} (R[t]/J_{t}) and, since S is finite over R[t], our quotient S/J_{t}S is finite over R[t]/J_{t}. But there is the inclusion R[t]/J_{t} ↪ S/J induced by the given one R[t] ↪ S and we have just proved that S/J is finite over R. Hence, R[t]/J_{t} is finite over R and we have done. □

Let {𝔪_{i}}_{i∈ℑ} be the system of all maximal ideals lying over J′. It is finite because of the lemma. Let M be the corresponding multiplicative system. We see that ε^{−1}(𝔪_{R}) lies above I ⊃ J′, so it is one of the 𝔪_{i}, and ε^{−1}(𝔪_{R}) ⊂ ∪_{i∈ℑ}𝔪_{i}.

2. Now we are interesting in the properties of localizations taken via the multiplicative system M.

Lemma 2. For any localization S_{g}, g ∈ M, i) the quotient S_{g}/(f) is finite over R; ii) there exists an extension S_{h} of S_{g}, h ∈ M, such that S_{h} has a structure of a finite algebra over R[t′] for some specially chosen t′ ∈ S_{h}.

Proof. i) Since (f) ⊃ J, it is enough to show that S_{g}/JS_{g} is finite over R. By construction of M we know that g ∈ M is invertible in S/J, therefore, S_{g}/JS_{g} = (S/J)_{g} = S/J is finite over R. ii) Indeed, it is the reformulation of Lemma 8.2 [PO2]. □

Now show that the extension S_{h}/R satisfies properties S1, S2 and S3 of subsection 1.1. The property S1 follows from i) and ii). The existence of augmentation S2 can be

conclude from Remark 3. The property S3 is the consequence of the fact that the localization of a smooth algebra is again smooth and the fact that the element h is coprime with the kernel of augmentation I.

3. Apply now axiom E to the data (R →^{i} S, A →^{i_{S}} S, S →^{ε} R, M). We get that there exists the localization S_{g}, g ∈ M, the finite etale extension e : S_{g} → S̃, the augmentation ε̃: S̃ → R and the natural transformation Φ : 𝔉 → 𝔉_{R} between two functors 𝔉 and 𝔉_{R} restricted to the category of S̃-algebras satisfying properties E.(b) and E.(c).

By the previous step, we can find an extension S_{h}, h ∈ M, of S_{g} with the property that S_{h}/R satisfies conditions S1, S2, S3 of subsection 1.1.

Consider now the canonical tensor product diagram:

S_{h} ⟶^{i_{1}} S_{h} ⊗_{S_{g}} S̃
↑  ↑^{i_{2}}
S_{g} ⟶_{e}  S̃

where i_{1} : s_{h} ↦ s_{h} ⊗ 1 and i_{2} : s̃ ↦ 1 ⊗ s̃ are the canonical inclusions.

Set S̃_{h} = S_{h} ⊗_{S_{g}} S̃. We claim that the extension S̃_{h}/R satisfies properties S1, S2, S3. Indeed, we already know that the extension S_{h}/R does satisfy. Define the augmenta-

tion map as ε̃_{h} : s_{h} ⊗ s̃ ↦ ε(s_{h})ε̃(s̃), thus, we have checked S2. Since S̃_{h}/S_{h} is the finite etale, the properties S1 and S3 hold.

Clearly, we have the commutative diagram:

(∗′)  \begin{matrix}S̃ & ⟶^{i_{2}} & S̃_{h} \\ ε̃↓ & & ↓ε̃_{h} \\ R & & R \end{matrix}

In contrast with the extension S_{h}/R for the extension S̃_{h}/R by E.(c) we have the commutative diagram

(∗∗′)  \begin{matrix}𝔉(S̃_{h}) & ⟶^{Φ(S̃_{h})} & 𝔉_{R}(S̃_{h}) \\ ^{}^{ε̃_{h}^{∗}}↓ & & ↓^{}^{ε̃_{h}^{∗}} \\ 𝔉(R) & & 𝔉_{R}(R) \end{matrix}

4. Consider the modification of the diagram (∗∗∗) built up by using steps 1-3 for any triple (A, R, f):

𝔉(A_{f}) ⟶ 𝔉(S_{gf}) ⟶ 𝔉(S̃_{f}) ⟶ 𝔉(S̃_{hf}) ⟶^{Φ(S̃_{hf})} 𝔉_{R}(S̃_{hf})
↑  ↑  ↑  ↑  ↑
𝔉(A) ⟶^{i_{S}^{∗}∘i_{g}^{∗}} 𝔉(S_{g}) ⟶^{e^{∗}} 𝔉(S̃) ⟶^{i_{2}^{∗}} 𝔉(S̃_{h}) ⟶^{Φ(S̃_{h})} 𝔉_{R}(S̃_{h})
∥  _{ε^{∗}}↓  _{}_{ε̃^{∗}}↓  ^{}^{ε̃_{h}^{∗}}↓  ↓^{}^{ε̃_{h}^{∗}}
𝔉(A) ⟶_{i_{R}^{∗}} 𝔉(R)  𝔉(R)  𝔉(R)  𝔉_{R}(R)

where the lower squares are constructed as follows: the left side is induced by (∗) and by the commutative diagram from Remark 3; the next squares are induced by E.(b), (∗′) and (∗∗′), correspondingly. Hence, we get the desired 3 × 5 commutative diagram.

5. Repeat now the arguments used in the step 2 of the proof of Constant Case (see the end of subsection 1.3):

Let α ∈ 𝔉(A) and f be an element from 𝔭 such that α_{f} = 0.

Denote by αS̃_{h} the image in 𝔉_{R}(S̃_{h}) of α under the composition i_{S}^{∗} ∘ i_{g}^{∗} ∘ e^{∗} ∘ i_{2}^{∗} ∘ Φ(S̃_{h}). Since the image α_{f} in 𝔉(A_{f}) of α is trivial, then by the commutativity of the diagram the image in 𝔉_{R}(S̃_{hf}) of the element αS̃_{h} is trivial, too.

Hence, we can apply Specialization Lemma to the right column of our diagram and we get ε̃_{h}^{∗}(αS̃_{h}) = 0. By the commutativity of the diagram i_{R}^{∗}(α) = ε̃_{h}^{∗}(αS̃_{h}), thus, i_{R}^{∗}(α) = 0. And we have finished the proof of Non-Constant Case. □

3. Applications

We keep all notations and agreements used before. Let A be a smooth algebra over an infinite field k and let R be its local regular ring at some prime ideal 𝔭. Let K be a field of fractions of R.

In the present section we consider examples of some functors 𝔉 from the category of A-algebras to the category of abelian groups. Namely, the point is to show that such functor satisfies axioms C, TE, TA, TB and E from the previous sections. Then we can use our main theorem (see section 2)

Theorem (Non-Constant Case). Let 𝔉 be the functor on the category of A-algebras and let 𝔉_{R} be its restriction to the category of R-algebras. If 𝔉 and 𝔉_{R} satisfy axioms C, TE, TA, TB and E then the homomorphism 𝔉(R) → 𝔉(K) induced by the canonical inclusion is injective.

Hence, we get the injectivity of the map 𝔉(R) → 𝔉(K) (see the (a) statement of the theorems below). Finally, playing with a few long exact cohomology sequences and using the injectivity above we get the injectivity on the first cohomology level (see the (b) statement of the theorems below).

As a consequence, we will get two cases of Grothendieck’s Conjecture about principal homogeneous spaces [Gr] — the case of Special Linear group and the case of Special Unitary group.

Now let describe our special functors and results we are interested in (detailed proofs one will find in the corresponding subsections below):

Linear Case. Let A be some Azumaya algebra over the given local regular ring R (for the definition see subsection 3.1). Let Nrd : A^{∗} → R^{∗} denotes the reduced norm homomorphism. For any R-algebra T let A_{T} = A ⊗_{R} T be the extended Azumaya algebra over T.

Define the group scheme SL_{1,A} related to the Azumaya R-algebra A as

SL_{1,A} : T ↦ SL(A_{T}) = {a ∈ A_{T} | Nrd(a) = 1}.

Our aim is to show the result proved before in [PS]:

Theorem (Linear Case). Let A be an Azumaya algebra over a local regular ring R of geometric type. Then

(a) the homomorphism R^{∗}/Nrd(A^{∗}) ⟶ K^{∗}/Nrd(A_{K}^{∗}) is injective;

(b) the canonical map H1_{ét}(R, SL_{1,A}) ⟶ H1_{ét}(K, SL_{1,A_{K}}) on the first cohomology groups induced by the canonical inclusion is injective.

Proof. See subsection 3.2.

Observe that the statement (a) of the theorem corresponds to the case when the functor 𝔉 is defined as 𝔉 : T ↦ T^{∗}/Nrd(A_{T}^{∗}).

We recall that the conjecture of Grothendieck [Gr] states the triviality of the kernel of the canonical map

H1_{ét}(R, G) ⟶ H1_{ét}(K, G_{K})

for any reductive flat group scheme G over a regular semilocal ring R. Thus, the statement (b) of the theorem above proves the following assertion:

Corollary (Special Linear Case). The Grothendieck’s conjecture is true for the group scheme G = SL_{1,A} related to an Azumaya algebra A over a local regular ring R of geometric type.

In the same notation assume that there is the additional structure on our Azumaya algebra: Let (A, σ) be an Azumaya algebra with involution over R (for the definition see subsection 3.1). Further in the paper when the discussion is about the involution structure we assume that the characteristic of the base field k is different from 2.

We know that there can be three different types of involution: ortogonal, sympletic and unitary. It turns out that in the case of ortogonal and sympletic involution we can prove the same results as above by using well-known facts about quadratic forms (for the details see subsection 3.3). So the only interesting case for us is the unitary case:

Unitary Case. Let (A, σ) be an Azumaya algebra with unitary involution over R. It means that there is a tower A/C/R, where C is the center of A and C/R is an etale quadratic extension over R with restricted involution σ. Therefore, C_{K}/K is a separable quadratic extension of the corresponding fields of fractions.

Let U(A_{T}) = {a ∈ A_{T} | aa^{σ} = 1} be the unitary group of an algebra (A_{T} , σ) for any R-algebra T. We define the group scheme SU_{1,A} related to the Azumaya R-algebra A with unitary involution σ as

SU_{1,A} : T ↦ SU(A_{T}) = {a ∈ A_{T} | aa^{σ} = 1, Nrd(a) = 1}

Our goal will be to show:

Theorem (Unitary Case). Let (A, σ) be an Azumaya algebra with unitary involution over a local regular ring R of geometric type. Then

(a) the homomorphism U(C)/Nrd(U(A)) ⟶ U(C_{K})/Nrd(U(A_{K})) is injective;

(b) the kernel of the canonical map H1_{ét}(R, SU_{1,A}) ⟶ H1_{ét}(K, SU_{1,A_{K}}) is trivial.

Proof. See subsection 3.4.

Clearly, the statement (a) respects the functor 𝔉 : T ↦ U(C_{T})/Nrd(U(A_{T})), where C_{T} = C ⊗_{R} T is the center of the extended Azumaya algebra A_{T}.

Thus, the statement (b) of the theorem above proves the following assertion:

Corollary (Special Unitary Case). The Grothendieck’s conjecture is true for the group scheme G = SU_{1,A} related to an Azumaya R-algebra A with unitary involution over a local regular ring of geometric type.

Torsion Cases. Next two theorems represent independent interest. One can look on it as on the modification of the corresponding Linear and Unitary Case.

For the functor 𝔉 : T ↦ T^{∗}/Nrd(A_{T}^{∗})(T^{∗})^{d}, d ∈ ℕ, we have

Theorem (Linear Torsion Case). Let A be an Azumaya algebra over a local regular ring R of geometric type and d be some natural number. Then the homomorphism

R^{∗}/Nrd(A^{∗})(R^{∗})^{d} ⟶ K^{∗}/Nrd(A_{K}^{∗})(K^{∗})^{d}

is injective.

And for the functor 𝔉 : T ↦ U(C_{T})/Nrd(U(A_{T}))U(C_{T})^{d}, d ∈ ℕ, we have

Theorem (Unitary Torsion Case). Let (A, σ) be an Azumaya algebra with unitary involution over a local regular ring R of geometric type and d be some natural number. Then the homomorphism

U(C)/Nrd(U(A))U(C)^{d} ⟶ U(C_{K})/Nrd(U(A_{K}))U(C_{K})^{d}

is injective.

Proof. The torsion cases one can prove easily by following the proofs of the corresponding Linear and Unitary cases (see subsections 3.2 and 3.4).

3.1. The Properties of Azumaya algebras

Let us formulate some properties of Azumaya algebras which we are going to use. Mostly, one can find them in [Kn] and [PO2].

We recall that an Azumaya R-algebra over a ring R is an R-algebra A which satisfies the following two properties: it is finitely generated projective R-module and the canonical R-algebra homomorphism A ⊗_{R} A^{op} → End_{R}(A) is an isomorphism. Clearly, for any R-algebra T, the scalar extension A_{T} = A ⊗_{R} T of the algebra A is an Azumaya algebra over T.

We will use the reduced norm homomorphism Nrd_{R} : A^{∗} → R^{∗}. This homomorphism respects the scalar extensions. In the case where R is a field and T is its algebraic closure, the reduced norm homomorphism Nrd_{T} can be identified with the usual determinant det : GL_{d}(T) → T^{∗}; this identification is induced by an T -algebra isomorphism between A_{T} and the matrix algebra M_{d}(T) mentioned above.

By an Azumaya algebra with involution over a ring R we mean a pair (A, σ) consisting of an R-algebra A and an R-linear involution σ on A such that: i) A is an Azumaya algebra over its center C; ii) C is either R or an etale quadratic extension of R; and iii) C^{σ} = R. Observe that the involution σ commutes with the reduced norm map, i.e. Nrd_{C}(a^{σ}) = Nrd_{C}(a)^{σ} for any a ∈ A^{∗}.

When C = R we have involution of the first kind. We say that involution σ is of ortogonal (or sympletic) type if the dimension over R of the set of symmetric elements \{a ∈ A | a^{σ} = a\} of the algebra A equals \frac{n(n+1)}{2} (or \frac{n(n−1)}{2}), where n is the degree of the Azumaya algebra A (see [Kn]).

In the case when C is quadratic etale over R we have unitary involution (involution of the second kind).

Now we fix a local regular ring R = A_{𝔭} of a smooth k-algebra A. Then we have the following properties:

A1. Since an Azumaya R-algebra A is given by the finite number of generators and relations we can find the localization A_{g}, g ∈ A \ 𝔭, such that the algebra A come from some Azumaya A_{g}-algebra A_{g}, i.e. A = A_{g} ⊗_{A_{g}} R. On geometric language it means that we can extend an Azumaya algebra given at a point to some neighbourhood of this point.

By the same reasons as before we can state that if there is an isomorphism Ψ : A → B of Azumaya R-algebras then there exists the localization A_{g}, g ∈ A \ 𝔭, the Azumaya A_{g}-algebras A_{g} and B_{g}, where A = A_{g} ⊗_{A_{g}} R and B = B_{g} ⊗_{A_{g}} R, and the isomorphism Ψ_{g} : A_{g} → B_{g} of the Azumaya A_{g}-algebras such that Ψ = Ψ_{g} ⊗_{A_{g}} id_{R}.

Observe that the arguments above also work in the case of Azumaya algebras with involutions and when R is a semilocal regular ring, i.e. R is the localization at some finite number of prime ideals of A.

A2. We also will need in the following reformulation of the Proposition 7.1 of [PO2]:

Let O be some semilocal regular ring such that there is the inclusion R ↪^{i} O with the augmentation ε. Let (A_{O}, σ) and (B_{O}, τ) be two Azumaya algebras with involution over O, of the same rank. Assume that there exists an isomorphism ψ : (A_{R}, σ) → (B_{R}, τ). Then there exists a finite etale extension e : O → Õ, an augmentation ε̃: Õ → R of e over R and an isomorphism Ψ : (AÕ, σ̃) → (BÕ, τ̃) of extended Azumaya algebras with involutions over Õ such that the extension Ψ ⊗Õ id_{R} : (AÕ, σ̃) ⊗Õ R → (BÕ, τ̃) ⊗Õ R of Ψ via ε̃ coincides with ψ.

3.2. The Proof of Linear Case

Let R = A_{𝔭} be the local regular ring of the smooth k-algebra A and let A be the Azumaya algebra over R.

Let 𝔉 be the functor defined on the category of R-algebras as:

𝔉 : T ↦ T^{∗}/Nrd(A_{T}^{∗}).

First of all, by property A1, we may assume that our Azumaya algebra A over the ring R come from some Azumaya algebra over the localization A_{g}, for some g ∈ A \ 𝔭. Thus, our functor is defined on the category of A_{g}-algebras.

Since A_{g} is smooth k-algebra and R is again its local regular ring, we may write A instead of A_{g} and nothing will be changed. Hence, we will write A_{R} instead of A meaning that A_{R} is the scalar extension of the Azumaya A-algebra A via the canonical inclusion A ↪^{i_{R}} R. Thus, we may assume that our functor 𝔉 is defined on the category of A-algebras.

Show that the functor 𝔉 and its restriction 𝔉_{R} to the category of R-algebras satisfy axioms C, TE, TA, TB and E:

C. Observe that the functor G_{m} : T ↦ T^{∗} sending any A-algebra T to its group of units is continuous. Since A is finitely generated projective as the A-module, the functor of extension of scalars T ↦ A_{T} is continuous. Thus, the functor T ↦ Nrd(G_{m}(A_{T})) is continuous, too, and we get the required. □

E. Let we are under the hypothesis of the axiom E (see section 2): Let we have the R-algebra R →^{i} S, the inclusion i_{S} : A → S, the augmentation ε : S → R of i and the multiplicative system M.

For the functors 𝔉 and 𝔉_{R} restricted to the category of S-algebras we may write:

𝔉 : T ↦ T^{∗}/Nrd(A_{T}^{∗})  and  𝔉_{R} : T ↦ T^{∗}/Nrd(B_{T}^{∗}),

where for any S-algebra T, A_{T} is the extension of scalars of A via the inclusion A →^{i_{S}} S → T and B_{T} is the extension of scalars of A via A →^{i_{R}} R →^{i} S → T (in general, A_{T} and B_{T} are not isomorphic).

We will check axiom E in four steps:

First, by using A2 property we will produce the finite etale extension O →^{e} Õ of the localization O = M^{−1}S and the augmentation ε̃ : Õ → R, such that the extended Azumaya Õ-algebras AÕ and BÕ become equivalent via the isomorphism Ψ.

Secondly, we will show that the semilocal ring Õ is, indeed, the localization M^{−1}S̃ of some R-algebra S̃ which is finite etale over S.

After that, by using A1 we will find the localization S̃_{g}, g ∈ M, of S̃ such that the extended Azumaya S̃_{g}-algebras AS̃_{g} and BS̃_{g} are still equivalent via the isomorphism Ψ_{g} with Ψ = Ψ_{g} ⊗S̃_{g} Õ. Considering the extension e : S_{g} → S̃_{g} and the augmentation

ε̃: S̃_{g} → Õ →^{ε̃} R we get conditions (a) and (b) of axiom E.

We will end with showing condition (c) of axiom E. For this purpose by using the isomorphism Ψ_{g} : AS̃_{g} → BS̃_{g} we construct the natural equivalence Φ between functors 𝔉 and 𝔉_{R} restricted to the category of S̃_{g}-algebras. It turns out that all necessary properties for condition (c) are satisfied.

1. We introduce the semilocal ring O as the localization M^{−1}S of S. By Remark 3 we have the augmentation map ε : O → R for the inclusion R →^{i} S ↪^{i_{O}} O compatible with augmentation on S. Hence, there is the commutative diagram:

A ⟶^{i_{S}} S ⟶^{i_{O}} O
∥  ^{ε}↓  ↓^{ε}
A ⟶_{i_{R}} R  R

Consider the extended Azumaya O-algebras A_{O} and B_{O}. Since the extensions A_{R} and B_{R} of the algebras A_{O} and B_{O} via the augmentation ε coincide (see the diagram above), we are under the hypothesis of property A2 (our ψ is the identity).

So there exists the finite etale extension e : O → Õ, the augmentation ε̃: Õ → R for the inclusion R →^{i} S →^{i_{O}} O →^{e} Õ compatible with ε and the isomorphism Ψ : AÕ → BÕ of extended Azumaya algebras such that its extension Ψ ⊗Õ id_{R} via ε̃ is the identity.

2. By the properties of finite etale extensions there exists the localization S_{h}, h ∈ M, a finite etale extension e : S_{h} → S̃ such that Õ = O ⊗_{S_{h}} S̃. Since O = M^{−1}S_{h}, we have Õ = M^{−1}S̃. To simplify the notation we will write S instead of S_{h}. Thus, we get the commutative diagram:

S̃ ⟶_{via M}^{localization} Õ
e↑  ↑^{e}
S ⟶_{via M}^{localization} O

3. By the properties of finite extensions for the direct system of localizations {S_{g}}_{g∈M} with \underrightarrow{lim}_{g∈M} S_{g} = O we have an induced direct system \{S̃_{g}\}_{g∈M}, where S̃_{g} = S_{g} ⊗_{S} S̃, and the canonical map \underrightarrow{lim}_{g∈M} S̃_{g} → Õ is an isomorphism. Thus, one has the diagram

S̃ ⟶ S̃_{g} ⟶ Õ
e↑  ^{e}↑  ↑^{e}
S ⟶ S_{g} ⟶ O

where e is finite etale.

Since our Azumaya Õ-algebras AÕ and BÕ are isomorphic via Ψ, applying A1 to the case A = S̃ and R = Õ we get that there exists a localization S̃_{g}, g ∈ M, such that the Azumaya S̃_{g}-algebras AS̃_{g} and BS̃_{g} are isomorphic via Ψ_{g}.

Take the composition ε̃ : S̃_{g} → Õ →^{ε̃} R to be the augmentation on S̃_{g}. Clearly, it is compatible with ε. Since the isomorphism Ψ is got from Ψ_{g} by extension of scalars S̃_{g} → Õ and the extension of Ψ via ε̃: Õ → R is the identity, the extension of Ψ_{g} via the augmentation ε̃: S̃_{g} → R is the identity as well.

To simplify the notations we replace S̃_{g} by S̃ and Ψ_{g} by Ψ. So we have constructed:

a) the localization S_{g}, g ∈ M, and the finite etale extension S_{g} →^{e} S̃;

b) the augmentation ε̃ on S̃ compatible with ε.

Thus, we get conditions (a) and (b) of axiom E.

4. The last step is to check E.(c):

Since we have the isomorphism Ψ : AS̃ → BS̃ on Azumaya S̃-algebras, we have the isomorphism Ψ(T) : A_{T} → B_{T} of Azumaya T -algebras got by extension of scalars for any S̃-algebra T. By definition there is the commutative diagram:

A_{T}^{∗}  ⟶^{Ψ(T)}  B_{T}^{∗}
Nrd↓  ↓^{Nrd}
T^{∗} = Z(A_{T})^{∗} ⟶_{Ψ_{Z}(T)} Z(B_{T})^{∗} = T^{∗}

where Ψ_{Z}(T) is the restriction of Ψ(T) to the groups of units T^{∗} of the centers of the Azumaya algebras A_{T} and B_{T}. Thus, there is the isomorphism on the quotients

Φ(T) = \overline{Ψ_{Z}(T)} : T^{∗}/Nrd(A_{T}^{∗}) → T^{∗}/Nrd(B_{T}^{∗}).

and Φ : 𝔉 → 𝔉_{R} is the natural equivalence of the functors 𝔉 and 𝔉_{R} on the category of S̃-algebras.

Moreover, the extension of the isomorphism Φ(S̃) : 𝔉(S̃) → 𝔉_{R}(S̃) via the augmentation ε̃: S̃ → R is the identity, i.e. Φ(R) = id_{𝔉(R)} : 𝔉(R) → 𝔉_{R}(R).

Observe that in our concrete case the isomorphism Ψ_{Z}(T) is the identity. In general, it is not true. For instance, when we have Azumaya algebras with unitary involutions (see subsections 3.1 and 3.4) the isomorphism on the centers may not coincide with the identity.

This completes the check of axiom E. □

TE. Let T be an R-algebra finitely generated projective as the R-module. Thus, T is a semilocal ring. We define the transfer map

Tr_{R}^{T} : T^{∗}/Nrd(A_{T}^{∗}) ⟶ R^{∗}/Nrd(A_{R}^{∗})

to be the usual norm map N_{R}^{T} : T^{∗} → R^{∗} on the quotients.

To see that it is well-defined we need in:

Lemma 3. Let A be an Azumaya algebra over a semilocal ring R. Let T be an R-algebra finitely generated projective as the R-module and let N_{R}^{T} : T^{∗} → R^{∗} be its norm map. Then we claim that:

N_{R}^{T}(Nrd_{T} (A_{T}^{∗})) ⊂ Nrd_{R}(A^{∗}).

Proof. To prove this inclusion we are passing our reduced norms through K_{1}-groups.

By results of [Ba] (ch.V, Theorem 9.1) there is the commutative diagram:

A_{T}^{∗} ⟶^{i_{T}} K_{1}(A_{T}) ⟶^{N^{∗}} K_{1}(A) ⟵_{surj}^{i} A^{∗}
Nrd_{T} ↓  ^{Nrd}_{T}^{∗} ↓  ↓^{Nrd}_{R}^{∗}  ↓^{Nrd}_{R}
T^{∗}  T^{∗}  ⟶_{N} R^{∗}  R^{∗}

where N^{∗} is the norm map on K_{1}-groups; i^{∗} is the surjective homomorphism induced by inclusion A^{∗} = GL_{1}(A) ↪ K_{1}(A); and Nrd^{∗} is the reduced norm on K_{1} of an Azumaya algebra. If A splits, i.e. A = M_{d}(R), we can write our reduced norm as the composition:

Nrd^{∗} : K_{1}(M_{d}(R))

Morita equivalence

⟶ K_{1}(R)

⟶^{det}
R^{∗}.

We get the required inclusion since i is surjective. This completes the proof of Lemma 3 and the check of axiom TE. □

TA. The additivity follows from the corresponding property of the norm map: If T_{1} and T_{2} are finitely generated projective over R then for any (t_{1}, t_{2}) ∈ T_{1} × T_{2} we have N_{R}^{T_{1}×T_{2}}(t_{1}, t_{2}) = N_{R}^{T_{1}}(t_{1})N_{R}^{T_{2}}(t_{2}). Since 𝔉_{R}(T_{1} × T_{2}) = 𝔉_{R}(T_{1}) × 𝔉_{R}(T_{2}), we get the required. □

TB. Let S be any R[t]-algebra finitely generated projective as the R[t]-module. Observe that the functor G_{m} : T ↦ T^{∗} with the usual norm in the role of the transfer map satisfies homotopy invariance H and base changing TB′ properties, thus, it satisfies TB (see Remark 1).

To see the axiom TB look on the diagram induced by TB applying to G_{m}:

(∗)

G_{m}(S)  ⟶^{can_{0}^{∗}} G_{m}(S/(t))
can_{1}^{∗}↓  ↓^{N}_{0}
G_{m}(S/(t − 1)) ⟶_{N_{1}} G_{m}(R)

where can_{i}^{∗}, i = 0, 1, are induced by the canonical projections, and N_{i} denote the corresponding norm map.

The following two diagrams induced by the canonical projection

pr : G_{m}(T) = T^{∗} ⟶ T^{∗}/Nrd(A_{T}^{∗}) = 𝔉_{R}(T)

are commutative as well:

G_{m}(S) ⟶^{can_{i}^{∗}} G_{m}(S/(t − i)) ⟶^{N_{i}} G_{m}(R)
pr↓  ↓^{pr}  ↓^{pr}
𝔉_{R}(S) ⟶_{can_{i}^{∗}} 𝔉_{R}(S/(t − i)) ⟶_{Tr_{i}} 𝔉_{R}(R)

where Tr_{i}, i = 0, 1, denote the corresponding transfer map.

Thus, since the projection pr is surjective, we get the required diagram commutes:

𝔉_{R}(S)  ⟶^{can_{0}^{∗}} 𝔉_{R}(S/(t))
can_{1}^{∗}↓  ↓^{Tr}_{0}
𝔉_{R}(S/(t − 1)) ⟶_{Tr_{1}} 𝔉_{R}(R)

And we have checked that functors 𝔉 and 𝔉_{R} satisfy axioms C, TE, TA, TB, E. □

Applying now our Theorem (Non-Constant Case) to the functors 𝔉 and 𝔉_{R} we get the statement (a) of Linear Case:

Theorem. (a) The map R^{∗}/Nrd(A^{∗}) ⟶ K^{∗}/Nrd(A_{K}^{∗}) induced by the canonical inclusion is injective.

To prove the statement (b) let look on the short exact sequence of smooth group schemes:

1 ⟶ SL_{1,A} ⟶ GL_{1,A} ⟶^{Nrd} G_{m} → 1,

where GL_{1,A} : T ↦ A_{T}^{∗} for any R-algebra T.

It induces the long exact cohomology sequence:

1 ⟶ SL(A) ⟶ A^{∗} ⟶^{Nrd} R^{∗} ⟶ H1_{ét}(R, SL_{1,A}) ⟶ H1_{ét}(R, GL_{1,A}) ⟶ ⋯

Since R is local, the group H1_{ét}(R, GL_{1,A}) is trivial (see [Kn]).

The inclusion R ↪ K induces the natural map on our long cohomology sequence, so that the diagram

1 ⟶ SL(A) ⟶ A^{∗} ⟶^{Nrd} R^{∗} ⟶ H1_{ét}(R, SL_{1,A}) ⟶ 1
↓ ↓ ↓ ↓
1 ⟶ SL(A_{K}) ⟶ A_{K}^{∗} ⟶_{Nrd_{K}} K^{∗} ⟶ H1_{ét}(K, SL_{1,A_{K}}) ⟶ 1

commutes.

Taking the cokernels from the left side we get:

R^{∗}/Nrd(A^{∗}) ⟶^{≅} H1_{ét}(R, SL_{1,A})
↓  ↓
K^{∗}/Nrd(A_{K}^{∗}) ⟶_{≅} H1_{ét}(K, SL_{1,A_{K}})

Since the left vertical arrow is injective by (a), we get the statement (b):

Theorem. (b) The map H1_{ét}(R, SL_{1,A}) ⟶ H1_{ét}(K, SL_{1,A_{K}}) induced by the canonical inclusion is injective.

This completes the proof of Linear Case. □

3.3. Ortogonal and Sympletic Cases

Let (A, σ) be an Azumaya algebra with ortogonal involution over R (for the definition see subsection 3.1).

Let O(A_{T}) = {a ∈ A_{T} | aa^{σ} = 1} be the ortogonal group of an algebra (A_{T} , σ) for any R-algebra T. We define the group scheme SO_{1,A} related to the Azumaya R-algebra A with ortogonal involution σ as

SO_{1,A} : T ↦ SO(A_{T}) = {a ∈ A_{T} | aa^{σ} = 1, Nrd(a) = 1}.

For the field of fractions K we have O(K) = {x ∈ K | x^{2} = 1} = {±1}. Since R is the local ring of an affine variety over the field k, we get also O(R) = {±1}.

We would like to show the analogy of the statement (a) of Linear Case, i.e. the injectivity of the map

O(R)/Nrd(O(A)) ⟶ O(K)/Nrd(O(A_{K}))

induced by the canonical inclusion.

First of all, note that our map is always surjective and it is not injective if and only if Nrd(O(A)) = {1} and Nrd(O(A_{K})) = {±1}.

Let Nrd(O(A_{K})) = {±1} then by theorem of Knezer [BI] we conclude that A_{K} splits, i.e. A_{K} is the matrix algebra over K. Moreover, by theorem of Grothendieck [BI], the algebra A splits, too. Hence, we get Nrd(O(A)) = {±1} and our map must be injective anyway, indeed, it is an isomorphism.

Now look on the short exact sequence of smooth group schemes:

1 ⟶ SO_{1,A} ⟶ O_{1,A} ⟶^{Nrd} O → 1.

It induces the long exact cohomology sequence:

1 ⟶ SO(A) ⟶ O(A) ⟶^{Nrd} O(R) ⟶ H1_{ét}(R, SO_{1,A}) ⟶ H1_{ét}(R, O_{1,A}) ⟶ ⋯

The canonical inclusion R ↪ K induces the natural map on our long cohomology sequence, so that the diagram

1 ⟶ SO(A) ⟶ O(A) ⟶^{Nrd} O(R) ⟶ H1_{ét}(R, SO_{1,A}) ⟶ H1_{ét}(R, O_{1,A})
↓ ↓ ↓ ↓ ↓
1 ⟶ SO(A_{K}) ⟶ O(A_{K}) ⟶_{Nrd_{K}} O(K) ⟶ H1_{ét}(K, SO_{1,A_{K}}) ⟶ H1_{ét}(K, O_{1,A_{K}})

commutes.

Taking the cokernels from the left side we get:

1 ⟶ O(R)/Nrd(O(A)) ⟶ H1_{ét}(R, SO_{1,A}) ⟶ H1_{ét}(R, O_{1,A})
≅↓  ↓  ↓
1 ⟶ O(K)/Nrd(O(A_{K})) ⟶ H1_{ét}(K, SO_{1,A_{K}}) ⟶ H1_{ét}(K, O_{1,A_{K}})

Since the left vertical arrow is the isomorphism and the right vertical arrow has the trivial kernel by the result of [PO2], we get the triviality of the kernel of the middle vertical arrow.

Thus, we have proved the following assertion:

Theorem (Special Ortogonal Case). The Grothendieck’s conjecture is true for the group scheme G = SO_{1,A} related to an Azumaya algebra A with ortogonal involution over a local regular ring R of geometric type.

Special Sympletic Case of Grothendieck’s conjecture about principal homogeneous spaces follows immediately from [PO2].

3.4. The Proof of Unitary Case

We keep all notations and definitions used in the subsection 3.1. The most arguments of our discussion here are taken from the proof of Linear Case (subsection 3.2). The only difference is that we have an additional structure of unitary involution on our Azumaya algebras.

Thus, let (A, σ) be an Azumaya algebra with unitary involution over R.

Let 𝔉 be the functor defined on the category of R-algebras as:

𝔉 : T ↦ U(C_{T})/Nrd(U(A_{T})),

where U(C_{T}) = {c ∈ C_{T} | cc^{σ} = 1} is the unitary group of the center of A_{T}.

By the same arguments as in subsection 3.2, we may assume that our functor is given on the category of A-algebras.

Now the problem is to check the axioms C, TE, TA, TB and E.

C, E. These axioms can be proved following exactly to the corresponding proofs in subsection 3.2.

TE. The main difficulty is to show the existence of the transfer map for some finitely generated projective extension T/R. Indeed, we would like to see the norm homomorphism in the role of the transfer map again but there is no way to show the inclusion

N_{C}^{C_{T}} (Nrd_{C_{T}} (U(A_{T}))) ⊂ Nrd_{C}(U(A))

by using arguments with K_{1} (there is no well-defined norm map for the unitary K_{1}). The following important theorem gives us another possibility to do this:

Theorem (Norm Principle for the Unitary group). Let T be a semilocal ring with infinite residue fields of characteristic different from 2. Let (A_{T} , σ) be an Azumaya algebra with unitary involution σ over T. Let C_{T} be the center of A_{T}, so C_{T} /T is an etale quadratic extension with the restricted involution σ. Then the following equality holds:

Nrd_{C_{T}} (U(A_{T})) = Nrd_{C_{T}} (A_{T}^{∗})^{1−σ},

where c^{1−σ} = c(c^{σ})^{−1}, for any c ∈ C_{T}^{∗}.

Proof. See section 4 below.

Since the norm commutes with the involution we get that

N_{C}^{C_{T}} (Nrd_{C_{T}} (U(A_{T}))) = N_{C}^{C_{T}} (Nrd_{C_{T}} (A_{T}^{∗})^{1−σ}) =
N_{C}^{C_{T}} (Nrd_{C_{T}} (A_{T}^{∗}))^{1−σ} ⊂ Nrd_{C}(A^{∗})^{1−σ} = Nrd_{C}(U(A)),

where the inclusion follows from the Lemma 3 applied to the Azumaya algebra A over the semilocal ring C and extension C_{T} /C.

Hence, we can take the norm homomorphism as the transfer map and we get TE. □

TA. Since we have the identity 𝔉_{R}(T_{1} × T_{2}) = 𝔉_{R}(T_{1}) × 𝔉_{R}(T_{2}), additivity axiom TA holds. □

TB. To prove TB we consider the proof of this axiom in subsection 3.2 but now the diagram (∗) is induced by the functor R_{C/R}^{1}(G_{m}) : T ↦ U(C_{T}) coming from the short exact sequence of group schemes:

1 ⟶ R_{C/R}^{1}(G_{m}) ⟶ R_{C/R}(G_{m}) ⟶^{N_{R}^{C}} G_{m} ⟶ 1,

where R_{C/R} denotes Weil restriction.

We know that the functor R_{C/R}(G_{m}) : T ↦ C_{T}^{∗} satisfies TB and the involution is compatible with the norm map, hence, the kernel R_{C/R}^{1}(G_{m}) satisfies axiom TB as well. □

Summarizing our discussion we have proved that the functors 𝔉 and 𝔉_{R} satisfy axioms C, TE, TA, TB and E.

Applying now our theorem (Non-Constant Case) to the functors 𝔉 and 𝔉_{R} we get the statement (a) of Unitary Case:

Theorem. (a) The map U(C)/Nrd(U(A)) ⟶ U(C_{K})/Nrd(U(A_{K})) induced by the canonical inclusion is injective.

To show the statement (b) let look on the short exact sequence of smooth group schemes:

1 ⟶ SU_{1,A} ⟶ U_{1,A} ⟶^{Nrd} R_{C/R}^{1}(G_{m}) ⟶ 1,

where the U_{1,A} : T ↦ U(A_{T}) for any R-algebra T.

It induces the long exact cohomology sequence:

1 ⟶ SU(A) ⟶ U(A) ⟶^{Nrd} U(C) ⟶ H1_{ét}(R, SU_{1,A}) ⟶ H1_{ét}(R, U_{1,A}) ⟶ ⋯

Observe that the set H1_{ét}(R, U_{1,A}) is not trivial in general.

The canonical inclusion R ↪ K induces the natural map on our long cohomology sequence, so that the diagram

1 ⟶ SU(A) ⟶ U(A) ⟶^{Nrd} U(C) ⟶ H1_{ét}(R, SU_{1,A}) ⟶ H1_{ét}(R, U_{1,A})
↓ ↓ ↓ ↓ ↓
1 ⟶ SU(A_{K}) ⟶ U(A_{K}) ⟶_{Nrd_{K}} U(C_{K}) ⟶ H1_{ét}(K, SU_{1,A_{K}}) ⟶ H1_{ét}(K, U_{1,A_{K}})

commutes.

Taking the cokernels from the left side we get:

1 ⟶ U(C)/Nrd(U(A)) ⟶ H1_{ét}(R, SU_{1,A}) ⟶ H1_{ét}(R, U_{1,A})
↓  ↓  ↓
1 ⟶ U(C_{K})/Nrd(U(A_{K})) ⟶ H1_{ét}(K, SU_{1,A_{K}}) ⟶ H1_{ét}(K, U_{1,A_{K}})

Since the left vertical arrow is injective by (a) and the right vertical arrow has the trivial kernel by the main result of [PO2], we get the triviality of the kernel of the middle vertical arrow, i.e. we get:

Theorem. (b) The kernel of the map H1_{ét}(R, SU_{1,A}) ⟶ H1_{ét}(K, SU_{1,A_{K}}) induced by the canonical inclusion is trivial.

This completes the proof of Unitary Case. □

4. Norm Principle for the Unitary group

In this section we will prove the following theorem (see section 3 for the definitions):

Theorem (Norm Principle for the Unitary group). Let R be a semilocal ring with infinite residue fields of characteristic different from 2. Let (A, σ) be an Azumaya algebra with unitary involution σ over R. Let C be a center of A, so C is the etale quadratic R-algebra with the standard involution σ. Then the following equality holds:

Nrd_{C}(U(A)) = Nrd_{C}(A^{∗})^{1−σ},

where c^{1−σ} = c(c^{σ})^{−1}, for any c ∈ C^{∗}.

The constant case, i.e., when R is a field, was proved by A. Merkurjev in [Me] (the elementary proof of this result can be found in [BP]).

To simplify the proof we will assume that R is a local ring with maximal ideal 𝔪 and with infinite residue field k. Indeed, one can easily extend all arguments below to the semilocal case.

The proof consists of three steps:

First, we are going to prove the inclusion U(A) ⊂ A^{∗}^{1−σ} (see subsection 4.1). Thus, taking reduced norms, we will get Nrd(U(A)) ⊂ Nrd(A^{∗}^{1−σ}).

Afterwards, we will show the inclusion Nrd(V^{1−σ}) ⊂ Nrd(U(A)) (see subsection 4.2), where V is an open subset in A^{∗}, in some specially chosen topology.

For this we will introduce some kind of Zariski topology on the free R-module A. Then by using tricky arguments with the intersection of two free submodules of A, we get that almost all invertible elements have their reduced norms in the Nrd(U(A)). The precise description of this procedure is a bit technical and is contained in subsection 4.3.

We will finish with ‘approximation’ lemma which gives us the way to prove the inclusion Nrd(A^{∗}^{1−σ}) ⊂ Nrd(U(A)).

4.1. The Proof of the Inclusion U(A) ⊂ A^{∗}^{1−σ}

Let make some remarks and definitions on the structure of considered rings and unitary groups.

Consider the quotients C̄ = C/𝔪C and R̄ = k. By definition C̄ is the etale quadratic algebra over k with the standard involution σ. Therefore, C̄ is the quadratic field extension k(\sqrt{s}), s ∈ k^{∗}, of k or it is the product of two fields k × k.

Recall that U(C) = {c ∈ C | N(c) = cc^{σ} = 1}, where N : C^{∗} → R^{∗} is the norm map. And similar let U(C̄) = {c̄ ∈ C̄ | N̄(c̄) = c̄c̄^{σ} = 1}, where N̄ : C̄^{∗} → k^{∗} is the norm map for the etale quadratic k-algebra C̄. The canonical projection C ⟶ C̄ induces a map U(C) → U(C̄) and we denote by \overline{U(C)} its image. Observe that \overline{U(C)} = U(C̄). In fact by Hilbert 90 for local rings [Se] one has \overline{U(C)} = \overline{C^{∗}^{1−σ}} = C̄^{∗}^{1−σ} = U(C̄).

Now we are going to prove the inclusion U(A) ⊂ A^{∗}^{1−σ}: The main idea here is the same as in the proof of Hilbert 90.

Let a ∈ U(A). Write b_{c} = c + c^{σ}a, for every c ∈ C^{∗}. If one of these b_{c} is invertible, then ab_{c}^{σ} = a(c^{σ} + ca^{σ}) = b_{c}, hence, a = b_{c}(b_{c}^{σ})^{−1} ∈ A^{∗}^{1−σ}.

Denote l = c(c^{σ})^{−1} ∈ U(C), then b_{c} = c^{σ}(l + a). Thus, our aim is to show that for every a ∈ U(A) there exists l ∈ U(C) such that (l + a) ∈ A^{∗}.

The condition (l + a) ∈ A^{∗} is equivalent to the condition Nrd(l + a) = chr_{a}(−l) ∈ C^{∗}, where chr_{a}(t) ∈ C[t] is the reduced characteristic polynomial of a.

Since 𝔪C is the radical ideal of C, the element chr_{a}(−l) is invertible in C if and only if its reduction \overline{chr_{a}(−l)} modulo 𝔪C is invertible in C̄.

Thus, our aim is to prove:

Lemma 4. For every a ∈ U(A) there exists l̄ ∈ U(C̄) such that \overline{chr_{a}}(−l̄) ∈ C̄^{∗}, where \overline{chr_{a}}(t) ∈ C̄[t] denotes the reduction modulo 𝔪C of the reduced characteristic polynomial chr_{a}(t).

Proof. Assume that \overline{chr_{a}}(−l̄) is non-invertible for every l̄∈ U(C̄).

In case when C̄ is the field it means that \overline{chr_{a}}(−l̄) = 0 for every l̄ ∈ U(C̄). Hence, every element l̄∈ U(C̄) is the root of the polynomial \overline{chr_{a}}(t). Since the number of roots is finite we get that the unitary group U(C̄) must be finite.

Consider now the case when C̄ = k×k is the product of two fields. Since the involution σ in this case is just the permutation map (x, y) ↦ (y, x) we get that the unitary group U(C̄) consists of the points of the type (x, x^{−1}), x ∈ k^{∗}.

Clearly, C̄[t] = k[t] × k[t]. Therefore, a polynomial f ∈ C̄[t] is in fact just a pair (f′, f′′) of polynomials over k[t]. In particular \overline{chr_{a}}(t) is just a pair of polynomials chr′(t) and chr′′(t) over k.

By assumption the element \overline{chr_{a}}(−l̄) ∈ C̄ is non-invertible for every l̄∈ U(C̄). Thus, either chr′(x) = 0 or chr′′(x^{−1}) = 0 for every l̄= (x, x^{−1}) ∈ U(C̄).

Since the number of solutions of the equation chr′(x)chr′′(x^{−1}) = 0 is finite, the group U(C̄) is finite as well.

On the other hand we have the isomorphism U(C̄) ⟶ k^{∗}, (x, x^{−1}) ↦ x, and the group k^{∗} is infinite. Thus, we get contradiction. This completes the proof of Lemma 4 and the Inclusion. □

4.2. The Proof of the Equality Nrd(A^{∗}^{1−σ}) = Nrd(U(A))

Topology on R^{n}. We introduce topology on R^{n} by lifting the Zariski topology given on the affine space k^{n} via the map R^{n} →^{can} k^{n} induced by the canonical projection. Thus, the subbase of this topology consists of the preimages can^{−1}(V_{f}) of the main open subsets V_{f} = {x ∈ k^{n} | f(x) ≠ 0}, where f ∈ k[t_{1}, t_{2}, …, t_{n}].

Two important and obvious properties of this topology are: i) every finite system of open subsets has nonempty intersection; and ii) any polynomial map g : R^{l} → R^{m} is continuous.

For instance, to see ii) it is enough to look at the commutative diagram:

R^{l} ⟶^{g} R^{m}
can↓  ↓^{can}
k^{l} ⟶_{ḡ} k^{m}

where ḡ denotes the quotient of the polynomial map g. Since the vertical arrows and the down arrow are continuous, the upper arrow is continuous as well.

In the same way we get another important examples of continuous maps:

1. Let A be an R-algebra and a free R-module of rank n. Then the regular representation R^{n} = A →^{rpr} End_{R}A = M_{n}(R) = R^{n^{2}} is continuous.

2. Multiplication map M_{n}(R) × R^{n} →^{m} R^{n}, m : (M, x) ↦ Mx, is continuous.

3. Let GL_{n}(R) be the group of invertible R-matrices. Then GL_{n}(R) is open in M_{n}(R) and the inverse map GL_{n}(R) →^{inv} GL_{n}(R), inv : M ↦ M^{−1}, is continuous in the induced topology on GL_{n}(R).

The Proof of the Inclusion of Nrd(A^{∗}^{1−σ}) ⊂ Nrd(U(A)). Consider the Azumaya algebra A as the free R-module of rank 2m with the topology constructed above. Further we will always identify A with R^{2m}.

Define two free submodules of A of rank m:

A_{+} = {x ∈ A | x = x^{σ}} and A_{−} = {x ∈ A | x = −x^{σ}}.

It is easy to see that A is the direct sum of the R-modules A_{−} and A_{+}. Moreover, we have A_{−} = A_{+} \sqrt{b} and A_{+} = A_{−} \frac{\sqrt{b}}{b}.

Consider now the intersection aA_{−} ∩(R ·1_{A} ⊕A_{−}), for some chosen a ∈ A^{∗}. We claim that (we will prove this in the subsection 4.3) for almost all a there exists an invertible element in this intersection.

The last means that there exists a non-empty open V ⊂ A^{∗} such that for every a ∈ V there exist r ∈ R, u ∈ A_{−} and invertible element v ∈ A_{−} with the property r + u = av.

Take a ∈ V. By the property above we may write a = (r + u)v^{−1} and

Nrd(a^{1−σ}) = Nrd(−(r + u)v^{−1}(r − u)^{−1}v) = Nrd(−1)Nrd(\frac{r+u}{r−u}),

but the elements \frac{r+u}{r−u} and −1 lie in the unitary group U(A), thus, we get that

Nrd(a)^{1−σ} ∈ Nrd(U(A)).

The following ‘approximation’ lemma finishes our proof:

Lemma 5. Let V ⊂ A^{∗} is the open subset of A^{∗}, then for every a ∈ A^{∗} there exist v_{1}, v_{2} ∈ V such that a = v_{1}v_{2}.

Proof. By (3) the subset V^{−1} is open in A^{∗}, thus, aV^{−1} is open in A^{∗}. Since the intersection V ∩ aV^{−1} is nonempty, there are v_{1}, v_{2} ∈ V, such that v_{1} = av_{2}^{−1} and we have proved the lemma. □

Now let a ∈ A^{∗}, a = v_{1}v_{2}, where v_{1}, v_{2} ∈ V and Nrd(V)^{1−σ} ⊂ Nrd(U(A)), then

Nrd(a^{1−σ}) = Nrd(v_{1})^{1−σ}Nrd(v_{2})^{1−σ} ∈ Nrd(U(A)).

This completes the proof of Equality.

4.3. The Proof of Existence of Invertible Element We are fixing the basis of A over R:

A = A_{−} ⊕ A_{+} = \{\sqrt{b} · 1_{A}, \sqrt{b}e_{2}, \sqrt{b}e_{3}, …, \sqrt{b}e_{m}\} ⊕ \{1_{A}, e_{2}, e_{3}, …, e_{m}\}.

The element \sqrt{b} ∈ A_{−} in this basis is represented by the matrix ( \begin{matrix}0_{m} & E_{m} \\ bE_{m} & 0_{m} \end{matrix} ) .

Consider two continuous maps M_{2m}(R) →^{pr_{1}} M_{m−1}(R) and M_{2m}(R) →^{pr_{2}} M_{m−1}(R), where pr_{1} and pr_{2} are the projections:

M = (m_{ij})_{i=1,... ,2m}^{j=1,... ,2m} ⟶^{pr_{1}} (m_{ij})_{i=m+2,... ,2m}^{j=2,... ,m} = N,

M = (m_{ij})_{i=1,... ,2m}^{j=1,... ,2m} ⟶^{pr_{2}} (m_{ij})_{i=m+2,... ,2m}^{j=1} = c̄.

In other words, pr_{1} sends any matrix M to its left-down corner (m − 1) × m without the first column denoted by c̄, i.e., to the (m − 1) × (m − 1) matrix N. And pr_{2} sends the matrix M to the column c̄.

Consider now continuous map ψ : A →^{rpr} M_{2m}(R) →^{pr_{1}} M_{m−1}(R), where rpr is the representation of A as the R-module in our fixed basis.

Let V_{1} be the preimage of the open subset GL_{m−1}(R) ⊂ M_{m−1}(R) under the map ψ. The intersection V_{2} = V_{1} ∩ A^{∗} is the open subset in A^{∗} (the subset A^{∗} is open in A). Define the map ω as the composition:

ω : V_{2} ⟶^{(ρ,−pr_{2})} GL_{m−1}(R) × R^{m−1} ⟶^{m} R^{m−1} ⟶^{i} R^{2m} = A,

where the map ρ is the composition ρ : a ↦^{ψ} N ↦^{inv} N^{−1}, m : (N, c̄) ↦ Nc̄ = d̄ is the multiplication and i : d̄↦ (1, d̄, 0, …, 0)^{t} = v is the inclusion.

Clearly, ω is continuous, so the preimage V = ω^{−1}(A^{∗}) of the open subset A^{∗} ⊂ A is open, too. Since \sqrt{b} ∈ V, the subset V is not empty.

We claim that for any a ∈ V the product av, where v = ω(a), is the required invertible element. Indeed, by the very construction, v ∈ A_{−} and av is invertible. Consider the product av in our fixed basis:

rpr(a)ω(a) = ( \begin{matrix}a_{1,1} & ... & a_{1,m} & ... \\ ⋮ & & ⋮ \\ a_{m+1,1} & ... & a_{m+1,m} & ... \\ c̄ & & N & ... \end{matrix} ) ( \begin{matrix}1 \\ N^{−1}(−c̄) \\ 0 \end{matrix} ) = ( u_{1} ; ⋮ ; u_{m+1} ; c̄+NN^{−1}(−c̄) ) =

= ( u_{1} ; ⋮ ; u_{m+1} ; 0 ) ∈ A_{−} ⊕ R · 1_{A}, and we have finished.

5. The Case of a Local Regular Ring containing a Field

In this section we generalize all theorems of section 3 to the case of a local regular ring containing a field. Let R be a local regular ring containing a field k not necessary infinite and let K be its field of fractions.

Theorem I. Let A be an Azumaya algebra over a local regular ring R containing a field not necessary infinite. Then

(a) the homomorphism R^{∗}/Nrd(A^{∗}) ⟶ K^{∗}/Nrd(A_{K}^{∗}) is injective;

(b) the canonical map H1_{ét}(R, SL_{1,A}) ⟶ H1_{ét}(K, SL_{1,A_{K}}) on the first cohomology groups induced by the canonical inclusion is injective.

Theorem II. Let (A, σ) be an Azumaya algebra with unitary involution (see subsection 3.1) over a local regular ring R containing an infinite field of characteristic ≠ 2. Then

(a) the homomorphism U(C)/Nrd(U(A)) ⟶ U(C_{K})/Nrd(U(A_{K})) is injective;

(b) the kernel of the canonical map H1_{ét}(R, SU_{1,A}) ⟶ H1_{ét}(K, SU_{1,A_{K}}) is trivial.

Theorem III. Let A be an Azumaya algebra over a local regular ring R containing a field not necessary infinite and d be some natural number. Then the homomorphism

R^{∗}/Nrd(A^{∗})(R^{∗})^{d} ⟶ K^{∗}/Nrd(A_{K}^{∗})(K^{∗})^{d}

is injective.

Theorem IV. Let (A, σ) be an Azumaya algebra with unitary involution over a local regular ring R containing an infinite field of characteristic ≠ 2 and d be some natural number. Then the homomorphism

U(C)/Nrd(U(A))U(C)^{d} ⟶ U(C_{K})/Nrd(U(A_{K}))U(C_{K})^{d}

is injective.

We consider in details only the proof of Theorem I. Other theorems are proved similarly.

Let A be an Azumaya algebra over R. Let 𝔉 be the functor (subsection 3.2) defined on the category of R-algebras as:

𝔉 : T ↦ T^{∗}/Nrd(A_{T}^{∗}).

Our aim is to show

Theorem I. (a) The map 𝔉(R) → 𝔉(K) induced by the canonical inclusion is injective.

Proof. First, we consider the case when k is infinite field (compare with Proof of Theorem B, section 8, [PO1]). Then we prove the theorem for any finite field k (compare with section 4, [PO1]).

1. Assume k is infinite. By Popescu’s theorem ([PO1], section 7) R is a filtered direct limit of essentially smooth local k-algebras, i.e, R = \underrightarrow{lim} R_{i}, where R_{i} is a local regular ring of an affine smooth variety over k.

Since an Azumaya R-algebra A is given by the finite number of generators and relations we can find the index j such that the algebra A come from some Azumaya algebra A_{j} over R_{j}, i.e. A = A_{j} ⊗_{R_{j}} R.

Indeed, let all relations between generators e_{1}, e_{2}, …, e_{n} of A over R come from R_{j′} for some index j′. Thus, we get some R_{j′}-algebra A_{j′} with the same generators and relations, i.e. A = A_{j′} ⊗_{R_{j′}} R. Then we can find an index j ≥ j′ such that the determinant det(Tr(e_{k}e_{l})_{k,l}) is invertible in R_{j}, where Tr : A_{j} → R_{j} is the trace map. Hence, the R_{j}-algebra A_{j} becomes an Azumaya algebra over R_{j}.

Fix the index j. Thus, we may assume that the functor 𝔉 is given on the category of R_{j}-algebras. We may replace the filtered direct system of the R_{i} by the subsystem of all R_{i} with i ≥ j. Clearly we have R = \underrightarrow{lim}_{i≥j} R_{i}.

Let α ∈ 𝔉(R) be such that its image α_{K} in 𝔉(K) is trivial. Clearly, for a suitable f ∈ R the image α_{f} in 𝔉(R_{f}) of the element α is trivial.

For a suitable index k ≥ j choose a lift f_{k} of f in R_{k}. Replacing the filtered direct system of the R_{i}, i ≥ j, by the subsystem of all R_{i} with i ≥ k we still have R = \underrightarrow{lim}_{i≥k} R_{i}. We put , for every i ≥ k, f_{i} = ϕ_{ik}(f_{k}) where the ϕ_{ik} : R_{k} → R_{i} are the transition homomorphisms. It is easy to see that \underrightarrow{lim}_{i≥k}(R_{i})_{f_{i}} = R_{f}.

By the very definition the functor 𝔉 commutes with filtered direct limits (compare with the proof of axiom C in subsection 3.2), i.e. the canonical maps \underrightarrow{lim}_{i≥k} 𝔉(R_{i}) → 𝔉(R) and \underrightarrow{lim}_{i≥k} 𝔉((R_{i})_{f_{i}}) → 𝔉(R_{f}) are isomorphisms. Thus, we have

\underrightarrow{lim}_{i≥k} ker[𝔉(R_{i}) → 𝔉((R_{i})_{f_{i}})] = ker[𝔉(R) → 𝔉(R_{f})].

By Theorem (a) of Linear Case (section 3) for any i the map 𝔉(R_{i}) → 𝔉(K_{i}) induced by the canonical inclusion of local ring R_{i} to its field of fractions K_{i} is injective. Since the inclusion R_{i} ↪ K_{i}, i ≥ k, is factorized through (R_{i})_{f_{i}} the map 𝔉(R_{i}) → 𝔉((R_{i})_{f_{i}}) is injective as well.

Thus, we get that the left side of the relation above is trivial. By the very assumption the element α ∈ 𝔉(R) lies on the right side, therefore, α = 0.

Hence, the map 𝔉(R) → 𝔉(K) is injective.

2. Assume that k is finite field of characteristic p. Suppose that the injectivity above holds for essentially smooth local algebras over any infinite field.

Let p^{m} be the cardinality of the residue field R/𝔪 and s be any integer greater than 1 and prime with m. For any natural i let k_{i} be the field of degree s^{i} over k (in some algebraic closure of k).

Since degrees of k_{i} and R/𝔪 over k are coprime, k_{i} ⊗_{k} (R/𝔪) is a field for any i. Thus, R_{i} = k_{i} ⊗_{k} R is essentially smooth local k_{i}-algebra for any i. Let K_{i} be its field of fractions.

Let k̃ be the union of all k_{i}. Clearly, R̃ = k̃ ⊗_{k} R is essentially smooth local algebra over the infinite field k̃. Denote by K̃ its field of fractions.

Consider commutative diagram:

𝔉(R) ⟶ 𝔉(R_{i}) ⟶ 𝔉(R̃)
↓  ↓  ↓
𝔉(K) ⟶ 𝔉(K_{i}) ⟶ 𝔉(K̃)

Let α ∈ 𝔉(R) be such that α_{K} = 0 in 𝔉(K). Denote by α_{i}, α̃ images of α in 𝔉(R_{i}) and 𝔉(R̃), correspondingly. Thus, one has α̃_{K} = 0.

By assumption the map 𝔉(R̃) → 𝔉(K̃) is injective, therefore, α̃ = 0. Since the functor 𝔉 commutes with filtered direct limits, we can find an index j such that α_{i} = 0 for any i ≥ j.

Consider essentially smooth local k_{j}-algebra R_{j}. By definition it is finite and free over R, therefore, there is the transfer map Tr : 𝔉(R_{j}) → 𝔉(R) (for the construction see section 3.2, the proof of TE). Clearly, the composition 𝔉(R) → 𝔉(R_{j}) →^{Tr} 𝔉(R) is the multiplication by s^{i} the degree of k_{i} over k. Thus, one has s^{i}α = 0 for any integer s prime to m and for any integer i. It means that α = 0 in 𝔉(R). And we have finished. □

Theorem I. (b) The map H1_{ét}(R, SL_{1,A}) ⟶ H1_{ét}(K, SL_{1,A_{K}}) induced by the canonical inclusion is injective.

Proof. To prove this statement one has to repeat the corresponding arguments given in the proof of statement (b) of Linear Case (see the end of subsection 3.2).

In particular to construct the commutative diagram

1 ⟶ SL(A) ⟶ A^{∗} ⟶^{Nrd} R^{∗} ⟶ H1_{ét}(R, SL_{1,A}) ⟶ 1
↓ ↓ ↓ ↓
1 ⟶ SL(A_{K}) ⟶ A_{K}^{∗} ⟶_{Nrd_{K}} K^{∗} ⟶ H1_{ét}(K, SL_{1,A_{K}}) ⟶ 1

we don’t use the fact that R is essentially smooth over k. Indeed, it is enough for R to be just a local ring.

Thus, nothing will be changed and we have finished □

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St. Petersburg Branch, Steklov Mathematical Institute, St. Petersburg Department, Russian Academy of Sciences, Fontanka 27, 191011, St. Petersburg, Russia

E-mail address: kirill@pdmi.ras.ru