PURITY FOR SIMILARITY FACTORS

Ivan Panin

June 2003

Abstract. Let R be a regular local ring, K its field of fractions and A_{1}, A_{2} two Azumaya algebras with involutions over R. We show that if A_{1} ⊗_{R} K and A_{1} ⊗_{R} K are isomorphic over K, then A_{1} and A_{2} are isomorphic over R. In particular, if two quadratic spaces over the ring R become similar over K then these two spaces are similar already over R. The results are consequences of a purity theorem for similarity factors.

Introduction

Let R be a regular local ring, K its field of fractions. Let (A_{1}, σ_{1}) and (A_{2}, σ_{2}) be two Azumaya algebras with involutions over R (see right below for a precise definition). Assume that (A_{1}, σ_{1}) ⊗_{R} K and (A_{2}, σ_{2}) ⊗_{R} K are isomorphic. Are (A_{1}, σ_{1}) and (A_{2}, σ_{2}) isomorphic too? We show that this is true if R is a regular local ring containing a field of characteristic different from 2. If A_{1} and A_{2} are both the n × n matrix algebra over R and the involutions are symmetric then σ_{1} and σ_{2} define two quadratic spaces q_{1} and q_{2} over R up to similarity factors. In this particular case the result looks as follows: if q_{1} ⊗_{R} K and q_{1} ⊗_{R} K are similar then q_{1} and q_{2} are similar too.

Grothendieck [G] conjectured that, for any reductive group scheme G over R, rationally trivial G-homogeneous spaces are trivial. Our result corresponds to the case when G is the projective unitary group PU_{A,σ} for an Azumaya algebra with involution over R. If R is an essentially smooth local k-algebra and G is defined over k (we say that G is constant) Grothendieck’s conjecture has been proved in most cases: by Colliot-Thélène and Ojanguren [C-TO] for a perfect infinite field k and then by Raghunathan [R] for any infinite k. One notable open case is that

Typeset by AMS-TEX

The author thanks very much for the support the RTN-Network HPRN-CT-2002-00287 , the grant of the years 2001 - 2003 of the Russian Science Support Foundation at the Russian Academy of Science, the grant INTAS-99-00817 and the University of Lausanne.

of a finite base field. For a non-constant group G only few cases have been proved: when G is a torus, by Colliot-Thélène and Sansuc [C-TS], when G is the group SL_{1}(D) of norm one elements of an Azumaya R-algebra D, by Panin and Suslin [PS], when G is the unitary group U_{A,σ}, by Panin and Ojanguren [Oj-P1], when G is the special unitary group SU_{A,σ}, by Zainoulline [Z]. Recall as well that for semi-simple group schemes G over a discrete valuation ring the conjecture has been proved by Nisnevich in [Ni].

The paper is organized as follows. Section 1 contains a reduction of the main theorem (Th. 1.1) to a purity theorem for similarity factors (Th. 1.3). Section 2 is devoted to a theorem of Nisnevich and its Corollaries. The rest of the text is devoted to the proof of Theorem 1.3. The proof is given in §8. It is based on the Specialization Lemma (stated in §3 and proved in §4), the Equating Lemma (§5) and the Unramifiedness Lemma (§5).

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

§1. Rationally isomorphic Azumaya algebras with involutions are locally isomorphic

Let R be a regular local ring containing a field k (char(k) ≠ 2) and let K be its quotient field. By an R- Azumaya algebra with involution (A, σ) we mean (see [Oj-P1]) an R-algebra A which is an Azumaya algebra over its center Z(A) equipped with an involution σ : A → A^{op}, such that Z(A) is either R itself or an étale quadratic extension of R such that Z(A)^{σ} = R.

1.1. Theorem (Main). Let (A_{1}, σ_{1}) and (A_{2}, σ_{2}) be two Azumaya algebras with involutions over the ring R. If the Azumaya algebras with involutions (A_{1,K}, σ_{1,K}) and (A_{2,K}, σ_{2,K}) are isomorphic, then (A_{1}, σ_{1}) and (A_{2}, σ_{2}) are already isomorphic.

Reduction to a Purity Theorem. Since (A_{1,K}, σ_{1,K}) ≃ (A_{2,K}, σ_{2,K}), one concludes that the two Azumaya algebras A_{1,K} and A_{2,K} over Z_{K} are isomorphic. Thus A_{1} ≃ A_{2} (Z is regular semilocal as an étale quadratic extension of R). Therefore one may assume that A_{1} = A_{2}, Z_{1} = Z_{2} and we have two involutions σ_{1} and σ_{2} on the same algebra (A over Z and Z is a quadratic étale extension of R (or Z just coincides with R).

Now consider the composite A ⟶^{σ_{2}} A^{op} ⟶^{σ_{1}^{−1}} A. It is an Azumaya algebra isomorphism. Thus it is of the form Int(α) for an element α ∈ A^{∗}. Thus σ_{1}∘Int(α) = σ_{2} and α is symmetric with respect to σ_{1}. Therefore we have two hermitian spaces over (A, σ_{1}), namely (A, 1) and (A, α). Set

h_{1} = (A, 1) and h_{2} = (A, α).

Since (A_{K}, σ_{1,K}) is isomorphic to (A_{K}, σ_{2,K}), h_{1,K} is similar to h_{2,K} i.e. there exist an element a ∈ K^{∗} and an isometry a · h_{1,K} ≃ h_{2,K}.

We will prove (this suffices to prove the theorem) that there exists an element b ∈ R^{∗} such that b · h_{1} ≃ h_{2} over (A, σ_{1}). To find the desired element b ∈ R^{∗}, it suffices to find a similarity factor b_{𝔪} ∈ K^{∗} of the space h_{1,K} and a unit a_{𝔪} ∈ R^{∗} such that a = a_{𝔪} · b_{𝔪}. In fact, if b_{𝔪} ∈ K^{∗}, a_{𝔪} ∈ R^{∗} are the mentioned elements then one has a chain of relations (h_{2} ⊥ −a_{𝔪} · h_{1})_{K} ≃ h_{2,K} ⊥ −a_{𝔪} · b_{𝔪} · h_{1,K} = h_{2,K} ⊥ −a · h_{1,K} ≃ h_{2,K} ⊥ −h_{2,K}. Thus (h_{2} ⊥ −a_{𝔪} · h_{1})_{K} is hyperbolic and by the main theorem of [Oj-P1] the space h_{2} ⊥ −a_{𝔪} · h_{1} is hyperbolic, whence h_{2} ≃ a_{𝔪} · h_{1}. Therefore putting b = a_{𝔪} we get h_{2} ≃ b · h_{1} over (A, σ_{1}). It remains to find a similarity factor b_{𝔪} of h_{1,K} and a unit a_{𝔪} ∈ R^{∗} such that a = a_{𝔪} · b_{𝔪}. By the corollary of a theorem of Nisnevich below (§3, Cor. 3.2), for a height one prime ideal 𝔭 in R there exist elements b_{𝔭} ∈ K^{∗} and a_{𝔭} ∈ R^{∗} such that

(1) b_{𝔭} is a similarity factor of the space h_{1,K} and

(2) a = a_{𝔭} · b_{𝔭}.

Thus by the Purity Theorem (Theorem 1.2) there exist a similarity factor b_{𝔪} of h_{1,K} and a unit a_{𝔪} ∈ R^{∗} with a = a_{𝔪} · b_{𝔪}. So we have reduced Theorem 1.1 to the Purity Theorem. □

1.2. Theorem (Purity Theorem). Let R, K be as in Theorem 1.1. Let (A, σ) be an Azumaya algebra with involution over R and let h be the hermitian space (A, 1) over (A, σ). Let a ∈ K^{∗}. Suppose that for each prime ideal of height 1 𝔭 in R there exist a_{𝔭} ∈ R_{𝔭}^{∗}, b_{𝔭} ∈ K^{∗} with a = a_{𝔭} · b_{𝔭} and h_{K} ≃ b_{𝔭} · h_{K}. Then there exist b_{𝔪} ∈ K^{∗}, a_{𝔪} ∈ R^{∗} such that

(1) b_{𝔪} is a similarity factor of the space h_{K},

(2) a = a_{𝔪} · b_{𝔪}.

It is convenient for the proof to restate Theorem 1.2 it in a slightly more technical form. For that consider the similitude group scheme G = Sim_{A,σ} of the Azumaya algebra with involution (A, σ). Recall that for an R-algebra S the S-points of G are those α ∈ (A ⊗_{R} S)^{∗} for which α^{σ} · α ∈ S^{∗}. Further consider a group scheme morphism μ : G → 𝔾_{m} which takes a similitude α ∈ G(S) to its similarity factor μ(α) = α^{σ} · α ∈ S^{∗}. Finally for an R-algebra S consider the group F(S) = S^{∗}/μ(G(S)). For an element a ∈ S^{∗} we will often write ā for its class in F(S).

1.3. Theorem. Let R, K and (A, σ) be as in Theorem 1.2. Let a ∈ K^{∗}. If for each height 1 prime 𝔭 in R the class ā ∈ F(K) can be lifted in F(R_{𝔭}), then ā can be lifted in F(R).

Remark. Theorems 1.2 and 1.3 are equivalent. In fact, the group μ(G(R)) coincides with the group G_{R}(h) of similarity factors of the hermitian space h = (A, 1).

Remark. It is quite plausible that the method of [Z1] could be adapted to prove Theorem 1.3.

§2. A theorem of Nisnevich

Let R be a discrete valuation ring containing a field and let K be its quotient field. Let (A, σ) be an Azumaya algebra with involution over R. The following theorem is a consequence of a theorem of Nisnevich on principal G-bundles. ([Ni], Theorem ??).

2.1. Theorem (Nisnevich). Let h_{1} and h_{2} be two hermitian spaces over (A, σ). Suppose h_{1,K} is similar to h_{2,K}, then h_{1} is similar to h_{2}.

This Theorem is a particular case of the theorem of Nisnevich just mentioned, namely the case when G is the projective unitary group scheme P U_{h_{1}} over R.

2.2. Corollary. Let h_{1}, h_{2} be two hermitian spaces over (A, σ). Let a ∈ K^{∗} be such that h_{2,K} ≃ a · h_{1,K}. Then there exist an element b′ ∈ K^{∗} and a unit a′ ∈ R^{∗} such that

(1) b′ is a similarity factor of the space h_{1,K},

(2) a = a′ · b′.

Proof. By the theorem there exists a unit a′ ∈ R^{∗} such that a′ · h_{2} ≃ h_{1}. Thus one has a chain of relations

a · (a′)^{−1} · h_{1,K} ≃ a · h_{2,K} ≃ a^{2} · h_{1,K} ≃ h_{1,K} .

Therefore b′ = a · (a′)^{−1} is a similarity factor of the space h_{1,K} and a = a′ · b′. □

2.3. Corollary. The kernel of the map H^{1}(R, Sim_{A,σ}) → H^{1}(K, Sim_{A,σ}) is trivial.

Proof. The group scheme Sim_{A,σ} fits in an exact sequence of algebraic groups

0 → R_{Z/R}(𝔾_{m,Z}) → Sim_{A,σ} → PU_{A,σ} → 0

where R_{Z/R}(𝔾_{m,Z}) is the Weil restriction of the multiplicative group 𝔾_{m,Z}. By Hilbert’s Theorem 90 H^{1}(R, R_{Z/R}(𝔾_{m,Z})) = H^{1}(Z, 𝔾_{m,Z})) = 0. Thus the kernel of the map H^{1}(R, Sim_{A,σ}) → H^{1}(R, PU_{A,σ}) is trivial. On the other hand the kernel of the map H^{1}(R, PU_{A,σ}) → H^{1}(K, PU_{A,σ}) is trivial by Theorem 3.1. Thus the kernel of the map H^{1}(R, Sim_{A,σ}) → H^{1}(K, Sim_{A,σ}) is trivial as well, whence the Corollary.

§3. A Specialization Lemma

In this section we state a theorem which is one of the main ingredient in the proof of purity. The theorem itself will be proved in §5 below.

Let k be a field (char(k) ≠ 2) and let (A, σ) be an Azumaya algebra with involution over k (see Section 1 for the definition). Let G = Sim_{A,σ} be the similitude group of (A, σ) (see the end of Section 1 for the definition), and let μ : Sim_{A,σ} ⟶ 𝔾_{m} be a group morphism which takes a similitude α to its similarity factor μ(α) = α^{σ} · α. The group G coincides with the similitude group of the hermitian space (A, 1) = h.

3.1. Notation. For a commutative k-algebra S, set F(S) = S^{∗}/μ(G(S)). For an element u ∈ S^{∗} we shall write in this section \overline{u} for the image of u in F(S). Observe that μ(G(S)) = G_{S}(h⊗_{k} S) is the group of similarity factors of the hermitian space h ⊗_{k} S.

Let S be a k-algebra which is a Dedekind domain and let L be the quotient field of S. Let 𝔭 ⊆ S be a non-zero prime ideal in S and let S_{𝔭} be the corresponding local ring.

3.2. Definition. Let a ∈ L^{∗}. The element \overline{a} ∈ F(L) is said to be unramified at a prime 𝔭 if \overline{a} belongs to the image of the group F(S_{𝔭}) in F(L). In other terms, the element \overline{a} is unramified at 𝔭 if a = a_{𝔭} · b_{𝔭} for certain elements a_{𝔭} ∈ S_{𝔭}^{∗} and b_{𝔭} ∈ μ(G(L)). We denote by F_{un}(S) the subgroup in F(L) consisting of all those elements in F(L) which are unramified at each non-zero prime 𝔭 in S. Elements of F(S) are called S-unramified elements.

Let S ⊇ k[s] be a finite extension of the polynomial ring in one variable. Suppose S is a Dedekind domain. Let S_{1} = S/(s − 1)S and S_{0} = S/J, and let ϵ : S ⟶ k be an augmentation such that S/tS = S/Ker(ϵ) × S/J = k × S/J for certain ideal J in S. For an element v ∈ S we will write v_{1} and v_{0} for its images in S_{1} and S_{0} respectively. If furthermore g ∈ S be an element coprime as to (s − 1) so to (s), then the canonical map S ⟶ S_{i} is factorized as the composite S ⟶ S_{g} ⟶ S_{i}. In this case for an element v ∈ S_{g} we will write v_{1} and v_{0} for its images in S_{1} and S_{0} respectively. We will denote N_{S_{i}/k} : S_{i}^{∗} ⟶ k^{∗} the norm map.

3.3. Theorem (Specialization Lemma). Let S ⊇ k[s] be an integral extension of the polynomial ring in one variable k[s] and suppose S is a Dedekind domain and L its quotient field. Let f ∈ S be an element coprime to s and (s − 1). Let u ∈ S_{f}^{∗} be a unit. Suppose the element \overline{u} ∈ F(L) is S-unramified, i.e. \overline{u} belongs to the subgroup F_{un}(S). Then the following relation holds in the group F(k)

(*)  \overline{ϵ(u)} = \overline{N_{S_{1}/k}(u_{1})} · \overline{N_{S_{0}/k}(u_{0})^{−1}} .

3.4. Remark. This theorem is proved in §4 below. Now observe only that if u ∈ S^{∗}, then N_{S/k[s]}(u) ∈ k[s]^{∗} = k^{∗} and already ϵ(u) = N_{S_{1}/k}(u_{1}) · N_{S_{0}/k}(u_{0})^{−1}. So there is nothing to prove in this case. The trouble is that we do not assume u ∈ S^{∗}.

§4. Proof of Specialization Lemma

Let k be a field of characteristic different of 2 and let (A, σ) be an Azumaya algebra with involution over k (see Section 1 for the definition). Let h be the hermitian space (A, 1). We preserve in this section notation of §3.

Let K be a function field of an irreducible curve over k and let L ⊇ K be a finite field extension (separable). We will consider in this section discrete valuations of K and L which are trivial on k and they will be called valuations. For valuations x : K^{∗} ⟶ ℤ and y : L^{∗} ⟶ ℤ, we write y/x if y extends x. We will need completions to avoid dealing with semi-local Dedekind domains.

4.1. Notation. Let y be a valuation of L. Denote by L̂_{y} the completion of L with respect to y. Denote by O_{y} the ring of integers associated with y, i.e. O_{y} = {a ∈ L | y(a) ⩾ 0}. And denote by Ô_{y} the ring of y-integers in L̂_{y}, i.e. Ô_{y} = {a ∈ L̂_{y} | y(a) ⩾ 0}. We shall write k(y) for the residue field of y, i.e. k(y) = O_{y}/𝔪_{y} = Ô_{y}/𝔪̂_{y}.

If x and y are valuations of K and L respectively and y extends x, then O_{y} ⊇ O_{x} and Ô_{y} ⊇ Ô_{x} and the ring extension Ô_{y} ⊇ Ô_{x} is integral. Thus one has norm mappings N_{O_{y}/Ô_{x}} : Ô_{y}^{∗} ⟶ Ô_{x}^{∗} and N_{L_{y}/K̂_{x}} : L̂_{y}^{∗} ⟶ K̂_{x}^{∗} (we will use below a short notation N_{y/x} for both of these maps). There is the norm map N_{k(y)/k(x)} : k(y)^{∗} ⟶ k(x)^{∗} and two diagrams commute

\begin{matrix}Ô_{y}^{∗} & ⟶ & L̂_{y}^{∗} \\ ^{N_{y/x}}↓ & & ↓^{N_{y/x}} \\ Ô_{x}^{∗} & ⟶ & K̂_{x}^{∗} & , \end{matrix}  \begin{matrix}Ô_{y}^{∗} & ⟶ & k(y)^{∗} \\ ^{N_{y/x}}↓ & & ↓^{N_{k(y)/k(x)}^{i(y/x)}} \\ Ô_{x}^{∗} & ⟶ & k(x)^{∗} & , \end{matrix}

where i(y/x) = the length of O_{y}/𝔪_{x}O_{y} is the ramification index of y over x.

4.2. Remark. Let U_{A,σ} be the unitary group of the form h. It is an algebraic group over k such that for any k-algebra R the group of its R-points is the group {α ∈ (A ⊗_{k} R)^{∗} | α^{σ} · α = 1}. With the notation of §3 the group U_{A,σ} fits in an exact sequence of algebraic groups 1 ⟶ U_{A,σ} ⟶ Sim_{A,σ} ⟶^{μ} 𝔾_{m} ⟶ 1. This sequence of algebraic groups induces exact sequences of pointed sets (R is a domain, K is its quotient field)

1 ⟶ F(R) ⟶^{∂} H_{et}^{1}(R, U_{A,σ}) ⟶ H_{et}^{1}(R, Sim_{A,σ})
↓^{θ_{1}}  ↓^{θ_{2}}  ↓^{θ_{3}}
1 ⟶ F(K) ⟶^{∂} H_{et}^{1}(K, U_{A,σ}) ⟶ H_{et}^{1}(K, Sim_{A,σ}).

In the case of a Dedekind local ring R and its quotient field K the maps θ_{2} and θ_{3} have trivial kernels as well. This holds for θ_{2} by Corollary 2.3 and for θ_{3} by [Oj]. In particular, in this case the map θ_{1} : F(R) ⟶ F(K) has the trivial kernel and thus it is injective. Observe as well that for a field K the map F(K) ⟶^{∂} H_{et}^{1}(K, U_{A,σ}) is injective, i.e. (∂(a) = ∂(b)) ⟹ a = b.

4.3. Notation. Let y be a valuation of L and let i : L ⟶ L̂_{y} be the inclusion. Then by Remark 4.3 the map F(Ô_{y}) ⟶^{i_{∗}} F(L̂_{y}) is injective and we will identify F(Ô_{y}) with its image under this map. Set

F_{y}(L) = i_{∗}^{−1}(F(Ô_{y})).

The inclusions O_{y} ↪ L and O_{y} ↪ Ô_{y} induce a map F(O_{y}) ⟶ F_{y}(L) which is injective by Remark 4.3. Both groups are subgroups of F(L). The following lemma shows that F_{y}(L) coincides with the subgroup of F(L) consisting of all elements unramified at y.

4.4. Lemma. F(O_{y}) = F_{y}(L).

Proof. We only have to check the inclusion F_{y}(L) ⊆ F(O_{y}). Let a_{y} ∈ F_{y}(L) be an element. It determines the elements a ∈ F(L) and â ∈ F(Ô_{y}) which coincide when regarded as elements of F(L̂_{y}). We denote this common element in F(L̂_{y}) by â_{y}. Let ξ = ∂(a) ∈ H_{et}^{1}(L, U_{A,σ}), ξ̂ = ∂(â) ∈ H_{et}^{1}(Ô_{y}, U_{A,σ}) and ξ̂_{y} = ∂(â_{y}) ∈ H_{et}^{1}(L̂_{y}, U_{A,σ}). Clearly, ξ̂ and ξ both coincide with ξ̂_{y} when regarded as elements of H_{et}^{1}(L̂_{y}, U_{A,σ}). Thus one can glue ξ and ξ̂ to get a ξ_{y} ∈ H_{et}^{1}(O_{y}, U_{A,σ}) which maps to ξ under the map induced by the inclusion O_{y} ↪ L and maps to ξ̂ under the map induced by the inclusion O_{y} ↪ Ô_{y}.

We now show that ξ_{y} has the form ∂(a′_{y}) for a certain a′_{y} ∈ F(O_{y}). In fact, observe that the image ζ of ξ in H_{et}^{1}(L, Sim_{A,σ}) is trivia. As mentioned in Remark 4.3 the map H_{et}^{1}(O_{y}, Sim_{A,σ}) ⟶ H_{et}^{1}(L, Sim_{A,σ}) has the trivial kernel. Therefore the image ζ_{y} of ξ_{y} in H_{et}^{1}(O_{y}, Sim_{A,σ}) is trivial as well. Thus there exists an element a′_{y} ∈ F(O_{y}) with ∂(a′_{y}) = ξ_{y} ∈ H_{et}^{1}(O_{y}, U).

We now prove that a′_{y} coincides with a_{y} in F_{y}(L). Since F(O_{y}) and F_{y}(L) are both subgroups of F(L), it suffices to show that a′_{y} coincides with the element a in

F(L). By Remark 4.3 the map F(L) ⟶^{∂} H_{et}^{1}(L, U_{A,σ}) is \underline{injective}. Thus it suffices to check that ∂(a′_{y}) = ∂(a) in H_{et}^{1}(L, U_{A,σ}). This is indeed the case because ∂(a′_{y}) = ξ_{y} and ∂(a) = ξ, and ξ_{y} coincides with ξ when regarded over L. We have proved that a′_{y} ∈ F(O_{y}) coincides with a_{y} in F_{y}(L). Thus the inclusion F_{y}(L) ⊆ F(O_{y}) is proved, whence the lemma. □

4.5. Definition. Let y be a valuation of L. Define a specialization map

s(y) : F_{y}(L) ⟶ F(k(y))

as the composite F_{y}(L) ⟶ F(Ô_{y}) ⟶^{res_{y}} F(k(y)) of the map F_{y}(L) ⟶ F(Ô_{y}) induced by the map i_{∗} (see 4.4) and the map F(Ô_{y}) ⟶ F(k(y)) induced by the residue map Ô_{y} ⟶ k(y). (If we identify F_{y}(L) with F(O_{y}) by Lemma 4.5, then the map s(y) : F(O_{y}) ⟶ F(k(y)) coincides with the map induced by the map

O_{y} ⟶^{res_{y}} k(y)).

4.6. Lemma-Definition. Let K be a field containing the field k and let K ⊆ L be a finite field extension. Then the norm map N_{L/K} : L^{∗} ⟶ K^{∗} takes the group G_{L}(h) into the group G_{K}(h). Therefore the norm map N_{L/K} induces a map which we still denote by

N_{L/K} : F(L) ⟶ F(K).

Proof. The Scharlau norm principle [KMRT, loc. cit.] states that there is a natural inclusion N_{L/K}(G_{L}(h)) ⊆ G_{K}(h), whence the lemma. □

4.7. Lemma. Let x be a valuation of K and let y be a valuation of L extending x. Then the map N_{L̂_{y}/K̂_{x}} : F(L̂_{y}) ⟶ F(K̂_{x}) takes F(Ô_{y}) into F(Ô_{x}).

Proof. The desired inclusion follows from the commutativity of the diagram

\xymatrix{Ô_{y}^{∗} \ar[ddd]_{N_{y/x}} \ar@{->>}[dr] \ar[rrr] &  &  & L̂_{y}^{∗} \ar[dl] \ar[ddd]^{N_{y/x}} \\  & F(Ô_{y}) \ar@{^{(}->}[r] & F(L̂_{y}) \ar[d]^{N} \\  & F(Ô_{x}) \ar@{^{(}->}[r] & F(K̂_{x}) \\ Ô_{x}^{∗} \ar[rrr] \ar[ur] &  &  & K̂_{x}^{∗} , \ar[ul]}

the surjectivity of the map Ô_{y}^{∗} ⟶ F(Ô_{y}) and the injectivity of the map F(Ô_{x}) ⟶ F(K̂_{x}) (see Remark 4.3). □ 4.8. Notation. The map F(Ô_{y}) ⟶ F(Ô_{x}) will be still denoted by N_{y/x}. 4.9. Notation. Let x be a valuation of K. Set F_{x}(L) = ⋂_{y/x}F_{y}(L).

4.10. Lemma. Let x be a valuation of K. Then N_{L/K}(F_{x}(L)) ⊆ F_{x}(K). Proof. The desired inclusion follows from Lemma 4.8 and the commutativity of the diagram

\xymatrix{∏_{y/x}F(Ô_{y}) \ar[ddd]_{ΠN_{y/x}} \ar[rrr] &  &  & ∏_{y/x}F(L̂_{y}) \ar[ddd]^{ΠN_{y/x}} \\  & F_{x}(L) \ar@{^{(}->}[r] \ar@{-->}[ul] & F(L) \ar[d]^{N_{L/K}} \ar[ur] \\  & F_{x}(K) \ar@{^{(}->}[r] \ar[dl] & F(K) \ar@{-->}[dr] \\ F(Ô_{x}) \ar[rrr] &  &  & F(K̂_{x}) ,}

and the definition of F_{x}(K) (see 4.4). □ 4.11. Lemma. Let x be a valuation of K. Then the diagram commutes.

F_{x}(L) ⟶^{Πs_{y}} ∏_{y/x}F(k(y))
N_{L/K} ↓  ↓^{ΠN_{k(y)/k(x)}^{i(y/x)}}
F_{x}(K) ⟶^{s_{x}} F(k(x)),

where N_{k(y)/k(x)} : F(k(y)) ⟶ F(k(x)) is the norm map for the field extension k(y)/k(x) and N_{k(y)/k(x)}^{i(y/x)} is its i(y/x)-th power, where i(y/x) is the ramification index of y over x, i.e. i(y/x) = the length of O_{y}/𝔐_{x}O_{y}. Proof. Consider the diagram

F_{x}(L) ⟶ ∏_{y/x}F(Ô_{y}) \underrightarrow{Πres_{y}} ∏_{y/x}F(k(y))
N_{L/K} ↓  ↓^{ΠN_{y/x}}  ↓^{ΠN_{k(y)/k(x)}^{i(y/x)}}
F_{x}(K) ⟶^{i_{∗}} F(Ô_{x}) ⟶^{res_{x}} F(k(x))

and observe that the left square commutes. It remains to check that the right hand square commutes. To do this it clearly suffices to check the commutativity of

F(Ô_{y}) ⟶^{res_{y}} F(k(y))
N_{y/x}↓  ↓^{N_{k(y)/k(x)}^{i(y/x)}}
F(Ô_{y}) ⟶^{res_{x}} F(k(x)).

To see this we include it in a bigger one:

[figure: Ô_{y}^{∗}; ^{res^{y}}; N_{y/x}; II; ρ; I; k(y)^{∗}; N_{k(y)/k(x)}^{i(y/x)}; F(Ô_{y}); _{res_{y}}; N_{y/x}; V; F(k(y)); N_{k(y)/k(x)}^{i(y/x)} IV; F(Ô_{y}); _{res_{x}}; III; F(k(x)); Ô_{x}^{∗}; _{res_{x}}; k(x)^{∗}.]

The large square in this diagram commutes and squares I to IV commute as well and the map ρ is surjective. Thus square V commutes as well and the lemma is proved. □

4.12. Proposition. Let K = k(t) be the rational function field in one variable and F_{un}(k(t)) = ⋂_{x∈𝔸_{k}^{1}}F_{x}(k(t)). Then the canonical map

F(k) ⟶ F_{un}(k(t))

is an isomorphism.

Proof. Injectivity is clear, because the composite F(k) ⟶ F_{un}(k(t)) ⟶^{s_{0}} F(k) coincides with the identity (here s_{0} is the specialization map at the point zero defined in 4.6).

It remains to check the surjectivity. Let a ∈ F_{un}(k(t)). Then by Lemma 4.5 the element ∂(a) ∈ H_{et}^{1}(k(t), U_{A,σ}) is a class which for every x ∈ 𝔸_{k}^{1} belongs to the image of H_{et}^{1}(O_{x}, U_{A,σ}). Thus by a lemma of Harder [H], ∂(a) can be represented by an element ξ ∈ H_{et}^{1}(k[t], U_{A,σ}), where k[t] is the polynomial ring. By Harder’s theorem [H], the map H_{et}^{1}(k, U_{A,σ}) ⟶ H_{et}^{1}(k[t], U_{A,σ}) is an isomorphism. Then ξ = ρ(ξ_{0}) for an element ξ_{0} ∈ H_{et}^{1}(k, U_{A,σ}). Consider the diagram

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

where all the mapping are canonical and all the vertical arrows have trivial kernels. Since ξ goes to the trivial element in H_{et}^{1}(k(t), Sim_{A,σ}), one concludes that ξ_{0} goes to the trivial element in H_{et}^{1}(k, Sim_{A,σ}). Thus there exists an element a_{0} ∈ F(k) such that ∂(a_{0}) = ξ_{0}. Clearly, one has ϵ(a_{0}) = a (use the injectivity of the map F(k(t)) ⟶ H_{et}^{1}(k(t), U_{A,σ}) mentioned in Remark 4.3). □

4.13. Theorem. Let L ⊇ K = k(t) be a finite separable field extension and let F_{un}(L) = ⋂_{y/x, x∈𝔸_{k}^{1}}F_{y}(L). Then for an element a ∈ F_{un}(L) the following relation

holds:

(*)  Π_{y/0}N_{k(y)/k}(s_{y}(a)^{i(y/0)}) = Π_{y/1}N_{k(y)/k}(s_{y}(a)^{i(y/1)}).

Proof. By lemma 4.11, the element N_{L/K}(a) is in the group F_{un}(K). Now by Lemma 4.12, the left hand side of the relation (*) coincides with s_{0}(N_{L/K}(a)), where s_{0} : F_{un}(K) ⟶ F(k) is the specialization map (see Definition 4.6) at the point zero. The right hand side of (*) coincides with s_{1}(N_{L/K}(a)), where s_{1} is the specialization map at 1. By Proposition 4.13, there exists an element a_{0} ∈ F(k) whose image in F(k(t)) is equal to N_{L/K}(a) ∈ F(k(t)). Thus

s_{0}(N_{L/K}(a)) = s_{0}(a_{0}) = a_{0} = s_{1}(a_{0}) = s_{1}(N_{L/K}(a)).

The theorem is proved. □

4.14. Corollary. The Specialization Lemma (Theorem 3.3) holds.

Proof. We use notation of §3. Let S ⊇ k[s] be the integral extension of the polynomial ring in one variable and suppose (as in the hypothesis of the Specialization Lemma) that S is an integral Dedekind domain. Let L be the quotient field of S, K = k(s), and u ∈ S_{f}^{∗} for the element f from the hypotheses of the Specialization Lemma.

The element \overline{u} ∈ F(L) is S-unramified, i.e. \overline{u} ∈ F_{un}(S). Thus \overline{u} ∈ F_{un}(L). Theorem 5.14 shows that the relation

(**)  Π_{y/1}N_{k(y)/k}(s_{y}(\overline{u})^{i(y/1)}) = Π_{y/0}N_{k(y)/k}(s_{y}(\overline{u})^{i(y/0)})

holds in F(k). It remains to check that the left hand side of the relation (**) coincides with the element \overline{N_{S_{1}/k}(u_{1})} in F(k) and the right hand side of the relation (**) coincides with the element \overline{N_{S_{0}/k}(u_{0})} · \overline{ϵ(u)} in F(k).

Let S_{1,y} be the localization at y of the Artinian ring S_{1} = S/(s − 1)S. Clearly, the diagram

S_{1,y} ⟶^{p_{y}} k(y)
↑  ↑
S_{y} ⟶ Ŝ_{y},

(where all the mappings are the canonical ones) commutes. For an element v ∈ S_{1} let v_{y} be its image in S_{1,y}. Now Lemma 4.5 and Definition 4.6 show that the element p_{y}((u_{1})_{y}) coincides with the element s_{y}(u) in F(k(y)). Observe as well that S_{1} = ∏_{y/1}S_{1,y} and that the diagrams

\begin{matrix}S_{1}^{∗} & ⟶^{∼} & ∏_{y/1}S_{1,y}^{∗} \\ ^{N_{S_{1}/k}}↓ & & ↓^{ΠN_{S_{1,y} /k}} \\ k^{∗} & ⟶^{id} & k^{∗}, \end{matrix}  \begin{matrix}S_{1,y}^{∗} & ⟶^{p_{y}} & k(y)^{∗} \\ ^{N_{S_{1,y} /k}}↓ & & ↓^{N_{k(y)/k}^{i(y/1)}} \\ k^{∗} & ⟶^{id} & k^{∗}. \end{matrix}

commute. This proves the relation Π_{y/1}N_{k(y)/k}(s_{y}(\overline{u})^{i(y/1)}) = Π_{y/1}N_{S_{1,y}/k}((u_{1})_{y}) =

N_{S_{1}/k}(u_{1}) in F(k). The relation Π_{y/0}N_{k(y)/k}(s_{y}(\overline{u})^{i(y/0)}) = N_{S_{0}/k}(u_{0}) · ϵ(u) in F(k)

is proved similarly (use that S/sS = k×S_{0} and the map S ⟶ k is the augmentation ϵ : S ⟶ k). The Corollary is proved. □

§6. Two lemmas

Let k be an infinite field and O an essentially smooth local k-algebra.

5.1. Definition. A perfect triple (R⇄^{ϵ}_{i}O, f) over O consists of a commutative

O-algebra i : O ⟶ R, an augmentation map ϵ : R ⟶ O and an element f ∈ R which are subjected to the following conditions:

(1) ϵ ∘ i = id_{O}, (2) the O-algebra R is smooth at each prime 𝔭 containing Ker(ϵ),

(3) R is essentially k-smooth and R is domain,

(4) R/fR is a finitely generated O-module,

(5) there exists an element t ∈ R such that O[t] is the polynomial ring in one variable and R is a finitely generated O[t]-module.

5.2. Remark. The condition (5) shows that Spec R is a relative curve over Spec O. The condition (3) shows that R is a regular ring. Since O[t] is a regular ring as well, a theorem of Grothendieck [Eis, Corollary 18.17] together with (5) show that R is flat over O[t] and thus it is a finitely generated projective O[t]-module.

Let (R⇄^{ϵ}_{i}O, f) a perfect triple. Let (A, σ) be an Azumaya algebra with involution

over R and let (A_{0}, σ_{0}) = O ⊗_{R} (A, σ), where O is considered as an R-algebra by means of the augmentation ϵ.

5.3. Lemma (Equating Lemma). There exist a quasi-finite étale extension j̃ : R ↪ R̃ and a lifting ϵ̃ : R̃ ⟶ O of the augmentation ϵ (i.e. ϵ̃ ∘ j̃ = ϵ) and an isomorphism Φ : R̃ ⊗_{R} (A, σ) ⟶ R̃ ⊗_{O} (A_{0}, σ_{0}) of Azumaya algebras with

involutions over R̃ such that the triple (R̃ ϵ̃ ; ⇄ ; ĩO, f̃) with ĩ = j̃ ∘ i and f̃ = j̃(f) is

still perfect and the map O ⊗R̃ Φ : (A_{0}, σ_{0}) ⟶ (A_{0}, σ_{0}) is the identity.

Proof. The required quasi-finite étale extension j̃ : R ↪ R̃ and lifting ϵ̃: R̃ ⟶ O of the augmentation ϵ and the isomorphism Φ : R̃ ⊗_{R} (A, σ) ⟶ R̃ ⊗_{O} (A_{0}, σ_{0}) of Azumaya algebras with involutions are constucted using geometric terminology in [Oj-P1, Proof of 8.1].

To see this set X = Spec(R), U = Spec(O) and consider the morphisms p : X → U and Δ : U → X induced by the ring homomorphisms i and ϵ. Let q : X → U × 𝔸^{1} be the finite surjective U-morphism corresponding to the integral extension O[t] ⊂ R. Let Z ⊂ X be the vanishing locus of f.

Now consider certain scheme morphisms from [Oj-P1, Proof of 8.1]. Namely, consider the quasi-finite étale morphism X̃ → X which is the composition of the finite surjective etale morphism π : X̃ → W and the open inclusion W ⊂ X. Consider the section Δ̃ : U → X̃ and the isomorphism of Azumaya algebras with involutions Φ from [Oj-P1, Proof of 8.1]. Recall that Δ̃^{∗}(Φ) is the identity, Δ = π ∘ Δ̃, Δ(U) ⊂ W, Z ⊂ W, and that there is a finite surjective U-morphism r : W → U × 𝔸^{1}.

Let j̃ : R ↪ R̃ be the inclusion induced by X̃ → X and ϵ̃ : R̃ → O the O-augmentation induced by Δ̃ : U → X̃. We claim that j̃, ϵ̃ and Φ satisfy the Lemma.

In fact, O ⊗R̃ Φ = Δ̃^{∗}(Φ) is the identity. The relation ϵ̃ ∘ j̃ = ϵ follows from the equality Δ = π ∘ Δ̃ mentioned just above. It remains to check that the triple

(R̃ ϵ̃ ; ⇄ ; ĩO, f̃) is perfect. To check this note that ϵ̃ ∘ ĩ = ϵ̃ ∘ j̃ ∘ i = ϵ ∘ i = id_{O}.

The O-algebra R̃ is smooth at each prime containing Ker(Δ̃) because Δ = π ∘ Δ̃ (with π an étale morphism ) and p : X → U is smooth along Δ(U). The kalgebra R̃ is essentially smooth because the k-algebra R is essentially smooth and j̃ : R → R̃ is etale. The vanishing locus Z̃ ⊂ X̃ of f̃ is finite over U because Z ⊂ W and π : X̃ → W is finite. Since Z̃ is finite over U the O-module R/fR is finitely generated. It remains to check that there is a finite surjective U-morphism X̃ → U × 𝔸^{1}. For that consider the finite surjective morphism r : W → U × 𝔸^{1} and take the composition r ∘ π : X̃ → U × 𝔸^{1}. □

Let R be a commutative k-algebra and let (A, σ) be an Azumaya algebra with involution over R. Let G = Sim_{A,σ} be the similitude group of (A, σ) and let μ : G ⟶ 𝔾_{m} be a group homomorphism which takes a similitude α to its similarity factor μ(α) = α^{σ} · α. Observe that μ(G(S)) = G_{S}(h) from 3.1. □

5.4. Notation. For every commutative R-algebra S denote by F(S) the group S^{∗}/μ(G(S)). An R-algebra homomorphism S ⟶^{α} T clearly induces a group map F(S) ⟶^{α_{∗}} F(T). For an element u ∈ S^{∗} we shall write \overline{u} for its image in F(S). The homomorphism α_{∗} takes \overline{u} to \overline{α(u)}.

5.5. Definition. Let S be an R-algebra which is a domain with the quotient field K and let 𝔭 be a height 1 prime ideal in S. An element v ∈ F(K) is called unramified at 𝔭 iff v belongs to the image of F(S_{𝔭}) in F(K). An element v ∈ F(K) is called S-unramified if it is unramified at each height 1 prime 𝔭 in S.

5.6. Lemma (Unramifiedness Lemma). Let R and S be domains with quotient fields K and L respectively. Let R ⟶^{α} S be an injective flat homomorphism of finite type and let β : K ⟶ L be the induced inclusion of the quotient fields. Then for each localization T ⊃ S of S the map β_{∗} : F(K) ⟶ F(L) takes S-unramified elements to T -unramified elements.

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

F(K) ⟶ F(L)
↑  ↑
F(R_{𝔭}) ⟶ F(T_{𝔯})

shows that the class \overline{β(v)} is in the image of F(T_{𝔯}). Whence the class \overline{β(v)} ∈ F(L) is T -unramified. The Lemma follows. □

§6. Relative Specialization Lemma

Let O be a regular local ring containing an infinite field k and which is an

essentially smooth k-algebra. Let K be the quotient field of O. Let (R⇄^{ϵ}_{i}O, f) be a

perfect triple. Denote by ϵ_{K} : R_{K} = R⊗_{O} K → K the homomorphism ϵ⊗_{O} K. We will consider O and K as R-algebras via ϵ and ϵ_{K} respectively. So for an Azumaya algebra with involution (A, σ) over R it makes sense to speak about the groups F(O) and F(K) (see Definition 5.5).

6.1. Lemma (Relative Specialization Lemma). Let (R⇄^{ϵ}_{i}O, f) be a perfect

triple and (A, σ) an Azumaya algebra with involution over R. Let K be the quotient field of R and let u ∈ R_{f}^{∗} be a unit such that the class ū ∈ F(K) is R-unramified. If ϵ(f) ≠ 0 then the class \overline{ϵ_{K}(u ⊗ 1)} ∈ F(K) can be lifted to F(O).

Proof. Set (A_{0}, σ_{0}) = (O⊗_{R}A, O⊗_{R}σ), where O is an R-algebra by means of ϵ. Set (A_{0}, σ_{0}) = (R⊗_{O}A_{0}, R⊗_{O}σ_{0}), where R is regarded as an O-algebra by means of the map i. There are two Azumaya algebras with involutions (A, σ) and (A_{0}, σ_{0}) over R. Their scalar extensions (A, σ) ⊗_{R} O and (A_{0}, σ_{0}) ⊗_{R} O tautologically coincides because the composite map O ⟶^{i} R ⟶^{ϵ} O is the identity. Thus by the Equating Lemma 5.3 one can find a quasi-finite étale extension j : R ↪ R̃ and a lifting ϵ̃ : R̃ ⟶ O of the augmentation ϵ and an isomorphism Φ : (Ã, σ̃) ⟶ (Ã_{0}, σ̃_{0}) of

Azumaya algebras with involutions over R̃ such that (R̃ ϵ̃ ; ⇄ ; ĩO, j(f)) is still a perfect

triple and the isomorphism O ⊗R̃ Φ is the identity. Here (Ã, σ̃) = R̃ ⊗_{R} (A, σ), (Ã_{0}, σ̃_{0}) = R̃⊗_{O}(A_{0}, σ_{0}) and O is regarded as an R̃-algebra by means of ϵ̃: R̃ ⟶ O.

Denote by ϵ̃_{K} : R̃_{K} = R ⊗_{O} K → K the augmentation ϵ̃ ⊗_{O} K. Set f̃ = j(f) and ũ = j(u) ∈ R̃∗ ; f̃. Since ϵ̃_{K}(ũ ⊗ 1) = ϵ_{K}(u ⊗ 1) it suffices to check that the class

\overline{ϵ̃_{K}(ũ ⊗ 1)} can be lifted to F(O).

Let K̃ be the quotient field of R̃. By the Unramifiedness Lemma the class

\overline{ũ} ∈ F(K̃) is R̃-unramified. So replacing (R⇄^{ϵ}_{i}O, f), (A, σ) and u by (R̃ ϵ̃ ; ⇄ ; ĩO, j(f)),

(Ã, σ̃) and ũ we may assume that (R⇄^{ϵ}_{i}O, f), is a perfect triple, (A, σ) = R ⊗_{O}

(A_{0}, σ_{0}) for an Azumaya algebra with involution (A_{0}, σ_{0}) over O, and u ∈ R_{f}^{∗} is such that the class ū ∈ F(K) is R-unramified. We must check that the class \overline{ϵ_{K}(u_{K})} ∈ F(K) can be lifted in F(O).

Since the triple (R⇄^{ϵ}_{i}O, f) is perfect, the geometric presentation lemma [Oj-P,

Lemma 5.2] shows that one can choose an element s ∈ R such that the extension R ⊇ O[s] is finite, the ring O[s] is the polynomial ring in one variable over O and the following holds:

(1) (1 − s)R + fR = R,

(2) sR = Ker(ϵ)∩J for a certain ideal J and

(3) J + fR = R and

(4) the map R/sR ⟶ R/Ker(ϵ) × R/J = O × R/J is an isomorphism.

Since R and O[s] are both essentially smooth k-algebras (and thus regular rings) and since the extesion R over O[s] is finite, a theorem of Grothendieck [Eis, Corollary 18.17] shows that R is a flat O[s]-module. Therefore R is a finitely generated projective O-module. Thus R_{1} = R/(1−s)R and R_{0} = R/J are finitely generated projective O-modules.

Consider the elements u_{1} = u mod (1 − s)R_{f} in R_{1,f}^{∗} and u_{0} = u mod J_{f} in R_{0,f}^{∗}. By (1) and (3) one has R_{i} = R_{i,f} and thus u_{i} ∈ R_{i}^{∗} (i = 0, 1). Since R_{1} and R_{0} are finitely generated projective O-modules, there are the norm mappings N_{R_{i}/O} : R_{i}^{∗} ⟶ O^{∗} (i = 0, 1) given by (v ↦ det(mult. by v)). Set

ϕ(u) = N_{R_{1}/O}(u_{1}) · N_{R_{0}/O}(u_{0}^{−1}) ∈ O^{∗} ⊆ K^{∗} .

Claim. \overline{ϕ(u)} = \overline{ϵ_{K}(u_{K})} in the group K^{∗}/μ(G(K)) = F(K).

Since ϕ(u) ∈ O^{∗}, the Claim clearly completes the proof of purity. The rest of the section is devoted to the proof of the Claim.

Set R_{K} = K ⊗_{O} R and u_{K} = 1 ⊗ u ∈ R_{K}. Set R_{i,K} = K ⊗_{O} R_{i} and u_{i,K} = 1 ⊗ u_{i} ∈ R_{K} ∈ R_{i,K}^{∗}. Finally set R_{f,K} = R_{K,1⊗f}. Clearly it suffices to prove the relation

(†)  \overline{ϵ_{K}(1 ⊗ u)} = \overline{N_{R_{1,K}/K}(1 ⊗ u_{1})} · \overline{N_{R_{0,K}/K}(1 ⊗ u_{0})^{−1}}

in the group F(K). The relation (†) will be checked below in this proof applying the Specialization Lemma (Theorem 3.3) to the integral extension R_{K} ⊇ K ⊗_{O} O[s] = K[s] and the Azumaya algebra with involution (A_{0}, σ_{0}) ⊗_{O} K over K.

Check the hypotheses of the Specialization Lemma. Since R is regular domain and R_{K} is its localization R_{K} is a regular domain as well. Since R_{K} is an integral extension of the polynomial ring K[s], the dimension of R is one. A regular domain of dimension 1 is a Dedekind domain. \underline{Thus} R \underline{is a Dedekind domain.}

The class ū ∈ F(K) of the element u ∈ R_{f}^{∗} is R-unramified. Thus the class ū_{K} ∈ F(K) of the element u_{K} ∈ R_{f,K}^{∗} is R_{K}-unramified.

Now check that the element 1 ⊗ f ∈ R_{K} is coprime with both s and s − 1 in R_{K}. Recall the conditions (1) to (4) mentioned above in this proof. The element 1⊗f is coprime with (s − 1) by condition (1). The element ϵ_{K}(1 ⊗ f) = ϵ(f)_{K} is non-zero in K by the very assumption on f. The element 1 ⊗ f is coprime with the ideal J_{K} by condition (3). Thus 1 ⊗ f is coprime with s by condition (4). We already checked that the class ū is R_{K}-unramified. Thus by Theorem 3.3 the relation (†) holds in F(K). The Claim is proved. The Relative Specialization Lemma follows.

§7. Geometric case of the Purity Theorem

Under the notation of 5.4 and 5.5 the following theorem holds.

7.1. Theorem. Let O be a local, essentially smooth algebra over a field k and let K be its quotient field. Let (A, σ) be an Azumaya algebra with involution over O and let v ∈ K^{∗} be such that the class v̄ ∈ F(K) is O-unramified. Then v̄ can be lifted in F(O).

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

From now on and till the end of the proof of Theorem 8.1 we assume that O = O_{X,x} is the local ring of a closed point x of a smooth d-dimensional irreducible affine variety X over k.

Replacing X by a sufficiently small affine neighbourhood of x we may assume that

(1) the algebra with involution (A, σ) is defined over k[X] and is an Azumaya algebra with involution already over k[X],

(2) the element v is a unit in k[X]_{g} for certain nonzero element g ∈ k[X],

(3) the class v̄ ∈ F(K) is k[X]-unramified. We must prove that v̄ can be lifted in F(O).

By Quillen’s trick there exists a polinomial subalgebra k[t_{1}, t_{2}, …, t_{n}] in k[X] such that the algebra R = k[X] is finite over k[t_{1}, t_{2}, …, t_{n}], the algebra R is smooth over k[t_{1}, t_{2}, …, t_{n−1}] at the maximal ideal 𝔪 and the k[t_{1}, t_{2}, …, t_{n−1}]- module R/fR is finite. Set P = k[t_{1}, t_{2}, …, t_{n−1}], R = O ⊗_{P} R, consider ring homomorphisms j : R → R, i : O → R and ϵ : R → O given by j(a) = 1 ⊗ a, i(b) = b ⊗ 1 and ϵ(a ⊗ b) = ab respectively.

We claim that (R⇄^{ϵ}_{i}O, f) with f = j(f) is a perfect triple (see 6.1 for definition).

This is checked in [Oj-P] using geometric terminology. This perfect triple fits in the diagram

\xymatrix{R \ar[d]_{ϵ} & R \ar[dl]^{can} \ar[l]_{j} \\ O \ar[u]_{i}}

with the localization map can. Clearly can = ϵ ∘ j.

Set (A, σ) = R ⊗_{k[X]} (A, σ) and u = j(v) ∈ R_{f}^{∗}. Let K be the quotient field of R. By the Unramifiedness Lemma the class ū ∈ F(K) is R-unramified. Since ϵ(f) = ϵ(j(f)) = f is nonzero element of O we are under the hypotheses of the Relative Specialization Lemma. Thus the class \overline{ϵ}_{K}(u) ∈ F(K) can be lifted in F(O). It remain to note that

ϵ_{K}(u) = ϵ_{K}(j(v)) = v ∈ K.

Thus the class v̄ ∈ F(K) can be lifted in F(O).

Now suppose that k is finite. So O is a local essentially smooth k-algebra with maximal ideal 𝔪. Let v ∈ K^{∗} be such that the class v̄ ∈ F(K) is O-unramified. Let p^{m} be the cardinality of the algebraic closure of k in A/𝔪 and s be an odd integer greater than 2 and prime to m. For any i let l_{i} be the field (in some fixed algebraic closure of k) of degree s^{i} over k. Let l be the union of all l_{i}. Since l ⊗_{k} (O/𝔪) is still a field, R = l ⊗_{k} O is a local essentially smooth algebra over the infinite field l. Let L = l ⊗_{k} K be its field of fractions. The image v̄_{L} of v̄ in F(L) is R-unramified. In fact, let 𝔮 be a hight-one prime of R and 𝔭 = O ∩ 𝔮. By assumption v̄ is in the image of F(O_{𝔭}) and since O_{𝔭} → L factors through R_{𝔮} the class v̄_{L} is in F(R_{𝔮}) for every 𝔮. We can now find a finite subfield l′ of l, and for O′ = l′ ⊗_{k} O, a v′ ∈ O′ which maps to v̄_{L}. Let K′ be the field of fractions of O′. Further enlarging l′ we may assume that the images v̄ and v̄′ in F(K′) coincide. Consider the diagram

O^{∗} ⟶ (O′)^{∗} ⟶^{N} O^{∗}
↓  ↓  ↓^{α}
F(K) ⟶ F(K′) ⟶^{N̄} F(K).

where N̄ is the norm map (it is well-defined by the Scharlau norm principle). Since the composition of the horizontal maps is the identity, we have α ∘ N(v′) = v̄ in F(K). Thus v̄ is indeed in the image of O^{∗}. Theorem 7.1 is proved.

§8. Proof of the Purity Theorem

Proof of Theorem 1.3. Let k be the prime subfield of the ring R. By Popescu’s theorem [P], [Sw] R = \underrightarrow{lim}R_{α} (a filtered direct limit), where R_{α}’s are smooth kalgebras. We first observe that we may replace the direct system of the R_{α}’s by a system of essentially smooth local k-algebras. In fact, if 𝔪 is the maximal ideal of R, we can replace each R_{α} by (R_{α})_{𝔭_{α}} where 𝔭_{α} = 𝔪 ∩ R_{α}. Note that in this case the canonical morphisms ϕ_{α} : R_{α} → R are local and that every R_{α} is a regular local ring thus in particular a factorial ring.

Now let K be the field of fractions of R and, for each α, let K_{α} be the field of fractions of R_{α}. The ideal 𝔯_{α} = ker(ϕ_{α}) is prime. Set S_{α} = (R_{α})_{𝔯_{α}}. Note that the S_{α}’s form a direct system of regular local rings with K = \underrightarrow{lim}S_{α} (a filtered direct limit).

We may assume that there exists an index α and an Azumaya algebra with involution (A_{α}, σ_{α}) over R_{α} such that (A, σ) = (A_{α}, σ_{α}) ⊗_{R_{α}} R. Replacing the direct system of indeces α’s by the subsystem of indeces β satisfying β ≥ α we may assume that we are given with a direct system of Azumaya algebras with involutions (A_{α}, σ_{α}) over the R_{α}’s such that (A, σ) = \underrightarrow{lim}(A_{α}, σ_{α}).

Let G_{α} = Sim_{A_{α},σ_{α}}. Then one has G(R) = \underrightarrow{lim}G_{α}(R_{α}) and G(K) = \underrightarrow{lim}G_{α}(S_{α}). Let μ_{α} : G_{α} → 𝔾_{m} be the group morphism which takes a similitude to its similarity factor (see the Introduction).

Let ā ∈ F(K) be an R-unramified class. We may represent ā by a unit a ∈ R_{f}^{∗}, where 0 ≠ f ∈ R. Let f = p_{1}p_{2} … p_{n} be a prime decomposition of f in R. Since ā is R-unramified for every index i = 1, 2, …, n there exist elements h_{i} ∈ R − p_{i}R and a_{i} ∈ R_{h_{i}}^{∗} and g_{i} ∈ G(K) such that a = a_{i}μ(g_{i}).

We can now choose an index α, elements p_{α,i} and h_{α,i} with ϕ_{α}(p_{α,i}) = p_{i} and ϕ_{α}(h_{α,i}) = h_{i}. Set f_{α} = p_{α,1}p_{α,2} … p_{α,n}. Since ϕ_{α}(f_{α}) = f ≠ 0 and ϕ_{α}(h_{α,i}) = h_{i} ≠ 0 one has the inclusions R_{α,f_{α}} ⊂ S_{α} and R_{α,h_{α,i}} ⊂ S_{α}. Futher enlarging the index α we can choose a_{α} ∈ R_{α,f_{α}}^{∗} and elements a_{α,i} ∈ R_{α,h_{α,i}}^{∗} and g_{α,i} ∈ G_{α}(S_{α}) which are preimages of the a and the a_{i}’s and the g_{i}’s respectively. Having choosen these preimages consider the relations

a_{α} = a_{α,i}μ_{α}(g_{α,i})

in S_{α}. Since they hold over K, we may assume, after replacing α by some larger index, that they hold over S_{α}. We claim that the class ā_{α} ∈ F(K_{α}) is R_{α}-unramified.

To prove this note first that each p_{α,i} is prime. In fact, ϕ_{α}(p_{α,i}) = p_{i}, the element p_{i} is prime and ϕ_{α} is a local homomorphism of local factorial rings. Thus p_{α,i} is indeed prime. Since a_{α} ∈ R_{α,f_{α}}^{∗} and f_{α} = p_{α,1}p_{α,2} … p_{α,n} the class ā_{α} can be ramified at most at one of the p_{α,i}’s. However the relations a_{α} = a_{α,i}μ_{α}(g_{α,i}) with a_{α,i} ∈ R_{α,h_{α,i}}^{∗} and the fact that p_{α,i} does not divide h_{α,i} prove that the class ā_{α} is unramified at each the p_{α,i}. Thus the class ā_{α} is indeed R_{α}-unramified.

By purity for R_{α} there exists an a′_{α} ∈ R_{α}^{∗} such that ā′_{α} = ā_{α} in F(K_{α}). The exact sequence 1 ⟶ U_{A,σ} ⟶ Sim_{A,σ} ⟶^{μ} 𝔾_{m} ⟶ 1 of algebraic group schemes over S_{α} shows that the kernel of the boundary map ∂ : F(S_{α}) → H^{1}(S_{α}, U_{A,σ}) is trivial. The Main Theorem of [Oj-P1] states that the kernel

ker[H^{1}(S_{α}, U_{A,σ}) → H^{1}(K_{α}, U_{A,σ})]

is trivial. Thus F(S_{α}) injects into F(K_{α}) and ā′_{α} = ā_{α} already in F(S_{α}). The commutative diagram

R_{α} ⟶^{ϕ_{α}} R
↓  ↓
S_{α} ⟶ K .

shows that \overline{ϕ_{α}(α′)} = ā in F(K). This completes the proof of Theorem 1.2.

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