SK_{1} OF AZUMAYA ALGEBRAS OVER HENSEL PAIRS

ROOZBEH HAZRAT

Abstract. Let A be an Azumaya algebra of constant rank n over a Hensel pair (R, I) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK_{1}(A) coincides with its reduction SK_{1}(A/IA). This generalizes the result of [6] to nonlocal Henselian rings.

Let A be an Azumaya algebra over a ring R of constant rank n. There exists an étale faithfully flat splitting ring R ⊆ S for A, i.e., A ⊗_{R} S ≅ M_{n}(S). This provides the notion of the reduced norm (and reduced trace) for A ([10], III, §1). Denote by SL(1, A) the set of all elements of A with reduced norm 1. SL(1, A) is a normal subgroup of A^{∗}, the invertible elements of A (see Saltman [14], Theorem 4.3). Since the reduced norm map respects the scaler extensions, it defines the smooth group scheme SL_{1,A} : T → SL(1, A_{T}) where A_{T} = A ⊗_{R} T for an R-algebra T. Consider 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}^{∗} and G_{m}(T) = T^{∗} for an R-algebra T. This exact sequence induces the long exact étale cohomology

(1) 1 ⟶ SL(1, A) ⟶ A^{∗} ⟶^{Nrd} R^{∗} ⟶ H_{et}^{1}(R, SL(1, A)) ⟶ H_{et}^{1}(R, GL(1, A)) → ⋯

Let A′ denote the commutator subgroup of A^{∗}. One defines the reduced Whitehead group of A as SK_{1}(A) = SL(1, A)/A′ which is a subgroup of (non-stable) K_{1}(A) = A^{∗}/A′. Let I be an ideal of R. Since the reduced norm is compatible with extensions, it induces the map SK_{1}(A) → SK_{1}(Ā), where Ā = A/IA. A natural question arises here is, under what circumstances and for what ideals I of R, this homomorphism would be a mono or/and epi and thus the reduced Whitehead group of A coincides with its reduction. The following observation shows that even in the case of a split Azumaya algebra, these two groups could differ: consider the split Azumaya algebra A = M_{n}(R) where R is an arbitrary commutative ring. In this case the reduced norm coincides with the ordinary determinant and SK_{1}(A) = SL_{n}(R)/[GL_{n}(R), GL_{n}(R)]. There are examples such that SK_{1}(A) ≠ 1, in fact not even torsion. But in this setting, obviously SK_{1}(Ā) = 1 for Ā = A/mA where m is a maximal ideal of R (for some examples see Rosenberg [13], Chapter 2).

If I is contained in the Jacobson radical J(R), then IA ⊂ J(A) (see, e.g., Lemma 1.4 [4]) and (non-stable) K_{1}(A) → K_{1}(Ā) is surjective, thus its restriction to SK_{1} is also surjective.

It is observed by Grothendieck ([5], Theorem 11.7) that if R is a local Henselian ring with maximal ideal I and G is an affine, smooth group scheme, then H_{et}^{1}(R, G) → H_{et}^{1}(R/I, G/IG) is an isomorphism. This was further extended to Hensel pairs by Strano [15]. Now if further R is a semilocal ring then H_{et}^{1}(R, GL(1, A)) = 0, and thus from the sequece (1) it follows

(2)  \xymatrix{ &  & (1 + IA)A′/A′ \ar[r] \ar[d] & 1 + I \ar[d] \\ 1 \ar[r] & SK_{1}(A) \ar[r] \ar[d] & K_{1}(A) \ar[r]^{Nrd} \ar[d] & R^{∗} \ar[r] \ar[d] & H_{et}^{1}(R, SL(1, A)) \ar[r] \ar[d]^{≅} & 1 \\ 1 \ar[r] & SK_{1}(Ā) \ar[r] & K_{1}(Ā) \ar[r]^{Nrd} \ar[d] & R̄^{∗} \ar[r] \ar[d] & H_{et}^{1}(R̄, SL(1, Ā)) \ar[r] & 1 \\  &  & 1 & 1}

The aim of this note is to prove that for the Hensel pair (R, I) where R is a semilocal ring, the map SK_{1}(A) → SK_{1}(Ā) is also an isomorphism. This extends the result of [6] to non-local Henselian rings.

Recall that the pair (R, I) where R is a commutative ring and I an ideal of R is called a Hensel pair if for any polynomial f(x) ∈ R[x], and b ∈ R/I such that f̄(b) = 0 and f̄′(b) is invertible in R/I, then there is a ∈ R such that ā = b and f(a) = 0 (for other equivalent conditions, see Raynaud [12], Chap. XI).

In order to prove this result, we use a recent result of Vasertein [17] which establishes the (Dieudonnè) determinant in the setting of semilocal rings. The crucial part is to prove a version of Platonov’s congruence theorem [11] in the setting of an Azumaya algebra over a Hensel pair. The approach to do this was motivated by Suslin in [16]. We also need to use the following facts established by Greco in [3, 4].

Proposition 1 ([4], Prop. 1.6). Let R be a commutative ring, A be an R-algebra, integral over R and finite over its center. Let B be a commutative R-subalgebra of A and I an ideal of R. Then IA ∩ B ⊆ \sqrt{IB}.

Corollary 2 ([3], Cor. 4.2). Let (R, I) be a Hensel pair and let J ⊆ \sqrt{I} be an ideal of R. Then (R, J) is a Hensel pair.

Theorem 3 ([3], Th. 4.6). Let (R, I) be a Hensel pair and let B be a commutative R-algebra integral over R. Then (B, IB) is a Hensel pair.

We are in a position to prove the main Theorem of this note.

Theorem 4. Let A be an Azumaya algebra of constant rank m over a Hensel pair (R, I) where R is a semilocal ring with m invertible in R. Then SK_{1}(A) ≅ SK_{1}(Ā) where Ā = A/IA.

Proof. Since for any a ∈ A, \overline{Nrd_{A}(a)} = NrdĀ(ā), it follows that there is a homomorphism ϕ : SL(1, A) → SL(1, Ā). We first show that ker ϕ ⊆ A′, the commutator subgroup of A^{∗}. In the setting of valued division algebras, this is the Platonov congruence theorem [11]. We shall prove this in several steps. Clearly ker ϕ = SL(1, A) ∩ 1 + IA. Note that A is a free R-module (see [1], II, §5.3, Prop. 5) . Set m = n^{2}.

SK_{1} OF AZUMAYA ALGEBRAS 3

1. The group 1 + I is uniquely n-divisible and 1 + IA is n-divisible.

Let a ∈ 1 + I. Consider f(x) = x^{n} − a ∈ R[x]. Since n is invertible in R, f̄(x) = x^{n} − 1 ∈ R̄[x] has a simple root. Now this root lifts to a root of f(x) as (R, I) is a Hensel pair. This shows that 1 + I is n-divisible. Now if (1 + a)^{n} = 1 where a ∈ I, then a(a^{n−1} + na^{n−2} + ⋯ + n) = 0. Since the second factor is invertible, a = 0, and it follows that 1 + I is uniquely n-divisible.

Now let a ∈ 1 + IA. Consider the commutative ring B = R[a] ⊆ A. By Theorem 3, (B, IB) is a Hensel pair. On the other hand by Prop. 1, IA ∩ B ⊆ \sqrt{IB}. Thus by Cor. 2, (B, IA ∩ B) is also a Hensel pair. But a ∈ 1 + IA ∩ B. Applying the Hensel lemma as in the above, it follows that a has a n-th root and thus 1 + IA is n-divisible.

2. Nrd_{A}(1 + IA) = 1 + I.

¿From compatibility of the reduced norm, it follows that Nrd_{A}(1 + IA) ⊆ 1 + I. Now using the fact that 1 + I is n-divisible, the equality follows.

3. SK_{1}(A) is n^{2}-torsion.

We first establish that N_{A/R}(a) = Nrd_{A}(a)^{n}. One way to see this is as follows. Since A is an Azumaya algebra of constant rank n, then i : A ⊗ A^{op} ≅ End_{R}(A) ≅ M_{n^{2}}(R) and there is an étale faithfully flat S algebra such that j : A ⊗ S ≅ M_{n}(S). Consider the following diagram

\xymatrix{A ⊗ A^{op} ⊗ S \ar[r]^{i⊗1} \ar[d] & End_{R}(A) ⊗ S \ar[r]^{≅} & End_{S}(A ⊗ S) \ar[r]^{≅} & M_{n^{2}}(S) \ar@{.>}[d]^{ψ} \\ A^{op} ⊗ A ⊗ S \ar[r]^{1⊗j} & A^{op} ⊗ M_{n}(S) \ar[r]^{≅} & M_{n}(A^{op} ⊗ S) \ar[r]^{≅} & M_{n^{2}}(S)}

where the automorphism ψ is the compositions of isomorphisms in the diagram. By a theorem of Artin (see, e.g., [10], §III, Lemma 1.2.1), one can find an ètale faithfully flat S algebra T such that ψ ⊗ 1 : M_{n^{2}}(T) → M_{n^{2}}(T) is an inner automorphism. Now the determinant of the element a ⊗ 1 ⊗ 1 in the first row is N_{A/R}(a) and in the second row is Nrd_{A}(a)^{n} and since ψ ⊗ 1 is inner, thus they coincide.

Therefore if a ∈ SL(1, A), then N_{A/R}(a) = 1. We will show that a^{n^{2}} ∈ A′. Consider the sequence of R-algebra homomorphism

f : A → A ⊗ A^{op} → End_{R}(A) ≅ M_{n^{2}}(R) ↪ M_{n^{2}}(A)

and the R-algebra homomorphism i : A → M_{n^{2}}(A) where a maps to aI_{n^{2}}, where I_{n^{2}} is the identity matrix of M_{n^{2}}(A). Since R is a semilocal ring, the Skolem-Noether theorem is present in this setting (see Prop. 5.2.3 in [10]) and thus there is g ∈ GL_{n^{2}}(A) such that f(a) = gi(a)g^{−1}. Also, since A is a finite algebra over R, A is a semilocal ring. Since n is invertible in R, by Vaserstein’s result [17], the Dieudonnè determinant extends to the setting of M_{n^{2}}(A). Taking the determinant from f(a) and gi(a)g^{−1}, it follows that 1 = N_{A/R}(a) = a^{n^{2}}c_{a} where c_{a} ∈ A′. This shows that SK_{1}(A) is n^{2}-torsion.

4. Platonov Congruence Theorem: SL(1, A) ∩ 1 + IA ⊆ A′.

Let a ∈ SL(1, A) ∩ 1 + IA. By (1), there is b ∈ 1 + IA such that b^{n^{2}} = a. Then Nrd_{A}(a) = Nrd_{A}(b)^{n^{2}} = 1. By (2), Nrd_{A}(b) ∈ 1 + I and since 1 + I is uniquely n-divisible, Nrd_{A}(b) = 1, so b ∈ SL(1, A). By (3), b^{n^{2}} ∈ A′, so a ∈ A′. Thus ker ϕ ⊆ A′ where ϕ : SL(1, A) → SL(1, Ā).

It is easy to see that ϕ is surjective. In fact, if ā ∈ SL(1, Ā) then 1 = NrdĀ(ā) = \overline{Nrd_{A}(a)} thus, Nrd_{A}(a) ∈ 1 + I. By (1), there is r ∈ 1 + I such that Nrd_{A}(ar^{−1}) = 1 and \overline{ar^{−1}} = ā. Thus ϕ is an epimorphism. Consider the induced map ϕ̄: SL(1, A) → SL(1, Ā)/Ā′. Since I ⊆ J(R), and by (3), ker ϕ ⊆ A′ it follows that ker ϕ̄ = A′ and thus ϕ̄: SK_{1}(A) ≅ SK_{1}(Ā). □

Let R be a semilocal ring and (R, J(R)) a Hensel pair. Let A be an Azumaya algebra over R of constant rank n and n invertible in R. Then by Theorem 4, SK_{1}(A) ≅ SK_{1}(Ā) where Ā = A/J(R)A. But J(A) = J(R)A, so Ā = M_{k_{1}}(D_{1}) × ⋯ M_{k_{r}}(D_{r}) where D_{i} are division algebras. Thus SK_{1}(A) ≅ SK_{1}(Ā) = SK_{1}(D_{1}) ⋯ × SK_{1}(D_{r}).

Using a result of Goldman [2], one can remove the condition of Azumaya algebra having a constant rank from the Theorem.

Corollary 5. Let A be an Azumaya algebra over a Hensel pair (R, I) where R is semilocal and the least common multiple of local ranks of A over R is invertible in R. Then SK_{1}(A) ≅ SK_{1}(Ā) where Ā = A/IA.

Proof. One can decompose R uniquely as R_{1} ⊕⋯⊕R_{t} such that A_{i} = R_{i} ⊗_{R} A have constant ranks over R_{i} which coincide with local ranks of A over R (see [2], §2 and Theorem 3.1). Since (R_{i}, IR_{i}) are Hensel pairs, the result follows by using Theorem 4. □

Remarks 6. Let D be a tame unramified division algebra over a Henselian field F, i.e., the valued group of D coincide with valued group of F and chr(F̄) does not divide the index of D (see [18] for a nice survey on valued division algebras). Jacob and Wadsworth observed that V_{D} is an Azumaya algebra over its center V_{F} (Theorem 3.2 in [18] and Example 2.4 in [8]). Since D^{∗} = F^{∗}U_{D} and V_{D} ⊗_{V_{F}} F ≃ D, it can be seen that SK_{1}(D) = SK_{1}(V_{D}). On the other hand our main Theorem states that SK_{1}(V_{D}) ≃ SK_{1}(D̄). Comparing these, we conclude the stability of SK_{1} under reduction, namely SK_{1}(D) ≃ SK_{1}(D̄) (compare this with the original proof, Corollary 3.13 [11]).

Now consider the group CK_{1}(A) = A^{∗}/R^{∗}A′ for the Azumaya algebra A over the Hensel pair (R, I). A proof similar to Theorem 3.10 in [6], shows that CK_{1}(A) ≅ CK_{1}(Ā). Thus in the case of tame unramified division algebra D, one can observe that CK_{1}(D) ≅ CK_{1}(D̄).

For an Azumaya algebra A over a semilocal ring R, by (1) one has

R^{∗}/Nrd_{A}(A^{∗}) ≅ H_{èt}^{1}(R, SL(1, A)).

If (R, I) is also a Hensel pair, then by the Grothendieck-Strano result,

R^{∗}/Nrd_{A}(A^{∗}) ≅ H_{èt}^{1}(R, SL(1, A)) ≅ H_{èt}^{1}(R̄, SL(1, Ā)) ≅ R̄^{∗}/NrdĀ(Ā^{∗}).

However specializing to a tame unramified division algebra D, the stability does not follow in this case. In fact for a tame and unramified division algebra D over a Henselian field F with the valued group Γ_{F} and index n one has the following exact sequence (see [7], Theorem 1):

1 ⟶ H^{1}(\overline{F} , SL(1, \overline{D})) ⟶ H^{1}(F, SL(1, D)) ⟶ Γ_{F} /nΓ_{F} ⟶ 1.

Acknowledgement. I would like to thank IHES, where part of this work has been done in Summer 2006.

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Dept. of Pure Mathematics, Queen’s University, Belfast BT7 1NN, United Kingdom

  E-mail address: r.hazrat@qub.ac.uk