On the non-neutral component of outer forms of the orthogonal group
aa r X i v : . [ m a t h . AG ] J un ON THE NON-NEUTRAL COMPONENT OF OUTER FORMS OFTHE ORTHOGONAL GROUP
URIYA A. FIRST ∗ Let A be a central simple algebra over a field F of characteristic not 2, and let σ : A → A be an orthogonal involution (see [10, Ch. I] for the definitions). LetO( A, σ ) denote the group of elements u ∈ A with u σ u = 1. The reduced norm mapNrd A/F : A → F restricts to a group homomorphism, Nrd A/F : O(
A, σ ) → {± } ;its kernel, SO( A, σ ), is the special orthogonal group of (
A, σ ). Both O(
A, σ ) andSO(
A, σ ) can be regarded as the F -points of algebraic groups over F , denoted O ( A, σ ) and SO ( A, σ ), respectively.The question of whether O(
A, σ ) has elements of reduced norm − O ( A, σ ), whichis an SO ( A, σ )-torsor, has an F -point. It is well-known that such an F -point existsif and only if [ A ], the Brauer class of A , is trivial in the Brauer group of F , denotedBr F ; see [8, Lemma 2.6.1b], for instance.In this note, we generalize this result to Azumaya algebras with orthogonalinvolutions over semilocal commutative rings. Given a commutative ring R , recallthat an R -algebra A is called Azumaya if A is a finitely generated projective R -module and A ( m ) := A ⊗ R ( R/ m ) is a central simple ( R/ m )-algebra for everymaximal ideal m ∈ Max R . In this case, an R -linear involution σ : A → A is calledorthogonal if its specialization σ ( m ) := σ ⊗ R id R/ m is orthogonal for all m ∈ Max R .See [9, III. §
5, III. §
8] or [6] for an extensive discussion. Note also that Nrd
A/R takesO(
A, σ ) to µ ( R ) := { ε ∈ R : ε = 1 } . We prove: Theorem 1.
Let R be a commutative semilocal ring with ∈ R × , let A be anAzumaya R -algebra and let σ : A → A be an orthogonal involution. Then O( A, σ ) contains elements of reduced norm − if and only if [ A ] = 0 in Br R . In the process, we prove another result of independent interest:
Theorem 2.
Let R , A and σ be as in Theorem 1. Then the natural map SO(
A, σ ) → Y m ∈ Max R SO( A ( m ) , σ ( m )) is surjective. This was proved by Knebusch [7, Satz 0.4] when A is a matrix algebra over R .The surjectivity of O( A, σ ) → Q m ∈ Max R O( A ( m ) , σ ( m )) may fail under the sameassumptions (Example 7).Applications of both theorems to Witt groups of Azumaya algebras with involu-tion appear in [5].We show that the “if” part of Theorem 1 is false if R is not assumed to besemilocal, see Example 9. As for the “only if” part of Theorem 1, we ask: ∗ University of Haifa
E-mail address : [email protected] . Date : June 5, 2020.
Question 3.
Let R be a commutative ring with ∈ R × , let A be an Azumaya R -algebra, and let σ : A → A be an orthogonal involution. Suppose that O( A, σ ) contains elements of reduced norm − . Is it the case that [ A ] = 0 ? We expect that the answer is “yes”. By Theorem 1, a counterexample, if itexists, will have the remarkable property that [ A ] = 0 while [ A ⊗ R S ] = 0 in Br S for every semilocal commutative R -algebra S . We do not know if Azumaya algebraswith this property exist. (Ojanguren [11] gave an example having this property forany local S , but in his example, [ A ⊗ R S ] remains nontrivial if S is taken to be thelocalization of R away from three particular prime ideals.) We further note thatthe answer to Question 3 is “yes” when R is a regular domain. Indeed, writing F for the fraction field of R , we observed that [ A ⊗ R F ] = 0 in Br F , and the mapBr R → Br F is injective by the Auslander–Goldman theorem [1, Theorem 7.2].1. Proof of Theorem 2
We shall derive Theorem 2 from a more general theorem addressing semilocalrings with involution. Recall that a ring A is called semilocal if A/ Jac A is semisim-ple aritinian, where Jac A denotes the Jacobson radical of A .Let ( A, σ ) be a ring with involution such that 2 ∈ A × . We let Skew( A, σ ) = { a ∈ A : a σ = − a } . Given y ∈ A and a ∈ Skew(
A, σ ) such that y σ y + a ∈ A × ,define s y,a = 1 − y ( y σ y + a ) − y σ ∈ A. Consider the ( σ, f : A × A → A given by f ( x, y ) = x σ y ;here, A is viewed as a right module over itself. Identifying End A ( A ) with A via ϕ ϕ (1 A ), one easily checks that the isometry group of f is U ( A, σ ) := { u ∈ A : u σ u = 1 } . Moreover, the elements s y,a are precisely the 1-reflections of f in the sense of [12, §
1] or [4, § s y,a ∈ U ( A, σ ) for all y and a as above ([12, Proposition 1.3]or [4, Proposition 3.3]). This can also be checked by computation.Suppose that A is semisimple artinian. We define a subgroup U ( A, σ ) of U ( A, σ )as follows: Assume first that A is simple artinian. By the Artin–Wedderburntheorem, A ∼ = M n ( D ), where D is a division ring with center K . We then define U ( A, σ ) := (cid:26)
SO(
A, σ ) D = K and σ is orthogonal U ( A, σ ) otherwise . Next, if A is not simple but ( A, σ ) is simple as a ring with involution, then there is asimple artinian ring B and an isomorhism A ∼ = B × B op under which σ correspondsto ( x, y op ) ( y, x op ) ( x, y ∈ B ). We then set U ( A, σ ) := U ( A, σ ) . Finally, when A is an arbitrary semisimple artinian ring, there exists an essentiallyunique factorization ( A, σ ) = Q ti =1 ( A i , σ i ) such that each factor ( A i , σ i ) fits intoexactly one of the previous two cases (see [12, p. 486], for instance). We then define U ( A, σ ) = t Y i =1 U ( A i , σ i ) . Example 4.
Suppose that A is an Azumaya algebra over a finite product of fields R = Q ti =1 K i and σ : A → A is an orthogonal involution. Then U ( A, σ ) =SO(
A, σ ). Indeed, writing (
A, σ ) = Q i ( A i , σ i ) with A i a central simple K i -algebra,we reduce into checking that U ( A i , σ i ) = SO( A i , σ i ). This follows from the defi-nition if [ A i ] = 0 (in Br K i ), and from [8, Lemma 2.6.1b] if [ A i ] = 0. N THE ORTHOGONAL GROUP 3
Theorem 5.
Let ( A, σ ) be a semisimple artinian ring with involution such that ∈ A × . Then the subgroup of U ( A, σ ) generated by the elements s y,a with y ∈ A , a ∈ Skew(
A, σ ) and y σ y + a ∈ A × contains U ( A, σ ) .Proof. We observed above that the elements s y,a are precisely the reflections of a( σ, f : A × A → A . The theorem is therefore a special case of[4, Theorem 5.8(ii)] (see also Remark 2.1 in that source). (cid:3) Theorem 6.
Let ( A, σ ) be a semilocal ring with involution such that ∈ A × . Write A = A/ Jac A , denote the quotient map A → A by a a and let σ : A → A begiven by a σ = a σ . Then the image of the map u u : U ( A, σ ) → U ( A, σ ) contains U ( A, σ ) .Proof. By Theorem 5, every element of U ( A, σ ) is a product of elements of theform s ˜ y, ˜ a with ˜ y ∈ A , ˜ a ∈ Skew(
A, σ ), ˜ y σ ˜ y + ˜ a ∈ A × . It is therefore enough toprove that there exists u ∈ U ( A, σ ) with u = s ˜ y, ˜ a . Choose y, a ∈ A with y = ˜ y and a = ˜ a . Replacing a with ( a − a σ ), we may assume that a ∈ Skew(
A, σ ). Since y σ y + a = ˜ y σ ˜ y + ˜ a ∈ ( A ) × , we have y σ y + a ∈ A × . We may therefore take u := s y,a , which clearly satisfies u = s ˜ y, ˜ a . (cid:3) Now we can prove Theorem 2.
Proof of Theorem 2.
Write J = Jac R . We first observe that Jac A = JA . Indeed, A/JA ∼ = A ⊗ ( R/J ) is Azumaya over
R/J , which is a product of fields, so
A/JA is semisimple artinian, meaning that JA ⊇ Jac A . On the other hand JA ⊆ Jac A because A is finitely generated as an R -module [9, Ch. II, Corollary 4.2.4].Now, using the notation of Theorem 6, A = A/ Jac A ∼ = A ⊗ ( R/ Jac R ) = A ⊗ Q ti =1 ( R/ m i ) ∼ = Q ti =1 A ( m i ), so we may identify Q ti =1 A ( m i ) with A . Underthis identification, Q i σ ( m i ) corresponds to σ , so we need to prove that the naturalmap u u : SO( A, σ ) → SO(
A, σ ) is surjective.Let v ∈ SO(
A, σ ). By Theorem 6 and Example 4, there exists u ∈ O( A, σ ) = U ( A, σ ) such that u = v . We claim that u ∈ SO(
A, σ ). Indeed, write α =Nrd A/R ( u ) ∈ R . Then α = 1, and so (1 − α ) is an idempotent. Since Nrd( v ) = 1in R/J , the image of (1 − α ) in R/J is (1 −
1) = 0. As J contains no nonzeroidempotents, it follows that (1 − α ) = 0, or rather, α = 1. (cid:3) Example 7.
The assumptions of Theorem 2 do not guarantee that O(
A, σ ) → Q m ∈ Max R O( A ( m ) , σ ( m )) is surjective in general. As a trivial counterexample onecould take R to be any non-local semilocal domain and note that O( R, id) = {± } while | Q m ∈ Max R O( R ( m ) , id R ( m ) ) | >
2. A counterexample with local R can beconstructed as follows. Take take R to be the localization of Z at 5 Z , let A be thequaternion Azumaya algebra R h i, j | i = j = − , ij = − ji i and let σ : A → A bethe orthogonal involution fixing i and j . Let m = 5 R denote the maximal ideal of R and let v be the image of 3 i in A ( m ). One readily checks that v ∈ O( A ( m ) , σ ( m ))and Nrd( v ) = −
1. However, v cannot be lifted to an element of O( A, σ ). Indeed, ifsuch a lift existed, it would have reduced norm −
1, but one can check directly thatelements of A have non-negative reduced norms.2. Proof of Theorem 1
Lemma 8.
Let A be an Azumaya algebra of constant degree d over a semilocalring R with ∈ R × and let σ : A → A be an orthogonal involution. If [ A ] = 0 ,then there exists u ∈ O( A, σ ) with u = 1 and reduced characteristic polynomial ( t + 1)( t − d − . In particular, Nrd
A/R ( u ) = − . ON THE ORTHOGONAL GROUP
Proof.
Since [ A ] = 0, we may assume that A = End R ( Q ) for some finitely generatedprojective R -module Q of rank d . By [13, Theorem 4.2a] (or, alternatively, [3,Proposition 4.6]), there exist δ ∈ µ ( R ), a rank-1 projective R -module L , and aunimodular L -valued bilinear form g : Q × Q → L satisfying g ( x, y ) = δg ( y, x )and g ( ax, y ) = g ( x, a σ y ) for all x, y ∈ Q , a ∈ A . (Here, unimodularity means that x g ( x, − ) : Q → Hom R ( Q, L ) is bijective.) Since R is semilocal and L has rank1, L ∼ = R , so we may assume L = R . Moreover, δ = 1 because σ is orthogonal;see [9, p. 170], for instance. Now, choose a vector x ∈ Q with g ( x, x ) ∈ R × —to see its existence, check it modulo Jac R and take an arbitrary lift. Writing P = { y ∈ Q : g ( x, y ) = 0 } , we have Q = xR ⊕ P and rank P = d −
1. Thereflection u := ( − id xR ) ⊕ id P is the required element. (cid:3) Proof of Theorem 1.
By writing R as a product of connected semilocal commu-tative rings and working over each factor separately, we may assume that R isconnected. Thus, d := deg A is constant on Spec R .That [ A ] = 0 implies the surjectivity of Nrd A/R : O(
A, σ ) → {± } has beenverified in Lemma 8, so we turn to prove the converse.Let u ∈ O( A, σ ) be an element with Nrd
A/R ( u ) = −
1. We let m , . . . , m t denotethe maximal ideals of R and set A i = A ( m i ), σ i = σ ( m i ). We further let u i denotethe image of u in A i .Since A carries an involution fixing R , we have [ A ] = [ A op ] = − [ A ]. By a theoremof Saltman [14], we also have d [ A ] = 0 in Br R , so [ A ] = 0 if d is odd. We maytherefore assume that d is even.If there exists 1 ≤ i ≤ t such that [ A i ] = 0 in Br( R/ m i ), then Nrd( u i ) = 1 by[8, Lemma 2.6.1b], which is impossible (because 2 ∈ R × ). Thus, [ A i ] = 0 for all i . Now, by Lemma 8, for every 1 ≤ i ≤ t , there is v i ∈ O( A i , σ i ) with reducedcharacteristic polynomial equal to ( t + 1)( t − d − .Observe that Nrd( u − i v i ) = 1, hence u − i v i ∈ SO( A i , σ i ). By Theorem 2, thereexists w ∈ SO(
A, σ ) such that the image of w in A i is u − i v i for all i . Writing v := uw ∈ O( A, σ ), we see that the image of v in A i is v i for all i .Let f = f v ∈ R [ t ] denote the reduced characteristic polynomial of v . Then f ≡ ( t + 1)( t − d − mod m i for all 1 ≤ i ≤ t . Note that σ : A → A preserves the reduced characteristicpolynomial (use [9, III. § R -algebra S such that ( A ⊗ S, σ ⊗ id S ) is isomorphic to M d ( S ) with the transposeinvolution). Since v − = v σ , this means that f v − = f v σ = f , so f (0) − t d f ( t − ) = f in R [ t, t − ]. Substituting t = − f (0) = Nrd A/R ( v ) = − d is even, we get − f ( −
1) = f ( − f ( −
1) = 0 and t + 1 | f .Write f = ( t + 1) g and g = ( t + 1) r + α , where g, r ∈ R [ t ] and α = g ( − f ≡ ( t + 1)( t − d − mod m i , we have g ≡ ( t − d − mod m i and α ≡ ( − d − mod m i . As this holds for all i , we have α ∈ R × . Thus, α − g − α − ( t + 1) r = α − α = 1 . Put e = α − g ( v ) and e ′ = − α − ( v + 1) r ( v ). Then e + e ′ = 1 A and ee ′ = e ′ e = 0(because ( v + 1) g ( v ) = f ( v ) = 0). Thus, e = e ( e + e ′ ) = e . Let e i denote the imageof e in A i . Then e i = ( − − d g ( v i ) = ( − − d ( v i − d − has rank one. This meansthat eAe is a projective R -algebra of rank 1, so eAe ∼ = R . Since eAe ∼ = End A ( eA ),we have [ A ] = [ eAe ] = [ R ] = 0 [9, Proposition III.5.3.1]. (cid:3) Example 9.
The “if” part of Theorem 1 is false if R is not assumed to be semilocal.Indeed, take R to be an integral domain with 2 ∈ R × admitting a non-principal N THE ORTHOGONAL GROUP 5 invertible fractional ideal L (we view L as a subset of the fraction field of R ). Define A = (cid:20) R L − L R (cid:21) and let σ : A → A be the involution given by (cid:2) a bc d (cid:3) σ = [ d bc a ]. To see that A is Azumaya over R and σ is orthogonal, observe that there is an isomorphism A ∼ = End R ( R ⊕ L ) under which σ is the adjoint to the unimodular symmetricbilinear form f : ( R ⊕ L ) × ( R ⊕ L ) → L given by f ( (cid:2) r ℓ (cid:3) , (cid:2) r ℓ (cid:3) ) = r ℓ + r ℓ . Thisalso shows that [ A ] = 0 in Br R .Straightforward computation shows that elements of O( A, σ ) of determinant − (cid:2) x − x (cid:3) , where x ∈ L . If such an element exists, then x − R ⊆ L − ,or rather, L ⊆ xR . Since x ∈ L , this means that L = xR , contradicting ourassumption that L is not principal. Thus, O( A, σ ) = SO(
A, σ ) and Nrd
A/R :O(
A, σ ) → µ ( R ) is not surjective.We remark that if R ⊕ L ∼ = M ⊕ M for some invertible fractional ideal M , thenwe also have A ∼ = End R ( M ⊕ M ) ∼ = M ( R ). Such examples exist, e.g., take R to bea Dedekind domain with class group containing an element [ M ] of order 4 (use [2],for instance) and let L = M . References [1] Maurice Auslander and Oscar Goldman. The Brauer group of a commutative ring.
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