Quadratic descent of hermitian forms (with an erratum)
aa r X i v : . [ m a t h . R A ] F e b Erratum to “Quadratic descent of hermitianforms” [Math. Nachr. 292 (2019), no. 10,2294–2299.]
A.-H. NokhodkarFebruary 26, 2020
Abstract
There exists a mistake in the proof of Theorem 4.2. We present a newproof of this theorem, which shows that the main results of the paper arestill true.
In [3, § K/F , a system ofquadratic forms q = ( q , · · · , q m ) over K has a descent to F if and only if every q i has a descent to F . This is incorrect in general. For example, let K/F be a separable quadratic extension and let α ∈ K \ F . The 2-fold system ofquadratic forms ( h i , h α i ) over K has no descent to F , while both forms h i and h α i can be descended to F . This claim is used in the proof of Proposition 3.2,which in turn plays a critical role in the proof of the implication ( ii ) ⇒ ( i ) ofTheorem 4.2. Hence, the given proof of Theorem 4.2 is not true. We presenta proof of the implication ( ii ) ⇒ ( i ) of Theorem 4.2 with a different approach,which does not depend on Proposition 3.2. Since Proposition 3.2 was only usedin the proof of Theorem 4.2, one can remove this proposition, with no effect onthe other results of the paper. Throughout this note, (
D, θ ) is a finite dimensional division algebra with in-volution of the first kind over a field F and λ = ±
1. We also fix
K/F as aseparable quadratic extension such that D K is a division ring.Let q = ( q , · · · , q n ) be a system of quadratic forms over K and let s : K → F be a nonzero F -linear map. We call the system of quadratic forms s ∗ ( q ) := ( s ∗ ( q ) , · · · , s ∗ ( q n )) over F the transfer of q . Similarly, one can definethe transfer of a hermitian form as follows: let ( V, h ) be a λ -hermitian spaceover ( D, θ ) K . Extend s to a D -linear map s D : D K → D and define s ∗ ( h ) : V × V → D via s ∗ ( h )( u, v ) = s D ( h ( u, v )) for all u, v ∈ V. s ∗ ( h ) is a λ -hermitian form over ( D, θ ). It is easy to see that if h isregular then so is s ∗ ( h ). It is also easily verified that if h is metabolic then sois s ∗ ( h ) (see [1, p. 362]). Lemma 1.
Let ( V, h ) be a λ -hermitian form over ( D, θ ) K and let s : K → F be a nonzero F -linear map. Then s ∗ ( q h, B K ) = q s ∗ ( h ) , B for every basis B of Sym λ ( D, θ ) .Proof. Write B = { u , · · · , u m } for some u , · · · , u m ∈ Sym λ ( D, θ ). It sufficesto show that s ∗ ( q u i ⊗ h, B K ) = q u i s ∗ ( h ) , B for all i = 1 , · · · , m . Fix an index 1 i m and a vector v ∈ V . Write K = F ( η ) for some η ∈ K . Since Sym λ (( D, θ ) K ) =Sym λ ( D, θ ) ⊗ K , identifying D with a subalgebra of D K , one can write h ( v, v ) = α + βη , where α, β ∈ Sym λ ( D, θ ). Write α = P mj =1 a j u j and β = P mj =1 b j u j , for some a j , b j ∈ F , j = 1 , · · · , m . Extend π i : Sym λ ( D, θ ) → F to a K -linearmap ( π i ) K : Sym λ (( D, θ ) K ) → K . Then s ∗ ( q u i ⊗ h, B K )( v ) = s (( π i ) K ( h ( v, v )))= s (( π i ) K ( P mj =1 ( a j u j + b j u j η )))= s ( a i + b i η ) = a i s (1) + b i s ( η )= π i ( P mj =1 a j u j s (1) + P mj =1 b j u j s ( η ))= π i ( αs (1) + βs ( η )) = π i ( s D ( α + βη ))= π i ( s D ( h ( v, v ))) = π i ( s ∗ ( h )( v, v )) = q u i s ∗ ( h ) , B ( v ) . We recall that a regular even λ -hermitian form is metabolic if and only if itis hyperbolic (see [2, Ch. I, (3.7.3)]). Lemma 2.
Let ( V, h ) be a regular even λ -hermitian form over ( D, θ ) K and let s : K → F be a nonzero F -linear map with s (1) = 0 . Then h has a descent to ( D, θ ) if and only if s ∗ ( h ) is hyperbolic.Proof. Observe first that s ∗ ( h ) is even and regular. Suppose that ( V, h ) ≃ ( V ′ , h ′ ) K for some λ -hermitian space ( V ′ , h ′ ) over ( D, θ ). We may identify V ′ ⊆ V and D ⊆ D K . The assumption s (1) = 0 then implies that s D | D = 0,hence, s ∗ ( h ) | V ′ × V ′ = 0. Since dim D V ′ = dim D V , one concludes that s ∗ ( h )is metabolic and therefore hyperbolic.To prove the converse, write h ≃ h ⊥ h , where h is anisotropic and h is hyperbolic. Clearly, h has a descent to ( D, θ ). Hence, it suffices to showthat h has a descent to ( D, θ ). Since h is metabolic, s ∗ ( h ) is metabolicand therefore hyperbolic. It follows that s ∗ ( h ) is also hyperbolic, because s ∗ ( h ) ≃ s ∗ ( h ) ⊥ s ∗ ( h ). Since h is anisotropic and the map s D is nonzero with s D | D = 0, one concludes that h represents a nonzero element α ∈ D . By [2, Ch.I, (3.6.2)], there exists a regular (even) λ -hermitian form h ′ over ( D, θ ) K suchthat h ≃ h α i K ⊥ h ′ . Since h α i K has a descent to ( D, θ ), the above argumentshows that s ∗ ( h α i K ) is hyperbolic, hence s ∗ ( h ′ ) is also hyperbolic. The resultnow follows by induction on dim D h .As in [5, Ch. 9], we say that a system ( V, q ) of quadratic forms over F is metabolic if there exists a subspace L of V with dim F L > dim F V such that q | L = 0. 2 emma 3. Let ( V, q ) be a system of quadratic forms over K and let s : K → F be a nonzero F -linear map with s (1) = 0 . If q has a descent to F then s ∗ ( q ) ismetabolic.Proof. Write (
V, q ) ≃ ( V ′ , q ′ ) K , where ( V ′ , q ′ ) is a system of quadratic formsover F . Considering V ′ as an F -subspace of V , one has s ∗ ( q ) | V ′ = 0. Sincedim F V ′ = dim F V , the system s ∗ ( q ) is metabolic.We are now able to correct the proof of Theorem 4.2 of [3] as follows: Proof of implication ( ii ) ⇒ ( i ) of [3, (4.2)]. Let s : K → F be a nonzero F -linear map with s (1) = 0. If q h, B K has a descent to F then s ∗ ( q h, B K ) ismetabolic by Lemma 3. Hence, q s ∗ ( h ) , B is also metabolic by Lemma 1. Theassumption D = F or λ = − F = 2 or D = F . Itfollows from [4, (4.2)] that s ∗ ( h ) is metabolic and therefore hyperbolic. Theresult now follows from Lemma 2. Acknowledgements.
The author is grateful to Uriya First, who pointed outthe mistake in the proof of Proposition 3.2. This research is partially supportedby the University of Kashan under the grant number 890193/2.
References [1] E. Bayer-Fluckiger, H. W. Lenstra, Forms in odd degree extensions and self-dual normalbases.
Amer. J. Math. (1990), no. 3, 359–373.[2] M.-A. Knus,
Quadratic and Hermitian Forms Over Rings , Grundlehren der Mathema-tischen Wissenschaften, vol. , Springer-Verlag, 1991.[3] A.-H. Nokhodkar, Quadratic descent of hermitian forms.
Math. Nachr. (2019), no.10, 2294–2299.[4] A.-H. Nokhodkar, Hermitian forms and systems of quadratic forms.
Doc. Mat. (2018), 747–758.[5] A. Pfister, Quadratic forms with applications to algebraic geometry and topology . Lon-don Mathematical Society Lecture Note Series, . Cambridge University Press, Cam-bridge, 1995.
A.-H. Nokhodkar, [email protected] ,Department of Pure Mathematics, Faculty of Science, University of Kashan, P. O.Box 87317-51167, Kashan, Iran. r X i v : . [ m a t h . R A ] F e b Quadratic descent of hermitian forms
A.-H. Nokhodkar
Abstract
Quadratic descent of hermitian and skew hermitian forms over divisionalgebras with involution of the first kind in arbitrary characteristic isinvestigated and a criterion, in terms of systems of quadratic forms, isobtained. A refined result is also obtained for hermitian (resp. skewhermitian) forms over a quaternion algebra with symplectic (resp. or-thogonal) involution.
Mathematics Subject Classification:
Keywords:
Hermitian form, quadratic descent, system of quadratic forms,quaternion algebra.
The classical Springer theorem asserts that if
K/F is a field extension of odddegree, then the canonical homomorphism of Witt rings r K/F : W ( F ) → W ( K )is injective. A theorem proved by Rosenberg and Ware [10] states that if K/F isa Galois extension of odd degree in characteristic not two, then r K/F : W ( F ) → W ( K ) Gal(
K/F ) is an isomorphism. The image of r K/F was studied further in [11]for extensions of odd degree in characteristic not two and a descent propertyof Pfister forms was obtained. Also, for purely inseparable extensions, somedescent properties of the associated Witt group of hermitian forms of a centralsimple algebra with involution over K were studied in [1].In this work, we investigate quadratic descent of hermitian and skew hermi-tian forms over division algebras with involution of the first kind in arbitrarycharacteristic. Our study is based on some constructions of [7]. For λ = ± h be a λ -hermitian form over a division algebra with involution of the firstkind ( D, θ ). For every basis B of the set of λ -symmetrized elements of ( D, θ ),a system of quadratic forms q h, B was associated to h in [7]. This system is ageneralization of the so-called Jacobson’s trace form defined in [4] (see also [12]).It was shown that, except for certain special cases, the system q h, B determinesthe isotropy and metabolicity of h , as well as the isometry class of h (see [7, § h , in terms of q h, B , to have a quadratic descent. Ourmain result is Theorem 4.2 which states that for a separable quadratic extension K/F , a division F -algebra with involution of the first kind ( D, θ ) and an even λ -hermitian space ( V, h ) over (
D, θ ) K the following statements are equivalent(except for the case where D = F and λ = − i ) h has a descent to ( D, θ ). 1 ii ) There exists a basis B of Symd λ ( D, θ ) for which q h, B has a descent to F .In § λ -hermitian forms overquaternion division algebras with involution of the first kind whose correspond-ing systems of quadratic forms reduce to quadratic forms. For a field extension K/F and a quaternion division algebra with involution of the first kind (
Q, σ )over K , we will find in Theorem 5.2 a criterion for the aforementioned hermi-tian forms over ( Q, σ ) to have descents to F . This criterion is stronger thanTheorem 4.2, in the sense that the underlying algebra with involution ( Q, σ ) isnow defined over K , while the pair ( D, θ ) K in Theorem 4.2 was extended from( D, θ ) over F . Let A be a central simple algebra over a field F . An involution on A is anantiautomorphism of A of order two. An involution σ on A is said to be of thefirst kind if it restricts to the identity on F . Otherwise, it is said to be of the second kind . For a central simple algebra with involution ( A, σ ) and λ = ± λ ( A, σ ) = { x ∈ A | σ ( x ) = λx } , Symd λ ( A, σ ) = { x + λσ ( x ) | x ∈ A } . Note that Sym λ ( A, σ ) ⊆ Symd λ ( A, σ ) with equality if char F = 2.Let K/F be a field extension and let (
A, σ ) be a central simple algebra withinvolution over K . We say that ( A, σ ) has a descent to F if there exists a centralsimple algebra with involution ( A ′ , σ ′ ) over F such that ( A, σ ) ≃ ( A ′ , σ ′ ) K ,where ( A ′ , σ ′ ) K := ( A ′ K , σ ′ K ) is the K -algebra with involution obtained from( A ′ , σ ′ ) by extending scalars to K , i.e., A ′ K = A ′ ⊗ F K and σ ′ K ≃ σ ′ ⊗ id.Let ( A, σ ) be a central simple algebra with involution of the first kind overa field F . According to [6, (2.1)] for every splitting field L of A , the involution σ L becomes adjoint to a symmetric or antisymmetric bilinear form b over L .We say that σ is symplectic if b is alternating and orthogonal otherwise. Also,the involution σ is said to be of type of type −
1) if b is symmetric(resp. antisymmetric). Note that an involution of the first kind over a field ofcharacteristic different from 2 is of type 1 if and only if it is orthogonal.Let ( D, θ ) be a finite dimensional division algebra with involution over a field F and let λ = ±
1. Let V be a finite dimensional right vector space over D . A λ -hermitian form on V is a bi-additive map h : V × V → D satisfying h ( uα, vβ ) = θ ( α ) h ( u, v ) β and h ( v, u ) = λθ ( h ( u, v )) for all u, v ∈ V and α, β ∈ D . The pair( V, h ) is called a λ -hermitian space over ( D, θ ). If λ = 1 (resp. λ = − h isalso called a hermitian (resp. skew hermitian ) form . For a subspace W of V ,the orthogonal complement of W (with respect to h ) is defined as W ⊥ h = { v ∈ V | h ( v, w ) = 0 for all w ∈ W } . The form h is called regular if V ⊥ h = { } . A λ -hermitian form h over ( D, θ ) iscalled even if h ( v, v ) ∈ Symd λ ( D, θ ) for all v ∈ V .Let ( D, θ ) be a division algebra with involution over F and let ( V, h ) be a λ -hermitian space over ( D, θ ). If
K/F is a finite extension for which D K is a2ivision algebra then there exists a λ -hermitian space ( V K , h K ) over ( D, θ ) K ,where V K = V ⊗ F K and h K : V K × V K → D K is induced by h K ( u ⊗ α, v ⊗ β ) = h ( u, v ) ⊗ αβ for u, v ∈ V and α, β ∈ K .Let K/F be a field extension and let (
D, θ ) be a division algebra with invo-lution of the first kind over K . For λ = ±
1, let (
V, h ) be a λ -hermitian spaceover ( D, θ ). We say that h has a descent to F if there exist a division algebrawith involution ( D ′ , θ ′ ) over F and a λ -hermitian space ( V ′ , h ′ ) over ( D ′ , θ ′ )such that ( D, θ ) ≃ ( D ′ , θ ′ ) K and ( V, h ) ≃ ( V ′ K , h ′ K ). Further, if ( D ′′ , θ ′′ ) is adivision algebra with involution over F satisfying ( D, θ ) ≃ ( D ′′ , θ ′′ ) K , we saythat h has a descent to ( D ′′ , θ ′′ ) if there exists a λ -hermitian space ( V ′′ , h ′′ ) over( D ′′ , θ ′′ ) such that ( V, h ) ≃ ( V ′′ K , h ′′ K ). Let V be a finite dimensional vector space over a field F of arbitrary character-istic. A quadratic form on V is a map q : V → F for which (i) q ( αv ) = α q ( v )for all α ∈ F and v ∈ V ; (ii) the map b q : V × V → F given by b q ( u, v ) = q ( u + v ) − q ( u ) − q ( v ) is a bilinear form. The pair ( V, q ) is called a quadraticspace over F . The orthogonal complement of a subspace W of V is defined as W ⊥ = { v ∈ V | b q ( v, w ) = 0 for all w ∈ W } . We say that q is regular if V ⊥ = { } . The form q is called isotropic if thereexists a nonzero vector v ∈ V such that q ( v ) = 0 and anisotropic otherwise. Let K/F be a field extension and let (
V, q ) be a quadratic space over F . We saythat ( V, q ) (or the quadratic form q itself) has a descent to F if there exists aquadratic space ( V ′ , ρ ) over F such that ( V, q ) ≃ ( V ′ K , ρ K ), where ρ K : V ′ K → K is the quadratic form satisfying ρ K ( v ⊗ α ) = α ρ ( v ) for all α ∈ K and v ∈ V ′ .An m -fold system of quadratic forms on V is an m -tuple q = ( q , · · · , q m ),where every q i : V → F is a quadratic form. The system q may be identifiedwith a quadratic map q : V → F m (see [9, p. 132]). This quadratic map inducesa polar map b q : V × V → F m defined by b q ( u, v ) = q ( u + v ) − q ( u ) − q ( v ) . The system q is called regular if there is no nonzero vector v ∈ V such that b q ( u, v ) = 0 ∈ F m for all u ∈ V . We say that q is totally regular if every q i isregular. Two systems of quadratic forms q : V → F m and q ′ : V ′ → F m aresaid to be equivalent if there exists a linear isomorphism f : V → V ′ such that q ′ ( f ( v )) = q ( v ) for every v ∈ V . In this case, we write ( V, q ) ≃ ( V ′ , q ′ ) or simply q ≃ q ′ .Let K/F be a field extension. Let V be a finite dimensional vector spaceover K and let q : V → K m be a quadratic map. We say that q has a descent to F m if there exist a vector space V ′ over F and a quadratic map ρ : V ′ → F m such that q ≃ ρ K , where ρ K : V ′ K → K m is the quadratic map induced by ρ K ( v ⊗ α ) = ρ ( v ) α for v ∈ V ′ and α ∈ K . Note that if q = ( q , · · · , q m ) then q has a descent to F m if and only if every q i has a descent to F .Let K/F be a finite field extension and let s : K → F be an F -linearfunctional. If ( V, b ) is a bilinear space over K the transfer s ∗ ( b ) of b is asymmetric bilinear form on V , as a vector space over F , defined by s ∗ ( b )( u, v ) =3 ( b ( u, v )) for u, v ∈ V . Also, if q is a quadratic form on V , the transfer s ∗ ( q ) of q is a quadratic F -form on V whose polar form is s ∗ ( b q ) and satisfies ( s ∗ ( q ))( v ) = s ( q ( v )) for v ∈ V . Lemma 3.1.
Let
K/F be a separable quadratic extension and let s : K → F bea nonzero F -linear functional with s (1) = 0 . A regular quadratic form q over K has a descent to F if and only if s ∗ ( q ) is hyperbolic.Proof. Observe first that if char F = 2 then q is even dimensional. Hence, the‘only if’ implication follows from [3, (34.4)] if char F = 2 and the exactness ofthe sequence in [3, (34.9)] at I q ( K ) if char F = 2. To prove the converse, write q ≃ q an ⊥ k H , where q an is anisotropic, H is the hyperbolic plane and k is anonnegative integer. By [3, (20.1)], we have s ∗ ( q ) ≃ s ∗ ( q an ) ⊥ s ∗ ( k H ). Also, s ∗ ( k H ) is hyperbolic by [3, (20.5)]. Hence, s ∗ ( q an ) is also hyperbolic. The proofsof [3, (34.4)] (if char F = 2) and [3, (34.9)] (if char F = 2) show that q an has adescent to F . Hence, q has a descent to F , proving the converse implication. Proposition 3.2.
Let
K/F be a separable quadratic field extension and let q ≃ ρ ⊥ ϕ be an isometry of totally regular quadratic maps over K m . If q and ρ have descents to F m , then so does ϕ .Proof. Write q = ( q , · · · , q m ), ρ = ( ρ , · · · , ρ m ) and ϕ = ( ϕ , · · · , ϕ m ) for somequadratic forms q i , ρ i and ϕ i over K , i = 1 , · · · , m . Then all q i and ρ i havedescents to F . We want to show that for i = 1 , · · · , m , ϕ i has a descent to F .Fix an index i with 1 i m .Let s : K → F be a nonzero F -linear functional with s (1) = 0. Since q i ≃ ρ i ⊥ ϕ i , by [3, (20.1)] we have s ∗ ( q i ) ≃ s ∗ ( ρ i ) ⊥ s ∗ ( ϕ i ). Also, s ∗ ( q i ) and s ∗ ( ρ i ) are hyperbolic by [3, (34.9)], which implies that s ∗ ( ϕ i ) is also hyperbolic.The result now follows from Lemma 3.1. We recall here some constructions from [7].Let (
D, θ ) be a finite dimensional division algebra with involution of the firstkind over a field F . Let ( V, h ) be a λ -hermitian space over ( D, θ ), where λ = ± B = { u , · · · , u n } of Sym λ ( D, θ ) over F and let { π , · · · , π n } be its dual basis of Hom(Sym λ ( D, θ ) , F ). For i = 1 , · · · , n , define a map q u i h, B : V → F via q u i h, B ( v ) = π i ( h ( v, v )) . Let q h, B = ( q u h, B , · · · , q u n h, B ). As observed in [7], considering V as a vector spaceover F , the map q h, B is a system of quadratic forms over F . If h is even, wewill consider B as a basis of Symd λ ( D, θ ), instead of Sym λ ( D, θ ). Proposition 4.1.
Let ( D, θ ) be a finite dimensional division algebra with in-volution of the first kind over a field F and let ( V, h ) be an even λ -hermitianspace over ( D, θ ) , where λ = ± . Suppose that either D = F or λ = − . If h isregular then for every basis B of Symd λ ( D, θ ) the system q h, B is totally regular. roof. In view of [6, (2.6)], the hypothesis D = F or λ = − λ ( D, θ ) = { } . Hence, the result follows from [7, (3.5)].Let ( D, θ ) be a finite dimensional division algebra with involution of the firstkind over a field F . Let K/F be a finite extension such that D K is a divisionring. Then for every F -basis B of Symd λ ( D, θ ), the set { u ⊗ | u ∈ B} is a K -basis of Symd λ (( D, θ ) K ) = Symd λ ( D, θ ) ⊗ K . We denote this basis by B K . Theorem 4.2.
Let ( D, θ ) be a finite dimensional division algebra with involu-tion of the first kind over a field F . Let K/F be a separable quadratic extensionsuch that D K is a division ring. Set λ = ± and suppose that either D = F or λ = − . For a regular even λ -hermitian space ( V, h ) over ( D, θ ) K the followingstatements are equivalent. ( i ) h has a descent to ( D, θ ) . ( ii ) There exists a basis B of Symd λ ( D, θ ) for which q h, B K has a descent to F .Proof. If h has a descent h ′ to ( D, θ ), then for every basis B of Symd λ ( D, θ ) wehave q h, B K ≃ ( q h ′ , B ) K by [7, (3.4)]. This proves the implication ( i ) ⇒ ( ii ).Conversely, suppose that ( V, q h, B K ) ≃ ( V ′ K , q ′ K ), where B = { u , · · · , u m } is a basis of Symd λ ( D, θ ), V ′ is a vector space over F and q ′ = ( q ′ , · · · , q ′ m )is a quadratic map on V ′ . We use induction on the right dimension dim D K V of V over D K . The assumption D = F or λ = − v ∈ V such that h ( v, v ) = 0, i.e., q h, B K ( v ) = 0. As q h, B K ≃ q ′ K it follows that q ′ ( v ) = 0 for some v ∈ V ′ . Let h : v D × v D → D be the λ -hermitian formsatisfying h ( v , v ) = m X i =1 q ′ i ( v ) u i ∈ Symd λ ( D, θ ) . We may identify ( v ⊗ D K with a subspace of V , so that h ( v ⊗ , v ⊗
1) = m X i =1 q u i ⊗ h, B K ( v ⊗ u i ⊗
1) = m X i =1 q ′ i ( v )( u i ⊗
1) = h ( v , v ) ⊗ . Hence, h | ( v ⊗ D K × ( v ⊗ D K ≃ ( h ) K . If dim D K V = 1 then h is a descent of h and we are done. Otherwise, let W = (( v ⊗ D K ) ⊥ h ⊆ V and h ′ = h | W × W . Since h and ( h ) K are regular, we have h ≃ ( h ) K ⊥ h ′ by [5, Ch. I, (3.6.2)],which implies that q h, B K ≃ q ( h ) K , B K ⊥ q h ′ , B K . Note that q ( h ) K , B K ≃ ( q h , B ) K ,so q ( h ) K , B K has a descent to F . By Proposition 4.1, the systems q h, B K , q ( h ) K , B K and q h ′ , B K are all totally regular. It follows from Proposition 3.2 that q h ′ , B K has a descent to F . The pair ( W, h ′ ) is therefore an even λ -hermitian space over( D, θ ) K such that q h ′ , B K has a descent to F . By induction hypothesis, h ′ has adescent to ( D, θ ). Hence, h ≃ ( h ) K ⊥ h ′ has a descent to ( D, θ ), proving theresult.
Remark 4.3.
The implication ( ii ) ⇒ ( i ) in Theorem 4.2 does not hold if D = F and λ = −
1. Indeed, if char F = 2 then Symd λ ( D, θ ) = { } and the system q h, B K is trivial. Hence, q h, B K does not give any information about h .5uppose now that char F = 2. Then Sym λ ( D, θ ) = F . Write K = F ( η ) forsome η ∈ K with η + η = δ ∈ F \ ℘ ( F ), where ℘ ( F ) = { x + x | x ∈ F } .Suppose that the field F satisfies F = F and choose an element a ∈ F \ F .Consider the diagonal bilinear form h = h , aδ + aη i over K with the diagonalbasis { v, w } , hence h ( v, v ) = 1 , h ( w, w ) = 1 + aδ + aη and h ( v, w ) = h ( w, v ) = 0 . Then the determinant of h is (1 + a + a δ ) F × ∈ F × /F × . Let N K/F : K → F be the norm of K over F . Since N K/F (1 + aδ + aη ) = 1 + a + a δ / ∈ F ,h has no descent to F . However, taking B = { } , the system q = q h, B K is thequadratic form associated to h (which is totally singular, i.e., its polar form iszero). We have q ( v ) = 1 and q ( ηv + (1 + η ) w ) = 1 + aδ ∈ F , implying that q hasa descent to F . This proves the claim. Note that this example also shows thatthe implication ( ii ) ⇒ ( i ) in Theorem 4.2 is not necessarily valid for non-evenhermitian forms. Let
K/F be a quadratic separable extension with the nontrivial automorphism ι . For a central simple algebra A over K , define the conjugate algebra ι A = { ι x | x ∈ A } with operations ι x + ι y = ι ( x + y ) , ι x ι y = ι ( xy ) and ι ( αx ) = ι ( α ) ι x, for x, y ∈ A and α ∈ K . Let s : ι A ⊗ K A → ι A ⊗ K A be the ι -semilinear mapinduced by s ( ι x ⊗ y ) = ι y ⊗ x for x, y ∈ A . The corestriction cor K/F ( A ) of A isdefined as cor K/F ( A ) = { x ∈ ι A ⊗ K A | s ( x ) = x } . By [6, (3.13 (4))], cor
K/F ( A ) is a central simple algebra over F . Lemma 5.1.
Let
K/F be a separable quadratic extension and let ( Q, σ ) be aquaternion algebra with involution of the first kind over K . ( i ) If σ is symplectic then ( Q, σ ) has a descent to F if and only if cor K/F ( Q ) splits. ( ii ) If σ is orthogonal then ( Q, σ ) has a descent to F if and only if cor K/F ( Q ) splits and there exists u ∈ Symd − ( Q, σ ) such that u ∈ F . Furthermore,if these conditions are satisfied then there exists a descent ( Q ′ , σ ′ ) of ( Q, σ ) such that u ∈ Symd − ( Q ′ , σ ′ ) .Proof. ( i ) According to [6, (2.22)] and [13, Ch. 8, (9.5)], Q has a descent to F ifand only if cor K/F ( Q ) splits. By [6, (2.21)], the unique symplectic involution on Q is the canonical involution. Hence, if cor K/F ( Q ) splits then for every descent Q ′ of Q , one has ( Q, σ ) ≃ ( Q ′ , γ ) K , where γ is the canonical involution on Q ′ .It follows that ( Q, σ ) has a descent to F if and only if cor K/F ( Q ) splits.( ii ) If char F = 2, the claim follows from [2, (2.4)] and its proof. Otherwise,it can be found in [8, (4.7)]. 6et ( Q, σ ) be a quaternion division algebra with involution of type ε overa field F and set λ = − ε . Then dim F Symd λ ( Q, σ ) = 1 by [6, (2.6)]. Choosea basis B = { u } of Symd λ ( Q, σ ). Let (
V, h ) be an even λ -hermitian space on( Q, σ ). Then the quadratic map q h, B reduces to a quadratic form, which wedenote it by q h,u . Note that the form q h,u is uniquely determined, up to a scalarfactor. Also, if σ is symplectic then Symd ( Q, σ ) = F . Taking u = 1, the form q h,u coincides with the Jacobson’s trace form of h , introduced in [4]. We willsimply denote this from by q h .Let K/F be a quadratic separable extension and let (
D, θ ) be a divisionalgebra with involution of the first kind over K . If ( D, θ ) has a descent ( D ′ , θ ′ )to F then Theorem 4.2 gives a criterion for hermitian spaces over ( D, θ ) to havea descent to ( D ′ , θ ′ ). In the case where ( D, θ ) is a quaternion division algebrawith involution of type ε over K and λ = − ε , using Lemma 5.1, we state thefollowing criterion for λ -hermitian spaces over ( D, θ ) to have a descent to F . Theorem 5.2.
Let
K/F be a separable quadratic extension and let ( Q, σ ) be aquaternion division algebra with involution of type ε over K . Set λ = − ε . Let ( V, h ) be a regular λ -hermitian space over ( Q, σ ) . ( i ) If σ is symplectic then h has a descent to F if and only if cor K/F ( Q ) splitsand q h has a descent to F . ( ii ) If σ is orthogonal then h has a descent to F if and only if cor K/F ( Q ) splitsand q h,u has a descent to F for some u ∈ Symd − ( Q, σ ) with u ∈ F .Proof. ( i ) If h has a descent to F then there exists a quaternion F -algebrawith involution ( Q ′ , σ ′ ) and a hermitian form h ′ on ( Q ′ , σ ′ ) such that ( Q, σ ) ≃ ( Q ′ , σ ′ ) K and h ≃ h ′ K . By Lemma 5.1 ( i ), cor K/F ( Q ) splits. We also have q h ≃ ( q h ′ ) K , proving the ‘only if’ implication.Conversely, if cor K/F ( Q ) splits then ( Q, σ ) has a descent ( Q ′ , σ ′ ) to F , thanksto Lemma 5.1 ( i ). Since q h has a descent to F , the result follows from Theorem4.2 by taking B = { } .( ii ) Suppose that h has a descent to F . Then there exists a quaternion F -algebra with involution ( Q ′ , σ ′ ) and a skew hermitian form h ′ on ( Q ′ , σ ′ )such that ( Q, σ ) ≃ ( Q ′ , σ ′ ) K and h ≃ h ′ K . By [6, (3.13 (5))], cor K/F ( Q ) splits.Let u ′ ∈ Symd − ( Q ′ , σ ′ ) be a nonzero element. Considering the isomorphism( Q, σ ) ≃ ( Q ′ , σ ′ ) K as an identification, we have u := u ′ ⊗ ∈ Symd − ( Q, σ ), u ∈ F and q h,u ≃ ( q h ′ ,u ′ ) K .Conversely, suppose that cor K/F ( Q ) splits and ( V, q h,u ) ≃ ( V ′ , q ′ ) K , where u ∈ Symd − ( Q, σ ) is an element with u ∈ F and ( V ′ , q ′ ) is a quadraticspace over F . By Lemma 5.1 ( ii ), ( Q, σ ) has a descent ( Q ′ , σ ′ ) to F with u ∈ Symd − ( Q ′ , σ ′ ). As Symd − ( Q, σ ) is one-dimensional, the result followsfrom Theorem 4.2 by taking B = { u } . References [1] E. Bayer-Fluckiger, D. A. Moldovan, Descent properties of Hermitian Witt groups ininseparable extensions.
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A.-H. Nokhodkar, [email protected] ,Department of Pure Mathematics, Faculty of Science, University of Kashan, P. O.Box 87317-51167, Kashan, Iran.,Department of Pure Mathematics, Faculty of Science, University of Kashan, P. O.Box 87317-51167, Kashan, Iran.