The discriminant Pfister form of an algebra with involution of capacity 4
Karim Johannes Becher, Nicolas Grenier-Boley, Jean-Pierre Tignol
aa r X i v : . [ m a t h . R A ] A ug THE DISCRIMINANT PFISTER FORM OF AN ALGEBRAWITH INVOLUTION OF CAPACITY FOUR
KARIM JOHANNES BECHER, NICOLAS GRENIER-BOLEY,AND JEAN-PIERRE TIGNOL
Abstract.
To an orthogonal or unitary involution on a central simple algebraof degree 4, or to a symplectic involution on a central simple algebra of degree8, we associate a Pfister form that characterises the decomposability of thealgebra with involution. In this way we obtain a unified approach to knowndecomposability criteria for several cases, and a new result for symplectic in-volutions on degree 8 algebras in characteristic 2.
Classification (MSC 2010):
Keywords:
Central simple algebra, involution, decomposability, ´etale algebra,quadratic form, isotropy, hyperbolicity, discriminant, cohomological invariant,composition formula, characteristic two Introduction
The decomposability of central simple algebras with involution into tensorproducts of quaternion algebras with involution has been in the focus of muchstudy, motivated notably by the analogy between decomposable involutions andquadratic Pfister forms; see [3], [4], and the references in [24]. When the char-acteristic is different from 2, vanishing of the first cohomological invariant yieldsa necessary condition for decomposability, which is also sufficient for algebras oflow degree; see [24]. The aim of this article is to establish in arbitrary charac-teristic a decomposability criterion for algebras with involution of low degree interms of a canonically associated quadratic Pfister form. (We refer to Section 4for the precise notion of total decomposability for algebras with involution.)
Main Theorem.
Let F be a field. Let n ∈ { , , } . Let A be an F -algebra with dim F A = 2 n +3 and σ an F -linear involution on A such that the following holds,depending on n : Date : 20 August, 2020.This work was supported by the FWO Odysseus Programme (project
Explicit Methods inQuadratic Form Theory ), funded by the Fonds Wetenschappelijk Onderzoek – Vlaanderen. Thethird author acknowledges support from the Fonds de la Recherche Scientifique–FNRS underCDR grants 1.5054.12F, J.0014.15, J.0149.17, J.0159.19. Work on this paper was initiatedin 2010 while the first and the third author were, respectively, Fellow and Senior Fellow of theZukunftskolleg, Universit¨at Konstanz, whose hospitality is gratefully acknowledged. n = 1 : char F = 2 , A is a central simple F -algebra of degree and σ is an or-thogonal involution on A . n = 2 : Either A is a central simple K -algebra of degree over some quadraticfield extension K of F and σ is an F -linear involution on A such that σ | K = id K , or ( A, σ ) ≃ ( A × A op , sw ) , where A is a central simple F -algebra of degree and where sw is the involution given by ( x, y ) ( y, x ) ,hence in either case σ is unitary. n = 3 : A is a central simple F -algebra of degree and σ is a symplectic involutionon A .Then ( A, σ ) is associated with a quadratic n -fold Pfister form P σ over F whichhas the following characteristic property:For any field extension F ′ /F , the F ′ -algebra with involution ( A, σ ) F ′ , obtainedfrom ( A, σ ) by scalar extension to F ′ , is totally decomposable if and only if P σ ishyperbolic over F ′ . In most of the cases the Main Theorem gives a reinterpretation of some previ-ously known decomposability criteria in terms of quadratic Pfister forms. Hereour principal aim is to handle these different cases by a new uniform approach.In the case where n = 3 and char F = 2 the result is itself novel.Note that when n = 1 we assume that char F = 2. In this case a criterion fordecomposability was established by Knus–Parimala–Sridharan [19]. In [20] thesame authors proved an analogous statement in arbitrary characteristic. Thiscriterion could also be formulated in terms of a bilinear 1-fold Pfister form (givenby the determinant of the involution, see [18, § D = A × A ; see [18, (15.12)]. This criterioncould be formulated in terms of a quadratic 1-fold Pfister form.The case n = 1 is the least difficult and interesting one, but at the same timeit would be the most cumbersome to cover if characteristic 2 were included, inview of the necessary distinction between orthogonal involutions and quadraticpairs. For this reason we decided to include the case n = 1 only when char F = 2,mainly in order to highlight the analogy with the other two cases.In the case n = 2, a criterion for decomposability was first obtained byKarpenko–Qu´eguiner [16]. Their result is stated in terms of the discriminant al-gebra, and it is obtained using the exceptional isomorphism A = D . The 2-foldPfister form P σ turns out to be the norm form of the F -quaternion algebra thatis Brauer equivalent to the discriminant algebra of ( A, σ ), see Proposition 8.2.For the case n = 3 a criterion for decomposability was established in characteris-tic different from 2 by Garibaldi–Parimala–Tignol [14] in terms of a cohomologyclass of degree 3, which gives the first nontrivial cohomological invariant of ( A, σ ).This class is given by the cup-product corresponding to the 3-fold Pfister form P σ , see Proposition 8.4. DISCRIMINANT PFISTER FORM 3
What unites the three cases in the Main Theorem, in spite of the differentdimensions of the algebra, is the fact that biquadratic ´etale subalgebras on which σ restricts to the identity are maximal for this property, by [6, Theorem 4.1].The core of the proof of the Main Theorem is carried out in Section 7. Theconstruction of the quadratic form P σ is inspired by the treatment in [14] of thesymplectic case in characteristic different from two. It further relies on a peculiarproperty of certain biquadratic ´etale subalgebras, which was first used by Rost–Serre–Tignol [22] to define a cohomological invariant of degree 4 for central simplealgebras of degree 4.A priori the construction depends on the choice of a biquadratic ´etale F -subalgebra L of A contained in the space Symd ( σ ) = { x + σ ( x ) | x ∈ A } , calleda neat biquadratic subalgebra of ( A, σ ). In Theorem 7.3 we show that such an L induces a decomposition Symd ( σ ) = L ⊕ W ⊕ W ⊕ W where each of the F -subspaces W , W , W is naturally endowed with a quadraticform (determined up to a similarity factor). Moreover, we show that these threequadratic forms are related by a composition formula. Hence they are similar toa quadratic Pfister form, which is determined by σ and L and which we denoteby P σ,L . In Proposition 7.6 we show that this form P σ,L is hyperbolic if and onlyif the algebra with involution decomposes along L (this notion is introduced inSection 4). We then prove in Proposition 7.10 that this Pfister form does notdepend on the choice of the biquadratic subalgebra L . Hence it is an invariantof σ , which we denote by P σ . The fact that P σ has the properties stipulated inthe Main Theorem is then achieved by Theorem 7.11.Showing the independence of the Pfister form P σ from the choices made in itsconstruction is the most delicate part of the proof of the Main Theorem. Thisis based on a reduction to the case where σ is hyperbolic, and hence it relieson a comprehensive study of the decomposability of algebras with hyperbolicinvolution. This is carried out in some more generality in Section 5, and thenspecialised in Section 6 to the situation of capacity 4.Our treatment also leads to a new result on the possible decompositions of atotally decomposable algebra with involution ( A, σ ) such as in the Main Theorem.In Theorem 7.8 we obtain that any biquadratic ´etale F -subalgebra of A to which σ restricts to the identity can be distributed nicely over the quaternion factors ofa certain decomposition of ( A, σ ). In Corollary 7.9 we further show that, when A is simple but contains zero-divisors (or when A is unitary of inner type and thesimple components contain zero-divisors), then a total decomposition of ( A, σ )can be found which contains a split F -quaternion algebra.In the final Section 8 we relate the quadratic Pfister forms arising in the threecases of the Main Theorem between each other and to other, previously knowninvariants. We further give some examples where the Pfister form invariant of analgebra with involution is computed explicitly. K.J. BECHER, N. GRENIER-BOLEY, AND J.-P. TIGNOL Algebras with involution
In this section we recall some basic facts and objects associated with involutionson central simple algebras. We recall the distinction of involutions into two kindsand further into three different types. We also include some notation from [6].Our main reference for involutions is [18].Let F always be a field. Let A be an F -algebra. We denote by Z ( A ) the centreof A . For an F -subalgebra B of A we denote by C A ( B ) the centraliser of B in A . Assume now that dim F ( A ) < ∞ . If A is simple, then Z ( A ) is a field and A is a central simple Z ( A )-algebra. In this case we denote respectively by deg A , ind A and exp A the degree, the index and the exponent of A , and we further set coind A = deg A ind A , and we call A split if ind A = 1. A central simple F -algebra ofdegree 2 is called an F -quaternion algebra .By an F -involution on A we mean an F -linear anti-automorphism σ : A → A such that σ ◦ σ = id A . Consider an F -involution σ on A . We set Sym ( σ ) = { x ∈ A | σ ( x ) = x } , Skew ( σ ) = { x ∈ A | σ ( x ) = − x } , Symd ( σ ) = { x + σ ( x ) | x ∈ A } . Using the F -linear map A → A, x x + σ ( x ) one obtains that dim F A = dim F Skew ( σ ) + dim F Symd ( σ ) . An F -algebra with involution is a pair ( A, σ ) of a finite-dimensional F -algebra A and an F -involution σ on A such that F = Z ( A ) ∩ Sym ( σ ) and such that A has no nontrivial two-sided ideals I with σ ( I ) = I . By an F -quaternion algebrawith involution we mean a pair ( Q, σ ) where Q is an F -quaternion algebra and σ is an F -involution on Q .For an ´etale extension L/F we denote by [ L : F ] the dimension of L as an F -vector space.In the sequel let ( A, σ ) be an F -algebra with involution. Then either Z ( A ) = F or Z ( A ) is a quadratic ´etale extension of F with nontrivial automorphism σ | Z ( A ) .We say that the involution σ isof the first kind if [ Z ( A ) : F ] = 1 , of the second kind if [ Z ( A ) : F ] = 2 . If Z ( A ) field, then A is a central simple Z ( A )-algebra. If ( A, σ ) is of the secondkind, then either Z ( A ) is a quadratic field extension of F , or Z ( A ) ≃ F × F .Involutions on central simple algebras are further distinguished into three types.We present this distinction in terms of the spaces Sym ( σ ) and Symd ( σ ). We set Sym ∗ ( σ ) = ( Symd ( σ ) if 1 ∈ Symd ( σ ), Sym ( σ ) otherwise. DISCRIMINANT PFISTER FORM 5
We call the involution σ orthogonal if dim F Sym ∗ ( σ ) > dim F A, unitary if dim F Sym ∗ ( σ ) = dim F A, symplectic if dim F Sym ∗ ( σ ) < dim F A. The property of σ to be orthogonal, unitary or symplectic is called the type of σ .A reduction to the split case shows that our definition of the type coincides withthe usual definition; see [18, § Z ( A ) ≃ F × F then we call ( A, σ ) unitary of inner type . (The term ismotivated by a corresponding notion for algebraic groups.) In this case we obtainthat ( A, σ ) ≃ ( A × A op , sw ) for a central simple F -algebra A , its oppositealgebra A op and the switch involution sw : A × A op → A × A op , given by sw ( a , a ) = ( a , a ) (see [18, (2.14)]). In this situation we set deg A = deg A , ind A = ind A and coind A = coind A .Notions and properties for an involution or for a central simple algebra shallalso be employed for the algebra with involution as a pair. So for example wesay that ( A, σ ) split if ind A = 1.2.1. Proposition.
Set d = deg A . Then dim F A = [ Z ( A ) : F ] · d and dim F Sym ∗ ( σ ) = d ( d +1)2 if σ is orthogonal ,d if σ is unitary , d ( d − if σ is symplectic . Proof:
This follows from the definitions together with [18, (2.6)] or [18, (2.17)],depending on whether σ is of the first or second kind. (cid:3) Remarks. (1) We have
Symd ( σ ) ⊆ Sym ( σ ) and this is an equality unless char F = 2 and ( A, σ ) is of the first kind. (See [18, (2.17)] for the case where char F = 2 and σ unitary.)(2) We have 1 / ∈ Symd ( σ ) if and only if σ is orthogonal and char ( F ) = 2. Or-thogonal involutions in characteristic two are peculiar. When we need to excludethis case, we will simply assume that 1 ∈ Symd ( σ ).3. Capacity
Let (
A, σ ) be an F -algebra with involution. Following [6, Section 5], we call an F -subalgebra L of A neat in ( A, σ ) or a neat subalgebra of ( A, σ ) if L is an ´etale F -algebra contained in Sym ( σ ) and such that A is free as a left L -module and,for all primitive idempotents e of L , the F -algebras with involution ( eAe, σ | eAe )have the same degree and the same type; this type coincides with the type of σ .3.1. Remark.
In this article we mostly avoid the case of orthogonal involutions incharacteristic 2, which is the most troublesome in the study of stable subalgebras,
K.J. BECHER, N. GRENIER-BOLEY, AND J.-P. TIGNOL as demonstrated in [6]. In particular, in the cases which we consider the definitionof neat subalgebra simplifies as follows: A commutative F -subalgebra L of A is neat in ( A, σ ) if L is ´etale, consists of σ -symmetric elements, and A is free as aleft (or right) L -module. In particular it then follows that, if L is neat in ( A, σ )and L is a free module over some subalgebra L ′ , then L ′ is also neat in ( A, σ ).Following [6], we define cap ( A, σ ) = (cid:26) deg A if σ is orthogonal or unitary, deg A if σ is symplectic,and we call this integer the capacity of ( A, σ ). By [6, Theorem 4.1] we have cap ( A, σ ) = max { [ L : F ] | L ´etale F -algebra with L ⊆ Sym ( σ ) } . Furthermore, by [6, Proposition 5.6], every ´etale F -subalgebra L of A containedin Sym ( σ ) and with [ L : F ] = cap ( A, σ ) is neat in (
A, σ ).3.2.
Proposition.
Let L be an ´etale F -subalgebra of A contained in Sym ( σ ) .Assume that there exists a σ -stable central simple F -subalgebra B of A such that L ⊆ B and C B ( L ) = L . Then L is neat in ( A, σ ) .Proof: Let C = C A ( B ), which is a simple σ -stable F -subalgebra of A . We set σ B = σ | B and σ C = σ | C . Then ( B, σ B ) and ( C, σ C ) are F -algebras with involutionsuch that ( A, σ ) = (
B, σ B ) ⊗ ( C, σ C ). Since C B ( L ) = L , we have that σ B isorthogonal and [ L : F ] = deg ( B ) = cap ( B, σ B ). It follows by [18, (2.23)] that σ C is of the same type as σ , and consequently cap ( A, σ ) = cap ( B, σ B ) · cap ( C, σ C ).We choose an ´etale F -subalgebra M of ( C, σ C ) with [ M : F ] = cap ( C, σ C ).Then LM is an ´etale extension of F contained in Sym ( σ ) with[ LM : F ] = [ L : F ] · [ M : F ] = cap ( B, σ B ) · cap ( C, σ C ) = cap ( A, σ ) . Hence LM is neat in ( A, σ ), by [6, Proposition 5.6]. Since LM is free as an L -module, it follows by [6, Lemma 5.8] that L is neat in ( A, σ ). (cid:3) We recall the most basic examples of involutions.3.3.
Examples. (1) The identity map id F is the unique orthogonal involution on F , viewed as acentral simple F -algebra.(2) Consider a quadratic ´etale extension K/F and let can
K/F denote the nontriv-ial F -automorphism of K . Then ( K, can K/F ) is an F -algebra with unitaryinvolution.(3) Let Q be an F -quaternion algebra. The unique symplectic involution on Q is given by x Trd A ( x ) − x , where Trd Q : Q → F denotes the reducedtrace form of Q . We denote this involution by can Q and call it the canonicalinvolution on Q .By an F -algebra with canonical involution we mean an F -algebra with involu-tion of one of the three types in Examples 3.3. DISCRIMINANT PFISTER FORM 7
Proposition.
The following are equivalent: ( i ) ( A, σ ) is an F -algebra with canonical involution. ( ii ) Sym ∗ ( σ ) = F . ( iii ) cap ( A, σ ) = 1 .Proof:
This follows from Proposition 2.1 and Examples 3.3. (cid:3) Decomposability
In this section we discuss decompositions of algebras with involution that arecompatible with a given multiquadratic ´etale subalgebra. We further recall thenotion of an algebra with involution being totally decomposable.In this article all tensor products of algebras and vector spaces are taken over F and simply denoted by ⊗ .Let ( A, σ ) be an F -algebra with involution. We call ( A, σ ) totally decomposable if, for some n ∈ N , there exist F -quaternion algebras with involution ( Q i , σ i ) for1 i n such that ( A, σ ) ≃ ( Z ( A ) , σ | Z ( A ) ) ⊗ n O i =1 ( Q i , σ i ) ;if Z ( A ) = F , then this simply means that ( A, σ ) ≃ N ni =1 ( Q i , σ i ). The degree ofany totally decomposable F -algebra with involution is a power of 2.Let r ∈ N and let B , . . . , B r be central simple F -subalgebras of A . We call B , . . . , B r independent if the F -subalgebra of A generated by B ∪ . . . ∪ B r , is F -isomorphic to the tensor product B ⊗ · · · ⊗ B r ; this is equivalent to havingthat B , . . . , B r are centralizing one another, that is, for any i, j ∈ { , . . . , n } with i = j , we have xy = yx for all x ∈ B i and y ∈ B j .4.1. Proposition.
Let n ∈ N be such that cap ( A, σ ) = 2 n . Then the followingare equivalent: ( i ) ( A, σ ) is totally decomposable. ( ii ) There exist n independent σ -stable F -quaternion subalgebras of A . ( iii ) There exist independent σ -stable F -quaternion subalgebras Q , . . . , Q n − of A such that σ | Q i is orthogonal for i n − . ( iv ) There exist independent σ -stable F -quaternion algebras Q , . . . , Q n of A such that σ | Q i is orthogonal for i n .Proof: The implication ( i ) ⇒ ( ii ) is trivial.( ii ⇒ iii ) This implication follows by induction on n , starting with the trivialcases where n
1. For the induction step, it suffices to observe that a tensorproduct of two F -quaternion algebras with involution can up to isomorphism berearranged to have an orthogonal involution on at least one of the two factors.See e.g. [5, Proposition 5.5] for a proof in arbitrary characteristic for this fact. K.J. BECHER, N. GRENIER-BOLEY, AND J.-P. TIGNOL ( iii ⇒ iv ) Assume that Q , . . . , Q n − are independent σ -stable F -subalgebrasof A such that σ | Q i is orthogonal for 1 i n −
1. Let B = Q · · · Q n − and C = C A ( B ). We set σ B = σ | B , σ C = σ | C and σ i = σ | Q i for 1 i n −
1. Then(
C, σ C ) is an F -algebra with involution, and we have( B, σ B ) ≃ n − O i =1 ( Q i , σ i ) and ( A, σ ) ≃ ( B, σ B ) ⊗ ( C, σ C ) . Since σ , . . . , σ n − are orthogonal, so is σ B , by [18, (2.23)]. Hence the typesof the involutions σ and σ C coincide, and we have that cap ( B, σ B ) = 2 n − and cap ( A, σ ) = cap ( B, σ B ) · cap ( C, σ C ). Since cap ( A, σ ) = 2 n = 2 · cap ( B, σ B ), weobtain that cap ( C, σ C ) = 2. Hence there exists an ´etale quadratic extension K of F contained in Sym ( σ C ), and by [6, Corollary 6.6], K is contained in a σ -stable F -quaternion subalgebra Q n of C . Then Q , . . . , Q n are independent and σ | Q n is orthogonal.( iv ⇒ i ) Assume that Q , . . . , Q n are independent σ -stable F -subalgebras of A such that σ | Q i is orthogonal for 1 i n . We set B = Q · · · Q n , C = C A ( B ), σ B = σ | B , σ C = σ | C and σ i = σ | Q i for 1 i n . Then( B, σ B ) ≃ n O i =1 ( Q i , σ i ) and ( A, σ ) ≃ ( B, σ B ) ⊗ ( C, σ C ) . Since σ , . . . , σ n are orthogonal, σ B is orthogonal as well, by [18, (2.23)]. Weconclude that cap ( B, σ B ) = 2 n = cap ( A, σ ) and that (
C, σ C ) is an F -algebrawith involution with cap ( C, σ C ) = 1, thus an algebra with canonical involution,by Proposition 3.4. If σ is symplectic, then C is an F -quaternion algebra, andotherwise C = Z ( A ). Therefore ( A, σ ) is totally decomposable. (cid:3)
We retrieve the following well-known fact, which is trivial in the orthogonalcase, and which is contained in [18, (2.22)] in the unitary case, and in [18, (16.16)]in the symplectic case.4.2.
Corollary. If cap ( A, σ ) = 2 , then ( A, σ ) is totally decomposable.Proof: This is the implication ( iii ⇒ i ) in Proposition 4.1 for n = 1. (Alterna-tively the statement is obtained directly from [6, Corollary 6.6], which was alsoused in the proof of ( iii ⇒ iv ) in Proposition 4.1.) (cid:3) In order to investigate decomposability of (
A, σ ), we first try to establish theexistence of an appropriate neat subextension and then to obtain criteria for thedecomposability of (
A, σ ) along this subextension – in a sense to be defined.Let L be an ´etale F -subalgebra of A contained in Sym ( σ ). We say that ( A, σ ) decomposes along L or is decomposable along L if we have [ L : F ] = 2 r for some r ∈ N and there exist independent σ -stable F -quaternion subalgebras Q , . . . , Q r of A such that Q i ∩ L is a quadratic F -algebra for 1 i r . Note that in thiscase L is necessarily neat in ( A, σ ), by Proposition 3.2.
DISCRIMINANT PFISTER FORM 9
Corollary.
For n ∈ N such that cap ( A, σ ) = 2 n , the following are equivalent: ( i ) ( A, σ ) is totally decomposable. ( ii ) ( A, σ ) decomposes along some neat F -subalgebra L with [ L : F ] = 2 n − . ( iii ) ( A, σ ) decomposes along some neat F -subalgebra M with [ M : F ] = 2 n .Proof: This is immediate from Proposition 4.1. (cid:3) Hyperbolic involutions and quaternion factors
Let (
A, σ ) be an F -algebra with involution. We call the involution σ isotropic if σ ( x ) x = 0 for some x ∈ A \ { } , and anisotropic otherwise. We call σ metabolic ifthere exists e ∈ A such that e = e , σ ( e ) e = 0 and dim F eA = dim F A . We furthercall σ hyperbolic if there exists e ∈ A such that e = e , σ ( e ) = 1 − e . Note thatany hyperbolic involution is metabolic and any metabolic involution is isotropic.Note further that any algebra with involution which is either split symplectic orunitary of inner type is hyperbolic. We recollect from [10, Lemma A.3] and [11,Proposition 4.10] the following fact.5.1. Proposition.
The involution σ is hyperbolic if and only if σ is metabolicand ∈ Symd ( σ ) .Proof: If σ is hyperbolic, then for e ∈ A with e = e and σ ( e ) = 1 − e , we obtainthat 1 = e + σ ( e ) ∈ Symd ( σ ) and σ ( e ) e = 0, so in particular σ is metabolic.Assume now that σ is metabolic and 1 ∈ Symd ( σ ). If char ( F ) = 2, then itfollows by [11, Proposition 4.10] that σ is hyperbolic. If char ( F ) = 2, then thecondition that 1 ∈ Symd ( σ ) says that σ is not orthogonal, and it follows by [10,Lemma A.3] that σ is hyperbolic. (cid:3) We are going to characterise the hyperbolicity of the involution σ by the ex-istence of certain σ -stable F -quaternion subalgebras. The following statementprovides the basis to this approach.5.2. Proposition. If cap ( A, σ ) = 2 , then σ is either anisotropic or metabolic.Proof: If (
A, σ ) is unitary of inner type or split symplectic, then σ is hyperbolic.Assume now that ( A, σ ) is neither unitary of inner type nor split symplectic.Since cap ( A, σ ) = 2 it follows that dim F I = dim F A for every nontrivial rightideal I of A . Any right ideal of A is generated by an element e with e = e . Hence,if there exists x ∈ A \ { } such that σ ( x ) x = 0, then we choose e ∈ A with e = e and xA = eA , and obtain that σ ( e ) e = 0, and furthermore dim F eA = dim F A ,because xA is a nontrivial right ideal of A . (cid:3) The following statement is a variation of [9, Theorem 2.2], without restrictionon the characteristic.5.3.
Proposition.
Assume that cap ( A, σ ) is even and ∈ Symd ( σ ) . Then thefollowing are equivalent: ( i ) σ is hyperbolic and coind A is even. ( ii ) There exists a split σ -stable F -quaternion subalgebra Q ⊆ A such that σ | Q is orthogonal and metabolic. ( iii ) There exist elements u, v ∈ A such that u = u , v = 1 , uv + vu = v , σ ( u ) = 1 − u + uv and σ ( v ) = − u + v − uv .Proof: ( iii ⇒ ii ) Assume that u, v ∈ A satisfy the relations in ( iii ). Then Q = F ⊕ uF ⊕ vF ⊕ uvF is a σ -stable F -quaternion subalgebra of A . Note that σ | Q is not the canonical involution on Q , because u = u and σ ( u ) = 1 − u . Hence σ | Q is orthogonal. Since σ ( u ) u = (1 − u + uv ) u = uvu = u ( v − uv ) = ( u − u ) v = 0 , we have that σ | Q is isotropic. As Q is an F -quaternion algebra, it follows byProposition 5.2 that σ | Q is metabolic.( ii ⇒ i ) Let Q be a σ -stable F -quaternion subalgebra of A such that σ | Q isorthogonal and metabolic. Let C be the centralizer of Q in A . Then C is σ -stableand ( A, σ ) ≃ ( Q, σ | Q ) ⊗ ( C, σ | C ) . Since σ | Q is metabolic, it follows that Q is split and σ is metabolic. Hence coind A is even, and as 1 ∈ Symd ( σ ), we obtain by Proposition 5.1 that σ is hyperbolic.( i ⇒ iii ) Suppose that ( A, σ ) is hyperbolic and coind A is even. Since cap ( A, σ )is even as well, there exists an F -algebra with involution ( B, τ ) of the same typeas (
A, σ ) and such that A ≃ M ( F ) ⊗ B . The matrices u = ( ) and v = ( ) in M ( F ) satisfy the relations in ( iii ) with respect to the involution σ ′ = Int ( u + v ) ◦ t on M ( F ), where t denotes the transposition involution on M ( F ). As in theproof of ( iii ⇒ ii ) we obtain that ( M ( F ) , σ ′ ) is metabolic. It follows that( M ( F ) , σ ′ ) ⊗ ( B, τ ) is metabolic. Since σ ′ ⊗ τ is of the same type as σ , we havethat 1 ∈ Symd ( σ ′ ⊗ τ ) and conclude by Proposition 5.1 that ( M ( F ) , σ ′ ) ⊗ ( B, τ )is hyperbolic. It follows from [18, (12.35)] that all algebras with hyperbolicinvolution of the same type and with the same underlying algebra are isomorphic.Since A ≃ M ( F ) ⊗ B , we conclude that ( A, σ ) ≃ ( M ( F ) , σ ′ ) ⊗ ( B, τ ). Hence A contains elements u and v satisfying the equations in ( iii ) with respect to σ . (cid:3) Corollary.
Let K be a separable field extension of F contained in Sym ( σ ) .Let C = C A ( K ) and assume that coind C and cap ( C, σ | C ) are even and σ | C ishyperbolic. Then there exists a σ -stable split F -quaternion subalgebra Q of A such that σ | Q is orthogonal and metabolic and such that K ⊆ C A ( Q ) .Proof: By Proposition 5.3, there exist elements u, v ∈ C such that u = u , v = 1, uv + vu = v , σ ( u ) = 1 − u + uv and σ ( v ) = − u + v − uv . Then Q = F ⊕ F u ⊕ F v ⊕ F uv is a σ -stable F -quaternion subalgebra of A such that σ | Q is orthogonal and metabolic and such that K ⊆ C A ( Q ). (cid:3) DISCRIMINANT PFISTER FORM 11
In the following proof we interpret an involution as adjoint to a hermitian formand use the correspondence between the concepts of hyperbolicity for the twosorts of objects.5.5.
Proposition.
Let K be a neat subalgebra of ( A, σ ) such that K ≃ F × F .Assume that σ is hyperbolic. Then K is contained in a split σ -stable F -quaternionsubalgebra Q of A such that σ | Q is orthogonal and metabolic.Proof: We first assume that A is simple. In this case we may identify A with End D ( V ) for a finite-dimensional F -division algebra D and a finite-dimensional D -right vector space V . We fix an involution τ on D of the same kind as σ .By Proposition 5.1, since σ is hyperbolic, σ is not orthogonal if char F = 2.Note further that, if char F = 2 or if τ is unitary, then every hermitian or skew-hermitian form over ( D, τ ) is even, by [17, Chap. I, Lemma 6.6.1]. Using theidentification of A with End D ( V ), we therefore obtain by [18, (4.2)] that theinvolution σ is adjoint to a nondegenerate even hermitian or skew-hermitian form h : V × V → D with respect to τ .Let e and e ′ be the primitive idempotents in K . Hence e ′ = 1 − e and we havethat e and e ′ are the two projections given by a direct decomposition V = W ⊕ W ′ for two D -subspaces W and W ′ of V . Since e, e ′ ∈ K ⊆ Sym ( σ ) we obtain that W and W ′ are orthogonal to one another with respect to h . Hence, we have anorthogonal decomposition( V, h ) ≃ ( W, h | W ) ⊥ ( W ′ , h | W ′ ) . Since K is neat in ( A, σ ), we have that ( eAe, σ | eAe ) and ( e ′ Ae ′ , σ | e ′ Ae ′ ) have thesame degree and the same type. Since these F -algebras with involution corre-spond to the D -endomorphism algebras of W and of W ′ with their involutionsadjoint to the restrictions of h , we conclude that dim D W = dim D W ′ = dim D V and that the restrictions of h to W and W ′ have the same type as h , hermitianor skew-hermitian.Since σ is hyperbolic, so is ( V, h ). Since also ( W ′ , − h | W ′ ) ⊥ ( W ′ , h | W ′ ) is hy-perbolic and of the same dimension as ( V, h ), it follows by [23, Proposition 7.7.3]that (
V, h ) ≃ ( W ′ , − h | W ′ ) ⊥ ( W ′ , h | W ′ ). Since h is even, so is h | W ′ , and hence wemay apply Witt Cancellation [17, Chap. I, Proposition 6.4.5] and conclude that( W, h | W ) ≃ − ( W ′ , h | W ′ ) . Let g : W → W ′ be an isometry between h | W and − h | W ′ . Let f : V → V be the D -automorphism of V determined by f ( w + w ′ ) = g − ( w ′ ) + g ( w ) for w ∈ W and w ′ ∈ W ′ . It follows that σ ( f ) = − f , ef + f e = f and f = id V . We concludethat e and f generate a split F -quaternion subalgebra Q which is σ -stable andsuch that σ | Q is orthogonal. Furthermore, since f = 1 and σ (1 − f ) · (1 − f ) = (1 + f )(1 − f ) = 1 − f = 0 , we conclude that σ | Q is isotropic. Since Q is an F -quaternion algebra, it followsby Proposition 5.2 that σ | Q is metabolic. This concludes the proof for the casewhere A is simple.Suppose finally that ( A, σ ) is unitary of inner type. Hence, we may identify(
A, σ ) with ( B × B op , sw ) for some central simple F -algebra B . We obtain that K = { ( x, x ) | x ∈ K } for an F -algebra K contained in B , isomorphic to F × F and such that B is free as a K -left module. Hence coind B is even. Weuse a variation of the above argument, without involutions and hermitian forms.We identify B with End D ( V ) for a finite-dimensional F -division algebra D anda finite-dimensional D -right vector space V . As in the previous case the twoprimitive idempotents of K give rise to a decomposition V = W ⊕ W ′ where the D -subspaces W and W ′ of V are of the same dimension and therefore isomorphic.We fix a D -isomorphism g : W → W ′ and then define a D -automorphism f of V determined by letting f ( w + w ′ ) = g − ( w ′ ) + g ( w ) for w ∈ W and w ′ ∈ W ′ .Let e : V → W be the projection on the first component for the decomposition V = W ⊕ W ′ . Then f and e generate a split F -quaternion subalgebra Q of B . Wefix an orthogonal metabolic involution τ on Q . Then Q = { ( x, τ ( x )) | x ∈ Q } is a split F -quaternion subalgebra of A = B × B op containing K and stableunder σ = sw . Furthermore, ( Q, σ | Q ) ≃ ( Q , τ ), whereby σ | Q is orthogonal andmetabolic. (cid:3) Theorem.
Let K be a separable quadratic field extension of F containedin Sym ( σ ) and let C = C A ( K ) . Assume that σ is hyperbolic and that σ | C isanisotropic. Then K is contained in a σ -stable split F -quaternion subalgebra Q such that σ | Q is orthogonal and metabolic.Proof: Let γ be the nontrivial F -automorphism of K and C ′ = { x ∈ A | xk = γ ( k ) x for every k ∈ K } . By [6, Proposition 6.1] we have that σ ( C ) = C , σ ( C ′ ) = C ′ and A = C ⊕ C ′ .Let e ∈ A be such that e = e and σ ( e ) = 1 − e . We write e = v + w with v ∈ C and w ∈ C ′ . We have( v + w ) + ( vw + wv ) = e = e = v + w. Note that v , w ∈ C and vw, wv ∈ C ′ . As A = C ⊕ C ′ , we obtain that v + w = v and vw + wv = w. Furthermore σ ( v ) + σ ( w ) = σ ( e ) = 1 − e = (1 − v ) − w . As σ ( C ) = C and σ ( C ′ ) = C ′ , it follows that σ ( v ) = 1 − v and σ ( w ) = − w . We conclude that wv = w − vw = σ ( v ) w . DISCRIMINANT PFISTER FORM 13
We claim that there exists v ′ ∈ C with vv ′ = 1. Suppose x ∈ C is such that vx = 0. Then σ ( x ) · (1 − v ) = σ ( vx ) = 0, and therefore σ ( x ) = σ ( x ) v. It follows that σ ( x ) x = σ ( x ) vx = 0 . Since σ | C is anisotropic we conclude that x = 0. Therefore the F -linear map C → C given by multiplication with v from the left is injective. Since C is finite-dimensional, this map is also surjective. Hence there exists an element v ′ ∈ C with vv ′ = 1. Using that wv = σ ( v ) w we obtain that σ ( v ′ ) w = σ ( v ′ ) wvv ′ = σ ( v ′ ) σ ( v ) wv ′ = σ ( vv ′ ) wv ′ = wv ′ . As σ ( w ) = − w we conclude that σ ( wv ′ ) = − wv ′ and( wv ′ ) = σ (( wv ′ ) ) = σ ( v ′ ) wσ ( v ′ ) w = σ ( v ′ ) w v ′ = σ ( v ′ )( v − v ) v ′ = σ ( v ′ )(1 − v ) . As σ ( v ) = 1 − v we obtain that ( wv ′ ) = 1. Since wv ′ ∈ C ′ it follows that Q = K ⊕ Kwv ′ is a split σ -stable F -quaternion subalgebra. Since K is a quadratic´etale F -algebra and σ | K = id K , the involution σ | Q is orthogonal. Since σ (1 + wv ′ )(1 + wv ′ ) = (1 − wv ′ )(1 + wv ′ ) = 1 − ( wv ′ ) = 0 , we have that σ | Q is metabolic. (cid:3) Hyperbolicity in capacity four
Our study of algebras with involution of capacity 4 in Section 7 will cruciallyrely on the special case where the involution is hyperbolic, which we study in thissection. We start by showing that an algebra with hyperbolic involution in capac-ity four is decomposable along any neat quadratic subalgebra (Proposition 6.1),except in one special case. We will then show that any decomposable unitary orsymplectic involution of capacity four can be made hyperbolic by passing to thefunction field of some quadratic form of dimension at least five. This will allowus in Section 7 to show a certain Pfister form which we will attach to an algebrawith involution of capacity 4 is independent from certain choices which we makein its construction.Let (
A, σ ) be an F -algebra with involution.6.1. Proposition.
Assume that cap ( A, σ ) = 4 , exp A and σ is hyperbolic.Then ( A, σ ) is decomposable along every neat quadratic F -subalgebra of ( A, σ ) .Proof: We have 1 ∈ Symd ( σ ), because σ is hyperbolic. Let K be an arbitraryneat quadratic F -subalgebra of ( A, σ ). To prove the statement we need to showthat K is contained in a σ -stable F -quaternion subalgebra of A . When K ≃ F × F this already follows by Proposition 5.5, so we may assume that K is a field.Suppose first that A is not simple. In this case we may identify ( A, σ ) with( B × B op , sw ) for some central simple F -algebra B . Then K = { ( x, x ) | x ∈ K } for a separable quadratic field extension K of F contained in B . Since wehave exp B = exp A
2, it follows by a theorem of Albert, [18, (16.2)], that K is contained in an F -quaternion subalgebra Q of B . We fix an orthogonalinvolution τ on Q with τ | K = id K . Then Q = { ( x, τ ( x )) | x ∈ Q } is an F -quaternion subalgebra of A = B × B op containing K and Q is stable under theinvolution σ = sw . Furthermore, ( Q, σ | Q ) ≃ ( Q , τ ), whereby σ | Q is orthogonal.Assume now that A is simple. Let C = C A ( K ) and σ C = σ | C . Then C issimple and ( C, σ C ) is a K -algebra with involution with cap ( C, σ C ) = 2. If σ C isanisotropic, then we obtain the desired conclusion by Theorem 5.6. Assume nowthat σ C is isotropic. As C is simple and σ C is isotropic, coind C is even. As further cap ( C, σ C ) = 2, it follows by Proposition 5.2 and Proposition 5.1 that ( C, σ C ) ishyperbolic. Hence, by Corollary 5.4 it follows that C contains a σ -stable split F -quaternion subalgebra Q ′ such that ( Q ′ , σ | Q ′ ) is orthogonal and metabolic. Then D = C A ( Q ′ ) is a σ -stable F -subalgebra of A containing K . Let σ D = σ | D . Then( D, σ D ) is an F -algebra with involution with cap ( D, σ D ) = 2 and σ D is of thesame type as σ . Since K ⊆ Sym ( σ D ), it follows by [6, Corollary 6.6] that K iscontained in a σ -stable F -quaternion subalgebra Q of D , and hence of A . (cid:3) The following example explains why the condition on the exponent cannot beomitted in the statement of Proposition 6.1.6.2.
Example.
Let B be a central F -division algebra with exp B = deg B = 4.Then ( B × B op , sw ) is an F -algebra with unitary involution of capacity 4, whichis hyperbolic and indecomposable.We will use standard notation from [13] for diagonal quadratic forms in char-acteristic different from 2 and for nonsingular binary quadratic forms in arbitrarycharacteristic. We recall some quadratic form terminology from [13, (7.17)], inparticular concerning the radicals of a quadratic form and of its polar form.Let q : V → F be a quadratic form over F , defined on a finite-dimensional F -vector space V . We denote by b q the polar form of q given by V × V → F, ( x, y ) q ( x + y ) − q ( x ) − q ( y ) . We further set rad ( b q ) = { x ∈ V | b q ( x, y ) = 0 for all y ∈ V } rad ( q ) = { x ∈ rad ( b q ) | q ( x ) = 0 } and observe that these are F -subspaces of V with rad ( q ) ⊆ rad ( b q ). Moreover, if char F = 2 then q ( x ) = b q ( x, x ) for all x ∈ V and thus rad ( q ) = rad ( b q ). Wecall the quadratic form q regular if rad ( q ) = { } and nondegenerate if q is regularand dim F rad ( b q ) DISCRIMINANT PFISTER FORM 15
Lemma.
Assume that cap ( A, σ ) = 4 , σ is unitary or symplectic and ( A, σ ) is totally decomposable. Then there exists an F -subspace V of Sym ( σ ) with dim F V = (cid:26) if σ is symplectic , if σ is unitary , such that x ∈ F for all x ∈ V and such that q : V → F, x x is a nondegen-erate quadratic form. Furthermore, if for such an F -space V the correspondingform q is isotropic, then σ is hyperbolic.Proof: Assume first that σ is symplectic. Since ( A, σ ) is totally decomposable,we obtain that (
A, σ ) ≃ ( Q , can Q ) ⊗ ( Q , can Q ) ⊗ ( Q , can Q )for three F -quaternion algebras Q , Q , Q . For i = 1 , , u i , v i ∈ Q i with(6.3.1) u i , v i ∈ F × and u i v i + v i u i = (cid:26) char ( F ) = 2 , char ( F ) = 2 . Set V = (cid:26) ( F u ⊕ F v ⊕ F u v ) u ⊕ ( F u ⊕ F v ⊕ F u v ) v if char ( F ) = 2 ,F u ⊕ F v ⊕ F u ⊕ F v ⊕ F u ⊕ F v if char ( F ) = 2 . Then dim F V = 6 and V ⊆ Sym ( σ ), and one can easily check that x x definesa nondegenerate quadratic form V → F : In fact, letting a i = u i , b i = v i ∈ F × for i = 1 , ,
3, we obtain that q ≃ (cid:26) a h a , b , − a b i ⊥ b h a , b , − a b i if char ( F ) = 2 , [ a , b ] ⊥ [ a , b ] ⊥ [ a , b ] if char ( F ) = 2 . Assume now that σ is unitary. Then( A, σ ) ≃ ( K, σ K ) ⊗ ( Q , can Q ) ⊗ ( Q , can Q )for K = Z ( A ), σ K = σ | K and two F -quaternion algebras Q , Q . If char ( F ) = 2then let d ∈ F × be such that K ≃ F ( √ d ). Hence there exists w ∈ K such that K = F ⊕ F w and w = d . For i = 1 , u i , v i ∈ Q i satisfying (6.3.1). Set V = (cid:26) ( F u ⊕ F v ⊕ F u v ) u ⊕ F wv ⊕ F wu v if char ( F ) = 2 ,F u ⊕ F v ⊕ F u ⊕ F v ⊕ F if char ( F ) = 2 . Then dim F V = 5 and V ⊆ Sym ( σ ), and one can easily check that x x definesa nondegenerate quadratic form q : V → F : In fact, letting a i = u i , b i = v i ∈ F × for i = 1 ,
2, we obtain that q ≃ (cid:26) a h a , b , − a b i ⊥ db h , − a i if char ( F ) = 2 , [ a , b ] ⊥ [ a , b ] ⊥ h i if char ( F ) = 2 . After having chosen V and q case by case, the rest of the argument can begiven uniformly for all four cases. Note that q ( x + y ) − q ( x ) − q ( y ) = xy + yx for any x, y ∈ V .
Suppose now that q is isotropic. Fix x ∈ V \ { } such that q ( x ) = 0. Since q is nondegenerate, there exists y ∈ V \ { } such that xy + yx = 1. It easilyfollows that Q = F ⊕ F x ⊕ F y ⊕ F xy is a σ -stable F -quaternion subalgebra of A .Since xσ ( x ) = x = q ( x ) = 0, it follows that σ | Q isotropic, and since Q is an F -quaternion algebra, we conclude that σ | Q is metabolic. This implies that σ ismetabolic. As σ is symplectic or unitary, we obtain by Proposition 5.1 that σ ishyperbolic. (cid:3) Lemma.
Assume that ( A, σ ) is totally decomposable, cap ( A, σ ) = 4 and ∈ Symd ( σ ) . Let n = 1 if σ is orthogonal, n = 2 if σ is unitary, and n = 3 if σ is symplectic. There exists a field extension F ′ /F such that ( A, σ ) F ′ is hyperbolicand every anisotropic quadratic n -fold Pfister form over F remains anisotropicover F ′ .Proof: Assume first that σ is orthogonal. Then the hypothesis implies that char ( F ) = 2. It is well-known (see e.g. [18, Corollary 15.12] or [7, Proposition3.9]) that every tensor product of two F -quaternion algebras with orthogonalinvolution is isomorphic to a tensor product of two F -quaternion algebras withcanonical involution. In particular, since ( A, σ ) is totally decomposable, thereexists a σ -stable F -quaternion subalgebra Q of A on which σ restricts to thecanonical involution. Let F ′ /F be the function field of the Severi–Brauer varietyassociated with Q , or equivalently, the function field of the projective conic givenby the pure part of the norm form of Q . Then Q F ′ is split, whereby σ | Q F ′ ishyperbolic. Hence ( A, σ ) F ′ is hyperbolic. Since F is relatively algebraically closedin F ′ , every anisotropic 1-fold Pfister form over F stays anisotropic over F ′ .Suppose now that σ is unitary or symplectic. Since ( A, σ ) is totally decompos-able, by Lemma 6.3 there exists an F -subspace V of Sym ( σ ) with dim F V = (cid:26) σ is symplectic , σ is unitary , such that x ∈ F for all x ∈ V and such that V → F, x x is a nondegeneratequadratic form. We denote this quadratic form by ψ . Let F ( ψ ) be the functionfield of the projective quadric defined by ψ over F . Since ψ F ( ψ ) is isotropic, itfollows by Lemma 6.3 that the F ( ψ )-algebra with involution ( A, σ ) F ( ψ ) is hyper-bolic.Our first attempt for choosing F ′ is to take F ( ψ ). Suppose that this choice isnot satisfying the claim. Then there must exist an anisotropic quadratic n -foldPfister form π over F such that π F ( ψ ) is isotropic. Since quadratic Pfister formsare either anisotropic or hyperbolic, it follows that π F ( ψ ) is hyperbolic. By theSubform Theorem [13, (22.5)], this implies that ψ is similar to a subform of π .In particular, 5 dim ( ψ ) dim ( π ), whereby n = 3. Hence the involution σ issymplectic and dim ( ψ ) = 6. It follows that π is similar to ψ ⊥ β for a regular2-dimensional quadratic form β over F . Hence β is similar to the norm formof a separable quadratic extension K/F . Since π is anisotropic over F , so is DISCRIMINANT PFISTER FORM 17 β , whereby K is a field. Moreover β K is hyperbolic, and since π K is a Pfisterform containing a form similar to β K , it follows that π K is hyperbolic. By WittCancellation [13, (8.4)] we obtain that ψ K is hyperbolic.Let C ( ψ ) denote the Clifford algebra of ψ . Since ψ is nondegenerate, C ( ψ ) is acentral simple F -algebra with dim F C ( ψ ) = 2 dim ( ψ ) = 2 . Moreover, the definitionof ψ as the squaring map on an F -subspace of A gives rise to an F -algebrahomomorphism C ( ψ ) → A , which is injective, because C ( ψ ) is simple. Note that deg A = 8 as ( A, σ ) is symplectic of capacity 4. Hence dim F C ( ψ ) = dim F A < ∞ .We conclude that A ≃ C ( ψ ).In particular, as ψ K is hyperbolic, A K is split. Therefore ind A [ K : F ] = 2.Hence A is Brauer equivalent to an F -quaternion algebra Q . Since ( A, σ ) is totallydecomposable, it follows by [12, Theorem 6.1] that (
A, σ ) ≃ Ad ( ϕ ) ⊗ ( Q, can Q )for some bilinear 2-fold Pfister form ϕ over F .Let ρ be the quadratic 4-fold Pfister form over F given by the tensor productof ϕ with the norm form of Q . If ρ is isotropic, then since it is a quadraticPfister form, it is hyperbolic, and it follows by [12, Theorem 5.2] that ( A, σ ) ishyperbolic. Denoting by F ( ρ ) the function field of the projective quadric definedby ρ over F , we obtain in any case that ( A, σ ) F ( ρ ) is hyperbolic. On the otherhand, since quadratic Pfister forms are either anisotropic or hyperbolic, it followsby the Subform Theorem [13, (22.5)] that every anisotropic 3-fold Pfister formover F remains anisotropic over F ( ρ ). Hence we may take F ′ = F ( ρ ). (cid:3) Construction of the discriminant from a biquadratic algebra
In this section we study an F -algebra with involution ( A, σ ) with1 ∈ Symd ( σ ) and cap ( A, σ ) = 4 . The first hypothesis excludes the case where σ is orthogonal and char ( F ) = 2.The second hypothesis means that dim F A = 2 n +3 for n = σ is orthogonal,2 if σ is unitary,3 if σ is symplectic.We will attach to ( A, σ ) a quadratic form over F of dimension 2 n and study itsproperties. This form will yield a criterion for the decomposability of ( A, σ ). Theconstruction of a candidate for this form crucially relies on the existence of a neatbiquadratic F -subalgebra L of ( A, σ ), which was proven in [6, Theorem 7.4].By a biquadratic F -algebra we mean a commutative F -algebra of dimension 4which is isomorphic to a tensor product of two quadratic F -algebras. Let L bean ´etale biquadratic F -algebra, and for an F -automorphism γ , let L γ = { x ∈ L | γ ( x ) = x } . There are three F -automorphisms γ for which L γ is a quadratic ´etale extensionof F . Together with id L , they form a subgroup G of the automorphism group of L such that the fixed field L G is equal to F . (In other terms, L is a G -Galoisalgebra or a Galois F -algebra if G is clear from the context, see [18, (18.15)].) Werefer to G as the Galois group of L over F .7.1. Lemma.
Assume that cap ( A, σ ) = 4 . Let L be a neat biquadratic F -subalgebra of ( A, σ ) and let γ be a nontrivial element of the Galois group of L/F . Let s : L γ → F be an F -linear form with Ker ( s ) = F and let W = { x ∈ Symd ( σ ) | yx = xγ ( y ) for all y ∈ L } . Then dim F W = 2 n for n = log dim F A − , and q : W → F, x s ( x ) is a nondegenerate quadratic form over F . This form is isotropic if and only ifthere exists an element x ∈ W with x ∈ F × .Proof: We set K = L γ , C = C A ( K ), σ C = σ | C and V = C ∩ Symd ( σ ). Note that W is an F -subspace of V . In order to prove that q is nondegenerate, we relate itto a nondegenerate quadratic form e q on V . We will see that e q restricts to q on W and is Witt equivalent to q . For the definition of e q , we consider separately thecases where K is a field and where K is split.Assume first that K is a field. Then ( C, σ C ) is a K -algebra with involutionof the same type as ( A, σ ) and with cap ( C, σ C ) = 2, and L is a maximal neatsubalgebra of ( C, σ C ). We obtain by [6, Proposition 4.6, Proposition 6.1 andProposition 6.5] that V = Symd ( σ C ) = L ⊕ W, dim F W = dim F A, and that there exists a canonical nondegenerate quadratic form c : V → K withthe following properties: W is the orthogonal complement of L , the restriction c | L is the norm form N L/K of the quadratic ´etale extension
L/K , and c ( x ) = − x holds for all x ∈ W . It follows that the transfer e q = − ( s ◦ c ) : V → F is anondegenerate quadratic form over F such that e q ( x ) = s ( x ) for all x ∈ W ,whereby q is the restriction of e q to W . Moreover, W and L are orthogonal to oneanother with respect to e q , and since dim F V = dim F W + dim F L , it follows that W is the orthogonal complement of L in V with respect to e q . This implies that q is nondegenerate and e q = − ( s ◦ N L/K ) ⊥ q. Since L is a biquadratic F -algebra, the form N L/K is extended from a quadraticform defined over F . Hence s ◦ N L/K is hyperbolic and e q is Witt equivalent to q .Since s (1) = 0, it is clear that q is isotropic whenever there exists x ∈ W such that x ∈ F × . Conversely, assume that q is isotropic. Hence there exists y ∈ W \ { } such that q ( y ) = 0, whereby y ∈ Ker ( s ) = F . If y = 0, thenlet x = y , and otherwise, the regular quadratic form c | W : W → K , x x is DISCRIMINANT PFISTER FORM 19 isotropic and hence universal, so we can find an element x ∈ W with x = 1.This shows that q is isotropic if and only if there exists x ∈ W with x ∈ F × .Consider now the case where K ≃ F × F . We denote by e and e the twoprimitive idempotents of K . Since L is a biquadratic ´etale F -algebra, thereexists an F -automorphism γ ′ of L which is of order 2 and different from γ . Then γ ′ | K = id K , and it follows that γ ′ interchanges e and e . Since L = e L ⊕ e L ,we conclude by [6, Lemma 5.8] that K is neat in ( A, σ ). For i = 1, 2 we set E i = e i Ae i and denote by τ i the restriction of σ to E i . Then C = E ⊕ E andthe F -algebras E and E are isomorphic. Moreover, ( E , τ ) and ( E , τ ) are F -algebras with involution, and by [6, Proposition 3.1], the involutions τ and τ have the same type as σ . It follows that cap ( E , τ ) = cap ( E , τ ) = 2. For i = 1,2, since e i L is a quadratic ´etale F -subalgebra of E i contained in Sym ( τ i ) and withnontrivial automorphism γ | e i L , we obtain by [6, Proposition 4.6, Proposition 6.1and Proposition 6.5] that e i V e i = e i V = e i L ⊕ e i W, dim F e i W = dim F A, and that there exists a canonical nondegenerate quadratic form c ,i : e i V → e i F with the following properties: e i W is the orthogonal complement of e i L , the re-striction c ,i | e i L is the norm form N e i L/e i F of the quadratic ´etale extension e i L/e i F ,and c ,i ( x ) = − x holds for all x ∈ e i W . We consider the quadratic maps c : V → K , x c , ( e x ) + c , ( e x ) and e q = − ( s ◦ c ) : V → F . The re-striction of c to L = e L ⊕ e L is the norm form N L/K , and its restriction to W = e W ⊕ e W is the map x
7→ − x . The definition of W implies that W and L are orthogonal to one another with respect to e q . Since dim F V = dim F W + dim F L ,we obtain that e q = − ( s ◦ N L/K ) ⊥ q. Since
Ker ( s ) = F , we have that s ( e ) = − s ( e ) = α for some α ∈ F × , whereby e q ≃ − αc , ⊥ αc , . Therefore, the form e q is nondegenerate, and the same holdsfor q . Note further that dim F W = dim F e W + dim F e W = dim F A and that theform N L/K is extended from F . We conclude that s ◦ N L/K is hyperbolic and q isWitt equivalent to e q .Since Ker ( s ) = F , it is clear that q is isotropic whenever there exists x ∈ W such that x ∈ F × . Conversely, if q is isotropic, we may find w , w ∈ W with( w , w ) = (0 ,
0) and λ ∈ F such that ( e i w i ) = e i λ for i = 1 ,
2. If λ = 0, then for x = e w + e w ∈ W we have x = λ ∈ F × . If λ = 0, then for i = 1 ,
2, the form c ,i | e i W is isotropic and hence universal, so we may find w ′ i such that ( e i w ′ i ) = e i ,and then x = e w ′ + e w ′ ∈ W satisfies x = 1. (cid:3) Remark.
The construction of the form e q in the proof of Lemma 7.1 goesback to [14, Section 7], for the case where char F = 2 and σ is symplectic. Thereit also turns out, by a very different argument, that e q contains a hyperbolic planeand therefore is Witt equivalent to a smaller subform q . We will use in the sequelquite a different method than in [14] to determine the properties of q . Theorem.
Assume that cap ( A, σ ) = 4 and let n = log dim F A − . Let L be a neat biquadratic F -subalgebra of ( A, σ ) and let G = { id L , γ , γ , γ } be theGalois group of L/F . For i = 1 , , , we set W i = { x ∈ Symd ( σ ) | yx = xγ i ( y ) for all y ∈ L } and fix an F -linear form s i : L γ i → F with Ker ( s i ) = F , and obtain a quadraticform q i : W i → F, x s i ( x ) over F . There exists an n -fold Pfister form over F which is similar to q , q and q . This Pfister form is uniquely determined by L .Moreover, Symd ( σ ) = L ⊕ W ⊕ W ⊕ W . Proof:
For i = 1 ,
2, we choose u i ∈ L γ i such that s i ( u i ) = 1. Then in particular u , u / ∈ F . Let c = s (cid:0) γ ( u u ) + γ ( u u ) (cid:1) . Using (1 , u i ) as an F -basis of L γ i for i = 1 ,
2, it is easy to see that the two F -bilinear maps L γ × L γ → F given by( x, y ) s (cid:0) γ ( xy ) + γ ( xy ) (cid:1) and ( x, y ) c · s ( x ) · s ( y )coincide.For i = 1 ,
2, since L γ i is a quadratic ´etale F -algebra with non-trivial auto-morphism γ − i | L i and since u i / ∈ F , we obtain that γ − i ( u i ) = u i and therefore( u i − γ − i ( u i )) ∈ F × . It follows that ( u − γ ( u ))( u − γ ( u )) = 0. Hence u γ ( u ) + γ ( u ) u = u u + γ ( u ) γ ( u ) . Since γ ( u ) = u and γ ( u ) = u , we get that γ ( u u ) + γ ( u u ) / ∈ F . Thisshows that c = 0.Since the statement is invariant under scaling of the form q by an element of F × , we may replace s by c · s and assume that c = 1. We obtain that s ( γ ( xy ) + γ ( xy )) = s ( x ) s ( y ) for all x ∈ L γ , y ∈ L γ . We define an operation ∗ : A × A → A by letting x ∗ y = xy + yx for x, y ∈ A . (If char ( F ) = 2, then ∗ corresponds up to a factor 2 to the Jordan operation on A .)Note that for x, y ∈ Sym ( σ ) we have σ ( xy ) = yx and thus x ∗ y ∈ Symd ( σ ).Consider now x ∈ W and y ∈ W . Using that γ ◦ γ = γ = γ ◦ γ , we obtainfor any z ∈ L that z ( xy ) = zxy = xγ ( z ) y = xyγ ( γ ( z )) = xyγ ( z ) , whereby xy ∈ W . In the same way we obtain that yx ∈ W . Hence x ∗ y ∈ W .We claim that s (cid:0) ( x ∗ y ) (cid:1) = s ( x ) · s ( y ) . DISCRIMINANT PFISTER FORM 21
Then ( x ∗ y ) = ( xyxy + yxyx ) + xy x + yx y = ( xyxy + yxyx ) + γ ( y ) x + γ ( x ) y = ( xyxy + yxyx ) + γ ( x y ) + γ ( x y ) . Now, x ∗ y ∈ W , whereby ( x ∗ y ) ∈ L γ . Since γ ( x y ) + γ ( x y ) ∈ L γ , itfollows that xyxy + yxyx ∈ L γ . To prove the claim it now suffices to show that xyxy + yxyx ∈ F . Since x ∈ W , we have γ ( xyxy + yxyx ) · x = x · ( xyxy + yxyx ) = x yxy + xyxyx. Since y ∈ W , yx ∈ W , γ ◦ γ = γ and x ∈ L γ , we have yxyx = ( yx ) γ ( x ) y = γ ( γ ( x )) yxy = x yxy, hence γ ( xyxy + yxyx ) · x = yxyx + xyxyx = ( xyxy + yxyx ) · x and therefore (cid:0) γ ( xyxy + yxyx ) − ( xyxy + yxyx ) (cid:1) · x = 0 . Observe that, in any ´etale quadratic algebra with nontrivial automorphism ι ,elements of the form ι ( z ) − z are invertible if they are nonzero. Therefore, thelast equation yields γ ( xyxy + yxyx ) = xyxy + yxyx. It follows that xyxy + yxyx ∈ L γ ∩ L γ = F . This establishes the claim.Hence we have shown that ∗ : ( W , q ) × ( W , q ) → ( W , q )is a composition of nondegenerate quadratic forms in the sense of [18, p. 488]. Asthese forms are of dimension 2 n , it follows by [18, (33.18) and (33.27)] that theyare similar to one and the same quadratic n -fold Pfister form π over F .Since π is a Pfister form, it is uniquely determined up to isometry by itssimilarity class, which is also given by each of the forms q , q and q . Sincethe linear forms s i : L γ i → F for i = 1 , , Ker ( s i ) = F , it follows that π does also not depend on the choicesof s , s , s . Therefore π is determined up to isometry by L .Note that dim F Symd ( σ ) = 4+3 · n = dim F L + dim F W + dim F W + dim F W and L + W + W + W ⊆ Symd ( σ ). In order to show that Symd ( σ ) = L ⊕ W ⊕ W ⊕ W ,it therefore suffices to prove that the sum L + W + W + W is direct.We fix an element a ∈ L γ \ F of trace equal to 1 in the quadratic extension L γ /F . Then (2 a − ∈ F × and γ ( a ) = γ ( a ) = 1 − a . Hence for any x ∈ L + W we have ax + xa = 2 ax , whereas for any x ∈ W + W we obtain that ax + xa = x .Since 2 a − ∈ L × it follows that ( L + W ) ∩ ( W + W ) = 0. In the same waywe obtain that ( L + W ) ∩ ( W + W ) = 0. Together this implies that the sum L + W + W + W is direct. (cid:3) Remark. If char ( F ) = 2, then we may take in Theorem 7.3 for i = 1 , , Tr L γi /F as F -linear form s i : L γ i → F with Ker ( s i ) = F . With thiscanonical choice we obtain from the beginning of the proof that c = 1.For a neat biquadratic F -subalgebra L of ( A, σ ), we denote by P σ,L the qua-dratic Pfister form over F which is characterised in Theorem 7.3. We will seelater that this Pfister form does actually not depend on the choice of L . It isclearly functorial:7.5. Proposition.
Assume that cap ( A, σ ) = 4 . Let L be a neat biquadratic F -subalgebra of ( A, σ ) and let F ′ /F be a field extension. Then L ⊗ F ′ is a neatbiquadratic F ′ -subalgebra of ( A, σ ) F ′ and the associated Pfister form P σ F ′ ,L ⊗ F ′ isobtained by scalar extension to F ′ from P σ,L . Recall in this context that all tensor products are taken over F . Proof:
See [6, Proposition 5.5] for the fact that L ⊗ F ′ is neat in ( A, σ ) F ′ . Theremaining part follows directly from the definitions and from Theorem 7.3. (cid:3) We next give a criterion for the hyperbolicity of P σ,L .7.6. Proposition.
Assume that cap ( A, σ ) = 4 . Let L be a neat biquadratic F -subalgebra of ( A, σ ) . Let K be a neat quadratic F -subalgebra of ( A, σ ) containedin L . The following conditions are equivalent: ( i ) P σ,L is hyperbolic. ( ii ) ( A, σ ) is decomposable along L . ( iii ) ( A, σ ) is decomposable along K .Proof: Let G be the Galois group of L/F . Since K and L are neat in ( A, σ ),it follows that L is free as a K -module and hence K is the fixed subalgebra ofsome element of G . We write G = { id L , γ , γ , γ } with K = L γ . For i = 1, 2, 3we set W i = { x ∈ Symd ( σ ) | yx = xγ i ( y ) for all y ∈ L } and fix an F -linear form s i : L γ i → F with Ker ( s i ) = F , and we consider the quadratic form q i : W i → F , x s i ( x ). Then q , q and q are similar to the Pfister form P σ,L . Hence P σ,L is hyperbolic if and only if any of the forms q , q or q is isotropic, and in thiscase they are all isotropic.( i ⇒ ii ): Suppose that P σ,L is hyperbolic. Then q is isotropic, and byLemma 7.1, there exists an element x ∈ W with x ∈ F × . Then Q = L γ ⊕ L γ x is a σ -stable F -quaternion algebra contained in ( A, σ ). Since σ is the identityon L γ , the involution σ | Q is orthogonal. Let A ′ = C A ( Q ) and σ ′ = σ | A ′ . Itfollows that ( A ′ , σ ′ ) is an F -algebra with involution with cap ( A ′ , σ ′ ) = 2. Since K = L γ ⊆ A ′ , it follows by [18, Corollary 6.6] that K is contained in a σ -stable F -quaternion subalgebra Q of A ′ , and hence of A . Hence Q and Q are inde-pendent σ -stable F -quaternion subalgebras of A such that L = ( L ∩ Q ) · ( L ∩ Q ).Therefore ( A, σ ) is decomposable along L .( ii ⇒ iii ): This implication is obvious. DISCRIMINANT PFISTER FORM 23 ( iii ⇒ i ): Recall the quadratic form e q : C A ( K ) ∩ Symd ( σ ) → F defined in theproof of Lemma 7.1 as a transfer with respect to an F -linear form s : K → F .This form e q is Witt equivalent to q and hence similar to P σ,L . Hence to prove( i ) it suffices to show that e q is hyperbolic. Condition ( iii ) yields a decomposition A = Q ⊗ B with a σ -stable F -subalgebra B of A and a σ -stable F -quaternionsubalgebra of A containing K . For C = C A ( K ) we obtain that C = K ⊗ B. Let σ B = σ | B . Since σ is the identity on K , the involution σ | Q is orthogonal.Hence σ B has the same type as σ and cap ( B, σ B ) = 2. The quadratic form c on Symd ( σ C ) over K given by [6, Proposition 4.6] is extended from the correspondingquadratic form c on Symd ( σ B ) over F , hence its transfer with respect to s : K → F is hyperbolic. Since e q = − s ◦ c , it follows that ( i ) holds. (cid:3) Corollary.
Assume that cap ( A, σ ) = 4 . If σ is hyperbolic and exp A ,then P σ,L is hyperbolic for every neat biquadratic F -subalgebra L of ( A, σ ) .Proof: Let L be a neat biquadratic F -subalgebra of ( A, σ ). We fix a neatquadratic F -subalgebra K of ( A, σ ) contained in L . If σ is hyperbolic, thenProposition 6.1 shows that ( A, σ ) is decomposable along K , and hence P σ,L ishyperbolic by Proposition 7.6. (cid:3) Theorem.
Assume that ( A, σ ) is totally decomposable and cap ( A, σ ) = 4 .Then ( A, σ ) is decomposable along every neat biquadratic F -subalgebra of ( A, σ ) .Proof: Let L be a neat biquadratic F -subalgebra of ( A, σ ). By Lemma 6.4there exists a field extension F ′ /F such that ( A, σ ) F ′ is hyperbolic and suchthat every anisotropic n -fold Pfister form over F remains anisotropic over F ′ .Note that exp A F ′ exp A
2, because (
A, σ ) is totally decomposable. Since σ F ′ is hyperbolic, it follows by Proposition 7.5 and Corollary 7.7 that ( P σ,L ) F ′ ishyperbolic. By the choice of F ′ /F , we conclude that P σ,L is hyperbolic. Therefore( A, σ ) is decomposable along L , by Proposition 7.6. (cid:3) Corollary.
Assume ( A, σ ) is totally decomposable, cap ( A, σ ) = 4 and coind A is even. Then ( A, σ ) has a total decomposition involving a split F -quaternionalgebra with orthogonal involution.Proof: If σ is hyperbolic then the statement follows directly from Proposition 5.3.Hence we may assume that σ is not hyperbolic. Since coind ( A, σ ) is even, it followsfrom [6, Corollary 5.12] that (
A, σ ) contains a split quadratic neat F -subalgebra K . By [6, Theorem 6.10], K is contained in a biquadratic neat F -subalgebra L of ( A, σ ). It follows from Theorem 7.8 that (
A, σ ) is decomposable along L . Inparticular, K is contained in a σ -stable F -quaternion subalgebra Q of A . Since K is split, the quaternion algebra Q is split. Since σ is not hyperbolic, we conclude that σ | Q is not hyperbolic. Hence σ | Q is orthogonal, for the unique symplecticinvolution on a split quaternion algebra is hyperbolic. (cid:3) We are now in the position to show that the form P σ,L given by a neat bi-quadratic subalgebra L of ( A, σ ) is independent of the choice of this subalgebra.7.10.
Proposition.
Assume that cap ( A, σ ) = 4 and let n = log dim F A − . Thereexists a unique n -fold Pfister form π over F such that π ≃ P σ,L for every neatbiquadratic F -subalgebra L of ( A, σ ) .Proof: By [6, Theorem 7.4] there exists a neat biquadratic F -subalgebra L of( A, σ ). Consider another neat biquadratic F -subalgebra L ′ of ( A, σ ). We need toshow that P σ,L ′ = P σ,L .If ( A, σ ) is totally decomposable, then we obtain by Theorem 7.8 and Propo-sition 7.6 that P σ,L and P σ,L ′ are hyperbolic, whereby P σ,L ′ = P σ,L . Assumenow that ( A, σ ) is not totally decomposable. Then it follows by Proposition 7.6that P σ,L and P σ,L ′ are both anisotropic. Let F ′ denote the function field of theprojective quadric over F given by P σ,L . Then P σ,L becomes hyperbolic over F ′ .By Proposition 7.5 and Proposition 7.6, it follows that ( A, σ ) F ′ is decomposablealong L ⊗ F ′ . By Theorem 7.8, then ( A, σ ) F ′ is also decomposable along L ′ ⊗ F ′ .By Proposition 7.5 and Proposition 7.6, it follows that P σ,L ′ becomes hyperbolicover F ′ . Since P σ,L and P σ,L ′ are anisotropic n -fold Pfister forms and since F ′ is the function field of P σ,L over F , we conclude by the Subform Theorem [13,(22.5)] that P σ,L ′ = P σ,L . (cid:3) When cap ( A, σ ) = 4, we denote by P σ the quadratic Pfister form over F whichis characterised in Proposition 7.10 and we call it the discriminant Pfister formof ( A, σ ).7.11.
Theorem.
Assume that cap ( A, σ ) = 4 . The form P σ is hyperbolic if andonly if ( A, σ ) is totally decomposable. Moreover, for any field extension F ′ /F ,we have that ( P σ ) F ′ = P σ F ′ .Proof: By [6, Theorem 7.4], there exists a neat biquadratic F -subalgebra of( A, σ ). Hence the first part follows from Proposition 7.6 and Theorem 7.8. Thesecond part follows from Proposition 7.5. (cid:3)
This finishes the proof of our main theorem stated in the introduction.8.
Determination of the discriminant Pfister form
In this final section we relate the discriminant Pfister form of an algebra withinvolution of capacity 4 to other known invariants and we compute it in somespecial situations. We further aim to relate the notions of discriminant Pfisterform for the different types of algebras with involution of capacity 4. We will seethat an embedding between two such algebras with involution leads naturally toa factorisation relation between the corresponding Pfister forms.
DISCRIMINANT PFISTER FORM 25
In the sequel, we shall use the notion of quaternion K -algebra also in caseswhere K is an ´etale F -algebra, but not necessarily a field. More generally, quater-nion K -algebras can be defined over an arbitrary commutative ring K : see [17, p.4] for the general case or [2, p. 27] in the case where K is a semilocal ring (whichtherefore covers the case where K is an ´etale F -algebra). The Skolem-Noethertheorem, familiar in the case where K is a field, extends to the case where K isan ´etale F -algebra: this follows directly by applying it from the situation wherethe center is a field to the simple components of K (whose centers are the simplefactors of K , hence fields).8.1. Lemma.
Let ( A, σ ) be an F -algebra with unitary involution of capacity . Let L be a neat biquadratic F -subalgebra of ( A, σ ) with Galois group { id L , γ , γ , γ } and let Z = Z ( A ) . The following hold: (1) There exists w ∈ Sym ( σ ) ∩ A × such that wℓ = γ ( ℓ ) w for all ℓ ∈ L . (2) For every w as in (1) , we have P σ ≃ h , − Nrd A ( w ) i ⊗ N N/F for the ´etalequadratic F -algebra N = ( L γ ⊗ Z ) γ ⊗ σ | Z .Proof: First observe that, after fixing an element z ∈ Z \ F such that z + σ ( z ) = 1,every element a ∈ A can be decomposed as a = s + s z with s , s ∈ Sym ( σ ),given explicitly by s = (cid:0) z − σ ( z ) (cid:1) − (cid:0) a − σ ( a ) (cid:1) and s = a − s z . Thereforemultiplication in A induces an isomorphism of F -vector spaces Sym ( σ ) ⊗ Z → A .
This allows us to identify L ⊗ Z with the F -subalgebra LZ of A . We obtain that LZ is a Galois F -algebra: the nontrivial elements of its Galois group are σ | LZ and the maps γ i ⊗ id Z and γ i ⊗ σ | Z for i = 1 , , K = L γ . Let C = C A ( K ) = C A ( KZ ), which isa KZ -quaternion algebra and to which σ restricts as a unitary involution. Letfurther(8.1.1) D = { x ∈ C | σ ( x ) = can C ( x ) } . Then D is a K -quaternion algebra: if KZ is a field, then this follows by [18,(2.22)], using that K is the F -subalgebra of KZ fixed under σ | KZ . Clearly, LZ ⊆ C = DZ , and can C restricts to γ ⊗ id Z on LZ because KZ is the centerof C . Therefore the F -algebra M = ( LZ ) γ ⊗ σ | Z lies in D , and it is a quadraticGalois extension of K with Galois group generated by γ ⊗ id Z . By the Skolem–Noether Theorem, ( γ ⊗ id Z ) | M extends to an inner automorphism Int D ( y ) of D for some y ∈ D × . Since ( γ ⊗ id Z ) | M = id M , and D = M [ y ], it follows that y ∈ C ( D ) = K . Since ( γ ⊗ id Z ) | M = id M , we have y / ∈ C ( D ) = K , andas σ | D = can D and y ∈ K , we conclude that σ ( y ) = − y . As C = DZ and M Z = LZ , it follows that Int C ( y ) | L = γ .We pick an element z ′ ∈ Z × such that σ ( z ′ ) = − z ′ and set w = yz ′ . (If char ( F ) = 2 then we may take z ′ = 1.) Then σ ( w ) = w and w ∈ A × , and for any ℓ ∈ L we have w ℓw − = Int C ( y )( ℓ ) = γ ( ℓ ) . Hence w is a possible choice for an element w satisfying the conditions in (1).To prove part (2), we now consider an arbitrary element w as in (1) whilekeeping our choice for w . We set W = { x ∈ Sym ( σ ) | xℓ = γ ( ℓ ) x for all ℓ ∈ L } , and fix an F -linear form s : K → F with Ker ( s ) = F . Since σ is unitary, we have Sym ( σ ) = Symd ( σ ), hence q : W → F, x s ( x )is the quadratic form from Lemma 7.1, and by definition the Pfister form P σ issimilar to q . Since M = ( LZ ) γ ⊗ σ | Z , we have σ | M = ( γ ⊗ id Z ) | M = Int A ( w ) | M ,hence σ ( wm ) = σ ( m ) w = wm for all m ∈ M ,which proves that wm ∈ Sym ( σ ). Since moreover M centralizes L it follows that wM ⊆ W . As w ∈ A × we have dim F wM = dim F M = 4 = dim F W , so weconclude that W = wM . As w ∈ W ∩ A × , we may write w = w m for some m ∈ M , hence w = w m w m = σ ( m ) w m = σ ( m ) y z ′ m . Since y ∈ K , z ′ ∈ F and σ ( m ) m = N M/K ( m ) ∈ K , it follows that w ∈ K .For m ∈ M we have( wm ) = w σ ( m ) m = w N M/K ( m ) ∈ K .
We conclude that q is isometric to the quadratic form q ′ : M → F, m s (cid:0) w N M/K ( m ) (cid:1) . Note that M = ( LZ ) γ ⊗ σ | Z is a Galois F -algebra whose Galois group is generatedby σ | M = γ ⊗ id Z and γ ⊗ id Z . Since σ | K = id K , it follows that M = KN for N = M γ ⊗ id Z = ( L γ ⊗ Z ) γ ⊗ σ | Z and M is naturally isomorphic to K ⊗ N . Therefore the quadratic form N M/K extends N N/F . By Frobenius Reciprocity [13, (20.2), (20.3c)] it follows that q ′ isisometric to s ∗ ( h w i ) ⊗ N N/F . Set d = Nrd A ( w ). Since w / ∈ K and w ∈ K , weobtain that d = N K/F ( w ). Since s ∗ ( h i ) is hyperbolic, it follows from [13, (34.19)]that s ∗ ( h w i ) is similar to h , − d i . Therefore q ′ is similar to h , − d i ⊗ N N/F . Thisshows that the forms P σ and h , − d i ⊗ N N/F are similar, and since they are bothPfister forms, we conclude that they are isometric. (cid:3)
Proposition.
Let ( A, σ ) be an F -algebra with unitary involution of capacity . Then P σ ≃ Nrd Q for an F -quaternion algebra Q , which is Brauer equivalentto the discriminant algebra of ( A, σ ) . DISCRIMINANT PFISTER FORM 27
Proof:
We fix a neat biquadratic F -subalgebra L of ( A, σ ), whose existence isguaranteed by [6, Theorem 7.4]. We use the same notation as in the proof ofLemma 8.1. By [18, p. 129], the K -quaternion algebra D defined in (8.1.1) is thediscriminant algebra of C . By [10, Lemma 3.1(2)], the discriminant algebra of( A, σ ) is Brauer equivalent to the corestriction of D to F .The proof of Lemma 8.1 yields that D = M ⊕ M y = KN ⊕ KN y where M is a biquadratic Galois F -algebra and K and N are the subalgebras fixedby two different nontrivial elements of the Galois group of M/F and where y ∈ D × is such that Int D ( y ) extends the nontrivial K -automorphism of M , σ ( y ) = − y and y ∈ K . Hence D is isomorphic to the crossed product algebra ( KN/K, y )over K . Since M = KN , which is a free compositum over F , it follows by theprojection formula (see [15, Prop. 3.4.10]) that the corestriction of D is Brauerequivalent to the crossed product algebra Q = (cid:0) N/F, N K/F ( y ) (cid:1) over F , which isan F -quaternion algebra with norm form h , − N K/F ( y ) i ⊗ N N/F .By the proof of Lemma 8.1, after choosing z ′ ∈ C ( A ) × with σ ( z ′ ) = − z ′ andletting w = yz ′ , we obtain that P σ ≃ h , − Nrd A ( w ) i ⊗ N N/F . Since z ′ ∈ F × and Nrd A ( w ) = N K/F ( y ) z ′ , it follows that P σ is isometric to Nrd Q . (cid:3) We can now retrieve from Theorem 7.11 the criterion from [16, Section 3] fortotal decomposability of an algebra with unitary involution of capacity 4.8.3.
Corollary (Karpenko-Qu´eguiner) . An F -algebra with unitary involution ofcapacity is totally decomposable if and only if its discriminant algebra is split.Proof: Let (
A, σ ) be an F -algebra with unitary involution of capacity 4 and let D be its discriminant algebra. By Proposition 8.2, D is Brauer equivalent to an F -quaternion algebra Q such that P σ ≃ Nrd Q . It follows that D is split if andonly if P σ is hyperbolic, and by Theorem 7.11 this is equivalent to ( A, σ ) beingtotally decomposable. (cid:3)
When char F = 2, for an F -algebra with symplectic involution of degree amultiple of 8, a cohomological invariant ∆( A, σ ) ∈ H ( F, µ ) was defined in [14].In the case where deg A = 8 this invariant is related to the discriminant Pfisterform.For a 3-fold Pfister form π over a field F of characteristic different from 2,and for a , b , c ∈ F × such that π ≃ h , − a i ⊗ h , − b i ⊗ h , − c i , the cup product( a ) ∪ ( b ) ∪ ( c ) in H ( F, µ ) is an invariant of π , by [1, Satz 1.6], also called the Arason invariant of π .8.4. Proposition (Garibaldi-Parimala-Tignol) . Suppose that char F = 2 . Let ( A, σ ) be an F -algebra with symplectic involution with cap ( A, σ ) = 4 . Then ∆( A, σ ) is the Arason invariant of P σ . Furthermore ∆( A, σ ) = 0 if and only if ( A, σ ) is totally decomposable. Proof:
It is proven in [14, Proposition 8.1] that ∆(
A, σ ) is the Arason invariantof a 10-dimensional quadratic form of trivial discriminant and Clifford invariant:this is the form e q appearing the proof of Lemma 7.1. This form e q is Witt equiv-alent to the 8-dimensional quadratic form q in Lemma 7.1, and the Pfister form P σ is by definition similar to q , whereby its Arason invariant is the same as for q and e q . This relates the two invariants in the way as it is claimed here.The equivalence of the vanishing of ∆( A, σ ) with the decomposability of (
A, σ )is shown in [14, Section 9]; it can now alternatively be obtained from Theo-rem 7.11. In either way one relies on the fact that a quadratic 3-fold Pfister formis hyperbolic if and only if its Arason invariant is trivial, which follows from [1,Satz 5.6]. (cid:3)
We return to the situation where the field F is of arbitrary characteristic. Foran involution σ on an F -algebra A one defines Alt ( σ ) = { x − σ ( x ) | x ∈ A } . Recall from [18, (7.2)] that the discriminant of an orthogonal involution σ on acentral simple F -algebra A of even degree 2 m is the square class ( − m Nrd A ( y ) F × in F × /F × given by an arbitrary element y ∈ A × ∩ Alt ( σ ), and that there alwaysexists such an element.8.5. Lemma.
Let ( B, τ ) be an F -algebra with orthogonal involution of degree .Let d ∈ F × be such that the discriminant of τ is dF × . Then d is represented by N E/F for every neat quadratic F -subalgebra E of ( B, τ ) .Proof: Fix a neat quadratic F -subalgebra E of ( B, τ ). Then C = C B ( E ) isa quaternion E -algebra. We fix an element y ∈ C × ∩ Alt ( τ | C ). Since τ | C isorthogonal, we have y / ∈ E . On the other hand y ∈ E . Hence deg B = 4 =[ E [ y ] : F ], and it follows that Nrd B ( y ) = N E/F ( y ). By the choice of d , we obtainthat dF × = Nrd B ( y ) F × = N E/F ( y ) F × , which shows the claim. (cid:3) Proposition.
Let ( B, τ ) be an F -algebra with orthogonal involution of capac-ity . Let d ∈ F × be such that dF × is the discriminant of ( B, τ ) . The followinghold: (1) If char F = 2 , then P τ = h , − d i . (2) For any quadratic ´etale F -algebra Z , the discriminant Pfister form of the F -algebra with unitary involution ( B, τ ) ⊗ ( Z, can Z/F ) of capacity is givenby h , − d i ⊗ N Z/F . (3) For any F -quaternion algebra Q , the discriminant Pfister form of the F -algebra with symplectic involution ( B, τ ) ⊗ ( Q, can Q ) of capacity is given by h , − d i ⊗ Nrd Q where Nrd Q is the reduced norm form of Q .Proof: By [6, Theorem 7.4], (
B, τ ) contains a neat biquadratic F -subalgebra L .Let { id L , γ , γ , γ } be the Galois group of L viewed as a Galois F -algebra. We set K = L γ and fix an F -linear functional s : K → F with Ker ( s ) = F . We further DISCRIMINANT PFISTER FORM 29 set C = C B ( K ) and observe that C is a K -quaternion algebra containing L andthat τ restricts to an orthogonal involution on C . We fix y ∈ C × ∩ Alt ( τ | C ).Then y ∈ K and y ∈ B × ∩ Alt ( τ ), hence(8.6.1) can C ( y ) = − y, Nrd B ( y ) = N K/F ( y ) and dF × = Nrd B ( y ) F × . Moreover, since y ∈ Alt ( τ | C ) and L ⊆ Sym ( τ | C ), it follows that Trd C ( yL ) = 0,by [18, (2.3)]. As can C | L = γ | L , this implies that(8.6.2) yℓ = γ ( ℓ ) y for every ℓ ∈ L .(1) Recall that (only) for part (1) we assume that char F = 2. Thus, Symd ( τ ) = Sym ( τ ) and L = K ⊕ vK for some element v ∈ L × with v ∈ F × and γ ( v ) = − v .The latter implies that yv = − vy . Let W = { x ∈ Sym ( τ ) | xℓ = γ ( ℓ ) x for all ℓ ∈ L } . By definition, the Pfister form P τ is similar to the quadratic form q : W → F, x s ( x ) . As τ ( y ) = − y , τ ( v ) = v and yv = − vy , it follows that yv ∈ Sym ( τ ), andas v ∈ L , we conclude by (8.6.2) that yv ∈ W . Hence yvK ⊆ W , and since yv ∈ B × and therefore dim F yvK = dim F K = 2 = dim F W , we obtain that W = yvK . For x ∈ K , we have q ( yvx ) = s (cid:0) ( yv ) x (cid:1) = − v s ( y x ) . Therefore, q , hence also P τ , is similar to s ∗ ( h y i ). But s ∗ ( h y i ) is similar to h , − N K/F ( y ) i by [13, (34.19)], hence also to h , − d i by (8.6.1). We concludethat P τ is similar to h , − d i , and since both binary forms represent 1, it followsthat they are isometric.(2) For proving part (2), we set ( A, σ ) = (
B, τ ) ⊗ ( Z, can Z/F ) and fix an element z ∈ Z × such that can Z/F ( z ) = − z . (If char ( F ) = 2 we may choose z = 1.)Then yz ∈ Sym ( σ ) ∩ A × and (8.6.2) shows that yzℓ = γ ( ℓ ) yz for every ℓ ∈ L .Hence it follows from Lemma 8.1 that P σ ≃ h , − Nrd A ( yz ) i ⊗ N N/F for the ´etalequadratic F -algebra N = ( L γ ⊗ Z ) γ ⊗ can Z/F . Since z ∈ F × , we have by (8.6.1) Nrd A ( yz ) = Nrd A ( y ) z ∈ dF × , whereby h , − Nrd A ( yz ) i ≃ h , − d i . To completethe proof, it suffices to show that h , − d i ⊗ N N/F is isometric to h , − d i ⊗ N Z/F .For this we note that ( LZ ) γ ⊗ id Z is a Galois F -algebra with Galois group iso-morphic to ( Z / Z ) , and the quadratic F -subalgebras fixed under the nontrivialelements of the Galois group are N , Z and L γ . Hence N ⊗ L γ ≃ Z ⊗ L γ .If L γ splits, then N ≃ Z , hence N N/F ≃ N Z/F and the proof is complete. If L γ is a field, the isomorphism N ⊗ L γ ≃ Z ⊗ L γ implies that N N/F and N Z/F become isometric after scalar extension to L γ , hence by [13, (34.9)] the form N N/F ⊥ − N Z/F is Witt equivalent to a multiple of N L γ /F . But since L γ is aneat quadratic subfield of ( B, τ ), Lemma 8.5 implies that h , − d i ⊗ N L γ /F is hyperbolic. Hence h , − d i ⊗ ( N N/F ⊥ − N Z/F ) is hyperbolic. Therefore we have h , − d i ⊗ N N/F ≃ h , − d i ⊗ N Z/F .(3) Refreshing the notation, we set (
A, σ ) = (
B, τ ) ⊗ ( Q, can Q ), which is an F -algebra with symplectic involution. We fix a quadratic ´etale F -subalgebra Z of Q . The non-trivial F -automorphism of Z extends to Int Q ( j ) for some j ∈ Q × ,and we obtain that Q = Z ⊕ Zj and j ∈ F × . Let A ′ = C A ( Z ) = B ⊗ Z and note that A ′ is σ -stable. Weset σ ′ = σ | A ′ . Then ( A ′ , σ ′ ) is an F -algebra with unitary involution. Note that cap ( A, σ ) = cap ( A ′ , σ ′ ) = 4 and L is a neat biquadratic F -subalgebra of ( A ′ , σ ′ )and of ( A, σ ). By (2) we have P σ ′ ≃ h , − d i ⊗ N Z/F . We set W = { x ∈ Symd ( σ ) | xℓ = γ ( ℓ ) x for all ℓ ∈ L } and W ′ = { x ∈ Symd ( σ ′ ) | xℓ = γ ( ℓ ) x for all ℓ ∈ L } . Since σ ′ is unitary we have Symd ( σ ′ ) = Sym ( σ ′ ) and thus W ′ = W ∩ A ′ . By thedefinition, P σ is similar to the quadratic form q : W → F, x s ( x ), and P σ ′ issimilar to q ′ : W ′ → F, x s ( x ), which is the restriction of q to W ′ .We first look at the case where q ′ is isotropic. In this case P σ ′ and P σ areisotropic, and hence hyperbolic, because they are Pfister forms. Since P σ ′ ≃h , − d i ⊗ N Z/F , and since N Z/F is a subform of
Nrd Q , it follows that the 3-foldPfister form h , − d i ⊗ Nrd Q is hyperbolic, and hence isometric to P σ .We may now assume for the rest of the proof that q ′ is anisotropic. Note that Int A ( j ) commutes with σ ′ , because σ ( j ) = − j and j ∈ Z = C ( A ′ ). Moreover j ∈ C A ( L ). It follows that W ′ is preserved under Int A ( j ).Note that L ∩ W ′ = 0 and L ⊕ W ′ ⊆ C A ′ ( K ). However Sym ( τ ) C A ( K ),so in particular Sym ( τ ) = L ⊕ W ′ . Since dim F L ⊕ W ′ = 8 = dim F Sym ( τ ) and L ⊆ Sym ( τ ), we obtain that W ′ Sym ( τ ) = Sym ( σ ′ ) ∩ B . As W ′ ⊆ Sym ( σ ′ ), weconclude that W ′ B .Since Q = Z ⊕ jZ and W ′ ⊆ A ′ = C A ( Z ), it follows that there exists w ∈ W ′ such that jw = w j . Let w = jw j − − w ∈ W ′ . Then w = 0 and wj = jw − w j = jw + σ ( jw ) ∈ Symd ( σ ) . Moreover, wj ∈ W because w ∈ W ′ ⊆ W , j ∈ C A ( L ) and jw = − wj .We fix an element z ∈ Z \ F with z − z ∈ F . Then jzj − = 1 − z .For every w ′ ∈ W ′ we have(8.6.3) ( w ′ wj + wjw ′ ) z = (1 − z )( w ′ wj + wjw ′ ) . On the other hand, since x ∈ K for every x ∈ W , it follows that w ′ wj + wjw ′ = ( w ′ + wj ) − w ′ − ( wj ) ∈ K, DISCRIMINANT PFISTER FORM 31 whereby(8.6.4) ( w ′ wj + wjw ′ ) z = z ( w ′ wj + wjw ′ )because K ⊆ A ′ = C A ( Z ). By comparing (8.6.3) and (8.6.4), we obtain for every w ′ ∈ W ′ that w ′ wj + wjw ′ = 0. This proves that wj lies in the orthogonalcomplement of W ′ with respect to the quadratic form q .We set a = q ( w ). As w ∈ W ′ \ { } and q ′ = q | W ′ is anisotropic, we have that a ∈ F × . Since q | W ′ is similar to P σ ′ , we obtain that P σ ′ ≃ aq ′ . Similarly, since q is similar to P σ and represents a , we obtain that P σ ≃ aq .Set b = j . Then b ∈ F × and Nrd Q ≃ h , − b i ⊗ N Z/F . We further have q ( wj ) = s (cid:0) ( wj ) (cid:1) = s ( − bw ) = − bq ( w ) = − ab . Since wj lies in the orthogonal complement of W ′ with respect to q , it followsthat q ′ ⊥ h− ab i is a subform of q . Therefore P σ ′ ⊥ h− b i is a subform of P σ . Onthe other hand, having P σ ′ ≃ h , − d i ⊗ N Z/F and h , − b i ⊗ N Z/F ≃ Nrd Q , we alsohave that P σ ′ ⊥ h− b i is a subform of h , − d i ⊗ Nrd Q . Hence the quadratic 3-foldPfister forms P σ and h , − d i ⊗ Nrd Q share a common 5-dimensional subform. Inview of [13, Lemma 23.1] this readily yields that they are isometric. (cid:3) Remark.
In view of Proposition 8.2, one can derive part (2) of Proposi-tion 8.6 alternatively from the description of the Brauer class of the discriminantalgebra of (
B, τ ) ⊗ ( Z, can Z/F ) in [18, (10.33)].We round up by computing the discriminant Pfister form in some special casesof Proposition 8.6 where the algebra B is split.8.8. Examples.
Let B = End F ( V ) for some 4-dimensional F -vector space V . Let β : V × V → F be a nondegenerate symmetric bilinear form over F , and let ad β denote the adjoint involution on End F ( V ), which is determined by β ( u, f ( v )) = β ( ad β ( f )( u ) , v ) for all f ∈ End F ( V ) , u, v ∈ V. Let d ∈ F × be the determinant of β (determined up to a square factor). ApplyingProposition 8.6, we obtain the following results:(1) If char ( F ) = 2, then P ad β ≃ h , − d i .(2) For ( A, σ ) = (
End F ( V ) , ad β ) ⊗ ( Z, can Z/F ), where Z is a quadratic ´etale F -algebra, we obtain that P σ ≃ h , − d i ⊗ N Z/F .(3) Let Q be an F -quaternion algebra. For ( A, σ ) = (
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