"Cohomological invariants for orthogonal involutions on degree 8 algebras"
Quéguiner-Mathieu, Anne ; Tignol, Jean-Pierre
Abstract Using triality, we define a relative Arason invariant for orthogonal involutions on a -possibly division- central simple algebra of degree 8. This invariant detects hyperbolicity, but it does not detect isomorphism. We produce explicit examples, in index 4 and 8, of nonisomorphic involutions with trivial relative Arason invariant.
Document type : Article de périodique (Journal article)
Référence bibliographique
Quéguiner-Mathieu, Anne ; Tignol, Jean-Pierre. Cohomological invariants for orthogonal involutions on degree 8 algebras. In: Journal of K-Theory : k-theory and its applications to algebra, geometry, analysis and topology, Vol. 9, no. 2, p. 333-358 (2012)
DOI : 10.1017/is011006015jkt160
Available at: http://hdl.handle.net/2078.1/110505
[Downloaded 2019/05/08 at 12:52:58 ] J. K-Theory 9 (2012), 333–358 ©2011 ISOPP doi:10.1017/is011006015jkt160
Cohomological invariants for orthogonal involutions on degree 8 algebras
by
ANNE QUÉGUINER-MATHIEU AND JEAN-PIERRE TIGNOL
Abstract
Using triality, we define a relative Arason invariant for orthogonal involutions on a -possibly division- central simple algebra of degree 8. This invariant detects hyperbolicity, but it does not detect isomorphism. We produce explicit examples, in index 4 and 8, of non isomorphic involutions with trivial relative Arason invariant.
Key Words: Cohomological invariant, orthogonal group, algebra with involu- tion, Clifford algebra, half-neighbor, triality Mathematics Subject Classification 2010: 11E72, 11E81, 16W10
1. Introduction
The discriminant and the Clifford algebra are classical invariants of quadratic forms over a field F of characteristic different from 2. Up to similarity, the discriminant classifies quadratic forms of dimension 2, while the even part of the Clifford algebra classifies forms of dimension 4. In [2], Arason defined an invariant e3,foreven- dimensional quadratic forms with trivial discriminant and split Clifford algebra, 3 which has values in H .F; 2/. Again, this invariant is classifying in dimension 8, that is for forms similar to a 3-fold Pfister form. The main purpose of this paper is to study the same question for orthogonal involutions on central simple algebras of degree 8. To see the relation between quadratic forms and involutions, recall that every (nondegenerate) quadratic form q on an F -vector space V defines an adjoint involu- tion adq on the endomorphism algebra EndF V , and every orthogonal involution on EndF V has the form adq for some (nondegenerate) quadratic form q on V , uniquely determined up to a scalar factor, see [20, p. 1]. Therefore, orthogonal involutions on central simple algebras can be viewed as twisted forms (in the sense of Galois cohomology) of quadratic forms up to scalars. Analogues for orthogonal involutions of the discriminant and the (even) Clifford algebra were defined by Jacobson and
Both authors acknowledge support from Wallonie–Bruxelles International and the French government in the framework of “Partenariats Hubert Currien” 334 A.QUÉGUINER-MATHIEU &J.P.TIGNOL
Tits. By [20, (7.4) and § 15.B], the discriminant classifies orthogonal involutions on a given quaternion algebra, and the Clifford algebra classifies orthogonal involutions on a given biquaternion algebra. On the other hand, it was shown in [4, § 3.4] that the Arason invariant does not extend to orthogonal involutions if the underlying algebra is a degree 8 division algebra. In this paper, we define a relative Arason invariant for orthogonal involutions on a degree 8 (possibly division) central simple algebra A over F . Our construction, based on triality [20, § 42], is specific to this degree. Namely, we assign to any pair of involutions . ; 0/ on A with trivial discriminant and 0 3 Clifford invariant, a degree 3 cohomology class e3. = / 2 H .F; 2/=F ŒA , explicitly described as the Arason invariant of some quadratic form associated to . ; 0/, and uniquely defined up to a multiple of the Brauer class of A (see 2.4). This invariant commutes with scalar extensions, and is compatible with the Arason invariant for quadratic forms, meaning that if A is split, and the involutions and 0 0 0 are respectively adjoint to the 3-fold Pfister forms and ,thene3. = / D 0 e3. / e3. / (see 2.5). If the algebra A is endowed with an involution with trivial discriminant and Clifford invariant, it is isomorphic to a tensor product of three quaternion algebras by [20, (42.11)]. Hence, combining [25] and [19, 4.6], we get that the restriction map induces an injection 3 3 H F;Q=Z.2/ =F ŒA ! H FA;Q=Z.2/ ; where FA is a generic splitting field of A. Hence, the relative Arason invariant 0 e3. = / is uniquely determined by its prescribed value in the split case. In particular, it coincides with the relative Arason invariant that one may define using the Rost invariant for Spin groups (see [12, § 3] and [33, § 3.5]). The definition based on the Rost invariant, as opposed to the one proposed here, is also valid in higher degree. The resulting invariant is studied in [28], where we provide a formula for computing it in various cases. This invariant is not anymore uniquely determined by its value in the split case (see [12, Cor. 10.2]). It has values in 3 ˝2 3 H .F; 4 /=F ŒA , and need not in general be represented by a class of H .F; 2/ (see [28]). Coming back to the present paper, note that in degree 8, contrary to what happens for quadratic forms, the relative e3-invariant is not classifying. Theorem 3.1 0 below states that the involutions and have trivial relative e3 if and only if they are conjugate over any splitting field of A. As was noticed by Sivatski [29, Prop. 4], Hoffmann’s examples of non-similar 8-dimensional quadratic forms that are half- neighbors (see [14, § 4]) prove the existence of non-isomorphic involutions that become isomorphic over any splitting field of the algebra. Using this, we produce Cohomological invariants in degree 8 335
0 0 in § 4 explicit examples of non-conjugate involutions and , with e3. = / D 0, on algebras that have index 4 and 8. To classify involutions in degree 8, we define in 3.5 an additional relative 0 0 0 invariant e4. = /, associated to any pair of involutions . ; / with e3. = / D 0, 4 and which has values in a quotient of H .F; 2/. We prove that the totally decomposable involutions and 0 of A are isomorphic if and only if the relative 0 0 invariants e3. = / and e4. = / are both trivial. Even though the relative Arason invariant does not detect isomorphism, it detects hyperbolicity. This is explained in § 5, where we concentrate on algebras of index 4. As in Garibaldi [12, § 3], we define the Arason invariant of a totally decomposable involution to be its relative Arason invariant with respect to an arbitrary hyperbolic involution, and we prove it vanishes if and only if the involution is hyperbolic. We also provide explicit formulas for computing this invariant in terms of some decomposition of .A; /, and give necessary and sufficient conditions 3 under which it is represented by a single symbol of H .F; 2/ (see 5.3 and 5.8).
Notation Throughout this paper, we work over a base field F of characteristic different from 2. We refer to [20] and to [22] for background information on central simple algebras with involution and on quadratic forms. The invariants defined below are represented by cohomology classes of order 2. Hence, for every integer n 0, we use the notation
n n H .F / D H .Gal.Fsep=F /; 2/:
1 2 2 In particular, H .F / D F =F ,andH .F / D Br2.F / is the 2-part of the Brauer group of F .Foreverya 2 F we let .a/ 2 H 1.F / be the square class of a,andfor n a1,...,an 2 F we let .a1;:::;an/ 2 H .F / be the cup-product
.a1;:::;an/ D .a1/ .an/:
In particular, .a1;a2/ denotes the Brauer class of the quaternion algebra .a1;a2/F . In general, if A is an F -central simple algebra, we let ŒA be the Brauer class of A. If A has exponent 2, we view ŒA as an element in H 2.F /. This applies in particular to the Clifford algebra of any even-dimensional quadratic form. If L is a field extension of F ,weletAL D A ˝F L be the L-algebra obtained from A by extending scalars. On the other hand, NL=F denotes the corestriction n n 3 map from H .L/ to H .F /.IfA has exponent 2, the notation MA.L/ stands for the quotient of H 3.L/ defined by 3 3 MA.L/ D H .L/= L ŒAL ; 336 A.QUÉGUINER-MATHIEU &J.P.TIGNOL
3 where L ŒAL denotes the subgroup f.`/ ŒAL ; ` 2 L g H .L/.LetFA be the function field of the Severi–Brauer variety of A, which is a generic splitting field of A. Scalar extension from F to FA yields a group homomorphism
3 3 3 MA.F / ! MA.FA/ D H .FA/:
It follows from [25] and [19, 4.5 and 4.6] that this map is injective when A decomposes into a tensor product of three quaternion algebras, even though it is not injective in general, even for A of degree 8. We depart from the notation in [22] by letting hha1;:::;anii denote the n-fold Pfister form
hha1;:::;anii D h1; a1i h1; ani for a1,...,an 2 F .
For any quadratic form q over F ,weletd.q/ 2 F =F 2 be the (signed) discriminant of q, C.q/ its Clifford algebra, and C0.q/ the even part of C.q/.Ifq has even dimension, its Clifford invariant is the Brauer class of C.q/. If additionally it has trivial discriminant, then C0.q/ is a direct product of two central simple algebras over F , which are Brauer equivalent to C.q/.An8-dimensional quadratic form has trivial discriminant and Clifford invariant if and only if it is similar to a 3-fold Pfister form. If so, its Arason invariant is given, with our conventions, by e3 hha1;a2;a3ii D .a1;a2;a3/:
More generally, for any quadratic form q of even dimension and trivial discriminant, and any scalar a 2 F , the quadratic form hhaii ˝ q has trivial discriminant and Clifford invariant. Its Arason invariant is given by
e3.hhaii ˝ q/ D .a/ ŒC.q/ :
For any orthogonal involution on A,weletd. / 2 F =F 2 be its (signed) discriminant and C.A; / its Clifford algebra, as defined in [20, § 7 and § 8]. If q is an even-dimensional quadratic form on the F -vector space V , we denote by Adq the split algebra with involution .EndF .V /;adq/. It satisfies
d.adq/ D d.q/ and C.Adq/ D C0.q/:
Any involution with trivial discriminant has Clifford algebra isomorphic to a direct product of two F -central simple algebras
C.A; / D CC.A; / C .A; /: Cohomological invariants in degree 8 337
If moreover the degree of A is divisible by 4, then the Brauer classes of C˙.A; / have order 2 and satisfy
ŒCC.A; / C ŒC .A; / D ŒA :
We say that has trivial Clifford invariant if one of the two components is split, so that the other is Brauer equivalent to A; in the split case, this amounts to ŒC.q/ D 0. If A has degree 4,and has trivial discriminant, then C.A; / is a direct product Q1 Q2 of two quaternion algebras Q1 and Q2 over F . Moreover, by [20, (15.12)], we have .A; / D .Q1; / ˝ .Q2; /; where stands for canonical involutions. Note that such an .A; / can also be written as a product of two quaternion algebras with orthogonal involutions. For instance, the same computation as in [27, Proof of Lem. 6.1] shows that for any a;a0;ı;ı0 2 F , 0 0 0 0 0 0 .a;ı/F ; ˝ ..a ;ı /F ; Š ..aı ;ı/F ; ˝ ..a ı;ı /F ; ; (1) where and 0 are orthogonal involutions of discriminant .ı/ and .ı0/ respectively.
2. Relative Arason invariant
Throughout the paper, A denotes a central simple F -algebra of degree 8.We assume it is endowed with an orthogonal involution that has trivial discriminant and Clifford invariant. By [20, (42.11)], this precisely means that .A; / is totally decomposable, i.e. isomorphic to a tensor product of three quaternion algebras with involution. By the proof of [20, (42.11)] we actually have: Lemma 2.1 There exists a quadratic form ' over F , unique up to similarity, such that .C0.'/;can/ D .A; / .A; /: We call this form ' the quadratic form associated to .A; / by triality. Indeed, since A has degree 8, the canonical involution on C.A; / induces an orthogonal involution on each factor; the involution on the split factor is adjoint to a quadratic form '. Now, it follows from triality [20, (42.3)] (see also [27, 5.1]) that the even Clifford algebra of ' with canonical involution is .C0.'/;can/ D .A; / .A; /. Moreover, for any 8-dimensional form with even Clifford algebra .C0. /;can/ D .A; / .A; /, we have again by triality, .C.A; /; / D Ad .A; /. Hence Ad is isomorphic to Ad',sothat is similar to '. 338 A.QUÉGUINER-MATHIEU &J.P.TIGNOL
Example 2.2 Consider the algebra with involution
.A; / D .Q1 ˝ Q2 ˝ Q3; ˝ ˝ /; where Qi ,fori 2f1;2;3g,isanF -quaternion algebra, stands for canonical involutions, and is an orthogonal involution of Q3 of discriminant d. / D .ı/. We may then find 2 F such that ŒQ3 D .ı; /. Note that we also have Q3 Š . ; ı/F and the anticommuting pure quaternions of square and ı are skew-symmetric under .LetŒQ1 D .˛;ˇ/ and ŒQ2 D ."; /. We claim that the following quadratic form is associated to .A; / by triality:
' Dh1; ˛; ˇ;˛ˇ ; ˛ˇ ı;˛ˇı";˛ˇı ; ˛ˇı" i:
To see this, let .ei /1 i 8 be an orthogonal basis for ' corresponding to this diagonalisation. One may check that the elements i1 D e1e2, j1 D e1e3, i2 D e1e2e3e4, j2 D e1e2e3e5, i3 D e1e2e3e4e5e6, j3 D e1e2e3e4e5e7 and z D e1e2e3e4e5e6e7e8 of C0.'/ generate an algebra with involution isomorphic to
.˛;ˇ/F ˝ . ; ı/F ˝ ."; /F ˝ .F F/ endowed with the involution ˝ ˝ ˝ Id, which is nothing but .A; / .A; /. Hence, by dimension count, .C0.'/;can/ Š .A; / .A; /. Consider now another totally decomposable involution 0 on A, and denote by '0 the quadratic form associated to 0 by triality. Since C.'/ and C.'0/ are both Brauer-equivalent to A, the difference ' '0 belongs to I 3F . The forms ' and '0 are uniquely defined only up to a scalar factor. The following lemma shows that the 3 0 class in MA.F / of the Arason invariant of ' ' does not depend on these scalars: Lemma 2.3 For any , 0 2 F , we have
0 0 0 3 e3.h i' h i' / D e3.' ' / 2 MA.F /:
Proof: In the Witt ring WF of the field F ,wehave
h i' h 0i'0 D .' '0/ hh ii' Chh 0ii'0:
Since ', '0 both have trivial discriminant and Clifford algebra Brauer equivalent to A, each term of the sum on the right side has trivial discriminant and Clifford invariant. Hence we get
0 0 0 0 e3.h i' h i' / D e3.' ' / . / ŒA C . / ŒA ; and the lemma is proved. Cohomological invariants in degree 8 339
This leads to the following definition of a relative Arason invariant for totally decomposable involutions: Definition 2.4 Let A be a degree 8 algebra endowed with two totally decomposable involutions and 0.TheArason invariant of with respect to 0 is
0 0 3 e3. = / D e3.' ' / 2 MA.F /; where ' and '0 are quadratic forms respectively associated to and 0 by triality as in Lemma 2.1. This is a well defined invariant of the pair of involutions . ; 0/ by the previous lemma. It is functorial, since the Clifford algebra construction and the Arason invariant for quadratic forms commute with scalar extensions. Moreover, it readily 0 0 00 follows from the definition that e3. = / D e3. = /,andif is another totally decomposable involution on A,wehave
00 0 0 00 e3. = / D e3. = / C e3. = /: (2)
Let us now compute its value in the split case: Proposition 2.5 If A is split, and and 0 are respectively adjoint to the three-fold Pfister forms and 0,then
0 0 3 e3. = / D e3. / e3. / 2 H .F /:
Indeed, as explained in [27, 5.1], the algebra A is split if and only if the forms ' 0 0 and ' are similar to 3-fold Pfister forms and , and we then have D ad and 0 0 0 0 3 D ad 0 . Hence we get e3. = / D e3.' ' / D e3. / e3. / 2 H .F /. Remark 2.6 Let A be an algebra of exponent 2 and ' a quadratic form of even dimension and trivial discriminant such that C.'/ is Brauer equivalent to A. In [18], 3 3 Bruno Kahn explains how one can view e3.'FA / 2 H .FA/=H .F / as an invariant of the algebra A.Indegree8, we may choose ' of dimension 8, which amounts to the choice of a totally decomposable orthogonal involution of A. For two different choices ' and '0, the relative Arason invariant of the corresponding involutions defined above precisely is the part that is factored out in Bruno Kahn’s invariant. Example 2.7 Let us consider a tensor product of three quaternion algebras, A D 0 0 Q1 ˝ Q2 ˝ Q3 endowed with the involutions D ˝ ˝ and D ˝ ˝ , where stands for canonical involutions, and and 0 are orthogonal involutions of 0 Q3 of discriminant respectively equal to .ı/ and .ı /. By the common slot lemma [2, 0 0 Lemma 1.7], there exists 2 F such that ŒQ3 D .ı; / D .ı ; /. Since .ıı ; / D 340 A.QUÉGUINER-MATHIEU &J.P.TIGNOL
2 0 0 0 0 0 2 H .F /,wehave.ıı / ŒA D .ıı / ŒQ1 C .ıı / ŒQ2 , hence, .ıı /:ŒQ1 0 .ıı /:ŒQ2 modF ŒA . We claim that the relative Arason invariant is given by
0 0 0 3 e3. = / D .ıı / ŒQ1 D .ıı / ŒQ2 2 MA.F /:
The proof goes as follows. Denote ŒQ1 D .˛;ˇ/ and ŒQ2 D ."; /. By Example 2.2, the following quadratic forms are associated respectively to and to 0 by triality:
' Dh1; ˛; ˇ;˛ˇ ; ˛ˇ ı;˛ˇı";˛ˇı ; ˛ˇı" i '0 Dh1; ˛; ˇ;˛ˇ ; ˛ˇ ı0;˛ˇı0";˛ˇı0 ; ˛ˇı0" i:
0 To compute e3.' ' /, note that
' '0 Dh ˛ˇıihhıı0iih ; "; ;" i:
Moreover, since .ıı0; / D 0, is a similarity factor for hhıı0ii, and we get
' '0 Dh ˛ˇıihhıı0;"; ii;
0 0 0 so that e3.' ' / D .ıı ;"; / D .ıı / ŒQ2 .
3. The e4-invariant
As opposed to what happens for quadratic forms, the relative Arason invariant is not a complete invariant for totally decomposable involutions in degree 8. It only detects isomorphism after scalar extension to a splitting field of the algebra. More precisely, we have the following: Theorem 3.1 Let A be a degree 8 algebra endowed with two totally decomposable orthogonal involutions and 0. We consider quadratic forms ' and '0 respectively associated to and 0 by triality as in Lemma 2.1. The following are equivalent:
0 3 (i) e3. = / D 0 2 MA.F /;
(ii) Š 0 ; FA FA
0 (iii) For any splitting field L=F of the algebra A, we have L Š L;
(iv) There exists 2 F such that ' h i'0 is similar to a 4-fold Pfister form, i.e. ' and '0 are half-neighbours. Cohomological invariants in degree 8 341
Remark 3.2 Clearly, conditions (i) to (iv) imply that and 0 are isomorphic if the algebra A is split. From [21, Thm.3], this is still the case in index 2 (see § 5 for a detailed proof). Nevertheless, this is not true anymore in higher index. In § 4 below we provide explicit examples of non-isomorphic involutions having a trivial relative Arason invariant, mostly inspired from Hoffmann [14, § 4]. The underlying algebra has index 4 or 8. Proof: The implication (i) ) (iii) follows from the observation that in the split 0 0 case, e3. = / vanishes if and only if and are isomorphic, and (iii) ) (ii) is 0 0 clear. If (ii) holds, then e3. = /FA D 0, hence e3. = / D 0 because, as explained in the introduction, the restriction map induces an injection
3 3 MA.F / ! H .FA/: The equivalence of (ii) and (iv) is Sivatski’s Proposition 4 [29], which is based on Laghribi [21]. Alternately, we may directly prove that (i) is equivalent to (iv) as 0 follows. By definition, e3. = / D 0 if and only if there exists 2 F such that 0 e3.' ' / D . / ŒA :
0 0 0 Since . / ŒA D e3.hh ii' /, this amounts to e3.' h i' / D 0, i.e. ' h i' 2 I 4.F /. The forms ' and '0 both have dimension 8, hence this exactly means that ' h i'0 is similar to a 4-fold Pfister form, and concludes the proof. The quadratic forms ' and '0 are only defined up to similarity, and the scalar need not be unique in general. Thus, we need the following computation to define an e4-invariant: Lemma 3.3 Consider two totally decomposable involutions and 0 of A with 0 0 e3. = / D 0, and let ', ' and be as in Theorem 3.1. The subset 4 E4 Dfe4.hh ii'/,where 2 F satisfies . / ŒA D 0g H .F / is a subgroup of H 4.F /. Moreover, for any ; 0 such that h i' h 0i'0 2 I 4.F /,the 0 0 0 4 invariants e4.' h i' / and e4.h i' h i' / have the same image in H .F /=E4. Remark 3.4 Since ' has trivial discriminant and Clifford algebra Brauer equivalent to A, the form hh ii' is in I 4F if and only if . / ŒA D 0 2 H 3.F /. Moreover, the set E4 does not depend on the choice of ' in its similarity class, and since ' h i'0 modI 4F; we also have
0 E4 Dfe4.hh ii' /, where 2 F satisfies . / ŒA D 0g: 342 A.QUÉGUINER-MATHIEU &J.P.TIGNOL
Proof: Consider and 0 2 F satisfying hh ii' 2 I 4F and hh 0ii' 2 I 4F .Inthe Witt group W.F/,wehave
hh 0ii' Dhh ii' Ch ihh 0ii':
Hence hh 0ii' 2 I 4.F / and
0 0 e4.hh ii'/ D e4.hh ii'/C e4.hh ii'/:
4 This proves E4 is a subgroup of H .F /. Assume now that h i' h 0i'0 2 I 4.F /. Since
h i' h 0i'0 Dh ihh 0 ii' Ch 0i.' h i'0/; we get that hh 0 ii' 2 I 4F and
0 0 0 e4.h i' h i' / e4.' h i' / modE4:
4 0 Hence, in the situation of Theorem 3.1, the image in H .F /=E4 of e4.' h i' / does not depend on the choice of the quadratic forms ' and '0 associated to and 0, nor on the factor such that ' h i'0 2 I 4F . This leads to the following definition: Definition 3.5 Let A beadegree8 algebra, and and 0 be two totally decom- 0 0 posable orthogonal involutions on A such that e3. = / D 0.Welet' and ' be 0 the quadratic forms respectively associated to and by triality. The relative e4 invariant of with respect to 0 is defined by
0 0 4 e4. = / D e4.' h i' / 2 H .F /=E4; where is any scalar in F satisfying ' h i'0 2 I 4.F /. Again, this is a well defined and functorial invariant of the pair of involutions 0 0 . ; / if e3. = / D 0. Moreover, we have: Proposition 3.6 Let A be a degree 8 algebra. Two orthogonal totally decomposable involutions and 0 on A are isomorphic if and only if
0 0 e3. = / D 0 and e4. = / D 0: Cohomological invariants in degree 8 343
0 Proof: The direct implication is clear. Assume conversely that e3. = / D 0 and 0 4 e4. = / D 0. There exists 2 F such that hh ii' 2 I .F / and
0 4 e4.' h i' / D e4.hh ii'/ 2 H .F /:
0 4 Hence, the form h i' h i' also is in I .F / and has trivial e4. Since it has dimension 16, it is hyperbolic, and we get that ' and '0 are similar. By Lemma 2.1, this implies that and 0 are isomorphic.
In the following statement, we use the e4 invariant to show that some involutions are isomorphic. As opposed to this, we will use it in §4 to produce examples of non- isomorphic involutions with trivial relative Arason invariant. Proposition 3.7 Let A be a degree 8 algebra, endowed with two orthogonal involutions and 0. We assume that A contains a product of two quaternion 0 algebras Q1 ˝ Q2, with split centralizer, which is stable under and , and on which both involutions act as the product ˝ of the canonical involutions of Q1 0 0 and Q2. If so, the involutions and are isomorphic if and only if e3. = / D 0.
Proof: The hypothesis means that A decomposes as A D Q1 ˝ Q2 ˝ M2.F /,and the involutions are given by D ˝ ˝adhhıii and 0 D ˝ ˝adhhı0ii,forsome 0 ı;ı 2 F .WeletnQ1 Dhh˛;ˇii and nQ2 Dhh"; ii be the norm forms of Q1 D .˛;ˇ/F and Q2 D ."; /F . Since we are in the situation of Examples 2.2 and 2.7 with 0 0 additionally D 1, the forms ' D nQ1 Ch ˛ˇıinQ2 and ' D nQ1 Ch ˛ˇı inQ2 0 0 0 are respectively associated to and by triality, and e3. = / D .ıı / ŒQ2 2 3 0 MA.F /. To prove the result, it is enough to check that e4. = / vanishes as soon as 0 e3. = / does. 0 3 Assume that e3. = / D 0 2 MA.F /. There exists 2 F such that 0 0 .ıı / ŒQ2 D . / ŒA ,thatis.ıı / ŒQ2 D . / ŒQ1 . This means that the 0 quadratic forms hhıı iinQ2 and hh iinQ1 are isometric, and a direct computation then shows that ' h i'0 Dhh˛ˇı; ;˛;ˇii,sothat
0 4 e4. = / D .˛ˇı; ;˛;ˇ/ D .ı; / ŒQ1 2 H .F /=E4: By the common slot Lemma [2, Lemma 1.7], we may find 2 F such that
0 .ıı / ŒQ2 D . / ŒQ2 D . / ŒQ1 D . / ŒQ1 :
In particular, we have . / ŒA D . / ŒQ1 C . / ŒQ2 D 0, and the forms hh iinQ1 and hh iinQ2 are isometric. Hence, we have hh ii' Dhh˛ˇı; ;˛;ˇii,sothat
.˛ˇı; ;˛;ˇ/ D e4.hh ii'/ 2 E4;
0 0 which proves e4. = / D 0. Alternately, one may also directly check that ' h i' is hyperbolic, so that ' and '0 are similar and Š 0. 344 A.QUÉGUINER-MATHIEU &J.P.TIGNOL
4. Examples of non-isomorphic involutions with trivial relative Arason invariant
In this section, we construct examples of orthogonal involutions on central simple algebras of degree 8 that are not isomorphic in spite of the fact that their relative Arason invariant is trivial. (In particular, in view of Theorem 3.1, they become isomorphic over any splitting field of the algebra.) By Corollary 5.3 below, such examples do not occur if the index of the algebra is 2. We give two (related) examples: in Example 4.2 the algebra has index 4, and in Example 4.3 it has index 8. The construction below is directly inspired by Hoffmann’s examples of nonsimilar half-neighbors in [14, § 4]. Our first observation is a general result of independent interest: Lemma 4.1 Let and 0 be involutions (of any type) on a central simple F -algebra A.If and 0 are isomorphic after scalar extension to some purely transcendental extension of F , then they are isomorphic. We are grateful to the referee for the observation that this lemma readily follows from the Bruhat-Tits theory [7, §3], which shows that for any (connected) reductive group G, the natural map H 1.F;G/ ! H 1.F..t//;G/ has zero kernel. Proof: Suppose first that A is split. If and 0 are symplectic, there is nothing to prove since all the symplectic involutions on a split algebra are isomorphic. If and 0 are orthogonal or unitary, the lemma is equivalent to the statement that two quadratic or hermitian forms over F are similar if they are similar after scalar extension to a purely transcendental extension. This is easily proved for quadratic forms, see [14, Corollary 3.2], and the hermitian case reduces to the case of quadratic forms by associating to each hermitian form its trace form, which is a quadratic form. For the rest of the proof we may thus assume A is not split; in particular, F is infinite. Arguing by induction on the transcendence degree, it suffices to prove the lemma when E D F.x/ for some indeterminate x. Suppose there is an E- 0 0 0 isomorphism .A; /E Š .A; /E .Then and are of the same type, so D Int.s/ ı for some s 2 A such that .s/D s. By the Skolem-Noether theorem, the E-isomorphism is the inner automorphism Int.g/ of AE for some g 2 AE .Wehave
Int.g/ ı D 0 ı Int.g/ D Int.s .g/ 1/ ı ; hence Int.g/ D Int.s .g/ 1/, which means there is an element e 2 E such that
es D g .g/: (3) Cohomological invariants in degree 8 345
Multiplying each side by a suitable factor of the form f .f/ with f 2 FŒx ,we may assume e 2 FŒx , hence es 2 A ˝F FŒx . By the analogue for involutions of the Cassels–Pfister theorem (see [32]), we may also assume g 2 A ˝F FŒx .Let x0 2 F be an element that is not a root of e. Substituting x0 for the indeterminate in (3), we get g.x0/ 2 A and e.x0/ 2 F such that e.x0/s D g.x0/ g.x0/ :
This equation shows that g.x0/ is invertible since e.x0/s 2 A , and it follows that 0 Int g.x0/ is an isomorphism between .A; / and .A; /. We now turn to the construction of the examples mentioned at the beginning of this section. The starting point is a field k and a central simple k-algebra E with the following properties: E has degree 8 and exponent 2, and it contains p p p a triquadratic field extension M D k. a; b; c/ but does not have any tensor product decomposition into quaternion algebras of the form
E D .a;u/k ˝k .b;v/k ˝k .c;w/k for any u, v, w 2 k .(4)
Examples of such algebras E were used to construct indecomposable division algebras of degree 8 and exponent 2 (over larger fields) in [1] (see also [31]). They exist over rational function fields in one variable over Q or in two variables over C (see [9, § 5]). On the other hand, one may also take for E any indecomposable division k-algebra of degree 8 and exponent 2, as those constructed by Karpenko in [19, 3.1], since a theorem of Rowen [17, V,Thm.5.6.10] shows that any such algebra contains a triquadratic extension of the center. p Fix a k-algebra E as above, and let K D k. a/, viewed as a subfield of M E. The centralizer C K of K in E is a K-algebra of degree 4 and exponent 2 pE p containing M D K. b; c/ as a maximal subfield, hence it has a decomposition
CE K D .b;r/K ˝K .c;s/K for some r, s 2 K .(5)
It follows from the projection formula that its corestriction to k is Brauer-equivalent to b;NK=k.r/ k ˝k c;NK=k.s/ k:
On the other hand, CE K is Brauer-equivalent to E ˝k K,andNK=k.E ˝k K/ D E˝2 D 0 in the Brauer group of k, hence 2 b;NK=k.r/ C c;NK=k.s/ D 0 in H .k/.(6) 346 A.QUÉGUINER-MATHIEU &J.P.TIGNOL
Example 4.2 Our first example is over the field F1 D k.x;y/ of rational functions in two variables over k. We consider the tensor product of quaternion algebras
A1 D .a;x/F1 ˝F1 .b;y/F1 ˝F1 .c;1/F1 :
The algebra .c;1/F1 is split. Since x and y are indeterminates over k it is easy to see that .a;x/F1 ˝F1 .b;y/F1 is a division algebra, hence the index of A1 is 4.Let1, i1, j1, i1j1 (resp. 1, i2, j2, i2j2,resp.1, i3, j3, i3j3) denote the standard quaternion basis of the first (resp. second, resp. third) factor. Consider the orthogonal involutions 1 D Int.j 1/ ı on .a;x/ F1 , 1 D Int.j2/ ı on .b;y/F1 , and 1 D Int.i3/ ı on .c;1/F1 (so .c;1/F1 ; 1 Š Adhhcii). Let
0 1 D 1 ˝ 1 ˝ 1 and 1 D .Int.s/ ı 1/ ˝ 1 ˝ 1; where s is defined in (5) and is identified with an element of k.i1/. The involutions 1, Int.s/ı 1, 1,and 1 have discriminant respectively equal to x;NK=k.s/x;y and c modulo F 2. Applying (1), one may check that
.b;y/F1 ; 1 ˝ .c;1/F1 ; 1 Š .y;c/F1 ; ˝ .y;bc/F1 ; :
Hence, we may apply Example 2.7 with ˛ D y, ˇ D bc, D a, ı D x, ı0 D NK=k.s/x, " D y,and D c, and we get
'1 Dhhy;bcii hbcyihha;xii hbcxyihhy;cii;
0 '1 Dhhy;bcii hbcyihha;NK=k.s/xii hbcNK=k.s/xyihhy;cii; and 0 3 e3. 1= 1/ D .NK=k.s/;y;c/ in MA.F1/.
By (6) we have .NK=k.s/;c/ D .NK=k.r/;b/. Since .NK=k.r/;a/ D .c;1/ D 0,it follows that .NK=k.s/;y;c/ D NK=k.r/ .a;x/ C .b;y/ C .c;1/ D NK=k.r/ ŒA1 ;
0 so e3. 1= 1/ D 0. 0 To compute e4. 1= 1/, observe that we have
0 '1 hNK=k.r/i'1 DhhNK=k.r/;y;bcii hbcxyihhNK=k.rs/;y;cii: (7) 2 Now, (6) yields bc;NK=k.r/ D c;NK=k.rs/ in H .k/, hence
hhbc;NK=k.r/ii D hhc;NK=k.rs/ii: Cohomological invariants in degree 8 347
0 Substituting in (7), we obtain '1 hNK=k.r/i'1 DhhNK=k.r/;y;bc;bcxyii. Since 3 .y;bc;bcy/ D 0 in H .F1/, we finally obtain
0 0 4 e4. 1= 1/ D e4.'1 hNK=k.r/i'1/ D .NK=k.r/;bc;x;y/ in H .F1/=E4.
0 0 To prove that 1 and 1 are not isomorphic, we show that e4. 1= 1/ ¤ 0. Suppose the contrary; we may then find 2 F1 such that . / ŒA D 0 and .NK=k.r/;bc;x;y/ D e4 hh ii'1 :
The condition . / ŒA D 0 is equivalent to
3 . ;a;x/ D . ;b;y/ in H .F1/.(8)
Assuming this condition holds, we have hh ;a;xii D hh ;b;yii, hence
hh ii'1 Dhh ;y;bcii hbcyihh ;b;yii hbcxyihh ;y;cii:
Decomposing hh ;y;bcii D hh ;b;yii C hbihh ;y;cii, we obtain
hh ii'1 Dhhbcy; ;b;yii C hbihhcxy; ;y;cii; hence e4 hh ii'1 D .bcy; ;b;y/C .cxy; ;y;c/: Since .by;b;y/ D .cy;y;c/ D 0, we finally get e4 hh ii'1 D .c; ;bx;y/:
Thus, we need to derive a contradiction from (8) and
4 .NK=k.r/;bc;x;y/ D .c; ;bx;y/ in H .F1/.(9) b Extend scalars to F 1 D k..x//..y//. By [22, Cor. VI.1.3], the square classes in b m n F 1 are represented by elements of the form ux y where u 2 k and m, n D 0 or m n 1; we may thus assume D 0x y for some 0 2 k and some m, n D 0 or 1. b We use the residue maps with respect to the y-adic valuation on F 1 and the x-adic valuation on k..x// (see [11, p. 18]),
@ 3 b y 2 @x 1 H .F 1/ ! H k..x// ! H .k/:
If n D 1, the image of (8) under the composition @x ı @y is ( 0 if m D 0, .a/ D .b/ if m D 1. 348 A.QUÉGUINER-MATHIEU &J.P.TIGNOL
p p p Each case contradicts the fact that M D k. a; b; c/ is a triquadratic field extension. Hence n D 0, and we may assume 2 k..x//. The image of (8) under @y is . ;b/ D 0, therefore, 3 b 2 . ;a;x/ D 0 in H .F 1/ and . ;b/ D 0 in H k..x// . 3 3 b b Since the map H k..x// ! H .F 1/ induced by the inclusion k..x// F 1 is injective, it follows that the first equation also holds in H 3 k..x// .Ifm D 1,the 1 image of the second equation under @x yields .b/ D 0 in H .k/, a contradiction. Therefore, D 0 2 k and the image of the first equation under @x yields . 0;a/ D 0 in H 2.k/. Since the map H 2.k/ ! H 2 k..x// is injective, we thus have
2 . 0;a/ D . 0;b/ D 0 in H .k/. (10)
On the other hand, taking the image of (9) under @x ı @y yields
2 .NK=k.r/;bc/ D .c; 0/ in H .k/.
Since . 0;b/ D 0, it follows that
2 .NK=k.r/ 0;bc/ D 0 in H .k/. (11)
In view of (6), we also have
2 .NK=k.rs/ 0;c/ D 0 in H .k/. (12)
Since . 0;a/ D 0, we may find t 2 K such that 0 D NK=k.t/ and rewrite (10), (11), and (12) as
2 .NK=k.t/;b/ D .NK=k.rt/;bc/ D .NK=k.rst/;c/ D 0 in H .k/.
Quaternion K-algebras whose corestriction is trivial descend to k, by [31, Lemma 2.6]. Therefore, we may find u1, u2, u3 2 F such that
.t;b/K D .u1;b/k ˝k K; .rt;bc/K D .u2;bc/k ˝k K; .rst;c/K D .u3;c/k ˝k K:
Since the cup-product is bilinear, we have
2 .b;r/K C .c;s/K D .b;t/K C .bc;rt/K C .c;rst/K in H .K/, and it follows that
.b;r/K ˝K .c;s/K D .b;u1u2/k ˝k .c;u2u3/k ˝k K: Cohomological invariants in degree 8 349
Therefore, by (5), the centralizer CE K contains a k-algebra isomorphic to .b;u1u2/k ˝k .c;u2u3/k. The centralizer of this k-algebra in E is a quaternion algebra containing K, hence E has a decomposition
E D .a;u4/k ˝k .b;u1u2/k ˝k .c;u2u3/k for some u4 2 k . This contradicts our hypothesis on E (see (4)) and completes the proof that 0 e4. 1= 1/ is nonzero. Example 4.3 Continuing with the notation in Example 4.2, let z be an indeterminate over F1 and let F2 be the field of functions on the product of the projective quadrics over F1.z/ given by the quadratic forms
hhc;z;NK=k.r/ii and hhc;z;NK=k.s/ii: (13) p p These quadratic forms are isotropic over F .z/. z/, hence F . z/ is purely p 1 2 transcendental over F1.z/. z/, and also over F1.Let
D D .a;x/F1.z/ ˝F1.z/ .b;y/F1.z/ ˝F1.z/ .c;z/F1.z/ and A2 D D ˝F1.z/ F2: The algebra D is a division algebra since x, y, z are indeterminates over k.The 3 quadratic forms in (13) are in I F1.z/. Therefore, it follows from Merkurjev’s index reduction formula [10, 30.11] that any F .z/ division algebra remains division over 1 p F2; in particular, A2 is a division algebra. Note that since F2. z/ splits .c;z/F2 we have an obvious isomorphism p p A2 ˝F2 F2. z/ Š A1 ˝F1 F2. z/: (14)
Consider the orthogonal involutions 2 D Int.j1/ ı on .a;x/F2 , 2 D Int.j2/ ı on .b;y/F2 ,and 2 D Int.i3/ ı on .c;z/F2 , and define the following involutions on A2: 0 2 D 2 ˝ 2 ˝ 2 and 2 D .Int.s/ ı 2/ ˝ 2 ˝ 2: 0 0 Clearly, the isomorphism (14) carries 2 to 1 and 2 to 1,thatis
p p 0 p 0 p .A2; 2/F2. z/ Š .A1; 1/F2. z/ and .A2; 2/F2. z/ Š .A1; 1/F2. z/: p 0 Since F2. z/ is purely transcendental over F1, and since 1 and 1 are not 0 isomorphic, it follows from Lemma 4.1 that 2 and 2 are not isomorphic. Yet, 0 we claim that e3. 2= 2/ D 0. Again, this relative invariant can be computed by (1) and Example 2.7. The involutions 2 and 2 have respective discriminant y and c 2 modulo F , so the components of the Clifford algebra of 2 ˝ 2 have Brauer class .bc;y/ and .c;yz/. Therefore,
0 3 e3. 2= 2/ D .NK=k.s/;bc;y/ D .NK=k.s/;c;yz/ in MA2 .F2/. 350 A.QUÉGUINER-MATHIEU &J.P.TIGNOL
Now, we have .NK=k.s/;c/ D .NK=k.r/;b/ by (6), and
3 .NK=k.s/;c;z/ D .NK=k.r/;c;z/ D 0 in H .F2/
3 since the quadratic forms (13) split over F2. Hence in H .F2/,wehave
.NK=k.s/;c;yz/ D .NK=k.s/;c;y/ D .NK=k.r/;b;y/; and .NK=k.r/;b;y/ D .NK=k.r/ .a;x/ C .b;y/ C .c;z/ D NK=k.r/ ŒA2 : 0 Therefore, e3. 2= 2/ D 0.
5. Algebras of index dividing 4
Throughout this section, we assume that the algebra A has index at most 4. Thus, A Š EndD V for some central division F -algebra D of exponent 2 and some right D-vector space V of dimension 2, 4,or8. It follows that A carries orthogonal hyperbolic involutions, which are those that are isomorphic to involutions on EndD V adjoint to hyperbolic hermitian forms on V (with respect to a given orthogonal involution on D). Since all hyperbolic hermitian forms on V are isometric, all the orthogonal hyperbolic involutions on A are isomorphic. They are totally decomposable since they have trivial discriminant and Clifford invariant. Fixing an arbitrary orthogonal hyperbolic involution 0 on A,wedefinetheArason invariant of the totally decomposable orthogonal involution on A by
3 e3. / D e3. = 0/ 2 MA.F /:
In particular, if is hyperbolic, we have e3. / D e3. = 0/ D 0 since and 0 3 are isomorphic. Hence, it follows from (2) that e3. / 2 MA.F / does not depend on the particular choice we made for 0. Moreover for any totally decomposable orthogonal involutions , 0 on A their relative Arason invariant is given by
0 0 0 e3. = / D e3. / C e3. / D e3. / e3. /:
This equality is a particular case of the compatibility of the Rost invariant with twisting, which was proven by Gille in [13]. In particular, it also holds in higher degree for the relative invariant defined using the Rost invariant (see [28]). As an easy consequence of Theorem 3.1, we now prove that the Arason invariant does detect hyperbolicity: Corollary 5.1 The totally decomposable involution is hyperbolic if and only if e3. / D 0. Cohomological invariants in degree 8 351
Proof: We have already explained that e3. / is zero if is hyperbolic. To prove the converse, let us pick an arbitrary hyperbolic involution 0 on A, and consider quadratic forms ' and '0 associated respectively to and 0 by triality. If e3. / D e3. = 0/ is trivial, Theorem 3.1 provides 2 F such that ' Ch i'0 is similar to a 4-fold Pfister form. As explained in [27, 5.1], since 0 is hyperbolic, the quadratic form '0 is isotropic. Hence ' Ch i'0 is hyperbolic, ' and '0 are similar, and and 0 are isomorphic. Thus is hyperbolic. If A is split, and is adjoint to the 3-fold Pfister form , it follows from 2.5 that its Arason invariant is given by
3 e3. / D e3. / in H .F /:
3 In particular, e3. / is always represented by a single symbol of H .F /.Inthe sequel, we will show that this remains true for algebras of index 2, but not in index 4.
Algebras of index 2 Let us first assume that A has index 2, and pick a quaternion algebra Q Brauer- equivalent to A. As we already mentioned, the Arason invariant is classifying in this case. More precisely, we have: Theorem 5.2 Let be a totally decomposable orthogonal involution on the algebra A D M4.Q/. There exist , 2 F and an orthogonal involution on Q such that
.A; / Š Adhh ; ii ˝F .Q; /: (15)
When this isomorphism holds, we have
3 e3. / D . ; / d. / in MQ.F /. p 3 Conversely, if e3. / D . ; ;ı/ in MQ.F / for some , , ı 2 F such that F. ı/ splits A,then(15) holds for some orthogonal involution on Q with d. / D .ı/. Proof: The existence of a decomposition of the form (15) was proved by Becher [5, Thm. 2]. Starting from there, we may find ı and 2 F so that d. / D .ı/ and ŒQ D . ;ı/. Applying (1) to Adhh ii ˝.Q; /,weget .A; / D Adhh ii ˝ . ;ı/F ; ˝ . ;ı/F ; :
Since the involution 0 D adhh1ii ˝ ˝ is hyperbolic, Example 2.7 gives
3 e3. / D . / . ;ı/ D . ; / d. / 2 MA.F /: 352 A.QUÉGUINER-MATHIEU &J.P.TIGNOL
3 Conversely, supposep e3. / D . ; ;ı/ in H .F /=.F ŒQ /p for some , , ı 2 F such that F. ı/ splits A. Sincepe3. / splits over F. ı/, Corollary 5.1 shows that becomes hyperbolic over F. ı/. Using either the proposition in the Appendix of [26], or [3, 3.4(2)], one may easily check that .A; / decomposes as
.A; / Š Adq ˝F .Q; / for some 4-dimensional quadratic form q over F and some orthogonal involution on Q with d. / D .ı/. Since .Q; /FQ Š Adhhıii,wehave
.A; /FQ Š Adq˝hhıii :
On the other hand, e3. / is represented by . ; ;ı/. Hence .A; /FQ Š Adhh ; ;ıii, so that the forms q ˝hhıii and hh ; ;ıii are similar over FQ.Todeducea decomposition of the type (15), we use the same argument as in the proof of [5, Thm. 2]. Scaling if necessary, we may assume the form q represents 1. Since hh ; ;ıii is a Pfister form, the similarity then has to be an isometry, so that .hh ; ii q/hhıii D 0 in the Witt group of FQ. This precisely means that the hermitian form .hh ; ii q/ h1i over .Q; / is in the kernel of the scalar extension map of Witt groups W.Q; / ! W.FQ/. But this map is injective by [8] or [24, 3.3], hence the hermitian forms hh ; ii h1i and q h1i over .Q; / are isometric. Therefore, Adq ˝F .Q; / Š Adhh ; ii ˝F .Q; /; proving (15). From this we get: Corollary 5.3 Let A be a central simple F -algebra of degree 8 and index 2, endowed with totally decomposable orthogonal involutions and 0. 3 (1) The Arason invariant e3. / is represented in H .F /=.F ŒA / by a single 3-symbol in H 3.F /. 0 0 (2) The involutions and are isomorphic if and only if e3. / D e3. /.
Proof: It readily follows from Theorem 5.2 that e3. / is represented by a single 0 symbol. If and are totally decomposablep orthogonal involutions such that 0 e3. / D e3. /,fix , , ı 2 F such that F. ı/ splits A and
0 3 e3. / D e3. / D . ; ;ı/ in MQ.F /. By Theorem 5.2 we have
0 .A; / Š Adhh ; ii ˝F .Q; / Š .A; / for some orthogonal involution on Q with d. / D .ı/, hence and 0 are isomorphic. Cohomological invariants in degree 8 353
Remark 5.4 The arguments developed in this subsection are still valid in higher degree, and enable us to define higher invariants, with values in
n n n 2 MA.F / D H .F /=H .F / ŒA ; where H n 2.F / ŒA denotes the subgroup fc ŒA ; c 2 H n 2.F /g H n.F /. Indeed, let .A; / be a totally decomposable algebra with orthogonal involution of degree 2n, Brauer equivalent to a quaternion algebra Q. By Becher’s theorem, it decomposes as .A; / D Ad ˝.Q; / for some .n 1/-fold Pfister form and some orthogonal involution on Q.OverFQ, the involution is adjoint to the form ˝hhıii, where .ı/ D d. /. Hence, n en. / D en 1. / d. / 2 MQ.F / is a well defined invariant of . Moreover, again by the very same arguments as in degree 8, two totally decomposable involutions and 0 of A are isomorphic if and 0 only if en. / D en. /. Alternately, this can also be deduced from Berhuy’s Theorem 13 in [6]. Indeed, one may easily check from [6, 2.4] that our invariant en. / coincides with his en;Q.h/ if is adjoint to the skew-hermitian form h over .Q; /.
Algebras of index 4 We now assume that A has index 4, and let D be a division algebra Brauer equivalent to A. As explained in [23, Thm. 1.1], for every totally decomposable orthogonal involution , there is a decomposition
.A; / D Adhh ii ˝.D; / for some 2 F and some orthogonal involution on D. Since .D; / need not be decomposable, we use the following refined decomposition to compute e3. /: Theorem 5.5 Let be a totally decomposable orthogonal involution on the index 4 algebra A D M2.D/. There exists a quadratic étale extension L=F , a quaternion algebra Q over L and 2 L such that
.A; / Š AdhhNL=F . /ii ˝NL=F .Q; /: (16)
When this isomorphism holds, we have 3 e3. / D NL=F . / ŒQ in MA.F /. 354 A.QUÉGUINER-MATHIEU &J.P.TIGNOL
Proof: The existence of such a decomposition is explained in [23, § 4], where we also compute a quadratic form ' associated to by triality. We sketch the argument for the reader’s convenience. Consider the quadratic étale extension L=F corresponding to the discriminant d. / 2 F =F 2. By [20, (15.7)], the Clifford algebra C.D; / is a quaternion algebra Q over L,and
.D; / Š NL=F .Q; /:
Hence, it only remains to prove that is the norm of some 2 L , or equivalently that . / d. / D 0 2 H 2.F /. By [30, 4.12] (see also [20, p.150]), . / d. / is the Brauer class of one component of the Clifford algebra C.A; /, while the other is . / d. /C ŒA . Since is totally decomposable, one of these classes is trivial. As A has index 4, it has to be the first one, and (16) is proved. For any quadratic form over L,welettr?. / be the transfer of associated to the trace map tr W L ! F , as defined in [22, VII.1.2]. From [23, Prop.2.1], one may easily check that each component of the Clifford algebra of tr?.hı inQ/, where nQ denotes the norm form of Q,isAdhhNL=F . /ii ˝NL=F .Q; / Š .A; /. Hence tr?.hı inQ/ is a quadratic form associated to by triality. Moreover, by the same argument, the quadratic form '0 D tr?.hıinQ/ is associated to the hyperbolic involution 0 D adhh1ii ˝ . Hence the Arason invariant e3. / D e3. = 0/ 2 3 MA.F / is represented by 3 e3.' '0/ D e3 tr?.hı ihh iinQ/ D NL=F . / ŒQ 2 H .F /:
Remark 5.6 If the algebra with involution .A; / decomposes as
.A; / D Adhh ii ˝.Q1; / ˝ .Q2; /; (17) then the formula easily follows from Example 2.7. Indeed, since the involution 0 Dhh1ii ˝ ˝ is hyperbolic, we have
3 e3. / D e3. = 0/ D . / ŒQ1 2 MA.F /:
On the other hand, the decomposition above is a decomposition as in (16), with L D F F , Q D Q1 Q2,and D . ;1/ 2 F F ,sothatNL=F . / ŒQ D . / ŒQ1 .
Example 5.7 By (1), the algebra with involution .A1; 1/ of Example 4.2 decom- poses as .A1; 1/ Š Adhhcii ˝.Q1; / ˝ .Q2; /; (18) where
Q1 D .ay;x/F1 and Q2 D .bx;y/F1 : Cohomological invariants in degree 8 355
Hence, the Arason invariant of 1 is given by
3 e3. 1/ D .c/ ŒQ1 D .c;ay;x/ 2 MA.F /:
Observe that if .A; / decomposes as in (17), then e3. / is represented by a single 3-symbol. The following proposition, which is a reformulation of Sivatski’s Proposition 2 in [29], shows that the converse also holds: Proposition 5.8 Let .A; / be a central simple F -algebra of degree 8 and index 4 with totally decomposable orthogonal involution, and let ' 2 I 2F be a quadratic form associated to by triality. The following are equivalent:
3 (i) e3. / can be represented by a single 3-symbol .˛;ˇ; / in H .F /;
(ii) ' Š 1 ? 2 for some quadratic forms 1, 2 of dimension 4 with trivial discriminant;
(iii) .A; / has a decomposition of the form Adhh ii ˝.Q1; / ˝ .Q2; / for some quaternion F -algebras Q1, Q2.
We refer the reader to [16, § 16], [15] and [29, § 2] for examples either of 8- dimensional quadratic forms with trivial discriminant and Clifford algebra of index 4 that do not decompose as in (ii), or totally decomposable degree 8 and index 4 algebras with involution that do not decompose as in (iii). They all prove that in index 4, the invariant e3. / is not always represented by a single 3-symbol. Remark 5.9 In [27], we constructed examples of algebras with involution that become hyperbolic over the function field of a conic and do not contain the corresponding quaternion algebra endowed with its canonical involution. Another 0 example can be derived from the triple .A1; 1; 1/ of Example 4.2. Indeed, since 0 0 3 e3. 1= 1/ D 0,wehavee3. 1/ D e3. 1/ 2 MA.F /. This invariant was computed in
Example 5.7, and becomes trivial over the field FQ1 . Hence, by Theorem 5.1, both 0 involutions 1 and 1 become hyperbolic over FQ1 . One of them, namely .A1; 1/ 0 does contain .Q1; / as a subalgebra with involution, by (18). Assume .A1; 1/ also 0 does. Then, the restriction of 1 to the centralizer Q2 ˝ M2.F / of Q1 in A1 is adjoint to a 2-dimensional skew hermitian form over .Q2; /, which, after scaling if necessary, diagonalizes as h1; c0i for some c0 2 F . Hence, we get
0 .A1; 1/ Š Adhhc0ii ˝.Q1; / ˝ .Q2; /: (19)
0 By Proposition 3.7, this implies that 1 Š 1. This is impossible since we proved in 0 Example 4.2 that the involutions 1 and 1 are not isomorphic. 356 A.QUÉGUINER-MATHIEU &J.P.TIGNOL
REFERENCES
1. S. A. Amitsur, L. H. Rowen and J.-P. Tignol, Division algebras of degree 4 and 8 with involution, Israel J. Math. 33 (1979), no. 2, 133–148. 2. J. Kr. Arason, Cohomologische Invarianten quadratischer Formen, J. Algebra 36 (1975), no. 3, 448–491. 3. E. Bayer-Fluckiger, D. B. Shapiro and J.-P. Tignol, Hyperbolic involutions, Math. Z. 214 (1993), no. 3, 461–476. 4. E. Bayer-Fluckiger, R. Parimala and A. Quéguiner-Mathieu, Pfister involutions, Proc. Indian Acad. Sci. Math. Sci. 113 (2003), no. 4, 365–377. 5. K. J. Becher, A proof of the Pfister factor conjecture, Invent. Math. 173 (2008), no. 1, 1–6. 6. G. Berhuy, Cohomological invariants of quaternionic skew-Hermitian forms, Arch. Math. (Basel) 88 (2007), no. 5, 434–447. 7. F. Bruhat and J. Tits, Groupes algébriques sur un corps local. Chapitre III. Compléments et applications à la cohomologie galoisienne. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 3, 671–698. 8. I. Dejaiffe, Formes antihermitiennes devenant hyperboliques sur un corps de dé- ploiement, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 2, 105–108. 9. R. Elman et al., Witt rings and Brauer groups under multiquadratic extensions. I, Amer. J. Math. 105 (1983), no. 5, 1119–1170. 10. R. Elman, N. Karpenko and A. Merkurjev, The algebraic and geometric theory of quadratic forms, Amer. Math. Soc., Providence, RI, 2008. 11. S. Garibaldi, A. Merkurjev and J.-P. Serre, Cohomological invariants in Galois cohomology, Amer. Math. Soc., Providence, RI, 2003. 12. S. Garibaldi, Orthogonal involutions on algebras of degree 16 and the Killing form of E8,inQuadratic forms—algebra, arithmetic, and geometry, 131–162, Contemp. Math. 493, Amer. Math. Soc., Providence, RI. 13. P. Gille, Invariants cohomologiques de Rost en caractéristique positive, K-Theory 21 (2000), no. 1, 57–100. 14. D. W. Hoffmann, Similarity of quadratic forms and half-neighbors, J. Algebra 204 (1998), no. 1, 255–280. 15. D. W. Hoffmann and J.-P. Tignol, On 14-dimensional quadratic forms in I 3, 8- dimensional forms in I 2, and the common value property, Doc. Math. 3 (1998), 189– 214 (electronic). 16. O.T. Izhboldin and N. A. Karpenko, Some new examples in the theory of quadratic forms, Math. Z. 234 (2000), 647–695. Cohomological invariants in degree 8 357
17. N. Jacobson, Finite-dimensional division algebras over fields, Springer, Berlin, 1996. 18. B. Kahn, Un invariant de degré 3 des algèbres centrales simples d’exposant 2, J. Ramanujan Math. Soc. 25 (2010) 355–358. 19. N. A. Karpenko, Torsion in CH2 of Severi-Brauer varieties and indecomposability of generic algebras, Manuscripta Math. 88 (1995), no. 1, 109–117. 20. M.-A. Knus et al., The book of involutions, Amer. Math. Soc., Providence, RI, 1998. 21. A. Laghribi, Isotropie de certaines formes quadratiques de dimensions 7 et 8 sur le corps des fonctions d’une quadrique, Duke Math. J. 85 (1996), no. 2, 397–410. 22. T. Y. Lam, Introduction to quadratic forms over fields, Amer. Math. Soc., Providence, RI, 2005. 23. A. Masquelein, A. Quéguiner-Mathieu and J.-P. Tignol, Quadratic forms of dimension 8 with trivial discriminant and Clifford algebra of index 4, Arch. Math. (Basel) 93 (2009), no. 2, 129–138. 24. R. Parimala, R. Sridharan and V. Suresh, Hermitian analogue of a theorem of Springer, J. Algebra 243 (2001), no. 2, 780–789. 25. E. Peyre, Galois cohomology in degree three and homogeneous varieties, K-Theory 15 (1998), no. 2, 99–145. 26. A. Quéguiner-Mathieu and J.-P. Tignol, Discriminant and Clifford algebras, Math. Z. 240 (2002), no. 2, 345–384. 27. A. Quéguiner-Mathieu and J.-P. Tignol, Algebras with involution that become hyper- bolic over the function field of a conic, Israel J. Math. 180 (2010), 317–344. 28. A. Quéguiner-Mathieu and J.-P. Tignol, Arason invariant for algebras with orthogonal involutions, in preparation. 29. A. S. Sivatski, Applications of Clifford algebras to involutions and quadratic forms, Comm. Algebra 33 (2005), no. 3, 937–951. 30. D. Tao, The generalized even Clifford algebra, J. Algebra 172 (1995), no. 1, 184–204. 31. J.-P. Tignol, Corps à involution neutralisés par une extension abélienne élémentaire, in The Brauer group (Sém., Les Plans-sur-Bex, 1980), 1–34, Lecture Notes in Math. 844, Springer, Berlin. 32. J.-P. Tignol, A Cassels-Pfister theorem for involutions on central simple algebras, J. Algebra 181 (1996), no. 3, 857–875. 33. J.-P. Tignol, Cohomological invariants of central simple algebras with involution, Quadratic forms, linear algebraic groups, and cohomology, 137–171, Dev. Math., 18, Springer, New York, 2010. 358 A.QUÉGUINER-MATHIEU &J.P.TIGNOL
ANNE QUÉGUINER-MATHIEU [email protected] LAGA - UMR 7539 du CNRS Université Paris 13 F-93430 Villetaneuse and UPEC, F-94010 Créteil France
JEAN-PIERRE TIGNOL [email protected] Zukunftskolleg, Universität Konstanz D-78457 Konstanz Germany and ICTEAM Institute Université catholique de Louvain B-1348 Louvain-la-Neuve Belgium Received: December 17, 2010