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The Descent of Biquaternion Algebras in Characteristic Two

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The Descent of Biquaternion Algebras in Characteristic Two The descent of biquaternion algebras in characteristic two Demba Barrya,b, Adam Chapmanc, Ahmed Laghribid aFacult´edes Sciences et Techniques de Bamako, BP: E3206 Bamako, Mali bDepartement Wiskunde–Informatica, Universiteit Antwerpen, Belgium cDepartment of Computer Science, Tel-Hai College, Upper Galilee, 12208 Israel dUniversit´ed’Artois, Facult´edes Sciences Jean Perrin, Laboratoire de math´ematiquesde Lens EA 2462, rue Jean Souvraz - SP18, 62307 Lens, France Abstract In this paper we associate an invariant to a biquaternion algebra B over a field K with a subfield F such that K=F is a quadratic separable extension and char(F) = 2. We show that this invariant is trivial exactly when B B0 ⊗ K for some biquaternion algebra B0 over F. We also study the behavior of this invariant under certain field extensions and provide several interesting examples. Keywords: Kato-Milne Cohomology, Cohomological invariants, Algebras with involution, Biquaternion algebras 2010 MSC: primary 11E81; secondary 11E04, 16K20, 19D45 1. Introduction Given a central simple algebra C over a field K with a subfield F, we say that C has a descent to F if there exists a central simple algebra C0 over F such that C C0 ⊗ K. When [K : F] = exp(C), one necessary condition for C to have a descent to F is that corK=F(C) ∼Br F, because if C C0⊗K ⊗ ⊗[K:F] ∼ ∼ then corK=F(C) = corK=F(C0 K) = C0 Br F, where Br denotes the Brauer equivalence. If K=F is a separable quadratic extension and Q is a quaternion algebra over K, then corK=F(Q) ∼Br F is a necessary and sufficient condition for Q to have a descent to F by [1, Chapter X, Theorem 21]. This fact does not generalize to biquaternion algebras. For example, take a division algebra A over F such that deg(A) = 8 and exp(A) = 2 which does not decompose as a tensor product of three quaternion algebras (e.g. [2]), take a separable quadratic field exten- sion K of F inside A (which exists by [28]) and B to be the centralizer of K in A. Then B ∼Br A⊗K and so corK=F(B) ∼Br F. However, if B B0 ⊗ K for some central simple algebra B0 over F, then A decomposes as the tensor product of three quaternion algebras, a contradiction. Therefore B has no descent to F. In [10], the invariant δ was associated to any biquaternion algebra B over K where K=F is a quadratic field extension and char(F) , 2. It was shown that the invariant is trivial exactly when B has a descent to F. This invariant proved to be a refinement of an invariant ∆ defined in [16, Section 11] for algebras of degree 8 and exponent 2, which is trivial when the algebra decomposes Email addresses: [email protected] (Demba Barry), [email protected] (Adam Chapman), [email protected] (Ahmed Laghribi) Preprint submitted to ? February 19, 2019 and is conjectured to be nontrivial otherwise. It was then used to show the existence of at least one algebra A of degree 8 and exponent 2 with nontrivial ∆(A). The main tool Barry used in order to construct this example was [10, Proposition 4.9], whose characteristic 2 analogue is given in this paper as Proposition 6.1. In this paper we present the characteristic 2 analogue for δ. We associate an invariant to any biquaternion algebra B over K where K=F is a separable quadratic extension and char(F) = 2. We show that this invariant is trivial exactly when B has a descent to F, and study its behavior under certain field extensions, proving (among other things) that it does not split over odd degree field extensions (see Proposition 5.1). This invariant is defined using the Kato-Milne cohomology groups, which form a characteristic 2 analogue to the classical Galois cohomology groups. 2. Kato-Milne cohomology m m 1 Throughout this paper F denotes a field of characteristic 2. Let ΩF = ^ ΩF be the space of 1 absolute m-differential forms over F, where ΩF is the F-vector space generated by the symbols dx, x 2 F, subject to the relations: d(x + y) = dx + dy; d(xy) = xdy + ydx 0 2 2 for any x; y 2 F. We set ΩF = F. Clearly, d(x y) = x d(y) for all x; y 2 F, and thus the map 1 F ! ΩF x 7! dx 2 2 2 m is F -linear, where F = fx j x 2 Fg. This map extends to the differential operator d : ΩF ! m+1 ΩF defined by ydx1 ^ · · · ^ dxm 7! dy ^ dx1 ^ · · · ^ dxm: Let B = fei j i 2 Ig be a 2-basis of F, which means that the set 8 9 >Y > < e"i j " 2 f0; 1g; and for almost all i 2 I;" = 0= > i i i > : i2I ; is a basis of F over F2. We choose an ordering on I. So the set ( ) dei1 deim ^ · · · ^ j ei1 ;:::; eim 2 B and i1 < i2 < ··· < im ei1 eim m is a basis of the F-vector space ΩF . The usual Frobenius map F ! F, x 7! x2, extends to a well-defined map, called the Frobenius operator, as follows: m m m−1 Φ : ΩF ! ΩF =dΩF X X dei1 eim 2 dei1 eim m−1 ci ^ · · · ^ 7! ci ^ · · · ^ + dΩF : ei1 eim ei1 eim i1<···<im i1<···<im 2 m m m−1 The Artin-Schreier operator } : ΩF ! ΩF =dΩF is defined by } = Φ − Id. m+1 Let H2 (F) and νF(m) denote the cokernel and the kernel of }, respectively. Usually we will m m consider the operator } as a map ΩF ! ΩF given by: X dei1 eim X dei1 eim ci ^ · · · ^ 7! }(ci) ^ · · · ^ : ei1 eim ei1 eim i1<···<im i1<···<im m−1 This map depends on the choice of the 2-basis, and it is well-defined modulo dΩF . With that we may take m+1 m m−1 m H2 (F) = ΩF =(dΩF + }(ΩF )): A famous result of Kato [23] asserts the following isomorphism: m+1 m m+1 fm+1 : H2 (F) ! I F ⊗ Wq(F)=I F ⊗ Wq(F) dx1 dxm a ^ · · · ^ 7! hhx1; ··· ; xmii ⊗ [1; a] x1 xm where ImF is the mth power of the fundamental ideal of the Witt ring W(F) of nondegenerate sym- metric F-bilinear forms, Wq(F) is the Witt group of non singular F-quadratic forms, hhx1; ··· ; xnii is the n-fold bilinear Pfister form h1; x1i ⊗ · · · ⊗ h1; xni, and [1; a] is the binary quadratic form 2 2 1 2 X + XY + aY . We have H2(F) F=}(F) and H2(F) 2Br(F) the 2-torsion of the Brauer group of F. Kato also proved in [23] that νF(m) coincides with the additive group generated by the differ- entials da1 ^ ::: ^ dam for a ; ··· ; a 2 F×, and it is isomorphic to the quotient ImF=Im+1F in a a1 am 1 m natural way. 3. An invariant for biquaternion algebras over a quadratic extension From now on K denotes a separable quadratic extension of F. We have K = F[α], where }(α) = a 2 F n }(F). Without loss of generality, we can assume a 2 F2. Because of the separability of the extension K=F, any 2-basis of F remains a 2-basis of K, and m m m thus we have ΩK = ΩF ⊕ αΩF for any nonnegative integer m. Let Tr : K ! F denote the trace map. This map extends to m m Tr∗ : ΩK ! ΩF w0 + αw1 7! w1 m for all w0; w1 2 ΩF . Since a 2 F2, we also have α 2 K2, and thus m m m dΩK = dΩF ⊕ αdΩF }(w0 + αw1) = }(w0) + aΦ(w1) + α℘(w1) m m m m m for all w0; w1 2 ΩF . Hence, Tr∗(dΩK) ⊂ dΩF and Tr∗(}(ΩK)) ⊂ }(ΩF ). Consequently, Tr∗ m+1 m+1 extends to a map from H2 (K) to H2 (F). Moreover, by [6, Corollary 2.5] (see also [13, Complex 34.20]), the group homomorphism n Tr∗ : Wq(K) −! Wq(F) induces by the trace map Tr : K ! F satisfies Tr∗(I K ⊗ Wq(K)) ⊂ n I F ⊗ Wq(F) for all n ≥ 0. This is a part of the following exact sequence: 3 Proposition 3.1 ([7, Cor. 6.5]). The following sequence is exact: n n resK=F n Tr∗ n 0 / I F ⊗ [1; a] / I F ⊗ Wq(F) / I K ⊗ Wq(K) / I F ⊗ Wq(F) / 0 where resK=F is the restriction map. m We will also need the following result describing the group I K ⊗ Wq(K). n Proposition 3.2 ([6, Corollary 2.4]). For any n ≥ 0,I K ⊗Wq(K) is generated by the Pfister forms × hha1;:::; an; b]] where a1;:::; an 2 F and b 2 K. m+1 m m+1 For any m ≥ 0, let us denote by e : I F ⊗ Wq(F) ! H2 (F) the group homomorphism given by: −1 m+1 ' 7! fm+1(' + I F ⊗ Wq(F)) where fm+1 is the Kato isomorphism given in the previous section. In particular, for any ' 2 m m+1 m+1 I F ⊗ Wq(F), we have e (') = 0 iff ' 2 I F ⊗ Wq(F). Proposition 3.3. The diagram n Tr∗ n I K ⊗ Wq(K) / I F ⊗ Wq(F) en+1 en+1 n+1 Tr∗ n+1 H2 (K) / H2 (F) is commutative for any n ≥ 0. n Proof. Let ' 2 I K ⊗ Wq(K). By Proposition 3.2 it suffices to verify the diagram commutativity × for ' = hha1;:::; an; b]] where a1;:::; an 2 F and b 2 K.
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