L'Enseignement Mathématique

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1 L'Enseignement Mathématique Lewis, D. W. EXACT SEQUENCES OF WITT GROUPS OF EQUIVARIANT FORMS L'Enseignement Mathématique, Vol.29 (1983) PDF erstellt am: Jun 10, 2009 Nutzungsbedingungen Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen für nicht-kommerzielle Zwecke in Lehre, Forschung und für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und unter deren Einhaltung weitergegeben werden. Die Speicherung von Teilen des elektronischen Angebots auf anderen Servern ist nur mit vorheriger schriftlicher Genehmigung des Konsortiums der Schweizer Hochschulbibliotheken möglich. Die Rechte für diese und andere Nutzungsarten der Inhalte liegen beim Herausgeber bzw. beim Verlag. SEALS Ein Dienst des Konsortiums der Schweizer Hochschulbibliotheken c/o ETH-Bibliothek, Rämistrasse 101, 8092 Zürich, Schweiz

2 EXACT SEQUENCES OF WITT GROUPS OF EQUIVARIANT FORMS by D. W. Lewis We construct two exact octagons i.e. circular eight-term exact séquences of Witt groups of forms invariant under the action of a finite group. When the group is trivial our octagons reduce to the two exact séquences obtained in [3]. See also [4]. We are indebted to Cl. Cibils and M. Kervaire for their suggestions to improve the original version of this paper. Let F be a skewfield, J an involution on F i.e. an anti-automorphism of period two. We allow the case of J being the identity if/ 7 is commutative. Let tc be a finite group. Définition. A form over (F, tc, J) is a map (j) :V x V->F, Fan Fn modulefinite dimensional over F, which is sesquilinear, hermitian symmetric with respect to J, and Tc-invariant in that <$>(gx, gy) = <\>(x, y) for ail g e tc, ail x, y e V. Our forms are assumed to be non-singular i.e. V -» F*, x -> (j)(x, ) is bijective for ail x e V, where F* is the F-dual of F We write W(F, tc, J) for the Witt group of non-singular forms over (F, tc, J), our définition of Witt group being as in [I]. (Remark the forms which hâve Witt class zéro are precisely those which are neutral i.e. which contain a submodule equal to its orthogonal complément. Note that we do not insist that this submodule be a direct summand as is required in another définition of Witt group which occurs in the literature. When char F does not divide tc then there is of course no différence between the two définitions of Witt group but in gênerai they will be différent.) No w let Kbea field, char K 2, and let Lbea quadratic extension of Kso that L = K{ï), i2i 2 = a for some ae K. L admits both the identity map and the map given by 1 = ias involutions. We will consider the groups W(K, n, 1), W{L, 7i, 1) and W{L, n, -). Also we write W_ tu, 1), W_ tc, 1) for the Witt groups of non-singular forms (j) defined as above except that now <\> is required to be skew-symmetric i.e. (y, x) = cj)(x, y) for ail x, ye V. Also we write W_ y (L, te, ) for the Witt group of skew-hermitian forms over L, i.e. <\>(y, x) = -<\>(x, y) for ail x, yev. Note that for tc =1, the groups

3 W_ 7i, 1), W_ i(l, 7i, 1) are trivial since the skew-symmetric forms are even dimensionalandclassified by rank alone [2, p. 334]. Note also that WL^L, n, ) is isomorphic to W(L, n, ) because if cj) is hermitian then i<\> is skew hermitianandvice versa. Let the trace maps T a : L-*K,a = 1, 2be deflned by where each r a e K. Thèse trace maps induce in an obvious way maps between Witt groups as follows : We dénote the last two maps by T\, T' 2 merely to distinguish them from the first two maps. Also we may use the tensor product in a natural way to defme maps and there are also maps given by tensor product together with multiplication by the élément i e L. E.g. given a form <\> :V x V-»K over (K, n, 1), U 2 (<\>) is the map given by for ail x, y e V, ail X, u g L. It is easily checked that ail thèse maps are well defined.

4 Theorem 1. There is an exact octagon of Witt groups Proof. We first show exactness of the portion i.e. we show that image of T x is the kernel of Uv Let (j) : V x V -* L represent an élément of W(L, n, ).To see that U^^is neutral as a form over (L, n, 1) we consider the subspace WofV (& K Las deflned by Clearly Wisan LTt-submodule and 2 dim x W= à\m K (V K L). We will show that W = Wl,W 1, orthogonal complément with respect to U^fy. Now if v, v' ev then is easily verified to be zéro using the sesquilinearity of (j) and the définitions of 7i, l/j.thus W a Wl.W 1. It dimension. follows that in fact W = W 1 since they hâve the same Next let i / : V x V -> K represent an élément of W(K, n, 1). We may assume v / is anisotropic by [I]. Now if U t ty is zéro in W(L, n, 1) then V K L contains a self-orthogonal L-submodule W This enables us to define an L-space structure on V as follows :

5 Observe that and that Wn(V ï) = 0 smce \ / is anisotropic. Thus V K L = (K i) W. It now follows that, given v e V, there exists a unique élément v' e V such that v (8) 1 + i/ (g) i g W Then define the operator 7 : V - F by J(t/) = v for each t; ek It is easily verifled that J is skew-adjoint, J2J 2 a and that J commutes with the Tc-action. Thus J can be used to give V an L7u-module structure, i e L operating as J on K Now define a form <\> : V x V -> L by for ail x, y g K Then (j) is a non-singular form over (L, tc, ) and T^ = v /. This proves exactness at W(K, te, 1). At the three points in the séquence exactness is proven by the same arguments Now consider the pièce If (j) :VxV-K represents an élément of W_ t (K, tc, 1) then we see that T' 2 U' 2 fy is neutral by looking at and checking that W = W L. for ail v,v'ev so that Wa Wl.W 1. Hence W= W 1 since Conversely if v /, representing an élément of W_ X (L, tc, 1), is such that T 2 v / is neutral then v / : V x V -> L, Van LTc-module, and there exists a J^Tc-module VF of V with W=WL, orthogonal complément with respect to jt 2 i /. Also

6 2 dir% W dim^ V. Defining (j> : W x W -> by (x, y) = i /(x, y) for x, ye W then H^ (g) X L = Fas LK-modules via the isomorphism Moreover U\((\)) = \J/ completing the proof of exactness at Wl X (L, 71, 1). For the three remaining points of the séquence, which each hâve U followed by T, the above arguments go through virtually unchanged. This complètes the proof. Now suppose we hâve a quaternion division algebra D over K, D = j j generated by i, 2 i2i j with = a,j 2 = b,ij = ji etc. We hâve involutions and A on D given by / = i,f= j and f = i,f = j respectively. Let L be the maximal subfield K(ï) of D. There are trace maps 7] : D -> L, i = 1,2 given by T i (z l + z 2 j) z1 where z{ z 2 & L, and thèse induce natural maps of Witt groups Also we hâve maps L/ l5 U\ given by the tensor product, U 2, U' 2 by the tensor product together with multiplication by the élément k = ij of D. E.g. given a form 4> : V x X -> L over (L, te, 1), U 2 (<\>) is the form K(g) L D x V L D D defined by (Beware that the position of k matters as D is not commutativeî).

7 Theorem 2. There is an exact octagon of Witt groups Proof. We need only modify the proof of theorem 1 slightly. Specifically j will play the rôle that i did in theorem 1. For example at the start of the proof we must put and later on the operator J is deflned in a similar fashion to that of theorem 1 except that we get J2J 2 = b leading to a Dît-module structure. The lack of commutativity of D causes no problem, although care must be taken in dealing with the maps U 2, U' 2. (See the comment above.) We leave the reader to check that with thèse modifications the proof goes through completely. Comment 1. When n = 1 the Witt groups W-^K, n, 1) and W-^L, n, 1) are trivial as we remarked earlier in this paper. Our séquences now reduce to those of [3]. Comment 2. Note that W_ 1 (L, n, )= W(L, n, ) for the reason stated earlier. Also W(D, n, A ) = W-^D, 7i, ) since forms hermitian with respect to A are équivalent to those skew-hermitian with respect to and vice versa. (The correspondence c ) <- i gives this since x= i~ l xi for ail xe D.) A conséquence of the above is that the two octagons each display an interesting symmetry

8 feature. In the "antipodal" position to W(F, n, J) in the octagon we always hâve Comment 3. Our proof is différent from that of [3] and it may well be possible that this new method of proof can also be used to generalize the séquences of [3] to the case when K is a commutative ring and L is some kind of Galois extension with Galois group cyclic of order two. REFERENCES [1] Cibils, Claude. Groupe de Witt d'une algèbre avec involution. U Enseignement Mathématique 29 (1983), [2] Jacobson, N. Basic Algebra. W. H. Freeman, San Francisco, [3] Lewis, D. W. New improved exact séquences of Witt groups. J. of Algebra 74 (1982), [4] A note on hermitian and quadratic forms. Bull. London Math. Soc. 11 (1979), (Reçu le 16 juillet 1982) D. W. Lewis Department of Mathematics University Collège Belfield Dublin 4 Ireland

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