process

136 M. KaroubiTheorem 6.2. 25 The (ungraded) twisted K-theory K .A/G.X/ is canonically isomorphicto the Grothendieck group of the category EG z .X/ l.Proof. One just repeats the argument in the proof of Theorem 2.6 in [31], where A is aClifford algebra C.V / and Z=2 plays the role of n . We simply “untwist” the actionof zG thanks to the formula (F) written explicitly in the proof of 2.6 (loc. cit.).For A D X A with A D M n .C/, the previous argument shows that K .A/G.X/ isa subgroup of the usual equivariant K-theory KG z .X/. From now on, we shall writeKG A .X/ instead of K.A/G.X/. Similarly, in the graded case (A D M n.C/ M n .C/ orM 2n .C/), we shall write KG A.X/ instead of KA G.X/. If X is a point and G is finite,K .A/G.X/ is just the K-theory of the semi-direct product G Ë A.Theorem 6.3. Let G be a finite group acting on the algebra of matrices A D M n .C/and let zG be the central extension G by n described in 6.1. Then, for X reduced to apoint, the group K .A/G.X/ D K.G Ë A/ is a free abelian group of rank the number ofconjugacy classes in G which split into n conjugacy classes in zG.Proof. We can apply the same techniques as the ones detailed in [31], §2.6/12 (forn D 2). By the theory of characters on zG, one is looking for functions f on zG (whichwe call of “linear type”) such that1. f.hgh 1 / D f.g/,2. f .x/ D f .x/ if is an n th root of the unity.The C-vector space of such functions is in bijective correspondence with the space offunctions on the set of conjugacy classes of G which split into n conjugacy classesof zG.Like the Brauer group of a space X, one may define in a similar way the Brauergroup Br.G/ of a finite group G by considering algebras A D M n .C/ as above with aG-action (see [24] for a broader perspective ; this is also a special case of the generaltheory of Atiyah and Segal mentioned at the end of §5). From the diagram written in6.1, one deduces a cohomology invariantw 2 .A/ 2 H 2 .GI n /and (via the Bockstein homomorphism) a second invariant W 3 .A/ 2 H 2 .GI S 1 / DH 2 .GI Q=Z/ D H 3 .GI Z/. It is easy to show that this correspondence induces awell-defined mapW 3 W Br.G/ ! H 3 .GI Z/:The following theorem is a special case of 5.9 in a more algebraic situation.25 There is an obvious generalization when A is infinite-dimensional. However, for our computations, werestrict ourselves to the finite-dimensional case.