process

140 M. KaroubiThe following theorem is a direct consequence of the previous considerations:Theorem 6.9. Let us now assume that A D M 2n .C/ is G-oriented as a graded algebra:in other words, there is an involutive element " of A of degree 0 which commutes withthe action of G and commutes (resp. anticommutes) with the elements of A of degree 0(resp. 1). Then, the graded K-theory GrK .A 0 /, with A 0 D GËA, is a finitely generatedfree module concentrated in degree 0. More precisely, one has GrK 0 .A 0 / D K.A 0 /and GrK 1 .A 0 / D 0. In particular, if V is an even-dimensional real vector space and ifA y˝ C.V / is G-oriented, we have (via the Thom isomorphism)KG A .V / D KA y˝C.V /G.P / D K.G Ë A y˝ C.V // and KG A .V ˚ 1/ D 0where P is a point. If we write A y˝ C.V / as an algebra of matrices M r .C/ with arepresentation of G and call zG the associated central extension by r , the rank ofKG A .V / is the number of conjugacy classes of G which split into r conjugacy classesin zG.In the abelian case, the following two theorems are related to results obtained byP. Hu and I. Kriz [26], using different methods.Theorem 6.10. Let us consider the algebra A D M n .C/ provided with an action ofan abelian group G and P a point. Then the ungraded twisted K-theory K .A/G.P / DK .A 0 /, with A 0 D G Ë A, is concentrated in degree 0 and is a free Z-module. If wetensor this group with the rationals and if we look at it as an R.G/ ˝ Q D QŒGmodule,it may be identified with R.G 0 / ˝ Q for a suitable subgroup G 0 of G. Inparticular, the rank of K 0 .A 0 / divides the order of G.Proof. The first part of the theorem is a consequence of the previous more generalconsiderations. As we have shown before, the algebra A 0 gives rise to the followingcommutative diagram:zG nSU.n/G PU.n/,the fibers of the vertical maps being n . The subset of elements Qg in zG n such that .Qg/splits into n conjugacy classes is just the center Z. zG n / of zG n (since G is abelian).Let us put n D .Z. zG n //. Then K.A 0 / may be written as KG A .P / where P is apoint. According to Theorem 6.10, this is the subgroup of the representation ring ofzG n generated by representations of linear type. At this stage, it is convenient to maken D1by extension of the roots of unity, so that we have an extension of G by Q=ZQ=Z zG G(the “linear type” finite-dimensional representations of zG are the same as the originallinear type finite-dimensional representations of zG n ). We call R. zG/ l / the associated