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M, '- M°'-t Al", M; 1-* ... is exact. As an interesting byproduct of our work on united K-theory, we obtain a Kunneth theorem. 12 Theorem. For spectra X and Y, there is a natural short exact sequence K*RTX ®CiZT K* RTl' >- K* RT(X A Y) -* TorCRT(K* RTX, K* RTE') This Kunneth exact sequence need not be splittable, since it expresses KOl(MZ/2 A MZ/2) as a nontrivial extension of Z/2 by Z/2.

E. Mahowald. The image of J in the EHP sequence. Annals of Mathematics, 116:65-112, 1982. [Mi181] H. R. Miller. On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space. Journal of Pure and Applied Algebra, 20:287-312, 1981. [Rav] D. C. Ravenel. A counterexample to the Telescope Conjecture. To appear. Ravenel : Report on the telescope conjecture 21 [Rav84] D. C. Ravenel. The Segal conjecture for cyclic groups and its consequences. American Journal of Mathematics, 106:415-446, 1984.

ATIYAH, Vector bundles and the Kunneth formula, Topology 1 (1962), 245-248. F. ATIYAH, K-theory and reality, Quart. J. Math. Oxford 17 (1966), 367-386. K. BoUSFIELD, The localization of spectra with respect to homology, Topology 18 (1979), 257-281. (Correction in Comm. Math. Hely. K. BOUSFIELD, On the homotopy theory of K-local spectra at an odd prime, Amer. J. Math. 107 (1985), 895-932. K. BOUSFIELD, A classification of K-local spectra, J. Pure Appl. Algebra. 66 (1990), 121-163. [10] B. MITCHELL, Rings with several objects, Adv.

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