Abstract. Let p be a prime, and denote by Sn the group of units in the maximal order of a cyclic division algebra over Qp of index n and Hasse invariant 1/n. The Morava Change of Rings Theorem says essentially that the cohomology of Sn with coefficients in a certain representation describes the Bousfield localization functor LK(n). This is the localization of stable homotopy theory with respect to the spectrum of the n-th Morava K-theory, K(n). The functors LK(n) play an important role in homotopy theory. At present the case n=1 is completely understood for all primes p. The next case, n=2, has been partially investigated for primes pThis paper will appear in the Proceedings of the American Mathematical Society.5. The functor LK(2)for small primes is harder to study because the group S2 is of infinite cohomological dimension. In this paper, we compute the cohomology of S2 at the prime 3 by resolving it by a free product Z/3*Z/3 and analyzing the "relation module."
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