Galois extensions of Lubin-Tate spectra

Andrew Baker (University of Glasgow)

Monday 14th January, 2008 16:00-17:00 214, Mathematics Building


John Rognes introduced a Galois theory of commutative S-algebras generalising the Galois theory of commutative rings due to Chase-Harrison-Rosenberg. Rognes proved that the sphere spectrum S has only split extensions - this is based on the algebraic fact that finite extensions of the integers Z must ramify. On the other hand, working Bousfield locally with respect to K(n), the n-th Morava K-theory at a prime p, it is know that the local sphere L_{K(n)}S has a large connected extension called the n-the Lubin-Tate spectrum, E_n; the group here is the Morava stabilizer group which is a group of units in a p-adic division algebra. I will describe recent work with Birgit Richter in which we show that every finite Galois extension of E_n splits, at least for odd primes. The proof involves three cases corresponding to the existence of a quotient of the Galois group which is either cyclic of order p, cyclic of prime order different from p, or non-abelian simple. To deal with the last case, we need to use the Feit-Thompson Theorem to deduce that the order is even.

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