The structure of Morava stabilizer algebras

Abstract

The purpose of this note is to prove some general theorems which will facilitate the computation of Ext*e.sp(BP , , v21BP./I.), where 1.=(p, vl, ..., v,_ 0 is the n-th invariant prime ideal in BP.. Specific calculations and applications to the Novikov spectral sequence will be exposed in [8] and 1-13]. This paper is a sequel to [4] in that we reprove some results of Morava ([10] and [11]) with more conventional algebraic topological methods. Our approach differs from those of Morava and Johnson-Wilson in that no use is made of any cohomology theories other than Brown-Peterson theory. Our results have the advantage of being more directly applicable to homotopy theoretic computations than Morava's were. Although none of his results are actually used here, this paper owes its existence to many inspiring and invaluable conversations with Jack Morava. I would also like to thank Haynes Miller, John Moore, Robert Morris, and Steve Wilson for their interest and help. In w 1 we use the change of rings theorem of [7] to show that computing the above mentioned Ext group is equivalent to computing the cohomology of a certain Hopf algebra S(n), which we call the Morava stabilizer algebra. We describe it explicitly using the results of [12]. In w we describe the relation of S(n) to a certain compact p-adic Lie group S. which Morava called the stabilizer group, as it was the isotropy group of a certain point in a scheme with a certain group action in [10]. This group has been studied to some extent by number theorists but we do not exploit this fact. Its basic cohomological properties were originally found by Morava and the author (very likely not for the first time) by application of the results of Lazard I-5]. The results of w 3, however, make no use of [5] or even of the existence of S., and w 3 is independent of w 2. We do however use this group theoretic interpretation to get a certain splitting (Theorem (2.12)) of S(n) when p does not divide n.