Roots of Unity and the Adams-Novikov Spectral Sequence for Formal A-Modules

Abstract

The cohomology of a Hopf algebroid related to the Adams-Novikov spectral sequence for formal A-modules is studied in the special case in which A is the ring of integers in the field obtained by adjoining pth roots of unity to ?2p, the p-adic numbers. Information about these cohomology groups is used to give new proofs of results about the E2 term of the Adams spectral sequence based on 2-local complex K-theory, and about the odd primary Kervaire invariant elements in the usual Adams-Novikov spectral sequence. One of the most powerful tools used in the computation of stable homotopy groups is the Adams-Novikov spectral sequence. The E2 term of this spectral sequence is a certain Ext group derived from a universal formal group law. In [R3] the corresponding Ext group for a universal formal A-module, for A the ring of algebraic integers in an algebraic number field, K, or its padic completion, was introduced and certain conjectures about these groups were formulated. One of these conjectures (concerning the value of ExtlI* ) was confirmed in [J] using a Hopf algebroid (i.e., a generalized Hopf algebra in which the left and right units need not agree), EAT, which generalizes the Hopf algebroid K*K of stable cooperations for complex K-theory. The present paper is concerned with the cohomology of EAT in the special case of A = p where C is a pth root of unity and Zp denotes the p-adic integers. We will show that in this case EAT is contained in an extension of Hopf algebroids EA )EA T EAT and that the cohomology of EAT can be completely described. This provides us with information about the cohomology of EAT via the Cartan-Eilenberg spectral sequence associated to this extension. Two applications of this result are presented. In the case p = 2, EAT can be identified with the 2-adic completion of the Hopf algebroid K*K(2) of stable cooperations for 2-primary complex K-theory. In this case the cohomology Received by the editors February 1, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 55T25; Secondary 55N22, 144L05. ? 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page