ROOTS OF UNITY AND THE ADAMS-NOVIKOV SPECTRAL SEQUENCE FOR FORMAL yl-MODULES

The cohomology of a Hopf algebroid related to the Adams-Novikov spectral sequence for formal ,4-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 Qp , 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 A"-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 yl-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 Ext1 ' * ) was confirmed in [J] using a Hopf algebroid (i.e., a generalized Hopf algebra in which the left and right units need not agree), EA T, which generalizes the Hopf algebroid KtK of stable cooperations for complex K-theory. The present paper is concerned with the cohomology of EAT in the special case of A = Z [Ç] where £ is a pth root of unity and Z denotes the p-adic integers. We will show that in this case EA T is contained in an extension of Hopf algebroids