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 A -modules is studied in the special case in which A A is the ring of integers in the field obtained by adjoining p p th roots of unity to Q ^ p {\widehat {\mathbb {Q}}_p} , the p p -adic numbers. Information about these cohomology groups is used to give new proofs of results about the E 2 {E_2} term of the Adams spectral sequence based on 2 2 -local complex K K -theory, and about the odd primary Kervaire invariant elements in the usual Adams-Novikov spectral sequence.