On the structure of the Hopf algebroid E ( n )* E ( n )

Abstract

This paper is devoted to giving a description of E ( n )* E ( n ). Ravenel and Hopkins have studied a certain completion, F ( n ), of E ( n ) and have showed that F ( n )* F ( n ) contains a subalgebra isomorphic to C ( S n , A ), the algebra of continuous functions on the n th Morava stabilizer group taking values in the ring of integers in an unramified degree n extension of the p -adic numbers. They also show that this completion is split with respect to this subalgebra, i.e. F ( n )* F ( n ) ≅ C ( S n , A )⊗ A F ( n )*. We display an injection of E ( n )* E ( n ) into C ( S n × S n , A ) which extends to the Hopkins-Ravenel completion and identify the image of their subalgebra as the subalgebra of functions which are invariant under translation by elements of S n . Using this we give a formula for the coaction of C ( S n , A ) on E ( n )* and another proof of the Hopkins-Ravenel splitting theorem.