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

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 nth 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)⊗_AF(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.