L-COMPLETE HOPF ALGEBROIDS 13 Example 3.6. Let G be a pro-p-group and suppose that G acts continuously (in the sense that the action map G×R −→ R is continuous) by ring automorphisms on R which are continuous with respect to the m-adic topology. Then (R, Mapc(G, R)) admits the structure of an L-complete Hopf algebroid, see the Appendix of [1] for details. This structure is dual to one on the twisted group algebra R[G]. Here m is invariant. If the residue field k has characteristic p, then (k, Mapc(G, R)/m) is the continuous dual of the pro-group ring k[G] = lim N G k[G/N], where N ranges over the finite index normal subgroups of G. Each finite group ring k[G/N] is local since its augmentation ideal is nilpotent, hence its only irreducible module is the trivial module. From this it easily follows that the dual Hopf algebra (k, Mapc(G, R)/m) is unipotent. In each of the examples we are interested in, there is a filtration G = G0 ⊃ G1 ⊃ · · · ⊃ Gk ⊃ Gk−1 ⊃ · · · by finite index normal subgroups Gk G satisfying k Gk = {1}, and the images of the natural maps Map(G/Gk,R) −→ Mapc(G, R) induced by the quotient maps G −→ G/Gk define a filtration with the properties listed in Lemma 3.3(a). 4. Unicursal Hopf algebroids The notion of a unicursal Hopf algebroid (A, Ψ) appeared in [13], see defini- tion A1.1.11. It amounts to requiring that for the subring AΨ = A Ψ A ⊆ A ⊗A A ∼ A we have Ψ = A ⊗AΨ A. If A is a flat AΨ-algebra then Ψ is a flat A-algebra. But the requirement that AΨ is the equalizer of the two homomorphisms A −→ A ⊗AΨ A is implied by faithful flatness, see the second theorem of [15, section 13.1]. Unicursal Hopf algebroids were introduced by Ravenel [13]. However, his lemma A1.1.13 has a correct statement for (b), but the statement for (a) appears to be incorrect. The proofs of (a) and (b) both appear to have minor errors or gaps. In particular the flatness of Ψ as an A-module is required. Therefore we pro- vide a slight modification of the proof given by Ravenel. Note that we work with left rather than right comodules. The formulation and proof, clarifying our earlier version, owe much to the comments of Geoffrey Powell and the referee, particularly the relationship to descent arguments based on faithful flatness. Lemma 4.1. Let (A, Ψ) be a unicursal Hopf algebroid where A is flat over AΨ. Let M be a left Ψ-comodule. Then there is an isomorphism of comodules M ∼ A ⊗AΨ (A Ψ M), where the coaction on the right hand comodule comes from the Ψ-comodule structure on A. In particular, if M is non-trivial then the primitive subcomodule A Ψ M is non-trivial.

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