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.

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.

13