18 ANDREW BAKER

Combining Theorems 5.4 and 3.2 we obtain our ﬁnal result in which we revert

to Z-gradings.

Theorem 5.6. The Hopf algebroid (K∗,K∗E) is unipotent, hence every ﬁnitely

generated comodule for the L-complete Hopf algebroid (E∗,E∗

∨E)

has a Landweber

ﬁltration.

Here it is crucial that we take proper account of the grading since the ungraded

Hopf algebra (K0,K0E) is not unipotent: this can be seen by considering the

comodule K0S2 which is not isomorphic to K0S0.

Appendix A. Representations of Galois Hopf algebroids

Twisted (or skew) group rings are standard algebraic objects. They were dis-

cussed for an audience of topologists in [1], and their duals as Hopf algebroids were

discussed. For a recent reference on their modules see [10]. Here we focus on the

special case of Galois extensions of ﬁelds, which is closely related to the unicursal

Hopf algebroids. In particular, the unicursal Hopf algebroids associated with K• in

Section 5 contain degree zero parts of this form.

Let k be a ﬁeld of positive characteristic char k = p and let A be a (ﬁnite

dimensional) commutative k-algebra which is a G-Galois extension of k for some

ﬁnite group G, where the action of γ ∈ G on x ∈ A is indicated by writing

γx.

This

means that

• AG = k,

• the A-algebra homomorphism

A ⊗k A −→

γ∈G

A; x ⊗ y →

(xγy)γ∈G

is an isomorphism, where the A-algebra comes from the left hand factor

of A.

The second condition is equivalent to the assertion that there is an isomorphism of

k-algebras

(A.1) A ⊗k A

∼

=

A ⊗k

kG∗,

where

kG∗

= Homk(kG, k)

is the dual group algebra.

The twisted group ring AG is the usual group ring AG as a left A-module, but

with multiplication deﬁned by

(a1γ1)(a2γ2) =

a1γ1

a2γ1γ2.

There is a natural k-linear map

AG −→ Endk A

under which aγ ∈ AG is sent to the k-linear endomorphism x → aγx. Another

consequence of the above assumptions is that this is a k-algebra isomorphism,

see [3].

If A = is a ﬁeld, then using the isomorphism of (A.1) we see that ⊗k is

isomorphic to G∗ as an -algebra. There is an associated ‘right’ action of on G∗

given by

(f · x)(γ) =

γxf(γ)

18

Combining Theorems 5.4 and 3.2 we obtain our ﬁnal result in which we revert

to Z-gradings.

Theorem 5.6. The Hopf algebroid (K∗,K∗E) is unipotent, hence every ﬁnitely

generated comodule for the L-complete Hopf algebroid (E∗,E∗

∨E)

has a Landweber

ﬁltration.

Here it is crucial that we take proper account of the grading since the ungraded

Hopf algebra (K0,K0E) is not unipotent: this can be seen by considering the

comodule K0S2 which is not isomorphic to K0S0.

Appendix A. Representations of Galois Hopf algebroids

Twisted (or skew) group rings are standard algebraic objects. They were dis-

cussed for an audience of topologists in [1], and their duals as Hopf algebroids were

discussed. For a recent reference on their modules see [10]. Here we focus on the

special case of Galois extensions of ﬁelds, which is closely related to the unicursal

Hopf algebroids. In particular, the unicursal Hopf algebroids associated with K• in

Section 5 contain degree zero parts of this form.

Let k be a ﬁeld of positive characteristic char k = p and let A be a (ﬁnite

dimensional) commutative k-algebra which is a G-Galois extension of k for some

ﬁnite group G, where the action of γ ∈ G on x ∈ A is indicated by writing

γx.

This

means that

• AG = k,

• the A-algebra homomorphism

A ⊗k A −→

γ∈G

A; x ⊗ y →

(xγy)γ∈G

is an isomorphism, where the A-algebra comes from the left hand factor

of A.

The second condition is equivalent to the assertion that there is an isomorphism of

k-algebras

(A.1) A ⊗k A

∼

=

A ⊗k

kG∗,

where

kG∗

= Homk(kG, k)

is the dual group algebra.

The twisted group ring AG is the usual group ring AG as a left A-module, but

with multiplication deﬁned by

(a1γ1)(a2γ2) =

a1γ1

a2γ1γ2.

There is a natural k-linear map

AG −→ Endk A

under which aγ ∈ AG is sent to the k-linear endomorphism x → aγx. Another

consequence of the above assumptions is that this is a k-algebra isomorphism,

see [3].

If A = is a ﬁeld, then using the isomorphism of (A.1) we see that ⊗k is

isomorphic to G∗ as an -algebra. There is an associated ‘right’ action of on G∗

given by

(f · x)(γ) =

γxf(γ)

18