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 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
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