10 ANDREW BAKER
Definition 2.6. Let (R, Γ) be a Hopf algebroid over a local ring (R, m) or an
L-complete Hopf algebroid.
The maximal ideal m R is invariant if = Γm. More generally, a
subideal I m is invariant if = ΓI.
An (R, Γ)-comodule M is discrete if for each element x M, there is a
k 1 for which mkx = {0}; if M is also finitely generated as an R-module,
then M is discrete if and only if there is a k0 such that mk0 M = {0}.
An (R, Γ)-comodule M is finitely generated if it is finitely generated as an
R-module.
If M is a (R, Γ)-comodule, then for any invariant ideal I, IM M is a subco-
module.
If (R, Γ) be a (possibly L-complete) Hopf algebroid for which m is invariant,
then (k, Γ/mΓ) is a Hopf algebroid over the residue field k. If a Γ-comodule is
annihilated by m then it is also a Γ/mΓ-comodule.
3. Unipotent Hopf algebroids
We start by recalling the notion of a unipotent Hopf algebra H over a field k
which can be found in [15]. This means that every H-comodule V which is a finite
dimensional k-vector space has primitive elements, or equivalently (by the Jordan-
older theorem) it admits a composition series, i.e., a finite length filtration by
subcomodules
(3.1) V = Vm Vm−1 · · · V1 V0 = {0}
with irreducible quotient comodules Vk/Vk+1

=
k. In particular, notice that k is
the only finite dimensional irreducible H-comodule. Reinterpreting H-comodules
as
H∗-modules
where
H∗
is the k-dual of H, this implies that
H∗
is a local ring,
i.e., its augmentation ideal is its unique maximal left ideal and therefore agrees
with its Jacobson radical.
Now given a Hopf algebroid (R, Γ) over local ring (R, m) with residue field
k = R/m and invariant maximal ideal m, the resulting Hopf algebroid (k, Γ/mΓ)
need not be a Hopf algebra. However, we can still make the following definition.
Definition 3.1. Let (k, Σ) be a Hopf algebroid over a field k. Then Σ is unipo-
tent if every non-trivial finite dimensional Σ-comodule V has non-trivial primitives.
Hence k is the only irreducible Σ-comodule and every finite dimensional comodule
admits a composition series as in (3.1).
In the next result we make use of Definition 2.6.
Theorem 3.2. Let (R, Γ) be a Hopf algebroid over a Noetherian local ring
(R, m) for which m is invariant and suppose that (k, Γ/mΓ) is a unipotent Hopf
algebroid over the residue field k. Let M be a non-trivial finitely generated discrete
(R, Γ)-comodule. Then M admits a finite-length filtration by subcomodules
M = M M
−1
· · · M1 M0 = {0}
with trivial quotient comodules Mk/Mk+1

=
k.
See [1] for a precursor of this result. We will refer to such filtrations as Landwe-
ber filtrations.
10
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