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 mΓ = Γm. More generally, a

subideal I ⊆ m is invariant if IΓ = 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-

H¨ 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

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 mΓ = Γm. More generally, a

subideal I ⊆ m is invariant if IΓ = 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-

H¨ 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