L-COMPLETE HOPF ALGEBROIDS 11

Proof. The proof is similar to that used in [1]. The idea is to consider the

descending sequence

M ⊇ mM ⊇ · · · ⊇

mkM

⊇ · · ·

which must eventually reach 0. So for some k0, mk0 M = 0 and mk0−1M = 0. The

subcomodule

mk0−1M

∼

=

mk0−1M/mk0

M

becomes a comodule over (k, Γ/mΓ) and so it has non-trivial primitives since

(k, Γ/mΓ) is unipotent, and these are also primitives with respect to Γ. Considering

the quotient M/PM, where PM is the submodule of primitives, now we can use

induction on the length of a composition series to construct the required ﬁltration.

Note that since R is local, its only irreducible module is its residue ﬁeld k which

happens to be a comodule.

Ravenel [13] introduced the associated Hopf algebra (A, Γ ) to a Hopf algebroid

(A, Γ). When the coeﬃcient ring A is a ﬁeld, the relationship between comodules

over these two Hopf algebroids turns out to be tractable as we will soon see.

Our next result provides a criterion for establishing when a Hopf algebra is

unipotent. We write ⊗ for ⊗k.

Lemma 3.3. Let (H, k) be a Hopf algebra over a ﬁeld.

(a) Suppose that

k = H0 ⊆ H1 ⊆ · · · ⊆ Hn ⊆ · · · ⊆ H

is an increasing sequence of k-subspaces for which H =

n

Hn and

ψHn ⊆ H0 ⊗ Hn + H1 ⊗ Hn−1 + · · · + Hn ⊗ H0.

Then H is unipotent. Furthermore, the Hn can be chosen to be maximal and satisfy

HrHs ⊆ Hr+s

for all r, s.

(b) Suppose that H has a ﬁltration as in (a) and let W be a non-trivial left H-

comodule which is ﬁnite dimensional over k and has coaction ρ: W −→ H ⊗ W .

Deﬁning

Wk =

ρ−1(Hk

⊗ W ) ⊆ W,

we obtain an exhaustive strictly increasing ﬁltration of W by subcomodules

{0} = W−1 ⊂ W0 ⊂ W1 ⊂ · · · ⊂ W = W.

Proof. (a) This is part of the theorem of [15, section 8.3]. The proof actually

shows that the ﬁltration by subspaces deﬁned in (b) is strictly increasing.

(b) The fact that Wk is a subcomodule follows by comparing the two sides of the

equation

(Id ⊗ ρ)ρ(w) = ρ(ψ ⊗ Id)ρ(w)

for w ∈ Wk. Thus if we choose a basis t1,...,td for Hk and write

ρ(w) =

j

tj ⊗ wj

for some wi ∈ Wk, then for suitable ai,r,s ∈ k we have

ψ(ti) =

r,s

ai,r,str ⊗ ts

11

Proof. The proof is similar to that used in [1]. The idea is to consider the

descending sequence

M ⊇ mM ⊇ · · · ⊇

mkM

⊇ · · ·

which must eventually reach 0. So for some k0, mk0 M = 0 and mk0−1M = 0. The

subcomodule

mk0−1M

∼

=

mk0−1M/mk0

M

becomes a comodule over (k, Γ/mΓ) and so it has non-trivial primitives since

(k, Γ/mΓ) is unipotent, and these are also primitives with respect to Γ. Considering

the quotient M/PM, where PM is the submodule of primitives, now we can use

induction on the length of a composition series to construct the required ﬁltration.

Note that since R is local, its only irreducible module is its residue ﬁeld k which

happens to be a comodule.

Ravenel [13] introduced the associated Hopf algebra (A, Γ ) to a Hopf algebroid

(A, Γ). When the coeﬃcient ring A is a ﬁeld, the relationship between comodules

over these two Hopf algebroids turns out to be tractable as we will soon see.

Our next result provides a criterion for establishing when a Hopf algebra is

unipotent. We write ⊗ for ⊗k.

Lemma 3.3. Let (H, k) be a Hopf algebra over a ﬁeld.

(a) Suppose that

k = H0 ⊆ H1 ⊆ · · · ⊆ Hn ⊆ · · · ⊆ H

is an increasing sequence of k-subspaces for which H =

n

Hn and

ψHn ⊆ H0 ⊗ Hn + H1 ⊗ Hn−1 + · · · + Hn ⊗ H0.

Then H is unipotent. Furthermore, the Hn can be chosen to be maximal and satisfy

HrHs ⊆ Hr+s

for all r, s.

(b) Suppose that H has a ﬁltration as in (a) and let W be a non-trivial left H-

comodule which is ﬁnite dimensional over k and has coaction ρ: W −→ H ⊗ W .

Deﬁning

Wk =

ρ−1(Hk

⊗ W ) ⊆ W,

we obtain an exhaustive strictly increasing ﬁltration of W by subcomodules

{0} = W−1 ⊂ W0 ⊂ W1 ⊂ · · · ⊂ W = W.

Proof. (a) This is part of the theorem of [15, section 8.3]. The proof actually

shows that the ﬁltration by subspaces deﬁned in (b) is strictly increasing.

(b) The fact that Wk is a subcomodule follows by comparing the two sides of the

equation

(Id ⊗ ρ)ρ(w) = ρ(ψ ⊗ Id)ρ(w)

for w ∈ Wk. Thus if we choose a basis t1,...,td for Hk and write

ρ(w) =

j

tj ⊗ wj

for some wi ∈ Wk, then for suitable ai,r,s ∈ k we have

ψ(ti) =

r,s

ai,r,str ⊗ ts

11