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
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