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, mk0M = 0 and mk0−1M = 0. The subcomodule mk0−1M ∼ mk0−1M/mk0M 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

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