since ψ(ti) ∈
Hi ⊗ Hk−i ⊆ Hk ⊗ Hk. Therefore
ti ⊗ ρ(wi) =
aj,r,str ⊗ ts ⊗ wj,
and comparing the coeﬃcients of the left hand ti, we obtain
aj,i,sts ⊗ wj ∈ Hk ⊗ W.
This shows that each wi ∈ Wk, so the coproduct restricted to Wk satisﬁes ρWk ⊆
H ⊗ Wk.
Example 3.4. Let p be an odd prime and let
P∗ = Fp[ζk : k 1]
be the (graded) polynomial sub-Hopf algebra of the mod p dual Steenrod algebra
A∗ with coaction
where ζ0 = 1. Then (P∗, Fp) is unipotent since the subspaces
P(n)∗ = Fp[ζk : 1 k n]
satisfy the conditions of Lemma 3.3. This shows that
is a local ring.
If p = 2, this also applies to the mod 2 dual Steenrod algebra and implies that
is a local ring.
For details on the next example, see the books by Ravenel and Wilson [13, 16].
Unfortunately the sub-Hopf algebra K(n)∗(E(n)) is commonly denoted K(n)∗K(n)
in the earlier literature, but at the behest of the referee we refrain from perpetuating
Example 3.5. Let p be an odd prime and let K(n) be the n-th p-primary
Morava K-theory. Then K(n)∗ = Fp[vn,vn −1], with vn ∈ K(n)2(pn−1). There is a
graded Hopf algebra over K(n)∗,
Γ(n)∗ = K(n)∗(E(n)) = K(n)∗[tk : k
t : 1),
where tk ∈ Γ(n)2(pk−1) and E(n) is a Johnson-Wilson spectrum. In fact Γ(n)∗ is a
proper sub-Hopf algebra of K(n)∗(K(n)). Using standard formulae, it follows that
Γ(n, m)∗ = K(n)∗(t1,...,tm) ⊆ Γ∗
satisfy the conditions of Lemma 3.3, therefore Γ(n, m)∗ is unipotent. When p =
2, the Hopf algebra Γ(n)∗ is also unipotent even though K(n) is not homotopy
Here is a major source of examples which includes the algebraic ingredients
used in  to prove the existence of a Landweber ﬁltration for discrete comodules
over the Hopf algebroid of Lubin-Tate theory. For two topologised objects X and
Y we denote the set of all continuous maps X −→ Y by