12 ANDREW BAKER

since ψ(ti) ∈

∑

i

Hi ⊗ Hk−i ⊆ Hk ⊗ Hk. Therefore

i

ti ⊗ ρ(wi) =

j,r,s

aj,r,str ⊗ ts ⊗ wj,

and comparing the coeﬃcients of the left hand ti, we obtain

ρ(wi) =

j,s

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

ψζn =

n

r=0

ζr ⊗

ζn−r,pr

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

P∗

is a local ring.

If p = 2, this also applies to the mod 2 dual Steenrod algebra and implies that

A

∗

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

that usage.

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

1]/(vntpn

− vn

p

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

the K(n)∗-subspaces

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

commutative.

Here is a major source of examples which includes the algebraic ingredients

used in [1] 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

Mapc(X,

Y ).

12

since ψ(ti) ∈

∑

i

Hi ⊗ Hk−i ⊆ Hk ⊗ Hk. Therefore

i

ti ⊗ ρ(wi) =

j,r,s

aj,r,str ⊗ ts ⊗ wj,

and comparing the coeﬃcients of the left hand ti, we obtain

ρ(wi) =

j,s

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

ψζn =

n

r=0

ζr ⊗

ζn−r,pr

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

P∗

is a local ring.

If p = 2, this also applies to the mod 2 dual Steenrod algebra and implies that

A

∗

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

that usage.

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

1]/(vntpn

− vn

p

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

the K(n)∗-subspaces

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

commutative.

Here is a major source of examples which includes the algebraic ingredients

used in [1] 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

Mapc(X,

Y ).

12