L-COMPLETE HOPF ALGEBROIDS 17

The right unit generates a second copy of K• in K•E and there is an element

θ0 =

ηL(u)−1ηR(u)

which satisﬁes the relation

θ0n−1

p

= 1.

The coproduct makes θ0 group-like,

ψ(θ0) = θ0 ⊗ θ0.

Now it is easy to see that K•E contains the unicursal Hopf algebroid

(5.1) K• ⊗Fp K• = F ⊗Fp F(u, θ0) = F ⊗Fp F[u,

θ0]/(upn−1

− 1,θ0

pn−1

− 1),

where θ0,u have degrees

¯,

0

¯

2 ∈ Z/2(pn − 1) respectively.

Ignoring the generator u and the grading, we also have the ungraded Hopf

algebroid

(F, F ⊗Fp F(θ0)) = (F, F ⊗Fp F[θ0]/(θ0

pn−1

− 1))

which is a subHopf algebroid of (F,K¯E).

0

Since F is a Galois extension of Fp with Galois group a quotient of Z, we obtain

a ring isomorphism

F ⊗Fp F

∼

=

F

Gal(F/Fp)∗.

If Gal(F/Fp) is ﬁnite this has its usual meaning, while if Gal(F/Fp) is inﬁnite we

have

F

Gal(F/Fp)∗

=

Mapc(Gal(F/Fp),

F).

Of course, if Gal(F/Fp) is ﬁnite this interpretation is still valid but then all maps

Gal(F/Fp) −→ F are continuous. In each case, we obtain an isomorphism of Hopf

algebroids

(5.2) F ⊗Fp F(θ0)

∼

= Map(Gal(F/Fp) Fpn

×

, F).

Now we consider the associated Hopf algebra over the graded ﬁeld K•,

K• ⊗F⊗Fp

F(u,θ0)

K•E = K•[θk : k 1]/(θk

pn

− θk : k 1)

whose zero degree part is

(5.3) F[θk : k 1]/(θk

pn

− θk : k 1)

∼

=

Mapc(Sn, 0

F).

The right hand side ﬁts into the framework of Example 3.6, so this Hopf algebra

over F is unipotent. Tensoring up with K• we have the following graded version.

Theorem 5.4. The Hopf algebra (K•,K• ⊗F⊗Fp

F(u,θ0)

K•E) is unipotent.

Remark 5.5. The identiﬁcation of (5.3) can be extended to all degrees of

K• ⊗F⊗Fp

F(u,θ0)

K•E. To make this explicit, we consider

Mapc(Sn,

Fur) with the

action of Gal(F/Fp) Fpn

×

induced from the action on Sn used in deﬁning Gn

nr

and

the F-semilinear action of Gal(F/Fp) Fpn

×

on

Fur

obtained by inducing up the

r-th power of the natural 1-dimensional representation of Fpn

×

. Then

[K• ⊗F⊗Fp

F(u,θ0)

K•E]2r

∼

=

Mapc(Sn,

Fur)Gal(F/Fp)Fpn

×

,

where the right hand side is the set of continuous Gal(F/Fp) Fpn

×

-equivariant

maps. This is essentially a standard identiﬁcation appearing in work of Morava

and others in the 1970’s.

17

The right unit generates a second copy of K• in K•E and there is an element

θ0 =

ηL(u)−1ηR(u)

which satisﬁes the relation

θ0n−1

p

= 1.

The coproduct makes θ0 group-like,

ψ(θ0) = θ0 ⊗ θ0.

Now it is easy to see that K•E contains the unicursal Hopf algebroid

(5.1) K• ⊗Fp K• = F ⊗Fp F(u, θ0) = F ⊗Fp F[u,

θ0]/(upn−1

− 1,θ0

pn−1

− 1),

where θ0,u have degrees

¯,

0

¯

2 ∈ Z/2(pn − 1) respectively.

Ignoring the generator u and the grading, we also have the ungraded Hopf

algebroid

(F, F ⊗Fp F(θ0)) = (F, F ⊗Fp F[θ0]/(θ0

pn−1

− 1))

which is a subHopf algebroid of (F,K¯E).

0

Since F is a Galois extension of Fp with Galois group a quotient of Z, we obtain

a ring isomorphism

F ⊗Fp F

∼

=

F

Gal(F/Fp)∗.

If Gal(F/Fp) is ﬁnite this has its usual meaning, while if Gal(F/Fp) is inﬁnite we

have

F

Gal(F/Fp)∗

=

Mapc(Gal(F/Fp),

F).

Of course, if Gal(F/Fp) is ﬁnite this interpretation is still valid but then all maps

Gal(F/Fp) −→ F are continuous. In each case, we obtain an isomorphism of Hopf

algebroids

(5.2) F ⊗Fp F(θ0)

∼

= Map(Gal(F/Fp) Fpn

×

, F).

Now we consider the associated Hopf algebra over the graded ﬁeld K•,

K• ⊗F⊗Fp

F(u,θ0)

K•E = K•[θk : k 1]/(θk

pn

− θk : k 1)

whose zero degree part is

(5.3) F[θk : k 1]/(θk

pn

− θk : k 1)

∼

=

Mapc(Sn, 0

F).

The right hand side ﬁts into the framework of Example 3.6, so this Hopf algebra

over F is unipotent. Tensoring up with K• we have the following graded version.

Theorem 5.4. The Hopf algebra (K•,K• ⊗F⊗Fp

F(u,θ0)

K•E) is unipotent.

Remark 5.5. The identiﬁcation of (5.3) can be extended to all degrees of

K• ⊗F⊗Fp

F(u,θ0)

K•E. To make this explicit, we consider

Mapc(Sn,

Fur) with the

action of Gal(F/Fp) Fpn

×

induced from the action on Sn used in deﬁning Gn

nr

and

the F-semilinear action of Gal(F/Fp) Fpn

×

on

Fur

obtained by inducing up the

r-th power of the natural 1-dimensional representation of Fpn

×

. Then

[K• ⊗F⊗Fp

F(u,θ0)

K•E]2r

∼

=

Mapc(Sn,

Fur)Gal(F/Fp)Fpn

×

,

where the right hand side is the set of continuous Gal(F/Fp) Fpn

×

-equivariant

maps. This is essentially a standard identiﬁcation appearing in work of Morava

and others in the 1970’s.

17