MORAVA ^-THEORIES AND LOCALISATION 17

is given by Theorem 4.3.34, which says that A(Uj) is the sum of the elements in a

certain unordered list Ay. These lists are used in such a way that we may ignore

any terms which have filtration less than that of £y. We start with the list

M y = {£y ® l , l ® * y } .

(This comes from Lemma 4.3.32, using the fact that k pn/(p — l).) We also recall

the Witt polynomials wj defined in Lemma 4.3.8. We are working modulo p so

we have wj = w?j, . In E°S(k) we must interpret this as wj =

FWJW~\J\w\j\.

We have set vn = 1 and killed all other v's, so vj = 0 unless J has the form

J = Jr = (n,... ,n) (with r terms), in which case vj = 1. With these observations,

Lemma 4.3.33 becomes

Ay = M y U {F-rWr(Ai-nrj) \ T 0}.

As Mij is invariant under the twist map, we see that the same holds for Ay, and

thus that E°S(k) is cocommutative. We also see that

A(£y) = Uj ® 1 + 1 ® tij + wi(U-n,j-i ® 1,1 ® *i-

n

,i-i) (mod t

r a

| r i - n).

Here w\ is given by

w1(xux2,...) = (53^? -

(X^X*)P)/^

t t

It is not hard to conclude that the Verschiebung is

V(Uj) = ti-

n

j_ i (mod t

r 5

| r i — n).

We also observe that the degree of Uj is less than that of tki whenever i k (this

follows easily from the definition in Theorem 6.3.1). It follows by induction on i

that each Uj lies in the image of V, so that V is surjective.

We now dualise, and conclude that EoS(k)* is a bicommutative connected graded

Hopf algebra for which the Frobenius map is injective. We can thus apply the Borel

structure theory [Spa66, Section 5.8][Bor53] to conclude that EoS(k)* is a formal

power series algebra. By looking at the Poincare series, we see that there must be n2

generators, in degrees equal to those of the elements tk+ij for 0 i,j n. (With a

little more work, one can check that the generators are dual to these elements.) •

We can now prove as promised that E* is Noetherian.

Proof of Theorem 2.24- According to Lemma 2.26, it is enough to check that 5* is

Noetherian. By Lemma 2.27, it is enough to check that EoS* is Noetherian. We

have an extension of Hopf algebras

S'(k) = Fp[tu... ,t*-i]/(t f - tj)^ S - » S(fe),

and thus an injective map of connected graded Hopf algebras EoS(k)*— EoS*.

It follows from the Milnor-Moore theorem (the dual of [Rav86, Corollary Al.1.20])

that EoS* is a free (as a left or right module) over EoS(k)*. The rank is just

dim(S"(fc)) oo. Thus, by Lemma 2.25, it is enough to check that E°S(k)* is

Noetherian, and this follows from Proposition 2.28. •

We next show that S* is local in a suitable sense. Let I be the kernel of the

augmentation map S* — K*.

Proposition 2.29. E* = lim S*/Jfc.