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.
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