16 M. HOVEY AND N. P. STRICKLAND
Lemma 2.26. Let R* be a graded algebra over K* (so that K* is central in i?*).
Write R = R*/(vn 1) andir: R* R for the projection map. If R is Noetherian,
then R* is Noetherian in the graded sense.
Proof It is enough to show that the map J H- TT J embeds the lattice of homoge-
neous ideals of i?* into the Noetherian lattice of ideals in i?, and thus enough to
show that the set of homogeneous elements in
7r_17rJ
is just J. As R* /J is a graded
module over the graded field if*, it is free, generated by elements e* of degree di
say. Suppose that a G i?* is homogeneous of degree d; then a = ]T\ aiVn ~
*"'Vn'ei
(mod J) where a; G F
p
, and a* is zero if the indicated exponent of vn is not an
integer. If 7r(a) G n(J) then
Yliaiir(ei) =
0, but it is clear that {7r(ei)} is a basis
for R/irJy so that a% 0 for all i and thus a G J.
Lemma 2.27. Ze£ R be ring, and {Is} a decreasing filtration such that Jo = R and
Islt Is+t- Suppose that R/Is is a finite set for all s, that R = lim R/Is, and
s
that the associated graded ring R' = EQR = Yls Is/Is+i is Noetherian. Then R is
Noetherian.
Proof. Let J be a left ideal in R. Then J' = n
s
( ^
n
^ ) / ( ^
n
^ + i )
1 S a
^
i(*eal *n
Rf.
It is thus finitely generated, so there are elements a* G Jnldi (for i 1,... , m
say) whose images generate J'. This means that for any element a G J f)Is there
are elements bi such that a = J2i
^iai (m°d
J fl Is+i). In other words, if if J
is the ideal generated by {ai,... , a
m
}, then Jfl J5 if + Jfl J5+i. It follows
easily that J = J fl Jo is contained in f]s(K + Js), which is the closure of K in the
evident topology given by the ideals J5. On the other hand, as R/Is is finite, we
see that R is J-adically compact and Hausdorff. As K is the image of an evident
continuous map
Rm
i?, we see that K is compact and thus closed. It follows
that J = K = (ai,... , am), which is finitely generated as required.
We next recall that for each k 0, the ideal (tj \ 0 j k) E* is a Hopf ideal,
so that £(&)* = S*/(tj | 0 j k) is a Hopf algebra, and E(fc)* is a quotient
Hopf algebra of S*. We also write S = E*/(vn -1) and S(k) = E(fc)*/(vn -1). We
write 5* = Hom(5,Fp) and S(k)* =
Hom(S,(fc),Fp).
Note that Ravenel [Rav86]
calls these objects E(n, &)*, 5(n,fc)and so on.
Proposition 2.28. J/fc pn/(p—l) then the Hopf algebra 5(&)* can be filtered so
that the associated graded ring is a commutative formal power series algebra over
F
p
on
n2
generators.
Proof. In this proof, all theorem numbers and so on refer to the book [Rav86]. Our
proposition is essentially RavenePs Theorem 6.3.3. That theorem appears to apply
to S rather than 5*, but this is a typo; this becomes clear if we read the preceding
paragraph. Some modifications are necessary to replace 5* by 5(&)*, and anyway
Ravenel does not give an explicit proof of his theorem, so we willfillin some details.
The Hopf algebra filtration of S given by Theorem 6.3.1 clearly induces a filtra-
tion on S(k). It is easy to see that
E°S(k) = T[tij\ikjGZ/n]
as rings, where T[t] =
T?P[t]/tp
and Uj corresponds to i? . There is therefore an
automorphism F on E°S(k) that takes Uj to Uj+i which has order n. Moreover,
this is a connected graded Hopf algebra (using the grading coming from the filtra-
tion, so that the degree of Uj is the integer dn^ of Theorem 6.3.1). The coproduct
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