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