THE BP <N> COHOMOLOGY OF ELEMENTARY ABELIAN GROUPS



                               N. P. STRICKLAND



                               1.Introduction

  In this paper we define three elements of a certain generalised cohomology ri*
 *ng BP <m; n>*BVk.
Here m, n and k are nonnegative integers with k+m  n+1, there is a fixed prime *
 *p not exhibited
in the notation, and Vk is an elementary Abelian p-group of rank k. We show tha*
 *t these elements
are equal; this is striking, because the three definitions are very different. *
 * The significance of
our equation is not yet entirely clear, but it makes contact with other work in*
 * the literature in a
number of fascinating ways.

  1.In the case where n = 1, m = 0 and k = 2, our result is closely related to *
 *[1, Theorem 4.2],
    which exhibits a relation in the connective complex K-theory group kU6B(Z=k*
 *)2 for all k.
    That result is in turn a key part of a proof (at least in special cases) of*
 * the main result
    of [5], which relates the Witten genus for spin manifolds [17] to the theor*
 *y of elliptic curves
    and elliptic spectra. The work described in the present paper started with *
 *an attempt to
    generalise this theorem in connective K-theory.
  2.Our methods also give an interesting filtration of the ring BP <n; m>*BVk. *
 *Again in the case
    n = 1, m = 0, k = 2 our filtration is compatible with a splitting of kU ^ B*
 *V2 discovered
    earlier by Ossa [11]. This was also a motivation for our investigations. It*
 * would clearly be
    interesting to improve our filtration to some kind of stable splitting in t*
 *he general case, but
    we have not succeeded as yet.
  3.One of our three definitions involves an iterated Bockstein map
                                                   n+1-1)=(p-1)-n-2
          q0n:= (BP <0>-q1!2p-1BP <1>q2-!:-:q:n!2(p             BP <n>):

    There is an analogous operation in Beilinson's motivic cohomology, which pl*
 *ays a central
    r^ole in Voevodsky's proof of the Milnor conjecture in algebraic K-theory [*
 *15]. We have no
    idea whether this fact is significant or not.
  4.By applying the (n + 2)'th space functor to the map q0nof spectra, we get a*
 * map

                     r :K(Z(p); n + 2) -!BP_<n>_2(pn+1-1)=(p-1):

    This map has appeared in a number of other places, for example the proof th*
 *at suspension
    spectra are harmonic [6]. Moreover, Ravenel, Wilson and Yagita have shown t*
 *hat BP *r is
    surjective, and thus given a nice description of BP *K(Z(p); n + 2) [12]. O*
 *ur results shed
    some interesting light on the nature of the map r.
  5.Our other two definitions make contact with the classical theory of Dickson*
 * invariants in
    H*(BVk; Fp) (see [16] for an exposition).
  6.Our proofs use the theory of multiple level structures [4], which generalis*
 *e Drinfel'd level
    structures on elliptic curves [2].  This theory was also central to the res*
 *ults of [4], which
    give a crude generalisation of the Hopkins-Kuhn-Ravenel generalised charact*
 *er theory for
    the rationalised Morava cohomology of classifying spaces of finite groups.

  There are other approaches to BP <n>*BVk and BP <n>*BVk, related to the Stein*
 *berg idempo-
tents and the Conner-Floyd conjecture [10, 7, 8]. We have not yet managed to fi*
 *nd any fruitful
interaction between our methods and these, but it is clearly an interesting pro*
 *ject to look for one.


                           2.Statement of results

  We now state our results in more detail.
                                      1
2                              N. P. STRICKLAND



  Fix a prime p and integers m  n, and define w = n + 1 - m. There is a BP -alg*
 *ebra spectrum
E = BP <m; n>with homotopy ring E* = Fp[vm ; : :;:vn] (or E* = Z(p)[v1; : :;:vn*
 *] if m = 0).
(We will recall a construction in Section 3.) Note that w is the Krull dimensio*
 *n of E*. We use
cohomological gradings, so that |vk| = -2(pk - 1) < 0.  Write Vk for the elemen*
 *tary Abelian
group Fkp. We will give a filtration of E*BVk by ideals, such that each quotien*
 *t is in a natural
way a finitely generated free module over a regular local ring. We will show th*
 *at E*BVk has no
vm -torsion when k < w. When k = w, we show that the vm -torsion is annihilated*
 * by the ideal
In+1 = (vm ; : :;:vn), and that it is a free module on one generator over the r*
 *ing Fp[[x0; : :;:xw-1]].
We give three very different formulae for this generator.
  Recall that the depth of a module M* over R* is the largest d such that there*
 * exists a sequence
{a0; : :;:ad-1} in R* which is regular on M*. We will need an analogous but dif*
 *ferent notion,
which we now define.

Definition 2.1.If M* is a module over E*, we define the chromatic depth of M* t*
 *o be the largest
integer d = cdepth(M*) such that the sequence (vm ; vm+1; : :;:vm+d-1) is regul*
 *ar on M*. Clearly
we have 0  cdepth(M*)  w.

  We next introduce some abbreviated notation for rings which will appear many *
 *times.

Definition 2.2.Given any algebra R* over BP *, and integers 0  i  j  k, we write

                    P (R*; j)= R*[[xt| 0  t <=j]]([p](xt) | 0  t < j)

                  P (R*; i;=j)R*[[xt| i  t=<(j]][p](xt) | i  t < j)

                 P (R*; i; j;=k)R*[[xt| i  t <=k]]([p](xt) | i  t < j):

Thus P (R*; j) = P (R*; 0; j) and P (R*; i; j) = P (R*; i; j; j).

  The most basic fact is as follows.

Proposition 2.3.If k  w then we have

             E*BVk = P (E*; k) = E*[[x0; : :;:xk-1]]=([p](x0); : :;:[p](xk-1))

and this has chromatic depth w - k.

  This is proved in Corollary 5.2 and Corollary 5.3.
  In the extreme case k = w we see that E*BVw has chromatic depth 0, so it must*
 * have some
vm -torsion. It is thus of interest to see what the torsion subgroup is. In ord*
 *er to state the answer,
we need some definitions.

Definition 2.4.Given __2 Fjp, we write [__](x_) = [0](x0) +F . .+.F[j-1](xj-1).*
 * We write
                                    Y                  m
                           OEj(t) =    (t -F [__](x_))p :

                                    _2Fjp
                               Q w-1
We also define ff = ff(m; n) =   j=0OEj(xj) 2 E*BVw. Note that ff is the produc*
 *t of the terms

[__](x_)pm, where __runs over elements of Fwpwhose last nonzero entry is one. T*
 *he degree of ff is
2pm (pw - 1)=(p - 1).

  In the following lemma, we regard BP <k>*= Z(p)[v1; : :;:vk] as a subring of *
 *BP <n>*in the
obvious way.
                                                                           P n
Lemma 2.5. There are unique series ssk(t) 2 BP <k>*[[t]]such that [p](t) =   k=*
 *0vkssk(t) 2

BP <n>*[[t]].  Moreover, the series ssk(t) is divisible by tpk, and is equal to*
 * tpk modulo (p; tpk+1)
or modulo In+1.

Proof.We give t degree two, which makes everythingPhomogeneous. Write ss0(t) = *
 *t. We know
that [p](t) = pt (mod t2) so [p](t) - v0ss0(t) =   k>1aktk say.  Each ak has st*
 *rictly negative
cohomological degree (because |t| = |[p](t)| = 2) so it lies in the ideal (v1; *
 *: :;:vn). Let fk(t) be
the sum of those monomials in [p](t)-v0ss0(t) that lie in BP <k>*[[t]], but do *
 *not lie in BP <k - 1>*[[t]].
              THE BP<N>COHOMOLOGY OF ELEMENTARY ABELIAN GROUPS              3



It is clear thatPfk(t) is divisible by vk, say fk(t) = vkssk(t). This gives ser*
 *ies ssk(t) 2 BP <k>*[[t]]
with [p](t) =   nk=0vkssk(t), as required. It is easy to see that they are uniq*
 *ue.
  Note that the degree of vkssk(t) is the same as the degree of [p](t), which i*
 *s 2, so ssk(t) has
degree 2pk. As |t| = 2 and |vi| < 0 for i > 0 we see that ssk(t) is divisible b*
 *y tpk. We thus have
ssk(t) = bktpk (mod tpk+1), say. One checks that |bk| = 0, so that bk 2 Z(p). G*
 *iven that [p](t) =

vktpk (mod v0; : :;:vk-1; tpk+1), we see that bk = 1 (mod p), and thus ssk(t) =*
 * tpk (mod p; tpk+1).
  Similarly, the image of ssk(t) modulo In+1 is a series in Fp[[t]]of degree 2p*
 *k, so it has the form

cktpk for some ck 2 Fp. Because [p](t) = vktpk (mod v0; : :;:vk-1; tpk+1), we s*
 *ee that ck = 1 and

ssk(t) = tpk (mod In+1).                                                       *
 *  |___|

Definition 2.6.We define ff0= ff0(m; n) to be the determinant of the square mat*
 *rix with entries
ssi(xj) for m  i  n and 0  j < w.

Definition 2.7.We shall see in Section 3 that there are cofibrations of spectra
            n-2        vn                        qn 2pn-1
          2p   BP <m; n>-!BP <m; n>-!BP <m; n - 1>-!     BP <m; n>:

This also works for n = m if we interpret BP <m; m - 1>as the mod p Eilenberg-M*
 *acLane spectrum
HFp. We let a0; : :;:aw-1 be the usual generators of H1(BVw; Fp) and we define

                    ff00= ff00(m; n) = qn . .q.m+1qm (a0a1: :a:w-1):

It is not hard to see that when g 2 Aut(Vw) we have det(g) 2 Fxpand g*ff00= det*
 *(g)ff00(which
makes sense because pff00= 0).

  Our main result is as follows.

Theorem 2.8.We have ff = ff0 = ff002 EdBVw, where d = 2pm (pw - 1)=(p - 1).  The
annihilator of vm on E*BVw is the same as the annihilator of In+1. It is a free*
 * module of rank
one over E*BVw=In+1 = P (Fp; w) = Fp[[x0; : :;:xw-1]]generated by ff.  It maps *
 *injectively to
H*(BVw; Fp).

  This will be proved after Proposition 5.13. Note that ff0and ff00appear to de*
 *pend on the choice
of generators vk, but in fact they do not, because they are equal to ff.
  The basic structure of the proof is as follows. It is quite easy to see that *
 *In+1ff0= In+1ff00= 0
and that ff, ff0, and ff00all have the same degree. We shall show in Section 3 *
 *that qk is compatible
up to sign with the Milnor Bockstein operation Qk in mod p cohomology. Given th*
 *is, classical
arguments about Dickson invariants show that ff = ff0= ff00(mod In+1). We will *
 *show using
an intricate argument inspired by the theory of multiple level structures [14, *
 *4] that ann(vm ) =
ann(In+1) = (ff). It follows for degree reasons that ff0= ff for some u 2 Fp, a*
 *nd we find that
 = 1 by reducing everything modulo In+1. The same argument shows that ff00= ff.


                           3.The spectra BP <m; n>

  For brevity, we will write MU for the p-local spectrum MU(p). Recall that MU *
 *can be made into
a strictly commutative ring spectrum (or "commutative S-algebra") in the founda*
 *tional setting
of [3], so we can construct a derived category DMU of strict MU-modules, and a *
 *category RMU of
ring objects in DMU  (referred to in [3] as MU-ring spectra). As usual, we let *
 *BP *be the largest
quotient ring of MU* over which the standard formal group law becomes p-typical*
 *.  We know
from [13] that (even when p = 2) there is a commutative ring BP in DMU  with ho*
 *motopy ring
BP *, and that this object is unique up to canonical isomorphism.              *
 *     k
  If p > 2 we choose once and for all a sequence of elements vk 2 BP2(pk-1)= BP*
 * -2(p -1)

such that [p](t) = vktpk (mod v0; : :;:vk-1; tpk+1). In particular, this forces*
 * v0 = p. Two popular
choicesPwould be to take vk to be the k'th Hazewinkel generator (so that [p](t)*
 * = expF(pt) +F
 F     pk                                                P F     pk
 k>0vkt  ) or the k'th Araki generator (so that [p](t) =   k0 vkt  ). In any ca*
 *se, we define
BP <m; n>*= BP *=(vi | i < m ori > n).  We know from [13, Theorem 2.6] that the*
 *re is a
commutative ring BP <m; n>in DMU  with homotopy ring BP <m; n>*, and that this *
 *object is
unique up to canonical isomorphism.
4                              N. P. STRICKLAND



  If p = 2 we do not have so much choice about the sequence of v's. Nonetheless*
 *, we know from [13,
Proposition 2.10] that there exist sequences of v's for which the rings BP <n>**
 *= BP *=(vi| i > n)
can be realised as the homotopy ring of a commutative ring object BP <n>, which*
 * is unique up
to canonical isomorphism (here we are writing BP <n>for the spectrum called BP *
 *<n>0in [13]).
Unfortunately, neither of the popular choices listed above work in this context*
 *.  We fix such
a sequence once and for all.  By the proof of [13, Theorem 2.13], we know that *
 *there is a
central BP <n>-algebra BPm<m; n>with homotopy ring BP <m; n>*= BP <n>*=Im and a*
 * deriva-
tion Qm-1 :BP <m; n>-! 2 -1BP <m; n>such that ab - ba = vm Qm-1(a)Qm-1(b).  It *
 *fol-
lows that when X is a space such that BP <m; n>*X is concentrated in even degre*
 *es, the oper-
ation Qn-1 is trivial on the ring BP <n; m>*X, and thus this ring is commutativ*
 *e. If BP <m; n>
and BP 0<m; n> are two such central BP <n>-algebras then either there is a uniq*
 *ue isomorphism
BP <m; n>' BP 0<m; n>, or there is a unique isomorphism BP <m; n>op' BP 0<m; n>.
  Next, recall from [13] that EKMM theory gives a map Qk:MU=vk -!2pk-1MU=vk in *
 *DMU
that is a derivation for any product on MU=vk. By smashing this over MU with BP*
 *=(vi | i <
m; i 6= k) we get a derivation Qk:P (m) -!2pk-1P (m). We can then smash this ov*
 *er MU with
MU=(vi| i > n) and then with MU=(vm ; : :;:vn) to get compatible derivations on*
 *nBP <m; n>and
HFp. It is also easy to see that there is a canonical map qn: BP <m; n - 1>-!2p*
 * -1BP <m; n>
and a ring map aen: BP <m; n>-!BP <m; n - 1>fitting into a cofibre sequence

           n-2        vn          aen            qn 2pn-1
         2p   BP <m; n>-!BP <m; n>-! BP <m; n - 1>-!     BP <m; n>


such that aenqn = Qn.
  We claim that our operation Qn on HFp is the same as the Milnor Bockstein ope*
 *ration QMnup
to sign. It is well-known that derivations in mod p cohomology are the same as *
 *primitive elements
in the Steenrod algebra HF*pHFp, and that the space of primitives in dimension *
 *2pn-1 is spanned
by QMn, so we have Qn = QMnfor some  2 Fp. We need to show that  = 1, so we nee*
 *d only
compute one nontrivial instance of Qn. We shall do this in the case p > 2. The *
 *case where p = 2
and n > 0 requires essentially only notational changes, and the case where p = *
 *2 and n = 0 can
be done with simple ad hoc arguments. Recall that HF*pBV1 = Fp[x]  E[a], where *
 *a is the usual
generator of HF1pBV1 and x = fia, which is the image of the usual generator of *
 *MU2CP 1 . It is
well-known that QMn(a) = xpn. Given this, our claim follows easily from Proposi*
 *tion 3.1 below.


Proposition 3.1.Let p be an odd prime, and let Cp be the cyclic group of p'th r*
 *oots of unity in
C, which we can identify with V1. Write

                                       n-1    2pn-1
                        X = cofibre(S2p   -! S     =Cp);


which is the 2pn-skeleton of BCp. Let qn be the Bockstein operation in the cofi*
 *bration


                   n-2      vn       ae          qn 2pn-1
                 2p   P (n) -! P (n) -!P (n + 1) -!      P (n):


Then there is a unique element b 2 P (n + 1)1X that hits the usual generator a *
 *2 HF1pX =
HF1pBCp.  Moreover, qnb = xpn, where x is the image in P (n)2BCp of the usual g*
 *enerator
x 2 MU2CP 1 .


  The sign ambiguity could be resolved by a careful analysis of conventions, wh*
 *ich we do not have
the patience to do.


                                                      __
Proof.For brevity, write m = pn - 1 and R = P (n) and R = P (n + 1), so we have*
 * a cofibration

                                     __ qn2m+1
                         2mR vn-!R ae-!R-!    R:
              THE BP<N>COHOMOLOGY OF ELEMENTARY ABELIAN GROUPS              5



We also write


                         P = CPm = S2m+1=S1

                          L= tautological bundle overP

                        P L= Thom space ofL ' CPm+1

                         Y = S2m+1=Cp

                         H = HFp:


Note that Y can also be thought of as the sphere bundle S(Lp) in the p'th tenso*
 *r power of L, or
as the (2m + 1)-skeleton of BV1 = S1 =Cp. Note also that


            H*X = P [x]  E[u]=(xm+2; axm+1) = Fp{1; a; x; : :;:axm ; xm+1}:

                                                                         __1
  An easy connectivity argument shows that there_is a unique element b 2 R X th*
 *at hits v 2
H1X. Indeed, let F be the fibre of the map R -!H, so that ss*F starts in dimens*
 *ion 2pn+1- 2.
The bottom cell of DX is in dimension -2pn, so the bottom cell of F ^ DX is in *
 *dimension
d = 2(p -_1)pn_- 2. This is strictly greater than_1 because p > 2 and n  0. It *
 *follows easily that

ss1(DX ^ R) = ss1(DX ^ H), or in other words R1X = H1X, so there is a unique el*
 *ement b as
described.
  We next consider the diagram


                                     P [

                                zLae     []zLp

                                 aeAE_____   p
                              P L    OE   PwL


Here zL and zLp are the zero-section inclusions, and OE is obtained in the obvi*
 *ous way from the
p'th power map of total spaces E(L) -! E(Lp). There is also a corresponding dia*
 *gram with P
replaced by CP 1. By applying R* to this, we get a diagram


                                   R*[[x]]
                               z*[]L    '*'z'*Lp
                              [ [
                        R*[[x]]uLu__________R*[[x]]uLp_OE*


Here uL and uLp are the Thom classes of L and Lp; the corresponding Euler class*
 *es are x and
[p](x). It is well-known that z*L(f(x)uL) = f(x)x and z*Lp(f(x)uLp) = f(x)[p](x*
 *), and it follows
easily (because <p>(x) = [p](x)=x is not a zero-divisor in R*[[x]]) that


                          OE*(f(x)uLp) = f(x)<p>(x)uL:


By mapping the CP mdiagram into the CP 1 diagram, we see that the same fomula h*
 *olds for the
map

                       p    *     m+1          *     m+1         * L
             OE*:R"*P L = (R [x]=x   ):uLp -!(R [x]=x   ):uL = "RP  :


In this context, however, we simply have <p>(x) = vnxm , so


                               OE*uLp = vnxm uL:


  We next apply the octahedral axiom to the maps S2m+1 -! Y -! P (which comes d*
 *own to
replacing the maps by cofibrations and using the fact that P=Y = (P=S2m+1)=(Y=S*
 *2m+1)). As
S2m+1 = S(L) and Y = S2m+1=Cp =pS(Lp), we see that the cofibres of S2m+1 -!P an*
 *d Y -! P
are homeomorphic to P Land P L. With these identifications,pthe map S2m+1 -! Y *
 *is the p'th
power map S(L) -!S(Lp), and thus the induced map P L-! P L is just OE. We there*
 *fore have an
6                              N. P. STRICKLAND



octahedral diagram



                               _____-P _____L
                                           J
                                      J]    J
                               oL c    J zL  J
                        r               J      J OE
                               AE   ss   J      J
                            S2m+1_________-PL
                                                 J
                            OE J         OE J     J
                        oLpc   OEJ       ss p JzLp JJ
                     J            J        L   J
                     J             J^           J^ AE
                       X ___________oe___________oecYP Lp
                             j              e



(A circled arrow U -!O  V means a map U -!V . The diagram can be made to look m*
 *ore like an
octahedron by liftingpup the outer three vertices and drawing in an extra arrow*
 * to represent the
composite je: P L -!O  X.) In particular, we seepthat the stable fibre of OE is*
 * just X.
  We next claim that the map e*:He1Y -! eH2P L is an isomorphism, and sends a t*
 *o the Thom
class uLp. To see this, let M be the restriction of L to the basepoint in P , s*
 *o M is just a one-
dimensional complex vectorpspace. The bottom cell of Y = S(Lp) is just thepcirc*
 *le S(Mp), and
the bottom cell of P L is the one-point compactification ofpMp, written SM  . T*
 *he restriction of
e to this bottom cell is the standard homeomorphism of SM  with the unreduced s*
 *uspension of
S(Mp), followed by the projection to the reduced suspension. The claim follows *
 *easily from this.
  We now consider the following diagram.

                       _____OE   p  _____-1je  ______-1r L
                -2P L       -2PwL        w X         w P
                   |           |            |           |
               xmu |        uLp|            |||c        |xmu
                  L|           |            ||          |   L
                   |u ________ |u____________u______ 2m+1|u
                 2mR     vn   wR     ae    Rw   qn  w    R

We see from the octahedron that the top line is a cofibration, and the bottom l*
 *ine is a cofibration
by construction. The left_hand square_commutes because OE*uLp = vnxm uL. It fol*
 *lows that there

exists a map c: -1X -!R (i.e. c 2 R1X) making the whole diagram commute.
  From our discussion of e* in cohomology and the commutativity of the middle s*
 *quare, we see
that the image of c in H1X is just a. By the uniqueness of b, we deduce that c *
 *= b. Thus, the
commutativity of the right hand square tells us that qnb is the image of xm uL *
 *2 R2m+2P Lin
R*X. Under the usual identification of P Lwith CP m+1, the element xm uL become*
 *s xm+1, and
the map X -!CP m+1 is a restriction of the usual map BCp -!CP 1. It follows tha*
 *t_qnb = xm+1
as claimed.                                                                   |*
 *__|


                           4.Commutative algebra

  In this section, we recall some basic ideas from commutative algebra.
  We will need to be a little more careful than is usual about the relationship*
 * between graded
and ungraded rings. For us, a graded ring R* will mean a sequence of Abelian gr*
 *oups Rk (for
k 2 Z) with product maps Ri Rj -! Ri+jwith the usual properties.  We will assum*
 *e that
Rk = 0 when k is odd, and that the product is commutative. This will apply to a*
 *ll rings that
we consider except for H*(BVk; Fp), and in that case we will not need to use th*
 *eLresults of
this section.  It is common to identify a graded ring R* with the ungraded ring*
 *   kRk.  We
shall not do this, for the following reason. There is an obvious way to interpr*
 *et the expression
R* = Z(p)[v1; : :;:vn][[x0; :a:;:xk-1]]s aPgraded ring (with |vk| = -2(pk - 1) *
 *and |xi| = 2).
Explicitly,PRk is the set of expressions   ffaffxff, where ff = (ff0; : :;:ffk-*
 *1)Lis a multiindex,

|ff| =   iffi and aff2 BP <n>-2|ff|. With this interpretation, the ring   kRk i*
 *s not the same as
              THE BP<N>COHOMOLOGY OF ELEMENTARY ABELIAN GROUPS              7


                                                                            P
the ungradedQring R0= Z(p)[v1; : :;:vn][[x0; :(:;:xk-1]]it does not contain   i*
 *0xi0, for example).
The set   kRk isLdifferent again. It is not clear how many of the nice ring-the*
 *oretic properties of
R0are shared by   kRk. For this reason, we prefer to work with graded rings as *
 *defined above.
Most theorems for ungraded rings have graded counterparts, which are proved by *
 *a straightforward
adaptation of the ungraded proofs. We will outline the results that we need, le*
 *aving most of the
task of adaptation to the reader.
  We will only consider homogeneous elements of R*, in other words elements of *
 *Rk for various
k. The word "module" will always mean "graded module", and similarly for ideals.
  Given an ideal I* in R*, let (IN )k denote the part of the N'th power of I in*
 * degree k. The cosets
a + (IN )k (for a 2 Rk) form a basis for a topology on Rk. We say that a topolo*
 *gy of this form is a
linear topology on R*, and we say that I* is an ideal of definition. If I* is o*
 *ne ideal of definition, it
is clear that another ideal J* is an ideal of definition for the same topology *
 *if I* contains a power
of J* and vice versa. We say that a linear topology is complete if Rk = lim -Rk*
 *=(IN )k for all k.
                                                                          N
We say that a homogeneous element is topologically nilpotent if some power of i*
 *t lies in I*. In the
Noetherian case, it is not hard to see that the topologically nilpotent element*
 *s form an ideal of
definition.
  If R* is a quotient of E*[[x0; : :;:xk-1]]then we give R* the complete linear*
 * topology de-
finedLby the ideal (x0; : :;:xk-1). InLthis context, we can consider the ungrad*
 *ed ring Tot(R*) =
lim - k(R=IN )k as a substitute for   kRk. This avoids the difficulty mentioned*
 * above: if
  N

                        R* = Fp[vm ; : :;:vn][[x0; : :;:xk-1]]

(in the usual graded sense) then

                      Tot(R*) = Fp[vm ; : :;:vn][[x0; : :;:xk-1]]

(in the usual ungraded sense). However, the relationship between properties of *
 *R* and those of
Tot(R*) is not as close as one might like, so we will not stress this point of *
 *view.
  We will need the following version of the Weierstrass preparation theorem.
                                            P
Proposition 4.1.Let R* be as above. Let y =   k0 akxk 2 R*[[x]]be a homogeneous*
 * element
such that ai is topologically nilpotent for i < d, and ad is a unit. Then R*[[x*
 *]]is freely generated
by {xi| i < d} as a module over R*[[y]], and thus R*[[x]]=y is freely generated*
 * by {xi| i < d} as a
module over R*.

Proof.We may assume that ad = 1. For any m  0, we can write m = ld + k with l  *
 *0 and
0  k < d, and then write wm = xkyl.  Let I* be the ideal of topological nilpote*
 *nts, so that
wm = xm (mod I*; xm+1). For any R*-module M* that is complete with respect to I*
 **, we can
define a map
                                 Y
                             M :    M* -!M*[[x]]
                                m0
           P                                              P
by M (b) =   mbm wm . Suppose that I*M* = 0, and that c =   m0 cm xm 2 M*[[x]].*
 * Given the
formPof wm mod I*, we see easily by induction that there is a unique sequence o*
 *f bm 's such that
c =   mk=0bm wm (mod xm+1) and thus that M  is an isomorphism. Moreover, if we *
 *have a pair
of modules N*  M* such that N and M=N are isomorphisms, then so is M (by a five*
 *-lemma
argument). It follows by induction that M=Ik is iso for all k, and thus by taki*
 *ng inverse limits
that M  is iso for all M*. In particular, R is an isomorphism. This means that *
 *for anyPseries
c(x) 2 R*[[x]], there are unique series b0(y); : :;:bd-1(y) 2 R*[[y]]such that *
 *c(x)_=   ibi(y)xi,
which proves the proposition.                                                  *
 *  |__|


  We say that a graded ring R* is local if it has only one maximal ideal, or eq*
 *uivalently if it has
an ideal such that every homogeneous element in the complement is invertible. I*
 *t is clear that
E* is local in this sense, with maximal ideal (vm ; : :;:vn).  Similarly, any q*
 *uotient of the ring
E*[[x0; : :;:xk-1]]is local, with maximal ideal (vm ; : :;:vn; x0; : :;:xk-1).
8                              N. P. STRICKLAND



  We say that a graded ring R* is Noetherian if every ideal is generated by a f*
 *inite set of homoge-
neous elements. Simple adaptations of the usual arguments in an ungraded contex*
 *t [9, Theorem
3.3] show that any quotient of E*[[x0; : :;:xk-1]]is Noetherian.
  We say that a graded ring R* is a domain if the product of two nonzero homoge*
 *neous elements
of R* is nonzero. This will hold if Tot(R*) is an ungraded domain, but unfortun*
 *ately we have not
been able to prove the converse. We say that an ideal P *in R* is prime if R*=P*
 * *is a domain,
and define the Krull dimension of R* to be the largest integer d such that ther*
 *e exists a chain
P0*< : :<:Pd*of prime ideals in R*.
  We say that a Noetherian graded local ring R* of Krull dimension d is regular*
 * if there is a
sequence of d homogeneous elements that generates the maximal ideal. Such a seq*
 *uence is called
a regular system of parameters; it is necessarily a regular sequence.  Simple a*
 *daptations of the
usual arguments in an ungraded context [9, Theorem 15.4] show that R* = E*[[x0;*
 * : :;:xk-1]]has
dimension w + k, so the sequence {vm ; : :;:vn; x0; : :;:xk-1} is a regular seq*
 *uence of parameters
and R* is regular. More generally, if R* is any graded regular local ring and x*
 * is a homogeneous
indeterminate then R*[[x]]is again a graded regular local ring.
  By graded versions of [9, Theorems 14.3 and 20.3], we see that a graded regul*
 *ar local ring is an
integral domain, with unique factorisation for homogeneous elements.
  The following result is a graded version of [9, Theorem 14.2], and can be pro*
 *ved in the same
way.

Theorem 4.2.Let R* be a graded regular local ring of dimension d, and {x0; : :;*
 *:xk-1} a se-
quence of homogeneous elements of R*. Then the following are equivalent:

 (a){x0; : :;:xk-1} is a subset of a regular system of parameters.
 (b)The images of {x0; : :;:xk-1} in m=m2 (where m is the unique maximal ideal *
 *of R*) are
    linearly independent over the graded field R*=m.
 (c)R*=(x0; : :;:xk-1) is a graded regular local ring of dimension d - k.

  We can also show that two weaker conditions are equivalent to each other.

Theorem 4.3.Let R* be a graded regular local ring of dimension d, and {x0; : :;*
 *:xk-1} a se-
quence of homogeneous elements of R*. Then the following are equivalent:

 (a){x0; : :;:xk-1} is a regular sequence.
 (b)R*=(x0; : :;:xk-1) has Krull dimension d - k.

Proof.This is a graded version of the equivalence (1),(3) in part (iii) of [9, *
 *Theorem 17.4], taking
into account the equation ht(I*) = dim(R*) - dim(R*=I*) from part (i) of that t*
 *heorem, and_the
fact that regular local rings are Cohen-Macaulay.                              *
 *       |__|



                         5. The structure of E*BVk

Proposition 5.1.Let R* be an algebra over E* of chromatic depth d > 0. Then [p]*
 *(x) is not a
zero-divisor in R*[[x]], and the ring R*[[x]]=[p](x) has chromatic depth at lea*
 *st d - 1.

Proof.Suppose that 0 6= f(x) 2 R*[[x]], say f(x) = axk (mod xk+1) with 0m6= a 2*
 * R*. AsmR* has
nonzero chromatic depth, we see that vm a 6= 0 and [p](x)f(x) = vm axk+p  (mod *
 *x1+k+p ) so
[p](x)f(x) 6= 0. This shows that [p](x) is not a zero-divisor in R*[[x]].
  Now suppose that d > 1, so that {vm ; vm+1} is regular on R*. Consider a seri*
 *es f(x) 2 R*[[x]]
such that vm f(x) is divisible by [p](x). We claim that f(x) is divisible by [p*
 *](x). Indeed, suppose
that vm f(x) = [p](x)g(x). We then have [p](x)g(x) = 0 in (R*=vm )[[x]]. Howeve*
 *r, vm+1 is not a
zero divisor in R*=vm , so [p](x) is not a zero divisor in (R*=vm )[[x]]by the *
 *previous paragraph. Thus
g(x) = 0 in R*=vm [[x]], or g(x) = vm h(x) in R*[[x]]say. This means that vm (f*
 *(x) - [p](x)h(x)) = 0
in R*[[x]], but vm is not a zero-divisor so f(x) = [p](x)h(x) as claimed. The c*
 *onclusion is that vm
is regular on R*[[x]]=[p](x). If d > 2 then we can replace R* by R*=vm and m by*
 * m + 1 and d by
d - 1 and run the same argument again. It follows by induction that R*[[x]]=[p]*
 *(x)_has_chromatic
depth at least d - 1.                                                          *
 *  |__|
              THE BP<N>COHOMOLOGY OF ELEMENTARY ABELIAN GROUPS              9



Corollary 5.2.Let R* be an algebra over E* of chromatic depth d > 0. Then for j*
 *  d and k  j
the sequence [p](x0); : :;:[p](xj-1) is regular in R*[[x0; : :;:xk-1]]and the q*
 *uotient P (R*; j; k) has
chromatic depth at least d - j.

Proof.This follows easily from the proposition, using the obvious fact that S*[*
 *[x]]has the_same
chromatic depth as S* for any algebra S* over BP <m; n>*.                      *
 *       |__|

Corollary 5.3.For k  w we have E*BVk = P (E*; k), and this has chromatic depth *
 *at least
w - k.

Proof.Write B = BZ=p and Z = CP 1, so E*Z = E*[[x]]and B is a circle bundle ove*
 *r Z with
Chern class [p](x). Thus Bix Zk-iis a circle bundle over Bi-1x Zk+1-iwith Chern*
 * class [p](xi).
This gives a long exact Gysin sequence

     . .-.E*(Bix Zk-i) - E*(Bi-1x Zk+1-i)  [p](xi)----E*-2(Bi-1x Zk+1-i) - . .:.

It follows easily by induction from Corollary 5.2 that these sequences are shor*
 *t exact, and_that
E*(Bix Zk-i) = P (E*; i; k). The case i = k gives the corollary.               *
 *          |__|

Definition 5.4.If k  w, we define A(k)* = A(m;Qn; k)* to be the largestmquotien*
 *t ring of
R* = P (E*; k) over which the series OEk(t) =   _2Fkp(t -F [__](x_))p  divides *
 *[p](t). We also write

 k(t) = [p](t)=OEk(t).
  In more detail, we note that t-F[__](x_) is a unit multiple of t-[__](x_), so*
 * OEk(t) is a unit multiple of
a monic polynomial of degree pm+k , whose lower coefficients are topologically *
 *nilpotent. It followsm+k
from Proposition 4.1 that R*[[t]]=OEk(t) is a free module over R* on generators*
 * 1; t; : :;:tp   -1.
                                     P pm+k-1
In particular, we can write [p](t) =   i=0   citi (mod [p](t)), for uniquely de*
 *fined coefficients
ci 2 R*.  We define A(k)* to be the quotient ring R*=(c0; : :;:cpm+k-1).  It al*
 *so follows from
Proposition 4.1 that OEk(t) is not a zero-divisor in A(k)*[[t]], so there is a *
 *unique series  k(t) 2
A(k)*[[t]]such that [p](t) = OEk(t) k(t).

Theorem 5.5.For k  w we have

 (a)A(k)* is a regular local ring in the graded sense.
 (b)There is a unique formal group law Fk over A(k)* for which OEk(t) is a homo*
 *morphism
    F -! Fk.                         _
 (c)There is a unique series k(t) = t k(t) over A(k)* such that [p]F(t) = k(OEk*
 *(t)).
 (d)If k > 0, we have A(k)* = A(k - 1)*[[xk-1]]= k-1(xk-1).

Moreover, when k = w we have A(w)* = P (Fp; w) = Fp[[x0; : :;:xw-1]]and In+1 = *
 *0 in A(w)*.

Proof.Suppose that k < w and that (a): :(:d) hold up to stage k (which is trivi*
 *al for k = 0).
Define B* = A(k)*[[xk]]= k(xk).
  We first claim that  k(xk) = 0 in A(k + 1)*. Indeed, we have
                                          p-1Y
                        OEk+1(t) = OEk(t)   OEk(t -F [i](xk));
                                          i=1
and this divides [p](t) = OEk(t) k(t) over A(k + 1)*.  As OEk(t) is a unit mult*
 *ipleQof a monic
polynomial, it is not a zero divisor in A(k + 1)*[[t]], so we conclude that   p*
 *-1i=1OEk(t -F [i](xk))
divides  k(t) in A(k + 1)*[[t]]. We now set t = xk to conclude that  k(xk) = 0 *
 *in A(k + 1)* as
claimed.
  It is also clear that OEk(t) divides OEk+1(t), so it divides [p](t) over A(k *
 *+ 1)*, so A(k + 1)* is an
algebra over A(k)*. This means that A(k + 1)* is a quotient of B* in a natural *
 *way.
  Next, we claim that B* is a regular local ring in the graded sense. For this,*
 * we write C* =
E*[[x0; : :;:xk]].  This is clearly a regular local ring, and B* = C*=( 0(x0); *
 *: :;: k(xk)).  By
Theorem 4.2 it suffices to check that the list

                 L = { 0(x0); : :;: k(xk); x0; : :;:xk; vm+k+1; : :;:vn}

is a regular system of parameters for C*. The length of L is n + k - m + 2 whic*
 *h is the same
as the Krull dimension of C*, so it is enough to check that L generates the max*
 *imal ideal of
10                             N. P. STRICKLAND


                         __*                                                   *
 *          __*
C*, or equivalently that C  = C*=(L) = Fp, or equivalently_that vm ; : :;:vm+k *
 *vanish in C , or

equivalently that [p](t) is divisible by_tpm+k+1over_C*. However, we know that *
 *C*_is_an algebra
over A(k)* so [p](t) = OEk(t) k(t) over_C*. Moreover, as x0; : :;:xk-1 vanish i*
 *n C* we see_from_
the definition that OEk(t)_= tpk+m in C *[[t]].  This means that [p](t) = tpk+m*
 * k(t) over C *, so
v0 = : :=:vk+m-1 = 0 in C* and vk+m =  k(0).  On the other hand, we have xk = 0*
 * and
 k(xk) = 0 so  k(0) = 0 so vk+m = 0 as required. This completes the proof that *
 *B* is a regular
local ring.
  It is clear from the above that {x0; : :;:xk; vm+k+1; : :;:vn} is a regular s*
 *ystem of parameters
for B*, and thus that the Krull dimension of B* is w. As B* is an integral doma*
 *in, it follows that
any proper quotient of B* has dimension strictly less than w. Thus, if we can s*
 *how that A(k + 1)*
has dimension w, we can deduce that the quotient map B* i A(k + 1)* is an isomo*
 *rphism.
  For the rest of the argument we need to separate the cases k+1 = w and k+1 < *
 *w. If k+1 = w
we argue as follows. It follows from the definition of A(w)* that A(w)*=In+1 is*
 * the largest quotient
of E*[[x0; : :;:xw-1]]=In+1 = P (Fp; w) over which OEw(t) divides [p](t). Howev*
 *er, in this context
[p](t) = 0 which is automatically divisible by OEw(t), so A(w)*=In+1 = P (Fp; w*
 *). This has the same
Krull dimension as B*, so we must have B* = A(w)* = A(w)*=In+1. In particular, *
 *we see that
In+1 = 0 in A(w)*, and thus that the formal group law F becomes F (s; t) = s + *
 *t in A(w)*[[s;.t]]
It follows that parts (b) and (c) of the theorem hold with Fw = F and w = 0.
  We now consider the case k + 1 < w, so that k < w - 1. For this, we use the t*
 *heory of multiple
level structures developed in [4, Section 5]; we will assume that the reader is*
 * familiar with this.
  Let bE*be the graded ring (E*[u; u-1]=(upn-1- vn))^In.  Write uk = vku1-pk 2 *
 *Eb0.  It is

easy to see that bE0= Fp[[um ; : :;:un-1]], or Zp[[u1; : :;:un-1]]if m = 0.  Mo*
 *reover, we have
bE*= bE0[u; u-1].

  Let F be the usual formal group law over E*, and define bF(s; t) = uF (s=u; t*
 *=u), so that bFis
a formal group law over bE0. Let G be the associated formal group over X = spf(*
 *Eb0), which has
height n and strict height m. This puts us in the context studied in [4].     n
  Now write J = In+(x0; : :;:xk)  A(k+1)*, and bA(k+1)* = (A(k+1)*[u; u-1]=(up *
 *-1-vn))^J.
It is clear from the definitions that bA(k+1)0 is just the ring OLevelm(Vk*+1;G*
 *)which classifies pm -fold

level-Vk*+1structures on G, and bA(k + 1)* = bA(k + 1)0[u; u-1]. We know from [*
 *4, Theorem 5.6]

that bA(k + 1)0 is an integral domain and a finitely generated free module over*
 * bE0. It is nonzero
because k + 1  n - m. It follows that bA(k + 1)* is a graded domain and a finit*
 *ely generated free
module on homogeneous generators over bE*.
  We now have a diagram as follows.


                             E*  _____A(kw+i1)*
                               |          |
                               |          |
                             c |          |d
                               |          |
                               |          |
                               |u         |u
                             Eb* _____Ab(kw+^1)*:-


For m  t  n we let bItbe the ideal in bE*generated by {vm ; : :;:vt-1}. This is*
 * clearly prime,
with bIt\E* = Itand 0 = Im < : :<:In. As bA(k+1)* is integral over bE*, the goi*
 *ng-up theorem [9,
Theorem 9.3] gives us a chain 0 = bJm< : :<:bJnof primes in bA(k+1)* such that *
 *bJt\Eb*= bIt. Let
Jtbe the preimage of bJtunder the map d: A(k + 1)* -!Ab(k + 1)*. It is clear th*
 *at the preimage of
this under the map i: E* -!A(k + 1)* is just It, so we have a chain of strict i*
 *nclusions of prime
ideals 0 = Jm < : :<:Jn. Let Jn+1 be the maximal ideal in A(k +1)*. As vn is a *
 *unit in bA(k +1)*
it is clear that vn 62 Jn, but vn 2 Jn+1 so we have a chain of strict inclusion*
 *s 0 = Jm < : :<:Jn+1.
This shows that A(k + 1)* has Krull dimension w, so B* = A(k + 1)* as explained*
 * earlier.
  We now define ^xi= uxi2 bA(k + 1)0, and for __2 Fk+1pwe define

                      [__](^x) = [0]Fb(^x0) +Fb. .+.bF[k]Fb(^xk):
              THE BP<N>COHOMOLOGY OF ELEMENTARY ABELIAN GROUPS             11



We also write            Y
             ^OEk+1(t) =   (t -Fb[__](^x))pm = upm+kOEk+1(t=u) 2 bA(k + 1)0[[t]*
 *]:

                         _

Clearly, the map __7! [__](x_) defines a pm -fold level-Vk*+1structure on G ove*
 *r spf(Ab(k + 1)0). If we
compose this with the projection Vm*xVk*+1-!Vk*+1we get an ordinary (1-fold) le*
 *vel-(Vm xVk+1)*

structure. It follows from [14, Proposition 32 and Corollary 33] that ^OEk+1(t)*
 * is a coordinate on a
quotient formal group of G, and that the kernel of the quotient map is containe*
 *d in the kernel of
p: G -!G. This means that there is a unique formal group law bFk+1defined over *
 *bA(k + 1)0 such
that ^OEk+1is a homomorphism bF-!Fbk+1, and that there is a unique power series*
 * ^k+1(t) defined
over bA(k + 1)0 such that [p]Fb(t) = ^k+1(^OEk+1(t)). We define Fk+1(s; t) = u-*
 *pkbFk+1(upks; upkt)

and k(t) = u-1^k+1(upkt). It is easy to deduce that Fk+1 is the unique formal g*
 *roup law over
bA(k + 1)* such that OEk+1 is a homomorphism F -! Fk+1, and that k+1 is the uni*
 *que series such

that [p]F(t) = k+1(OEk+1(t)).
  We next remark that A(k + 1)* is a Noetherian domain, so it is not hard to se*
 *e that the maps

              A(k + 1)* -!v-1nA(k + 1)* -!(v-1nA(k + 1)*)^J= bA(k + 1)*

are injective.
  Now write s0= OEk+1(s) and t0= OEk+1(t). We know from Proposition 4.1 that A(*
 *k + 1)*[[s; t]]
is a free module over the subringPA(k + 1)*[[s0;ot0]]n generators sitj for 0  i*
 *; j < pk+1.  We
can thus write OEk+1(F (s; t)) =   i;jFi;j(s0; t0)sitj for uniquely determined *
 *series Fi;j. Similarly,
bA(k + 1)*[[s;it]]s a free module over bA(k + 1)*[[s0;ot0]]n {sitj}, so the equ*
 *ation OEk+1(F (s; t)) =
P         0   0  i j                                                           *
 *             i j
 i;jFi;j(s ; t )s t  is the unique way to write OEk+1(F (s; t)) in terms of the*
 * generators s t . On the
other hand, we also have OEk+1(F (s; t)) = Fk+1(s0; t0) 2 bA(k + 1)*[[s;.t]]It *
 *follows that Fi;j= 0
for (i; j) 6= (0; 0) and that Fk = F0;0, so the series Fk is actually defined o*
 *ver A(k + 1)* rather
than bA(k + 1)*. It is clearly the unique formal group law over A(k + 1)* for w*
 *hich the series OEk+1
is a homomorphism F -! Fk. Similarly, we see that k+1(t) is defined over A(k + *
 *1)*, and it is
the unique series over A(k + 1)* for which [p](t) =_k+1(OEk+1(t)). By putting t*
 * = 0 we see that   __
k+1(t) is divisible by t, so it can be written as t k+1(t). This completes our *
 *induction step.   |__|

Proposition 5.6.The ring A(k)* has chromatic depth at least w - k.

Proof.The claim is that the sequence {vm ; : :;:vn-k} (of length w - k) is regu*
 *lar on A(k)*. By
Theorem 4.3, it is enough to check that the quotient A(k)*=(vm ; : :;:vn-k) has*
 * dimension at
most dim(A(k)*) - (w - k) = k.  Let B* be a graded integral domain which is a q*
 *uotient of
A(k)*=(vm ; : :;:vn-k). It is enough to show that every such B* has dimension a*
 *t most k, and
thus enough to show that the maximal ideal m of B* needs at most k generators. *
 *From now on
we work in B*.
  Write W = {__2 Fkp| [__](x_) = 0}. This is a subgroup of Vk*, of dimension d *
 *say. Note that
Aut(Fkp) = GLk(Fp) acts on A(k)* in a natural way. After applying a suitable el*
 *ement of this
group, we may assume that W is the evident copy of Fdpin Fkp, spanned by the fi*
 *rst d standard
basis vectors, so that x0 = : :=:xd-1 = 0 in B.  We write U for_the spaceQspann*
 *ed by the
remaining standard basis vectors, so that Fkp= W  U. Define OE(t) =  _2U (t -F *
 *[__](x_)), so that
         __    m+d
OEk(t) = OE(t)p   . As B* is an integral domain, we see that
                             __0      Y
                             OE(0) =     [__](x_) 6= 0
                                     _2U\0

and thus that ordtOEk(t) = pm+d, where ordtf(t) means the largest integer N suc*
 *h that tN divides
f(t).
  As v0 = : :=:vn-k_=_0 in B* we see that ordt[p](t)  pn+1-k. On the other hand*
 *, we have
ordt[p](t) = ordtk(OE(t)pm+d) = pm+d ordsk(s). Thus ordsk(s)  pw-d-k.
  Now consider the list

                        L = {xd; : :;:xk-1; vn+1-d; : :;:vn};
12                             N. P. STRICKLAND


                                                       __
so that L has length_k._It will be enough to show_that B = B=(L) = Fp, or equiv*
 *alently that
x0 = : :=:xk = 0 in B and v0 = : :=:vn = 0 in B. We already_have x0 = : :=:xd-1*
 * = 0 in
B and the remaining_x's_are in L so all x's vanish in B, as required. We next n*
 *ote that OEk(t)
becomes tpm+k over B, and [p](t) = k(OEk(t)),_and ords(k(s))  pw-d-k so [p](t) *
 *has height_at
least m + k + w - d - k = n + 1 - d over B._This means that v0 = : :=:vn-d = 0 *
 *over B and the
remaining v's are in L so they vanish in B also. Thus (L) = m, and m needs only*
 *_k_generators, as
required.                                                                     |*
 *__|

  We leave the proof of the following simple lemmas to the reader.

Lemma 5.7. Let R* be a ring, and let OE;   be elements of R* such that OE is no*
 *t a zero-divisor.
Then the annihilator of OE in R*=OE  is generated by  , and thus the ideal gene*
 *rated_by_OE in R*=OE 
is a free module of rank one over R*= .                                        *
 *    |__|

Lemma 5.8. For any elements OE and O in any ring R*, the annihilator ann(OEO; R*
 **) in R* is_the
preimage of ann(OE; R*= ann(O)) under the quotient map R* -!R*= ann(O).        *
 *        |__|
                                                Q
Definition 5.9.We_write OEifor OEi(xi) and Oj =  i<jOEi2 E*BVj  E*BVw. We also *
 *write  i

for  i(xi) =  i(OEi(xi)), so that OEi i= [p](xi). We also have E*BVk=( i| i < j*
 *) = P (A(j)*; j; k)
by Theorem 5.5(d).

Proposition 5.10.If j  k  w then the annihilator of Oj in E*BVk is precisely th*
 *e ideal
( 0; : :;: j-1). Thus, the ideal generated by Oj is a free module of rank one o*
 *ver P (A(j)*; j; k).

Proof.The proposition is trivial when j = 0 (so Oj = 1). We may thus assume the*
 * statement for Oj
and prove the one for Oj+1. As Oj+1= OEjOj, we see that the annihilator of Oj+1*
 *is the preimage of
the annihilator of OEj in S* = E*BVk= ann(Oj) = P (A(j)*; j; k). We write R* = *
 *P (A(j)*; j + 1; k),
so that S* = R*[[xj]]=OEj j. It follows from Lemma 5.7 that the annihilator of *
 *OEj in S* is just the
ideal generated by  j. Thus, we have

                    ann(Oj+1; E*BVk)= ann(Oj; E*BVk) + ( j)

                                   = ( 0; : :;: j-1) + ( j)

                                   = ( 0; : :;: j);

as required.                                                                  |*
 *___|

  The following corollary is immediate.

Corollary 5.11.If 0  j < k  w then the quotient (Oj)=(Oj+1) is isomorphic as a *
 *module over
E*BVk to P (A(j)*; j + 1; k)[[xj]]=OEj, which is (by Proposition 4.1) a free mo*
 *dule_of rank pmj over
P (A(j)*; j + 1; k).                                                           *
 *  |__|

Proposition 5.12.The ideal of vm -torsion elements in E*BVw is a free module ov*
 *er the ring
E*BVw=In+1 = Fp[[x0; : :;:xw-1]], generated by ff = Ow.  The map ann(vm ) ae E**
 *BVw i
E*BVw=In+1 is injective.

Proof.Take k = w in Corollary 5.11. We know from Proposition 5.6 that A(j)* has*
 * chromatic
depth at least w-j, and it follows from Corollary 5.2 that P (A(j)*; j+1; w) ha*
 *s chromatic depth at
least (w-j)-(w-(j +1)) = 1, so vm is regular on (Oj)=(Oj+1). It follows easily *
 *that vm is regular
on E*BVw=(Ow), and thus that the vm -torsion is contained in the ideal (Ow).  W*
 *e know from
Proposition 5.10 that the annihilator of Ow is the same as the kernel of the ma*
 *p E*BVw i A(w)*.
We also know from Theorem 5.5 that A(w)* = E*BVw=In+1 = Fp[[x0; : :;:xw-1]].  I*
 *t follows
immediately that the vm -torsion is precisely the ideal generated by Ow and tha*
 *t it is a free module
over E*BVw=In+1. It is easy to see from the definitions that Ow = ff. Moreover,*
 * modulo In+1 we
see that Ow is just the product of all linear polynomials of the form 0x0+ : :+*
 *:j-1xj-1+ xj for
some j < w. In particular, it is nonzero. As E*BVw=In+1 is an integral domain, *
 *it follows_easily
that the map (E*BVw=In+1):Ow ae E*BVw i E*BVw=In+1 is injective.               *
 *   |__|

  Our proof of Theorem 2.8 relies on some facts from the classical theory of Di*
 *ckson invariants [16].
For ease of reference, we give a swift proof of what we need.
              THE BP<N>COHOMOLOGY OF ELEMENTARY ABELIAN GROUPS             13



Proposition 5.13.In H*(BVk; Fp) = E[a0; : :;:ak-1]  Fp[x0; : :;:xk-1] we define
                                    Y X
                               fi =      ixi;
                                    u  i

where the product runs over nonzero sequences (0; : :;:k-1) 2 Fkpwhose last non*
 *zero entry is
one. We also define
                                       j
                         fi0= deti;j(xpi)


                        fi00m= Qm+k-1 : :Q:m(a0a1: :a:k-1)

Then fi00m= (fi0)pm = fipm.

Proof.Using column operations and the fact that (u + v)pj= upj+ vpj (mod p), we*
 * see easily
that g*fi0= det(g)fi0for all g 2 Aut(Vk). It is also clear that the element a0a*
 *1: :a:k-1transforms
in the same way, and thus that g*fi00m= det(g)fi00mfor all g and m.  It is imme*
 *diate from the
definition that fi0 lies in Fp[x0; : :;:xk-1] and that it is divisible by x0. I*
 *t is well-known that Qi
                                i
is a derivation with Qi(aj) = xpjand Qi(xj) = 0, and from this it follows that *
 *fi000is a sum of
                   Q    poe(i)
terms of the form   i<kxi   for various permutations oe. In particular, we see *
 *that fi000lies in
Fp[x0; : :;:xk-1] and that it is divisible by x0. As fi0and fi000arePdivisible *
 *by x0 and invariant under
SLk(Fp), we see that they are both divisible by each of the terms   iixiin the *
 *product formula
for fi. As these terms are inequivalent irreducibles and Fp[x0; : :;:xk-1] has *
 *unique factorisation,
we see that fi000and fi0are divisible by fi. It is easy to check that fi000=fi *
 *and fi0=fi have degree zero,
                                                              Q     pi
so they lie in Fp. By comparing coefficients for the monomial   i<kxi , we see *
 *that fi = fi0= fi000.
  Finally, we can define an (ungraded) ring map F :H*(BVk; Fp) -!H*(BVk; Fp) by*
 * F (ai) = ai m
and F (xi) = xpi. We then have F O Qi = Qi+1O F , from which it follows easily *
 *that (fi000)p__=
F mfi000= fi00m, which completes the proof of the proposition.                 *
 *            |__|

Proof of Theorem 2.8.Given Proposition 5.12, all that is left is to prove that *
 *ff = ff0= ff00.
  We first show that viff0= viff00= 0 for i = m; : :;:n.
  Recall that ff0is the determinant ofPthe matrix with entries ssi(xj) forPm  i*
 *  n and 0  j < w.
As we work modulo Im we have [p](t) =   ni=mvissi(t), so vissi(xj) = -  nk6=ivk*
 *ssk(xj) for all j.
This shows that when we multiply the i'th row of our matrix by vi, it becomes a*
 * linear combination
of the other rows, so the determinant becomes zero. It follows that viannihilat*
 *es the determinant
of the original matrix, in other words viff0= 0.
  Next, chooseielements vi 2 ss*MU lifting our elements vi 2 ss*BP .  We then h*
 *ave natural
maps vi:2p -2M -! M for all M 2 DMU .  Working in DMU  we see that viqi = 0 and*
 * thus
viqn : :q:m+1qm = 0. After applying the forgetful functor to the category of sp*
 *ectra, we conclude
that viff00= 0.
  In view of Proposition 5.12, it is now enough to check that ff, ff0and ff00ha*
 *ve the same image
modulo In+1, or equivalently the same imagemin H*(BVw;mFp). One can check from *
 *the definitions
that these are the same as the elements fip , (fi0)p  and fi00mof Proposition 5*
 *.13,_so they are all
the same, as required.                                                         *
 * |__|

  We conclude with a finer filtration of E*BVw.

Proposition 5.14.The ring E*BVw admits a finite filtration by ideals, whose quo*
 *tients are
finitely generated free modules over regular local rings of dimension w.

  This could be made more explicit but the bookkeeping would be tedious.

Proof.We actually prove that the ring P (A(i)*; j; k) admits such a filtration *
 *whenever i  j
k  w.  The case when i = j = 0 and k = w gives the proposition.  When j = k we *
 *have
P (A(i)*; j; k) = A(i)* so the claim follows from Theorem 5.5, so we work by in*
 *duction on k-j. We
have P (A(i)*; j; k) ' P (A(i)*; i; k+i-j), so we may assume that j = i. Note t*
 *hat P (A(i)*; i; k) =
P (A(i)*; i + 1;=k)[[xi]]OEi(xi) i(xi), so we can apply Lemma 5.7. This gives u*
 *s a two stage filtration
14                             N. P. STRICKLAND



of P (A(i)*; i; k) in which one quotient is P (A(i + 1)*; i + 1; k) and the oth*
 *er is a finitely_generated
free module over P (A(i)*; i + 1; k). The proposition follows by induction.    *
 *              |__|

  In the case where n = 1 and m = 0, the reduced E-cohomology of BV1 is a free *
 *module of
rank one over A(1)* = E*[[x]]=<p>(x), and we have A(2)* = Fp[[x0; x1]]= HF*p(CP*
 * 1 x CP 1).
Ossa has shown that E ^ BV2+ splits as an E-module as a wedge of copies of E, E*
 * ^ BV1 and
HFp ^ (CP 1 x CP 1)+. (He actually works with the connective K-theory spectrum *
 *kU, but it
is well-known that this splits p-locally as a wedge of copies of E. He also wor*
 *ks with BV1^ BV1
rather than BV2+ but again the translation is trivial.) One can check that the *
 *induced splitting
of E*BV2 splits our filtration of E*BV2.
  In more general cases, it is unclear what happens. The most plausible idea se*
 *ems to be that
there should be an BP <m; n>-algebra spectrum A(m; n; k) with homotopy ring A(m*
 *; n; k)* and
that BP <m; n>^ BVw should be a finite wedge of spectra of the form A(m0; n; k0*
 *) for various m0
and k0. However, much work remains to be done in this direction.

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