(x) = [p](x)=x is not a zero-divisor in R*[[x]]) that OE*(f(x)uLp) = f(x)

(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

(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