Completions of Z=(p)-Tate cohomology of periodic spectra
Matthew Ando, Jack Morava and Hal Sadofsky1
University of Virginia
Johns Hopkins University
University of Oregon
Email: "ma2m@faraday.clas.Virginia.edu, jack@math.jhu.edu, sadofsky@math.uoreg*
*on.edu"
Abstract
We construct splittings of some completions of the Z=(p)-Tate cohomology
of E(n) and some related spectra. In particular, we split (a completion of)
tE(n) as a (completion of) a wedge of E(n - 1)'s as a spectrum, where t is
shorthand for the fixed points of the Z=(p)-Tate cohomology spectrum (i.e.
the Mahowald inverse limit lim-k(P-k ^ E(n)) ). We also give a multiplicative
splitting of tE(n) after a suitable base extension.
AMS Classification numbers Primary: 55N22, 55P60
Secondary: 14L05
Keywords: Root invariant, Tate cohomology, periodicity, formal groups.
Submitted: September 5, 1997 (revised March 27 1998).
____________________________1
The authors were partially supported by the NSF.
1
1 Introduction
We fix a prime p, and an integer n. We use t to denote Mahowald's inverse
limit construction tE = lim-k(P-k ^ E) , where P-k stands for either RP-1kor
its analogue for BZ=(p) when p is odd; see [18] or [22]. This is an abbreviation
for the fixed points of the Greenlees-May Tate cohomology functor; we write
tE for what would be denoted in [4] by tZ=(p)(i*E)Z=(p). In particular if E is
a ring spectrum, then so is tE .
The starting point of this paper is a calculation on coefficients, which is by *
*now
well known (see Lemma 2.1). If E is a complex oriented ring spectrum in which
the series [p](x) is not a zero divisor in E*[[x]] = E*CP 1, then
ss-*(tE) = E*((x))=([p](x)):
where |x| = 2 and Ek = ss-k(E). Here we use the awkward grading to em-
phasize that the ring is related to E*CP 1 and E*BZ=(p). Graded more
conventionally, we have
ss*(tE) = E*((x))=([p](x))
where |x| = -2.
Throughout this paper we will use R((x)) to denote the ring of formal Laurent
series over R that are allowed to be infinite series in x, but only finite in x*
*-1 .
If E is a ring spectrum and x an indeterminateWin degree -d < 0, we write
E((x)) for the ring spectrum given by lim-i ji djE . The multiplication is
defined by the inverse limit of the obvious maps
_ _ _
( djE) ^ ( dkE) -! ldE: (1.1)
ji0 ki1 li0+i1
This gives ss*[E((x))] = E*((x)). (Note that the theoretical possibility of pha*
*n-
tom maps in (1.1 ) means the multiplicative structure there may not be unique,
but we are content to use whichever multiplicative structure might occur. Ac-
tually work of Hovey and Strickland shows that there are no phantom maps in
this situation [11], so the multiplication is defined uniquely.) Note also that*
* if
E is connective,
_ Y
lim- djE = djE: (1.2)
i ji j2Z
Q
In the succeeding, if we refer to a multiplication on j2ZdjE it will be the
Q
one coming from the equivalence (1.2 ), and the ring structure on j2ZdjR
2
for a connective graded ring R will be understood to be the one given by the
additive isomorphism with R((x)).
If M is a flat module over E*, we write M E for the spectrum representing
the homology theory
X 7! M E* E*(X):
This is a module spectrum over E (again, it will not matter for us that the
module structure is well-defined only up to phantom maps). We also use [p](x)
for the p-series of the formal group law over E given by the orientation. When
necessary we will decorate [p] with a subscript indicating the formal group law.
We'll work with a number of different spectra E , all closely related. The coho-
mology theory closest to BP is BP , the version of BP with singularities
satisfying BP * = Z(p)[v1; : :;:vn] where vi is the ith Araki generator. The
Johnson-Wilson theory E(n) = v-1nBP has the obvious coefficient ring ob-
tained from inverting vn 2 BP *
We'll also need to consider some simple variants of E(n). We list the theories
below and their coefficient rings. They are all flat over E(n) and are thus
determined by their coefficients.
E^ (n)= LK(n)E(n) so
E^ (n)*= Zp[[v1; : :;:vn-1]][vn; v-1n] = (E(n)*)^In-1
n-1
(En)*= E^ (n)*[u]=(up - vn)
and in later sections
n-1-1
E(n)[w]*= E(n)*[w]=(wp - vn-1)
The "^" is meant to suggest "complete." The reader should note that our
choice of En so that En;*= Zp[[u1; : :;:un-1]][u; u-1] conflicts with the common
convention that En denote the same theory extended by the Witt vectors of
Fpn. We make no use of that theory in this paper, so this should cause no
confusion. The multiplicative structure on E^ (n) is given by the composite
LK(n)E(n) ^ LK(n)E(n) ! LK(n)(LK(n)E(n) ^ LK(n)E(n)) =
LK(n)(E(n) ^ E(n)) ! LK(n)E(n):
The other two spectra are finite extensions of E^ (n), so the multiplicative
structure is determined in the obvious way from the coefficients.
3
Perhaps a few words of history are relevant here. Lin's theorem [15, 6] shows
that for any finite CW-spectrum X , the map
X ! lim-(P-k ^ X) = tX
k
is p-completion. This is not true [2, 1] for E = BP or various other BP -module
spectra, though there is some predictable behavior, which is quite different fr*
*om
that of finite complexes. In particular, tX is quite large in these cases.
By a lovely but simple argument involving the thick subcategory theorem, Ma-
howald and Shick [19] showed (for p = 2) that if X is a type n finite complex,
and v is a vn-self map, then t(v-1X) ' *. This is the starting point for a seri*
*es
of observations. For example, it turns out that tK(n) ' *; indeed [5] proves
that tG K(n) ' * for any finite group G. Hopkins has shown that if X is of
type n then t(LnX) ' * if X is type n (this is proved in [9]). Calculations of
Hopkins and Mahowald (including a proof when n = 1) lead them to conjecture
that if X is type n - 1, then
t(LnX) = Ln-1Xp^_ -1Ln-1Xp^: (1.3)
This is related to the chromatic splitting conjecture [8].
In light of (1.3), it seems worthwhile to investigate tE(n). In [5] it is shown
that tE(n) is vn-1 -periodic in an appropriate sense. Using this, one sees in
[9] that when X is finite, the Bousfield class of t(LnX) is compatible with the
conjecture (1.3) (for p = 2, this is also proved by different techniques in [14*
*]).
The goal of the present paper is to give as precise a description as possible
of tE(n) in terms of familiar vn-1 -periodic spectra like E(n - 1). We hope
that with the results of this paper in hand, it may be possible to make further
headway on (1.3).
The first step, in section 2, is a series of calculations of ss*tE in terms of *
*more
familiar objects for various E . It turns out to be useful to complete with res*
*pect
to In-1 = (p; : :;:vn-2). Since tE(n) is already E(n - 1) local, this amounts
to localizing with respect to K(n - 1). This completion is in some sense not
very dramatic; in particular if X is type n-1 as in (1.3), the completion leaves
t(LnX) unchanged. The main result, for the case of E(n), is
Theorem A (Proposition 2.11) There is an isomorphism of rings
^ ^
tE(n)* In-1= E(n - 1)*((x))In-1 (1.4)
4
Section 3 gives splittings of the spectra v-1ntBP and tE(n) in the manner
suggested by Theorem A. For tE(n) the result is
Theorem B (Theorem 3.9) There is a map of spectra
_
lim- 2iE(n - 1) ! tE(n)^In-1
i2N j-i
which after completion at In-1 (or equivalently after localization with respect
to K(n - 1)) induces the isomorphism of Theorem A on homotopy groups.
We emphasize that the map in Theorem B is not multiplicative, even though
the left hand side can be given a multiplication as in equation (1.1 ) and the
right hand side is a ring spectrum by [4], and Theorem A gives an isomorphism
of rings. Now the homotopy calculation in section 2 also shows that the obvious
formal group law over tE(n) has height n - 1, so one might view the map
E(n) ! tE(n)
as a sort of Chern character, the classical Chern character being the case n = *
*1.
To make sense of this, one ought to use the calculations of section 2 to constr*
*uct
a map of ring spectra.
To do this, we give an isomorphism between the natural formal group law over
tE(n) and the Honda formal group law of height n - 1. This can be done after
a dramatic base extension, but in section 4 we show that it can then be done in
a canonical way. Section 5 uses section 4 and the Lubin-Tate theory of lifts to
construct an isomorphism of ring spectra between an extended version of tE(n)
and an extended version of E(n - 1). A precise statement is Corollary 5.11.
The authors thank Johns Hopkins University, at which the conversations leading
to this paper took place, and Max-Planck-Institut f"ur Mathematik, at which
the last fragments were committed to paper.
2 Calculations on homotopy groups
The homotopy of tE for E complex oriented
We assume that E is complex oriented, and that [p](x) is not a 0 divisor
in E*[[x]]. In [4, x16] it is shown that tE is an inverse limit of E smashed
5
with Thom spectra over BZ=(p). In particular, [4, 16.3] shows that ss-*tE =
E*((x))=([p](x)) as a module over E*[[x]]=([p](x)) = E*(BZ=(p)+ ).
Now [4, 3.5] proves that the map
E*(BZ=(p)+ ) ! ss-*tE
is a map of rings. It follows that the element "x-1 " in ss-*tE = E*((x))=([p](*
*x))
satisfies x . x-1 = 1 in the ring ss*tE , so really is the inverse of x. From t*
*his
one concludes that
Lemma 2.1 There is an isomorphism of rings
ss-*(tE) = E*((x))=([p](x)): (2.2)
An isomorphism after completion
Recall R((x)) is S-1R[[x]] where S is the multiplicatively closed subset gener-
ated by x.Q If R is graded connected, and x has degree -2, then additively
R((x)) = k2Z2kR. As remarkedQin the introduction, wherever this notation
occurs, we will also consider k2Z2kR to have the ring structure given by this
isomorphism. Similarly,Qwhen E is a connectiveWring spectrum, we'll consider
the ring structure on k2Z 2kE = lim-iki 2kE as in (1.1 ).
The following observation motivates Conjecture 1.6 of [2] (which was corrected
as Conjecture 1.2 of [1]).
Proposition 2.3
ss*(tBP ) = BP *((x))=([p](x))
Y
' 2kBP p^* (2.4)
k2Z
' BP p^*((x)):
Proof The first line is equation (2.2). We wish to simplify the right hand
side.
For convenience, we introduce degree 0 elements
i-1
wi= vixp :
Then the ideal in the quotient above is generated by the relation
wn = -(p + a formal series inw1; :::; wn);
6
of the form
wn -(p + w1 + ::: + wn-1) (mod decomposables); (2.5)
by iterating this relation we conclude that
wn = Wn(w1; :::; wn-1) 2 Zp[v1; : :;:vn-1][[x]]
(note that we need the completion on the right since iterating the relation that
defines wn one gets p-adic coefficients, not integral ones). Thus the map
ss*(tBP ) ! BP p^*((x))
which sends vi to vi (i < n), x to x and vn to x-(pn-1)Wn(w1; :::; wn-1) is a
well defined ring map. There is a map
BP *((x)) ! ss*(tBP )
defined in the obvious way, that becomes an inverse map upon extending it to
BP p^*((x)). This extension exists, because the relation
[p](x) = 0
together with Araki's formula for the p-series [20, A2.2.4] allows us to write
n
px = [-1](v1xp +F . .+.Fvnxp ) (2.6)
where F is the formal group law on BP induced by the complex orientation
from BP . Now dividing (2.6) by x lets us to write any p-adic integer as a power
series in x. 2
The conjecture of [2, 1] is that the isomorphism (2.4)is the effect on homotopy
groups of an equivalence of spectra
Y
tBP ' 2kBP :
k2Z
This is proved for n = 1 in [2] and for n = 2 in [1] using the Adams spectral
sequence. Of course in general an isomorphism in homotopy doesn't give an
isomorphism of spectra, but it does if it is an isomorphism of MU*-modules
satisfying Landweber exactness. This suggests inverting vn to produce an iso-
morphism of spectra. We shall show that there is an isomorphism of the ex-
pected form after an appropriate completion, taking as our starting point the
7
spectra v-1n(tBP ) and tE(n). The first point is that these spectra are not
the same.
The map BP ! E(n) gives a map tBP ! tE(n), and since vn is a unit
on the right, this gives a map v-1ntBP ! tE(n). On homotopy we get
ss*(v-1n(tBP ))= v-1n[BP *((x))=([p](x))]
,! E(n)*((x))=([p](x))
= (tE(n))* = ss*t(v-1nBP ):
To see that the inclusion is proper, set r = |vn|=|v1| + 1 and notice that the
series
1 + v-1nvr1xp-1 + v-2nv2r1x2(p-1)+ : : :
is an element of E(n)*((x))=([p](x)) but not of v-1nBP *((x))=([p](x)). This
reflects the fact that t need not commute with direct limits (nor in fact does *
*it
generally commute with inverse limits).
We treat the case v-1n(tBP ) first.
We would like to extend the isomorphism of (2.4) to an isomorphism
ss*v-1ntBP ! v-1n-1BP p^*((x)): (2.7)
For the map to exist we need the image of wn to be a unit after inverting vn-1 .
This is certainly false for n > 1. Indeed, the image of wn modulo (v1; : :;:vn-*
*2)
is not a unit. We have
n-1-1
im (wn) = -(p + vn-1xp ) (mod v1; : :;:vn-2 and higher powers)of:x
Since vn-1xpn-1-1 is a unit in the given ring, the image of wn is a unit if and
only if the image of wn divided by vn-1xpn-1-1 is a unit. That element is
p
-(1 + ___________n-1+ other terms):
vn-1xp -1
The terms in the inverse of this Laurent series (in x) contain arbitrarily high
powers of _____p____v; then-1se series are not in the given ring.
n-1xp -1
Similarly, if the map existed, it could be an isomorphism only if vn-1 were a
unit in the domain after inverting vn. This also is not the case for n > 1, for
similar reasons. When n = 1 both of these conditions are met, the map above
exists, and is an isomorphism.
8
Proposition 2.8 If the map from (2.4) is completed at the ideal
In-1 = (p; v1; : :;:vn-2)
then there is an isomorphism
(ss*v-1ntBP )^In-1! (v-1n-1[BP p^*((x))])^In-1:(2.9)
Proof We follow the isomorphism of (2.4)
ss*(tBP ) = BP *((x))=([p](x))
' BP p^*((x))
by the inclusion
BP p^*((x)) ! (v-1n-1[BP p^*((x))])^In-1
into the module obtained by inverting vn-1 and then completing. Now note
that wn has image as in (2.5). So dividing by vn-1xpn-1 = wn-1 we see the
image of wn divided by wn-1 is -1 plus terms in the ideal In-1 and terms in
xBP p^*[[x]]. This is a unit, so since we've inverted vn-1 , the image o*
*f wn
is a unit. This allows us to extend our map to the domain given by inverting
vn. Since the range is complete, we can also extend to the completion.
On the other hand, a similar argument allows us construct the inverse map
from the inverse map of (2.4). 2
Now we shall show that the isomorphism in (2.9) is induced by a map of spectra
Y
(v-1ntBP )^In-1-!(v-1n-1 2kBP )^In-1:
Z
Both sides are ring spectra with obvious MU -module structures. They would
be isomorphic as MU -algebras by the Landweber exact functor theorem if we
could make them MU -algebras so that the coefficient isomorphism preserves
the map from MU*.
Let R be the ring spectrum on the right. In order to construct a map of ring
spectra inducing the isomorphism in (2.9), it is necessary that the FGL induced
by
v-1ntBP * ! R* ! R*=(p; : :;:vn-2)
9
be isomorphic to the "usual" FGL on R*=(In-1) (induced by BP * !
R*). We can't demonstrate such an isomorphism, but we can exhibit an iso-
morphism of spectra (that preserves neither the MU -module structure, nor the
multiplicative structure). We do this in section 3.
The situation for tE(n) is very similar. We can attempt to construct a map
tE(n)* ! E(n - 1)p^*((x)) (2.10)
as in (2.4), but we immediately run into the problem that wn, and hence vn,
does not go to a unit for n > 1. Also as before, vn-1 is not a unit on the left.
The solution is the same; after completing both sides at In-1 we can construct
an isomorphism.
Proposition 2.11 The map of (2.4) extends to an isomorphism
E(n)*((x))=[p](x)^In-1= E(n - 1)*((x))^In-1 (2.12)
where
ss*(tv-1nBP )^In-1= ss*tE(n)^In-1= E(n)*((x))=[p](x)^In-1
and
E(n - 1)*((x))^In-1= (v-1n-1BP )((x)))^In-1:
In the next section, we use this calculation to construct an isomorphism of
spectra. We are able to do this without showing the corresponding formal
group laws are isomorphic, so we do not get an isomorphism of MU -modules.
In section 5, we extend scalars suitably to construct an isomorphism of formal
group laws, yielding isomorphisms of MU -algebra spectra.
3 Structure of E(n) as an E(n - 1)-module spectrum
We begin with an algebraic observation.
Lemma 3.1 Let (A; m) be a complete local ring, k = A=m. Let M be an
A-module such that the map m x 7! mx induces isomorphisms
(mr=mr+1) k (M=mM) = mrM=mr+1M:
10
LetLI be an index set for a vector space basis of M=mM , so that M=mM '
Ik . Then there is a map
M "
A -!M
I
which is an isomorphism when completed at m.
Proof We construct a map as follows. Let {xff}; ff 2 I be a basis for M=mM
as a A=m-vector space. Let {xff} be lifts to M . Then we map IA to M by
"(1ff) = xff.
Our hypotheses imply that this map gives an isomorphism on the associated
graded with respect to the filtration induced by powers of m. This implies that
the completion of the map is an isomorphism. 2
Note that if the index set I in Lemma 3.1 is infinite, M will not generally be
isomorphic to a free A-module. For example if A = Zp and M = (N Zp)p^
then M is not a free Zp-module. To see this, observe that no free Zp-module
of infinite rank can be p-complete.
Next we recall the result proved in [10, Theorem 4.1]. The identification
BP*BP = BP*[t1; t2; : :]:gives a splitting
_
: BP ^ BP ' |R|BP (3.2)
R
where R ranges over multi-indices of non-negative integers (with only finitely
many positive coordinates) R = (r1; r2; r3; : :):, tR = tr11tr22: :a:nd
|R| = |tR | = 2(r1(p - 1) + r2(p2 - 1) + r3(p3 - 1) + : :)::
To build a map from right to left of (3.2), take |R|BP to BP ^ BP by using
the homotopy class tR 2 ss|R|(BP ^ BP ), smashing on the left with BP and
then multiplying the left pair of BP 's:
R ^1BP
BP ^ S|R|-1BP^t----!BP ^ BP ^ BP ----! BP ^ BP: (3.3)
The wedge over all R of these maps is an isomorphism on homotopy groups,
so is invertible, and is that inverse.
Theorem 4.1 of [10] states that the composite
_ ae_ _
BP -jR!BP ^ BP -! |R|BP -! |oeR|BP -! |oeR|BP (3.4)
R R2R R2R
11
is a homotopy equivalence after smashing with a type j spectrum and inverting
vj. The map ae is induced by leaving out wedge summands, and by the usual
reduction of ring spectra BP ! BP . We use the notation of [10], derived
from [12]: R is the set of multi-indices with the first j - 1 indices 0 , and
oeR = (pjej; pjej+1; : :):.
We shall use the following facts about MU -module spectra M :
M ^ LjZ = v-1jM ^ Z = M ^ v-1jZ (3.5)
when Z is a finite type j spectrum (which follows from [21, Theorem 1]);
M^Ij-1= lim-M ^ Z
Z
where Z runs through finite type j spectra, essentially by definition; and
LK(j)M = lim-(M ^ v-1jZ)= (v-1jM)^Ij-1:
Z
This last equality is by equation (3.5) combined with, say, the proof of [7,
Proposition 7.4] which verifies that LK(n)X = lim-Z(X ^ LnZ) as Z runs over
finite type n spectra.
So if vj is already a unit on M then
LK(j)M = M^Ij-1
and we will generally use the first notation rather than the second below.
Each map in (3.4) except jR is a map of left BP -modules. Recall Ij is the ideal
(p; : :;:vj-1) BP*. Since Ij is invariant, jR (Ij) = jL(Ij), and thus each map
in (3.4) is compatible on homotopy with the Ij-adic filtration. Let Z be a fini*
*te
type j spectrum; then smashing (3.4) with LjZ gives an equivalence, and thus
an E(j)-module structure on BP ^v-1jZ = BP ^LjZ or, by taking inverse lim-
its, a (possibly not unique) module structure on LK(j)BP = lim-ZBP ^ v-1jZ.
We prove the following proposition as a warm-up to our additive splitting of the
Tate cohomology spectrum. The construction is a general method for splitting
LK(j)F when F is a nice enough BP -module, given the splitting of LK(j)BP .
A theorem equivalent to Proposition 3.6 is proved as [10, Theorem 4.7].
Proposition 3.6 If n > j , there is a map
_
s : |VE|(j) -!LK(j)E(n) (3.7)
V
12
which gives an equivalence after completing with respect to (p; : :;:vj-1), or
equivalently, after localizing with respect to K(j). The index V runs through
the monomials in
Fp[vj+1; : :;:vn-1; v1n]:
Proof LK(j)E(n) is an LK(j)BP -module, and thus an E(j)-module. Let
m = Ij = (p; : :;:vj-1). Since
LK(j)E(n) = lim-(E(n) ^ v-1jZ) (3.8)
Z
where Z runs through finite type j spectra,
ss*LK(j)E(n) = Zp[[v1; : :;:vj-1]][v1j; vj+1; : :;:vn-1; v1n]:
Now since m is an invariant ideal mss*LK(j)BP is well-defined, whether we
think of m as an ideal in E(j)* acting on M = ss*LK(j)E(n) via the E(j)-
module action on LK(j)BP or as an ideal in BP* acting via the associated
localization map to ss*LK(j)BP .
We calculate that the associated graded to the m-adic filtration on M is
E0M = V |VF|p[[v0; : :;:vj-1]][v1j]
where V runs through monomials in Fp[vj+1; : :;:vn-1; v1n].
M satisfies the hypotheses of Lemma 3.1 (for the left m-structure which comes
from the map BP ! E(n) ! LK(j)E(n)), where
A = ss*E^ (j) = Zp[[v1; : :;:vj-1]][v1j]:
Now the splitting of LK(j)BP gives LK(j)E(n) an E^ (j)-structure, and thus
an associated A-action; the m-adic filtration is the same, so M also satisfies
the hypotheses of Lemma 3.1 for that m-adic filtration.
We can now make the usual homotopy theoretic argument: take generators of
M=mM and lift them to elements of M . Use the E(j) structure of LK(j)E(n)
to make maps E(j) ! LK(j)E(n) realizing these generators on the unit of the
ring spectrum E(j). This gives a map
_
|VE|(j) ! LK(j)E(n):
V
13
By Lemma 3.1, this map induces an isomorphism on homotopy groups after
completion with respect to (p; : :;:vj-1), i.e. after applying LK(j), which lea*
*ves
the right hand side unchanged. 2
We apply techniques similar to those used in Proposition 3.6 to prove the fol-
lowing.
Theorem 3.9 There is a map of spectra
_
lim- 2iE(n - 1) ! tE(n)^In-1
i2Nji
that becomes an isomorphism on homotopy groups after completion at In-1 (or
equivalently after localization with respect to K(n - 1)).
Proof We proceed as in the proof of Proposition 3.6. We've given M =
ss*tE(n)^In-1as a BP*-module in Proposition 2.11. It satisfies the hypotheses of
Lemma 3.1 with m = (p; : :;:vn-2) BP*, A = E^ (n-1)*. Since tE(n)^In-1=
LK(n-1)tE(n), we have an LK(n-1)BP -action and hence an E(n - 1)-action.
As above, the two available m-adic filtrations on ss*tE(n)^In-1(one from BP*,
the other from E(n - 1)* via the E(n - 1)-structure on LK(n-1)BP ) are the
same since m is an invariant ideal. We now proceed in a slightly different
manner. Note that
M
M=m = K(n - 1)*((x)) = ciK(n - 1)* (3.10)
i2I
for some indexing set I , since K(n - 1)* is a graded field. We could apply
Lemma 3.1 and proceed as before, but we would actually like better control over
our expression for tE(n)^In-1. In particular, the index set I in equation (3.10)
must be uncountable, but we would like to find a countable set of topological
generators for tE(n)^In-1. In fact, we'd like these generators to correspond to
the (positive and negative) powers of x.
To accomplish this, we first recall [4, Theorem 16.1] which states that tE =
lim-i[(BZ=(p))-i ^ E] , where is the usual complex line bundle. Recall also
that the Thom class of (BZ=(p))-i is in dimension -2i, and is not torsion.
So the cell in dimension -2i + 1 is not attached to the -2i cell generating the
Thom class, and the inclusion of the -2i + 1 cell, together with the unit of E
when E is a ring spectrum, gives
x-i+1 2 ss-2i+2(BZ=(p))-i ^ E:
14
Now, we take xj 2 ss2jtE(n)^In-1, and use the E^ (n - 1)-structure to construct
a sequence of maps
_
2jE^ (n - 1) -!tE(n)^In-1: (3.11)
j-i
We make a map -i by composing the map of (3.11) with the map
tE(n)^In-1! (BZ=(p))-(i+1) ^ E^In-1
given by [4, Theorem 16.1].
Taking inverse limits of the maps -i gives a map
_ f
lim-( 2jE^ (n - 1))-!tE(n)^In-1:
i ji
This map defines an isomorphism on the associated graded modules with respect
to m. It follows that f is an equivalence after completion, that is
_ 2j ^ ^ ^
lim-( E (n - 1))In-1= tE(n)In-1:
i ji
2
By a very similar argument one can prove the analog to Proposition 2.8:
Proposition 3.12 There is an equivalence of spectra
_
[v-1n-1lim-( 2jBP )]^In-1! (v-1ntBP )^In-1:
i ji
We leave the proof to the interested reader.
Given all the completions that occur in this section and in section 2, one might
hope that by using some other, already complete theory like E^ (n) or En, we
could prove a theorem with a simpler statement. This is unfortunately not the
case. There are similar results for these theories, but even if E is complete
with respect to (p; v1; : :;:vn-2), tE will generally not be, and so will need *
*to
be completed again. There are variants of Theorem 3.9 for these other spectra
as well, but the statement is not simpler.
15
4 A Honda coordinate on the formal group over tE
In this section we shall take p to be an odd prime and n > 1 to be an integer. *
*It
will ease the superabundance of superscripts to use the abbreviation q = pn-1 .
We define a number of formal group laws in this section that are used in the
remainder of the paper. For reference, we list them here.
G(n): a homogeneous formal group law of degree 2 on E(n)* induced by
the usual orientation of E(n).
Gn: a twist of the pushforward of G(n) to En;*by the element u. This
is homogeneous of degree 0.
F : a twist of the pushforward of G(n) to E(n)[w]* of G(n) by w. This
is also homogeneous of degree 0.
F0: the pushforward of F to the residue field of ss0T E .
H : the Honda formal group law of height n - 1 over Fp.
__
F: A formal group law introduced in the proof of Proposition 4.13 that
is shown in that proof to be the same as H .
We use the canonical orientation of E(n), which provides a coordinate so that
E(n)*(CP 1) = E(n)*[[x]];
and as usual, if is the multiplication on CP 1,
*(x) 2 E(n)*[[x; y]] = E(n)*[[x]] ^E(n)*[[y]] = E(n)*(CP 1 x CP 1)
is a formal group law which we'll denote G(n)(x; y).
P G(n) i
This formal group law has the feature that its p-series is given by in vixp :
Recall that
ss*En = Zp[[u1; : :;:un-1]][u; u-1]
with |ui| = 0 and |u| = 2. There is an isomorphism
E0n(CP 1) ~=ss0En[[t]]
16
in terms of which the coproduct on E0n(CP 1) is determined by the formula
t 7! Gn(s; t)
where Gn is the group law
Gn(s; t) def=uG(n)(u-1s; u-1t)
over ss0En.
^
Since vn-1 is a unit in ss* tE(n) In-1, we shall also consider the theory E(n)[*
*w]
obtained by adjoining an element w of degree 2 such that
wq-1 = vn-1 (4.1)
and then completing with respect to the ideal In-1 = (p; : :;:vn-2). We prefer
n-1)
this choice to the usual parameter u = v1=(pn (which gives En) because the
functor t will emphasize height n - 1 behavior instead of height n, and the
normalization we choose to make things 2-periodic leads to simpler statements
in section 4 and this section.
Equation (2.2)shows that there is an isomorphism
ss*tE(n)[w] ~=ss*E(n)[w]((x))=[p]G(n)(x):
Proposition 2.11 shows that vn-1 , and so also w, is a unit in the homotopy of
the In-1 -adic completion
i j^
T E def=tE(n)[w]
In-1
which is thus 2-periodic. If F denotes the formal group law
F (s; t) def=wG(n)(w-1s; w-1t) (4.2)
(which implies [p]F (s) = w[p]G(n)(w-1s)) and we introduce elements
i-1)
wi = viw-(p
y = wx
of degree zero, then the argument of Proposition 2.11 shows that there are
isomorphisms
h i^
ss0T E~= Zp[w1; : :;:wn-2; w1n]((y))=([p]F (y))
In-1
h i^
~= Zp[w1; : :;:wn-2]((y)) (4.3)
In-1
h i^
ss*T E~= Zp[w1; : :;:wn-2]((y)) [w; w-1]:
In-1
17
The group law F is defined over ss0E(n)[w] and hence over ss0T E ; it is p-
typical, and its p-series satisfies the functional equation
n-2 q pq
[p]F (t) = pt + w1tp + : :+:wn-2tp + t + wnt : (4.4)
F F F F F
We denote by F0 the image of the group law F in the residue field of ss0T E .
Proposition 4.5 The residue field of ss0T E is Fp((y)). The element wn of
ss0T E maps to -y(1-p)qin the residue field. The formal group law F0 has coef-
ficients in the subring Fp[y-1], and its p-series satisfies the functional equa*
*tion
[p]F0(t) = tq - y(1-p)qtpq: (4.6)
F0
Proof The statement about the residue field follows from equation (4.3).
Note that the image of equation (4.4)in the residue field is
[p]F0(t) = tq + wntpq;
F0
so equation (4.6)follows from the assertion that wn maps to -y(1-p)q.
Since [p]F0(y) = 0 we have
0 = yq + wnypq
F0
and so
yq = [-1]F0(wnypq) = -wnypq
wn = -y(1-p)q:
(F0 is p-typical and p is odd, so [-1]F0(t) = -t).
Finally, F0 is defined over the subring Fp[y-1] since F is actually defined over
the polynomial ring in w1; : :;:wn, and (1 - p)q is negative. 2
So F0 is a p-typical formal group law over Fp[y-1], of height n - 1 in the field
Fp(y) or Fp((y)). On the other hand, let H be the Honda law of height n - 1
over Fp, characterized by the fact that it is p-typical with p-series
[p]H (t) = tq:
Comparison with equation (4.6)shows that
F0 H mod y-1: (4.7)
18
Now Lazard [13, 3] proves that over the separable closure Fp(y)sep, there is an
isomorphism of formal group laws
F0 ~=H;
which is in general not at all canonical. In this section we show that there is*
* a
unique isomorphism : F0 ~=H which preserves equation (4.7)in a suitable
sense.
To make this precise, note that expanding a rational function as a power series
at infinity gives a map of fields
Fp(y) ! Fp((y-1))
which extends to
Fp(y)sep! Fp((y-1))sep:
Theorem 4.8 There is a unique isomorphism : F0 ! H of formal group
laws over Fp(y)sepsuch that the image of in Fp((y-1))sephas coefficients in
the image of
Fp[[y-1]] ! Fp((y-1))sep:
gives an isomorphism over Fp[[y-1]] satisfying
(t) t mod y-1:
We build up to the proof gradually; the proof itself appears after Proposition
4.13.
Lemma 4.9 There is a unique series o(t) 2 Fp[y-1][[t]] such that
[p]F0 = [p]H O o:
This power series has the properties that
o(t) t mod t2
o(t) t mod y-1:
19
Proof The equation which o must satisfy is
[p]F0(t) = o(t)q:
If
X
F0(s; t) = bijsitj
i;j
then the functional equation (4.6)becomes
X
[p]F0(t)= bij(tq)i(-y(1-p)qtpq)j
i;j
X
= bij(-y1-p)jqt(i+pj)q:
i;j
So we must show that bijhas a unique q throot. If it does then
X 1=q
o(t) = bij (-y1-p)jt(i+pj)
i;j
which shows that o(t) t mod t2.
If
X
G(n)(s; t) = aijsitj
i;j
then by definition
X
F (s; t) = aijw1-i-jsitj;
i;j
with bij= aijw1-i-j homogeneous of degree zero. The (p; v1; : :;:vn-2) reduc-
tion of aijis of the form
X
aij= cabvan-1vbn2 Fp[vn-1; vn] (4.10)
a;b
for coefficients cab depending on i; j . Substituting
vn-1 = wq-1
vn = wpq-1wn
= -wpq-1y(1-p)q;
20
into equation (4.10), one has
X
bij= w1-i-j cabwa(q-1)(-wpq-1y(1-p)q)b:
a;b
As bij is homogeneous of degree zero, the exponent of w in each term must
add to zero, and so
X
bij= cab(-y1-p)bq
a;b
X
b1=qij= cab(-y1-p)b
a;b
since cab2 Fp so c1=qab= cab. Thus one has
X
o(t) = rijt(i+pj); with
i;j
X
rij= cab(-y1-p)b+j
a;b
(recall the cab depend on i and j ), which shows that o(t) t mod y-1 as
well. 2
Proposition 4.11 There is a unique power series (t) 2 Fp[[y-1]][[t]] such that
(t) t mod t2;
(t) t mod y-1; and
[p]H O = O [p]F0
in Fp[[y-1]][[t]].
Proof If f in A[[t]] is a series with coefficients in a ring A of characteris*
*tic
p, let foebe the corresponding series with coefficients those of f , raised to *
*the
q -th power; thus
foe(tq) = (f(t))q:
In this notation, the equation supposedly satisfied by takes the form
oe= O ooe
21
where o 2 Fp[y-1][[t]] is the power series constructed in Lemma 4.9. If o-1 de-
notes the compositional inverse (not reciprocal) of o , and similarly o-oedenot*
*es
(o-1 )oe, then this equation can be rewritten in the form
= oeO o-oe
= [oeO o-oe]oeO o-oe= : :::
If is to be of the form
(t) t mod y-1;
then as r grows, oerwill converge to the identity in the (y-1)-adic topology,
and we must have
r -oer-1 -oe
= limr!1o-oe O o O . .O.o : (4.12)
On the one hand, if the limit exists, then it certainly intertwines the p-serie*
*s.
On the other hand, the limit exists: Lemma 4.9 asserts that
o(t) 2 t + y-1Fp[y-1][[t]];
so o-oer converges to the identity in the (y-1)-adic topology. The group of
formal power series under composition is complete with respect to the non-
archimedean norm defined by the degree of the leading term, so the infinite
composite (4.12)converges because the sequence of composita converges to the
identity. It is easy to see in addition that inherits the property
(t) t mod t2
from o . 2
Proposition 4.13 The power series is the unique strict isomorphism
F0 -! H
of formal group laws over Fp[[y-1]] such that
(t) t mod y-1:
Proof The uniqueness is already_contained in Proposition 4.11 (Note that
[p]__F(t) = tq.). Let us write F for the formal group law F0 where
F0 (x; y) = (F0(-1(x); -1(y))):
22
__
We need to show F = H: There is a canonical strict isomorphism
__
G ae-!F;
from a p-typical formal group law G, defined over Fp[[y-1]]. Indeed ae is given
by the formula [20, A2.1.23]
r __
X __F X F
ae(t) = [(r)]__F[1_r]__F iit; (4.14)
p-r1 i=1
where
(
0 r is divisible by a square
(r) =
(-1)k r is the product ofk distinct primes
and i is a primitive rth root of 1. We first claim that G = H ; then we shall
show that ae is the identity.
As G and H are both are p-typical, it suffices to show that [p]G = [p]H . Since
ae is a homomorphism of groups, one has
[p]__FO ae = ae O [p]G (4.15)
Using equation (4.14), one has
r __
X __F(r) X F
[p]__FO ae(t)= [___r]__F [p]__F(iit)
p-r1 i=1
r __
X __F(r) X F
= [___r]__F iiqtq
p-r1 i=1
= ae(tq)
since q = pn-1 and [p]__F(t) = tq: Thus
ae([p]G (t)) = ae(tq) = ae([p]H (t)):
Next we must show that ae is the identity. As
(t) t mod y-1
we have
__ -1
F0 F mod y :
23
But F0 is p-typical, so
X r
F0 i
[1_r]F0 i t = 0
i=1
for p - r > 1; it follows from equation (4.14)that
ae(t) t mod y-1: (4.16)
But equation (4.15)and [p]__F(t) = tq imply that
ae(tq) = ae(t)q;
P
if ae = i1 aeiti then we must have
aei= aeqi:
Together with equation (4.16)this implies ae1 = 1. For i > 1 we have aei 2
y-1Fp[[y-1]], so aei= 0. 2
Proof [Proof of Theorem 4.8] If
X
(t) = iti+1
i0
is any isomorphism from the group law F0 to the group law H over Fp(y)sep,
then it must satisfy the equation
[p]H O = O [p]F0:
By Lemma 4.9, we must have
oe= O ooe: (4.17)
Because
o(t) t mod t2;
the intertwining equation (4.17)can be rewritten inductively as a sequence of
generalized Artin-Schreier equations
qi- i= a polynomial in j's with j < i;
beginning with
q0- 0 = 0:
24
In particular, the i are all algebraic, and the Galois group of the extension
they generate acts by translating the solutions by an element of the field with
q elements.
The coefficients of satisfy the same equations in Fp((y-1)). Starting with
0 = 1, we may adjust each i by a Galois transformation so that its image in
Fp((y-1))sep is i. The resulting power series is an isomorphism of formal
group laws in Fp(y)sep, since it becomes one in Fp((y-1))sep.
The uniqueness of satisfying the hypotheses is a trivial consequence of the
uniqueness of in Proposition 4.11. 2
It is the field Fp((y)) rather than Fp((y-1)) which appears in the Tate homology
calculations. Expanding a rational function as a power series at zero gives an
embedding Fp(y) ! Fp((y)), which extends to an embedding
Fp(y)sep! Fp((y))sep:
Thus we have
Corollary 4.18 There is a unique strict isomorphism : F0 ! H , satisfying
1.the coefficients of are in the subfield Fp(y)sepand
2.the expansion of at y = 1 is a power series with coefficients in
Fp[[y-1]], congruent to the identity modulo y-1 .
.
5 A map of ring spectra
Lubin and Tate's theory of lifts
We recall briefly the deformation theory of Lubin and Tate [16]. Suppose that
F is a field of characteristic p.
Definition 5.1 A lift of F is a pair (A; i) consisting of
1.a Noetherian complete local ring A with residue field A0;
2.a map of fields i : F ! A0.
25
A map f : (A; i) ! (B; j) (or (A; i)-algebra) is a local homomorphism
A f-!B
such that j = f0 O i, where f0 : A0 ! B0 is induced by f .
We shall abbreviate (A; i) to A when i is clear from context.
Suppose that is a formal group law of finite height n over a field F of
characteristic p, that (A; i) is a lift of F, and that (B; j) is an (A; i)-alge*
*bra.
Definition 5.2 A deformation of to (B; j) is a pair (G; OE) consisting of
1.a formal group law G over B ;
2.an isomorphism of group laws OE : j* G0, where G0 denotes the group
law over B0 induced by G.
Two deformations (G; OE) and (G0; OE0) are ?-isomorphic if there is an isomor-
phism c : G ! G0 such that
OE0= c0 O OE : j* ! G00
in B0.
The set of ?-isomorphism classes of deformations to (B; j) is a functor from
(A; i)-algebras to sets; Lubin and Tate show that this functor is representable.
Namely, let
R = A[[u1; : :;:un-1]];
1 pi pi pi
Cpi(x; y)= __[(x + y) - x - y ];
p
and let (G; OE) be any deformation of to R such that
G0 =
(5.3)
G(s; t) s + t + uiCpi(x; y) mod u1; : :;:ui-1; and degree pi+ 1
for ffl a unit of A0 and 1 i n - 1. Lubin and Tate show that such defor-
mations exist; we shall call such a group law a Lubin-Tate lift of . Theorem
3.1 of [16] may be phrased as follows.
26
Theorem 5.4 If (G0; OE0) is a deformation of to an (A; i)-algebra (B; j),
then there is a unique map of (A; i)-algebras f : R ! (B; j) such that there is
a ?-isomorphism
(f*G; f*0OE) -c!(G0; OE0):
Moreover the ?-isomorphism is unique.
The map of ring theories
A classical theorem of I. Cohen (see [23]) asserts that for any field F of char-
acteristic p, there is an essentially unique complete discrete valuation ring CF
with maximal ideal generated by p and residue field F; in particular, CF is
Noetherian. If F is perfect, then CF is just the ring WF of Witt vectors of F,
but in general it is a subring of the Witt vectors, and (although it is a funct*
*or)
it is not very easily described. However, when the degree [F : Fp] is finite
(instead of 1 as in the perfect case), e.g. if F = Fp((y)), then the Cohen ring*
* is
still relatively tractable. In this case, for example, we have
Lemma 5.5 The Cohen ring of Fp((y)) is isomorphic to the p-adic completion
of Zp((y)). Any such isomorphism extends to an isomorphism
CFp((y))[[w1; : :w:n-2]] ~=ss0T E:
Proof The p-adic completion of Zp((y)) is a complete discrete valuation ring
with maximal ideal generated by p and residue field Fp((y)), so it is isomorphic
to the Cohen ring of Fp((y)). Indeed Schoeller [23, x8] shows that an isomor-
phism is given by a choice of p-base for Fp((y)) and representatives of that
p-base in (Zp((y)))^p.
Equation (4.3)reduces the second part of the lemma to the observation that
the rings
^ ^
Zp((y)) p[[w1; : :;:wn-2]] ~= Zp[[w1; : :;:wn-2]]((y)) In-1
are isomorphic. 2
Proposition 5.6 The formal group law F over ss0T E is a Lubin-Tate lift of
the group law F0 of height n - 1 over Fp((y)) to CFp((y))-algebras.
27
Proof It is a standard fact about G(n) that
G(s; t) s + t + viCpi(s; t) mod v1; : :;:vi-1; and degree pi+ 1
for 1 i n - 1. By the definition (4.2)of F , one then has
F (s; t) s + t + wiCpi(s; t) mod w1; : :;:wi-1; and degree pi+ 1
for 1 i n - 2. It follows that F satisfies equations (5.3)for the group law
= F0. 2
The formal group law Gn-1 over ss0En-1 has image H in Fp, so by Corol-
lary 4.18 the pair (CFp((y))sep^Gn-1; ) is a deformation of F0 to the CFp((y))-
algebra
CFp((y))sep^ss0En-1 ~=CFp((y))sep[[u1; : :;:un-2]]:
Zp
Theorem 5.4 provides a ring homomorphism
ss0T E -f!CFp((y))sep^ss0En-1 (5.7)
Zp
and an isomorphism of formal group laws
f*F -c!~CFp((y))sep^Gn-1 (5.8)
= Zp
such that c0 = .
Proposition 5.9 The map
CFp((y))sep^f : CFp((y))sep^CFp((y))ss0T E ! CFp((y))sep^ss0En-1
Zp
is an isomorphism.
Proof The ring on the left represents deformations of Fp((y))sep^F0 to
CFp((y))sep-algebras, while the ring on the right represents deformations of
Fp((y))sep^H to CFp((y))sepalgebras. The isomorphism of Corollary 4.18
induces an isomorphism between these functors. 2
There are isomorphisms
T E0(CP 1) ~=T E0[[t1]]
E0n-1(CP 1) ~=E0n-1[[t2]]
28
such that the formal group law F expresses the coproduct on T E0(CP 1),
and the formal group law Gn-1 expresses the coproduct on En-1 . A standard
argument [17] using Landweber's exact functor theorem gives
Theorem 5.10 There is a canonical map of ring theories
T E -! CFp((y))sep^En-1
Zp
whose value on coefficients is f , and whose value on T E0(CP 1) is determined
by the equation
(t1) = c(t2):
Here c is the isomorphism (5.8). 2
Corollary 5.11 CFp((y))sep^ gives an equivalence of complex oriented ring
Zp
spectra
CFp((y))sep^ T E ! CFp((y))sep^En-1:
CFp((y)) Zp
Composing the map from Theorem 5.10 with the maps
E(n) ! E(n)[w] ! T E
gives a canonical map of ring theories
E(n) ! CFp((y))sep^En-1;
Zp
our generalized Chern character.
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