TATE COHOMOLOGY LOWERS CHROMATIC BOUSFIELD
CLASSES
MARK HOVEY AND HAL SADOFSKY
December 7, 1994
Abstract.Let G be a finite group. We use the results of [5] to show that
the Tate homology of E(n) local spectra with respect to G produces E(n -*
* 1)
local spectra. We also show that the Bousfield class of the Tate homolog*
*y of
LnX (for X finite) is the same as that of Ln-1X.
To be precise, recall that Tate homology is a functor from G-spectra to
G-spectra. To produce a functor PG from spectra to spectra, we look at a
spectrum as a naive G-spectrum on which G acts trivially, apply Tate hom*
*ol-
ogy, and take G-fixed points. This composite is the functor we shall act*
*ually
study, and we'll prove that = when X is finite.
When G = p, the symmetric group on p letters, this is related to a
conjecture of Hopkins and Mahowald (usually framed in terms of Mahowald's
functor RP-1 (-)).
1.Introduction
We briefly recall the spectra that occur in Lin's proof of the Segal conjectu*
*re for
the group Z=(2). Embed Z=(p) into S1 and look at the pullback of the tautologic*
*al
(complex) line bundle over BS1 as a bundle over BZ=(p). Call this bundle .
We denote by P-2k the spectrum given by the Thom spectrum (BZ=(p))-k
when p = 2, or when p is odd the summand of that spectrum corresponding to
Bp. (We refer the reader to [18] for the definition of a Thom spectrum associat*
*ed
to a virtual bundle). P-2k has a cell in every dimension -2k and is the spectr*
*um
freqeuently called RP-12kwhen p = 2. When p is odd, P-2k has a cell in every
dimension congruent to 0 or -1 modulo q = 2p - 2 and -2k. P-2k is the same
as the spectrum denoted P-12kin [17], and constructed there by James periodicity
rather than via Thom spectra.
Lin's theorem [10] (Gunarwardena's theorem when p > 2 [1]) states that
(1) lim-(P-2k ^ X)= -1Xp^
k
when X is a finite spectrum. This inverse system of spectra (with some minor
alterations) is also what is used to define the root invariant (see [11] for p *
*= 2, or
more generally, [17]). As shorthand, we write P-1 (X) for lim-k(P-2k ^ X).
____________
1991 Mathematics Subject Classification. 55P60, 55P42; Secondary 55N91.
The authors were partially supported by the NSF.
1
2 MARK HOVEY AND HAL SADOFSKY
Mahowald and Ravenel [12] have conjectured a relationship between chromatic
periodicity and Mahowald's root invariant. There is a related conjecture by Hop*
*kins
and Mahowald that is more closely related to our concerns in this paper. Denote
Bousfield localization with respect to E(n) by Ln. They conjecture that
(2) P-1 (LnX) = -2Ln-1Xp^_ -1Ln-1Xp^
for X finite. (A word to the experts: this conjecture is connected with Hopkins*
*'s
chromatic splitting conjecture (see [7]) and at the time (2) was conjectured the
chromatic splitting conjecture was in too simple a form. In light of its curre*
*nt
corrected form as in [7], (2) is probably also too optimistic, though We expect*
* it is
true as stated when X has type n - 1.)
Greenlees and May put the P-1 construction in a more general context in [4].
There they define the Tate G spectrum associated to a G spectrum X, tG (X).
We are not concerned with equivariant spectra here, but we use tG to construct a
functor from (ordinary) spectra to spectra. We abuse notation and write i* for *
*the
functor which is the composite of the inclusion of ordinary spectra into the ca*
*tegory
of naive G-spectra (as the objects on which G acts trivially) with the left adj*
*oint
of the forgetful functor from G-spectra to naive G-spectra. So i* is a functor *
*from
ordinary spectra to G-spectra. Our other functor is the G-fixed point functor, *
*(-)G
which goes from G-spectra to spectra. We refer the reader to [9] for details. We
define
PG (X) = tG (i*X)G :
Then [4, 16.1] shows that
Pp (X) = P-1 (X):
(the left hand side needs to be localized at p when p is odd).
Henceforth we will only be concerned with the case where G is a finite group.
We now have a family of functors, one for each finite group G. Our main theorem
is the following.
Theorem 1.1. Let X be a finite spectrum. Then =
Here means the Bousfield class of the spectrum X as given in [2].
We also give a result about complex oriented vn-periodic spectra.
Theorem 1.2. If E is Landweber exact and vn-periodic then PZ=(p)E is Landweber
exact and vn-1-periodic. It follows generally that = .
Our proofs rely on [5, Theorem 1.1] which implies that PG (K(n)) ' *. We also
use two other results that are relatively well known. We use Ravenel's Proposi-
tion 1.34 from [14]:
(3) = _
where f is a self map of X, C(f) is the cofiber and T el(f) is the mapping tele-
scope. Finally, we use a theorem of Hopkins and Ravenel from [16] to show that
PG (K(n)) ' * implies PG (LnX) ' * when X is finite type n.
Using the interesting results of Mahowald and Shick in [13] one can show that
PZ=(2)(T el(X)) ' * where X is finite type n and T el(X) is the infinite mapping
telescope of X under a vn map. One can use this to deduce our theorems in the
special case G = Z=(2). Chun-Nip Lee has done this independently [8].
TATE COHOMOLOGY LOWERS CHROMATIC BOUSFIELD CLASSES 3
2.There is a finite type n spectrum X with LnX K(n)-nilpotent
Recall that X is said to be E-prenilpotent if LE X is E nilpotent; that is if*
* LE X
can be built up from cofibrations in a finite number of stages with cofibers re*
*tracts
of spectra of form E ^ Z. By [16, 8.3] there is a finite type 0 spectrum Y that*
* is
LnBP -prenilpotent.
By [16, Lemma 8.1.4]
=
which is in turn equal to by [14]. So LnY is LnBP -nilpotent, which give*
*s a
sequence of spectra
* = Y0 ! Y1 ! Y2 ! . .!.Yr = LnY
such that cofiber(Yi-1! Yi) is a retract of LnBP ^ Zifor some Zi. Now let M be
a finite type n spectrum with
BP*M = BP*=(pi0; : :;:vin-1n-1)
and such that M is a ring spectrum. (See [3] for the existence of such ring spe*
*ctra.)
Then BP ^ M = BP=(pi0; : :;:vin-1n-1), so by [15, Theorem 1],
LnBP ^ M = v-1nBP ^ M:
But v-1nBP ^M is made out of finitely many cofibrations with cofiber v-1nBP=In =
B(n). Now by [19, Remark 6.19], B(n) = K(n) ^ B for some B, so it follows that
LnBP ^ M is K(n)-nilpotent.
From this we see that LnBP ^Zi^M is K(n)-nilpotent, and deduce the following
lemma.
Lemma 2.1. There is a finite type n spectrum F with LnF K(n)-nilpotent.
Proof.Take F = Y ^ M. __|_|
3. PG (LnX ^ F ) ' * if F is type n.
We recall from [5] that tG (i*K(n)) ' * as a G-spectrum. It follows that
PG (K(n)) ' *. We record the following lemma.
Lemma 3.1. If X is K(n)-nilpotent then PG (X) ' *.
Proof.First note that since PG takes cofibrations to cofibrations, it suffices *
*to prove
that PG (K(n) ^ Z) ' * for any Z. But PG (R) is a ring spectrum when R is a ring
spectrum, and PG (N) is a module spectrum over PG (R) if N is a module spectrum
over R [4, Proposition 3.5]. It follows that PG (K(n) ^ Z) ' *. __|_|
Remark: The same proof shows that tG (X) ' * equivariantly if X is i*K(n)-
nilpotent in the category of G-spectra.
Corollary 3.2.If F is finite type n, then PG (LnX ^ F ) ' * for any spectrum X.
Proof.Let C be the category of finite spectra F such that PG (LnX ^ F ) ' * for*
* all
spectra X. C is a thick subcategory in the sense of [6]. It follows that if C \*
* Cn 6= OE
then Cn C. We recall that if a spectrum Y is K(n)-nilpotent, so is X ^ Y for
any spectrum X. Since LnX ^ F = X ^ LnF (Ln is smashing) by Lemma 3.1 and
Lemma 2.1, C \ Cn 6= OE. __|_|
4 MARK HOVEY AND HAL SADOFSKY
Remark: Since LnF = LK(n)F when F is finite type n, one might ask when
PG (LK(n)X) ' *. While we don't know the most general answer, this does not
hold in general for X finite. Using the methods of this paper, one can easily c*
*heck
that if X is finite, = .
4.PG (LnX) is E(n - 1)-local.
We use equation (3) inductively. We get
= _
= _ _ . . .
_ _ :
Now since the PG LnX ^ M(pi0; : :;:vin-1n-1) ' * by Corollary 3.2, we get
= _ _ . . .
_:
Since PG LnX is an LnS0 module, it follows that
v-1jPG LnX ^ M(pi0; : :;:vij-1j-1)=PG LnX ^ v-1jLnM(pi0; : :;:vij-1j-1)
= (PG LnX) ^ LjM(pi0; : :;:vij-1j-1):
Since = , we see that
(4) _ . ._. = :
Now since LnX is a LnS0-module, PG LnX is a PG LnS0 module. But PG LnS0
is self local since it is a ring spectrum [14], so by equation (4) PG LnS0 is E*
*(n - 1)-
local, hence so is PG LnX.
To finish the proof of Theorem 1.1 it remains to show the inequality in equa-
tion (4) is actually an equality when X = S0. In section 6 we use Theorem 1.2 to
do this.
5. PG Landweber exact vn-periodic theories.
In this section we prove Theorem 1.2. We will suppose that E is a complex
oriented homology theory. Without loss of generality, we assume that E is p-loc*
*al,
since we will be applying the functor PZ=(p)which gives the same value on X as
on X(p): Then we can assume E is oriented by a map from BP , so that we can
consider vi as an element of E*. We remind the reader that Ij = (p; v1; : :;:vj*
*-1),
and that for BP (and hence for any spectrum oriented from BP ),
2 pi
[p](x) = px +F v1xp +F v2xp +F . .+.Fvix +F . . .
where +F is the sum in the formal group law on E*.
We begin by remarking that for complex oriented E
ss*PZ=(p)E = E*((x))=([p](x))
TATE COHOMOLOGY LOWERS CHROMATIC BOUSFIELD CLASSES 5
where |x| = -2, E*((x)) denotes the ring of Laurent series over E* which have o*
*nly
finitely many terms involving negative powers of x, and [p](x) is the p-series.*
* It
follows that when [p](x) is not a zero divisor, we have a short exact sequence
.[p](x)
E*((x))----! E*((x)) ! ss*PZ=(p)E:
We now assume that E is vn-periodic Landweber exact. We define vn-periodic
almost as in [5, Definition 1.3]; E is vn-periodic if vn is a unit on E*=In and*
* in
addition E*=In 6= 0.
For each j n, we know that E*=Ij ! v-1jE*=Ij is injective by the hypothesis
of Landweber exactness. It follows that E*((x))=Ij ! v-1jE*((x))=Ij is injective
also. Now [p](x) is a unit in v-1jE*((x))=Ij since it is a power series with le*
*ading
term vjxpj, which is a unit. It follows that
.[p](x)
E*((x))=Ij ! v-1jE*((x))=Ij----! v-1jE*((x))=Ij
is injective, hence .[p](x)
E*((x))=Ij----! E*((x))=Ij
is also.
We examine the diagram of short exact sequences below, in which the bottom
row is the cokernel of the map between the top two rows.
0 ----! E*((x))=Ij -.[p](x)---!E*((x))=Ij----!(ss*PZ=(p)E)=Ij----! 0
? ? ?
.vj?y ?y.vj ?y.vj
0 ----! E*((x))=Ij -.[p](x)---!E*((x))=Ij----!(ss*PZ=(p)E)=Ij----! 0
?? ? ?
y ?y ?y
0 ----! E*((x))=Ij+1.[p](x)----!E*((x))=Ij+1----!(ss*PZ=(p)E)=Ij+1----!0
By the snake lemma applied to the first two rows (together with the observa-
tion that the first two vertical maps are injective) we see that vj is injectiv*
*e on
(ss*PZ=(p)E)=Ij.
We can also see that .[p](x) is not a unit on E*((x))=Ij unless vj is a unit *
*on E*,
therefore (ss*PZ=(p)E)=Ij 6= 0 unless j = n, and this last observation tells us*
* that
vn-1 is a unit on (ss*PZ=(p)E)=In-1.
We conclude that PZ=(p)E is Landweber exact, and that ss*(PZ=(p)E)=In-1 6= 0
while ss*(PZ=(p)E)=In = 0. It follows by using (3) as before (see [7, Corollary*
* 1.12],
that
= :
By using the maps of complex oriented ring spectra
E ! PG E ! PZ=(p)E
(when Z=(p) G) we also deduce that
= :
6 MARK HOVEY AND HAL SADOFSKY
6. Proof of Theorem 1.1.
We recall from section 4 that v-1jPG LnS0 ^ M(pi0; : :;:vij-1j-1) has the Bou*
*sfield
class of either a point or of K(j). So to show it has the class of K(j), we nee*
*d only
show that it is not contractible.
We pick i0; : :;:ij-1 so that M(pi0; : :;:vij-1j-1) is a ring spectrum. Then*
* we
observe that the map
S0 ! v-1jPG LnS0 ^ M(pi0; : :;:vij-1j-1) ! v-1jPZ=(p)LnS0 ^ M(pi0; : :;:vij-1j*
*-1) !
= -1 i ij-1
v-1jPZ=(p)E(n) ^ M(pi0; : :;:vij-1j-1)----! vj PZ=(p)E(n)=(p 0; : :;:vj-1) !
v-1jPZ=(p)E(n)=(p; : :;:vj-1)
is the unit of the ring spectrum v-1jPZ=(p)E(n)=(p; : :;:vj-1). This is non-zer*
*o if
j < n by Theorem 1.2. So none of the intervening spectra are contractible eithe*
*r.
For arbitrary finite X (instead of S0) just smash with X. Note that <->, PG (*
*-)
and localization commute with smashing with a finite spectrum.
Remark: The same proof can be iterated to draw the obvious conclusions about
PGk(LnS0).
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Department of Mathematics, M.I.T., Cambridge, MA 02139
E-mail address: hovey@math.mit.edu
Department of Mathematics, Johns Hopkins University, Baltimore MD 21230
E-mail address: hs@math.jhu.edu