VANISHING LINES IN GENERALIZED ADAMS SPECTRAL
SEQUENCES ARE GENERIC
M. J. HOPKINS, J. H. PALMIERI, AND J. H. SMITH
Abstract.We show that in the generalized Adams spectral sequence, the
presence of a vanishing line of fixed slope at some Er-term is a generic*
* property.
1.Introduction
Let E be a nice ring spectrum, let X be a spectrum, and consider the E-based
Adams spectral sequence converging to ss*X. In this note, we prove that, for any
number m, the property that the spectral sequence has a vanishing line of slope*
* m
at some term of the spectral sequence is generic.
Definition 1.1.We say that a spectrum X is E-complete if the inverse limit of
the Adams tower for X is contractible. (See Section 2 for a definition of the A*
*dams
tower.)
A property P of E-complete spectra is said to be generic if
o whenever Y is E-complete and Y satisfies P , then so does any retract of Y*
* ;
and
o if X ! Y ! Z is a cofibration of E-complete spectra and two of X, Y , and
Z satisfy P , then so does the third.
In other words, a property is generic if the full subcategory of all E-complete*
* spectra
satisfying it is thick.
Given a connective spectrum W , we write |W | for its connectivity.
We assume that our ring spectrum E satisfies the standard assumptions for con-
vergence of the E-based Adams spectral sequence_in other words, the assumptions
necessary for Theorem 15.1(iii) in [Ada74 , Part III]; see also Assumptions 2.2*
*.5(a)-
(c) and (e) in [Rav86 ].
Theorem 1.2. Let E be a ring spectrum as above, and consider the E-based Adams
spectral sequence E***(X) ) ss*(X). Fix a number m. The following properties of
an E-complete spectrum X are each generic:
(i)There exist numbers r and b so that for all s and t with s m(t - s) + b, *
*we
have Es;tr(X) = 0.
(ii)There exist numbers r and b so that for all finite spectra W with |W | = w
and for all s and t with s m(t - s - w) + b, we have Es;tr(X ^ W ) = 0.
Remark 1.3. (a)One usually draws Adams spectral sequences Es;trwith s on
the vertical axis and t - s on the horizontal; in terms of these coordinat*
*es,
the properties say that Es;tris zero above a line of slope m.
____________
Date: March 5, 1998.
1991 Mathematics Subject Classification. 55T15, 55P42.
Key words and phrases. Adams spectral sequence, vanishing line, generic.
1
2 M. J. HOPKINS, J. H. PALMIERI, AND J. H. SMITH
(b) Assuming that X is E-complete ensures that the spectral sequence converges,
which we need to prove the theorem. We do not need to identify the E2-term
of the spectral sequence, so we do not need to know that E is a flat ring
spectrum, for example.
We also mention one or two possible applications of the theorem. Since there
is a classification of the thick subcategories of the category of finite spectr*
*a (see
[Hop87 , HS , Rav92]), then if one is dealing with finite spectra X, one may be
able to identify all spectra with vanishing line of a given slope. For example*
*, in
the classical mod 2 Adams spectral sequence, since the mod 2 Moore spectrum
has a vanishing line of slope 1_2at the E2-term, then the mod 2n Moore spectrum,
and indeed any type 1 spectrum, has a vanishing line of slope 1_2at some Er-ter*
*m.
Similarly, any type n spectrum has a vanishing line of slope __1___2n+1-2at som*
*e Er-term
of the classical mod 2 Adams spectral sequence. Theorem 1.2 gives no control ov*
*er
the term r or the intercept b of the vanishing line.
Since the proof is formal, this theorem also applies in other stable homotopy
categories. The second author has used this result in an appropriate category of
modules over the Steenrod algebra to prove a version of Quillen stratification *
*for
the cohomology of the Steenrod algebra. See [Pala, Palb] for details.
2. Proof of Theorem 1.2
The difficulty in proving a result like Theorem 1.2 is that the Er-term of an
Adams spectral sequence does not have nice exactness properties if r 3_a cofi-
bration of spectra does not lead to a long exact sequence of Er-terms, for inst*
*ance.
So we prove the theorem by showing that the purported generic conditions are
equivalent to other conditions on composites of maps in the Adams tower, and
then we show that those other conditions are generic.
We start by describing the standard construction of the Adams spectral sequen*
*ce,
as found in [Ada74 , III.15],_[Rav86 , 2.2], and any number of other places. Gi*
*ven
a ring spectrum E, we let E denote the fiber of the unit map S0 -! E. For any
integer s 0, we let
__^s
FsX = E ^ X;
__^s
KsX = E ^ E ^ X:
We use these to construct the following diagram of cofibrations, which we call *
*the
Adams tower for X:
X _______F0X --g-- F1X --g-- F2X --g-- : :::
?? ? ?
y ?y ?y
K0X K1X K2X
This construction satisfies the definition of an "E*-Adams resolution" for X, as
given in [Rav86 , 2.2.1]_see [Rav86 , 2.2.9]. Note also that FsX = X ^ FsS0, and
the same holds for KsX_the Adams tower is functorial and exact.
Given the Adams tower for X, if we apply ss*, we get an exact couple and hence
a spectral sequence. This is called the E-based Adams spectral sequence. More
VANISHING LINES 3
precisely, we let
Ds;t1= sst-sFsX;
Es;t1= sst-sKsX:
If we let g :Fs+1X -! FsX denote the natural map, then g* = sst-s(g) is the map
Ds+1;t+11-!Ds;t1. Then we have the following exact couple (the pairs of numbers
indicate the bidegrees of the maps):
(-1; -1)
Ds;t1_____________________oDs+1;t+11e
@
@
(0; 0)@@ (1; 0)
@@R
Es;t1
This leads to the following rth derived exact couple, where Ds;tris the image of
gr-1*, and the map Ds+1;t+1r-!Ds;tris the restriction of g*:
(-1; -1)
Ds;tr_____________________oDs+1;t+1re
@
@
(r - 1; r -@1)@ (1; 0)
@@R
Es+r-1;t+r-1r
Unfolding this exact couple leads to the following exact sequence:
(2.1) : :-:!Es;t+1r-!Ds+1;t+1r-!Ds;tr-!Es+r-1;t+r-1r-!: :::
Fix a number m. With respect to the E-based Adams spectral sequence E***(-),
we have the following conditions on a spectrum X:
(1) There exist numbers r and b so that for all s and t with s m(t - s) + b, *
*the
map gr-1*:sst-s(Fs+r-1X) -!sst-s(FsX) is zero. (In other words, Ds;tr(X) =
0.)
(2) There exist numbers r and b so that for all s and t with s m(t - s) + b, *
*we
have Es;tr(X) = 0.
(3) There exist numbers r and b so that for all finite spectra W with |DW | = *
*-w
and for all s with s mw + b, then the composite W -! Fs+r-1X -! FsX is
null. (Here, DW denotes the Spanier-Whitehead dual of W .)
(4) There exist numbers r and b so that for all finite spectra W with |W | = w
and for all s and t with s m(t - s - w) + b, we have Es;tr(X ^ W ) = 0.
Each condition depends on a pair of numbers r and b, and we write (1)r;bto mean
that condition (1) holds with the numbers specified, and so forth.
Notice that if m = 0, then condition (3) says that Fs+r-1X -! FsX is a phantom
map whenever s b. If m = 0, then condition (1) says that Fs+r-1X -! FsX is a
ghost map (zero on homotopy) whenever s b.
Lemma 2.2. Fix numbers m, r, and b. We have the following implications:
(a) If r -m, then (1)r;b) (2)r;b+r-1. If r < -m, then (1)r;b) (2)r;b-m.
4 M. J. HOPKINS, J. H. PALMIERI, AND J. H. SMITH
(b) If r 1 - m, then (2)r;b) (1)r;b-m. If r < 1 - m, then (2)r;b) (1)r;b-r+1.
(c) If r -m, then (3)r;b) (4)r;b+r-1. If r < -m, then (3)r;b) (4)r;b-m.
(d) If r 1 - m, then (4)r;b) (3)r;b-m. If r < 1 - m, then (4)r;b) (3)r;b-r+1.
(Obviously, (3)r;b) (1)r;band (4)r;b) (2)r;b, but we do not need these facts.)
Proof.As above, we write g for the map Fs+1X -! FsX and g* for the map
Ds+1;t+11-!Ds;t1, so that Ds;tris the image of
gr-1*:sst-sFs+r-1X -! sst-sFsX:
(a): Assume that if s m(t - s) + b, then
gr-1*:sst-s(Fs+r-1X) -!sst-s(FsX)
is zero; i.e., Ds;tr= 0. In the case r -m, if s m(t - s) + b, then s + r
m((t+r-1)-(s+r))+b; so we see that Ds+r;t+r-1r= 0. By the long exact sequence
(2.1), we conclude that Es+r-1;t+r-1r= 0 when s m(t - s) + b. Reindexing, we
find that Ep;qr= 0 when p m(q - p) + b + r - 1; i.e., condition (2)r;b+r-1hold*
*s.
The case r < -m is similar; in this case, the long exact sequence implies that
Es;t+1r= 0.
(b): Assume that r 1 - m. If Es;tr(X) = 0 whenever s m(t - s) + b, then
Es+r-1;t+r-2r(X) = 0 when s m(t - s) + b. So by the exact sequence (2.1),
we see that Ds+1;tr-!Ds;t-1ris an isomorphism under the same condition. This
map is induced by g*: sst-s-1Fs+1X -! sst-s-1FsX, so we conclude that when
s m(t - s) + b, we have
lim-qsst-s-1FqX = Ds;t-1r;
lim-1qsst-s-1FqX = 0:
But by convergence of the spectral sequence, we know that lim-qsst-s-1FqX = 0,
so Ds;t-1r= imgr-1*= 0. Reindexing gives Dp;qr= 0 when p m(q + 1 - p) + b;
i.e., (2)r;bimplies (1)r;b+m.
If r < 1 - m, then a similar argument shows that Ds-r+1;t-r+1r= 0. __
Parts (c) and (d) are similar. |__|
It is easy to prove Theorem 1.2, once we have the lemma.
Proof of Theorem 1.2.The proofs of the genericity of the two statements are sim-
ilar, so we only prove that condition (i) is generic.
We know by Lemma 2.2 that condition (i) is equivalent, up to a reindexing, to
(*) There exist numbers r and b so that for all s and t with s m(t - s) + b, *
*the
map gr-1: Fs+r-1X -! FsX is zero on sst-s.
We show that this condition is generic. Since the Adams tower is functorial, if*
* Y
is a retract of X, then the Adams tower for Y is a retract of the Adams tower f*
*or
X. So if Fs+r-1X -! FsX is zero on sst-s, then so is Fs+r-1Y -! FsY . (Given
VANISHING LINES 5
St-s -!Fs+r-1Y , then consider
St-s ----! Fs+r-1Y ----! FsY
?? ?
yi ?yi
Fs+r-1X ----! FsX
?? ?
yj ?yj
Fs+r-1Y ----! FsY
Since sst-sFs+r-1X -! sst-sFsX is 0, then the map St-s -! FsX is null. But
St-s -!FsY factors through this map, and hence is also null.)
Given a cofibration sequence X -! Y -! Z in which X and Z satisfy conditions
(*)r;band (*)r0;b0, respectively, we show that Y satisfies (*)r+r0-1;max(b;b0-*
*r+1).
Consider the following commutative diagram, in which the rows are cofibrations:
Fs+r+r0-2X ----! Fs+r+r0-2Y ----! Fs+r+r0-2Z
?? ? ?
y ?yff ?yfi
Fs+r-1X ----! Fs+r-1Y ----! Fs+r-1Z
?? ? ?
y fl ?yffi ?y
FsX ----! FsY ----! FsZ
We assume that s m(t - s) + max(b; b0- r + 1), so that we have
s m(t - s) + b;
s + r - 1 m(t - s) + b0:
If we map St-s into this diagram, then since sst-sfi = 0, any map
St-s -!Fs+r+r0-2Y -ff!Fs+r-1Y
factors through Fs+r-1X. Since sst-sfl = 0, though, then the composite
St-s -!Fs+r+r0-2Y -ff!Fs+r-1Y -ffi!FsY
is null. *
* __
This shows that condition (*), and hence condition (i), is generic. *
* |__|
The same proof, in the case m = 0, also shows the following (using the langua*
*ge
of [Chr97]).
Corollary 2.3.If I is an ideal of maps that is part of a projective class, then*
* the
following property is generic for E-complete spectra X:
o There exist numbers r and b so that for all s b, the composite
gr-1: Fs+r-1X -! FsX
is in I.
6 M. J. HOPKINS, J. H. PALMIERI, AND J. H. SMITH
References
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[Rav86]D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, *
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Department of Mathematics, Massachusetts Institute of Technology, Cambridge,
MA 02139
E-mail address: mjh@math.mit.edu
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
E-mail address: palmieri@member.ams.org
Department of Mathematics, Purdue University, West Lafayette, IN 47907
E-mail address: jhs@math.purdue.edu