Self-maps of modules over
the Steenrod algebra
John H. Palmieri
April 1992
Journal of Pure and Applied Algebra 79 (1992) 281-291
Abstract
For a finite module M over the Steenrod algebra A, we prove
the existence of a non-nilpotent element in Exts;tA(M; M) "parallel to
the vanishing line." We use this result to give a proof of Margolis'
construction to kill Pts-homology groups, at all primes.
Let A be the mod p Steenrod algebra. It is well-known (see [8], for example)
that its dual A* is isomorphic (as an algebra) to F2[1; 2; : :]:when p = 2,
and to Fp[1; 2; : :]: E[o0; o1; : :]:when p is odd. Define the Milnor basis
of A to be the dualsto the monomial basis of A*; let Ptsbe the Milnor basis
element dual to pt, and, when p is odd, let Qt be dual to ot. It is easy to
check that when s < t then (Pts)p = 0, and for p odd, (Qt)2 = 0 for all t.
Therefore we call these elements differentials, and given any differential x
and any A-module M we can define the homology of M with respect to x
by:
ker Pts: M ! M
H(M; Pts) = _____________________s;
im (Pt)p-1 : M ! M
and for p odd
ker Qt : M ! M
H(M; Qt) = _________________:
im Qt : M ! M
1
Given a differential x, define its slope, slope(x), by
p|Pts|
slope(Pts) = _____;
2
and
slope(Qt) = |Qt|:
(This notation is motivated by Theorem 1.4.) The differentials are linearly
ordered by slope; say that a module M is type (also written M =
M) if and only if H(M; x) = 0 whenever slope(x) < m or slope(x) > n.
We prove the following theorem.
Theorem A Fix a differential x with slope(x) = m. Given a finite module
M = M with H(M; x) 6= 0, there is an element v 2 Extk;kmA(M; M)
for some k which is non-nilpotent under Yoneda composition.
This is a Steenrod algebra analog of the theorem of Hopkins and Smith that
any finite p-local spectrum X with K(n - 1)*(X) = 0 and K(n)*(X) 6= 0 has
a vn-map, a non-nilpotent self-map that induces an isomorphism on K(n)-
homology (see [4]). (In the appropriate setting, the map v in Theorem A
induces an isomorphism on x-homology.) Theorem A is a generalization of a
result used by Hopkins and Smith to prove the theorem for spectra. We ac-
tually prove Theorem A for a slightly larger collection of modules than finite
ones, namely for "stably finite" modules (see Section 1). In Section 3, for
each differential x we construct a stably finite module M = M
with H(M; x) 6= 0, solving the algebraic analog of the problem of construct-
ing a finite spectrum with a vn-map. Note that for each x one can also
construct a finite module M of this sort, by taking a tensor product of "pth
powers" of sub-Hopf algebras A(n) of A for particular n's, using Mitchell's
A-module structure. At the prime 2, for example, the module A(1) A(1)
(where is the doubling functor) is a module of type <7; 1>. See [10].
Using Theorem A, we give a simple proof of
Theorem B ([6 ] for p = 2) Given any bounded below module M and any
integer m 1,
(a) there exists a bounded below module N = N and a map f : M !
N so that H(f; x) is an isomorphism if slope(x) m;
2
(b) there exists a bounded below module L = L<1; m - 1> and a map g :
L ! M so that H(g; y) is an isomorphism if slope(y) < m.
In other words, we describe how to kill off homology groups for initial and
terminal intervals of differentials in the linear ordering by slope. Theorem B
is an important step in developing a chromatic picture for modules over the
Steenrod algebra (see [10]).
All of the main results in this paper hold when A is a sub-Hopf algebra of
the Steenrod algebra, as long as one restricts attention to the differentials
contained in A. For convenience, we state and prove everything in terms
of the full Steenrod algebra. For the more general case some changes are
needed; in particular, Proposition 1.9 will not hold, so Proposition 1.10 will
only be valid for s 0. The reader can work out the necessary changes in
the proofs of the main results.
In Section 1 we review some properties of the x-homology groups and the
stable category of modules over the Steenrod algebra; we also introduce the
category of stably finite modules. In Section 2 we prove Theorem A, and in
Section 3 we prove Theorem B.
I would like to extend my thanks to my thesis advisor Haynes Miller, for
teaching me about the Steenrod algebra, and to Mike Hopkins, for teaching
me about self-maps, and how to use them to prove results like Theorem B.
1 Preliminaries
We need a few facts about the Pts- and Qt-homology groups defined above.
For the differentials x with x2 = 0, given a short exact sequence of modules,
there is a long exact sequence in homology; there is a variant (which we will
not use explicitly) when xp = 0, for p > 2. We have the following two basic
results; in [6] Margolis proves both of these at the prime 2, and his proofs
extend easily to the general case. Given two A-modules, their tensor product
is an A-module via the diagonal action. There is not a K"unneth formula for
x-homology in general, but the following weaker result does hold.
3
Proposition 1.1 (19.18 in [6 ], for p = 2) Let L, M, and N be bounded
below A-modules, and let x 2 A be a differential.
(a) If H(M; x) = 0 or H(N; x) = 0, then H(M N; x) = 0.
(b) If f : M ! N induces an isomorphism on x-homology, then so does
f 1L : M L ! N L.
Proposition 1.2 (19.16 in [6 ], for p = 2) Given a differential x 2 A and
an inverse system of modules Mff, with {|Mff|} bounded below, lim1Mff= 0,
and lim1H(Mff; x) = 0, then
H(lim Mff; x) = lim H(Mff; x):
We also recall these two much deeper results:
Theorem 1.3 ([1 ], [9 ])Let M be a bounded below A-module. Then M is
free if and only if H(M; x) = 0 for all differentials x 2 A.
Theorem 1.4 ([2 ], [7 ])If M = M is a bounded below finite type
module, then Exts;tA(M; Fp) = 0 if t < ms - c, where c = e - |M| with e
independent of M.
Remark 1.5 The statement of the main theorem in [7] is not this precise
regarding the form of the intercept c; the proof is, though.
Henceforth, the slope of a line in Ext s;twill refer to its slope in (s; t)-
coordinates; for example, the line t = ms - c in Theorem 1.4 has slope m.
Similarly, when we speak about phenomena "above" or "below" such a line,
we mean in these coordinates; hence, Theorem 1.4 says that Exts;tA(M; Fp) is
0 below a line of slope m. Note that this language conflicts with the usual
pictures of Ext s;t, in (t - s; s)-coordinates, where a line with slope m in our
sense would be written s = __1_m-1(t - s) + d, and Exts;tA(M; Fp) would be 0
above this line.
4
It is convenient to work in the stable category of A-modules, as defined in
[6]: the objects of this category are bounded below A-modules, and the
morphisms are given by a bigraded group, written {M; N}*;*A. Define
{M; N}0;tA = Hom tA(M; N)= '
= Hom 0A(M; tN)= ' ;
where f ' 0 ("f is stably trivial") if and only if f factors through a projecti*
*ve
module. To define {M; N}s;tAwith s 6= 0, we need more notation: given M,
define an A-module M by the short exact sequence
0 ! M ! A M ! M ! 0;
and let sM = (s-1M) (and 0M = M by convention). Then define
( 0;t
{sM ; N}A if s 0,
{M; N}s;tA= -s 0;t
{M; N }A if s 0:
We say that M is stably equivalent to N, written M ' N, if there exist
f : M ! N and g : N ! M with f O g ' 1N and g O f ' 1M .
Remark 1.6 (1) Note that if there is a short exact sequence of A-modules
0 ! K ! P ! M ! 0
with P projective, then K ' M. We will use M to denote any of
these equivalent modules.
(2) For a differential x 2 A with x2 = 0, then using the long exact sequence
in homology and Theorem 1.3, for any k we have
H(kM; x) ~=k.slope(x)H(M; x);
for a general differential x, we have
H(2kM; x) ~=2k.slope(x)H(M; x):
(3) To build sM for s = 1; 2; 3; : :,:one constructs a projective resolution
for M. So the next result should not be surprising.
5
Proposition 1.7 ([6 ], 14.8) If M and N are bounded below A-modules and
s > 0, then Exts;tA(M; N) ~={M; N}s;tA.
In fact, the collection of stable maps satisfies most of the nice homological
properties of Ext. We will make free use of these without statement or proof;
see Chapter 14 in [6] for a reference.
We need a generalization of "finite module" suitable for use with the stable
category. We certainly want to include all finite modules; we also need to
include modules which are stably equivalent to finite ones. We also want
to be able to take kernels and cokernels. Motivated by work related to the
nilpotence theorem (see [3], for example), we make the following definitions.
Call a property P on the category of bounded below A-modules stably generic
if the following conditions are satisfied:
(a) If M N satisfies P , then so does M.
(b) If 0 ! L ! M ! N ! 0 is a short exact sequence of A-modules, and
two of L, M, and N satisfy P , then so does the third.
(c) If M is stably equivalent to N and N satisfies P , then M satisfies P .
(d) If M satisfies P , then kM satisfies P for all k 2 Z.
Let F be the smallest sub-category of the stable category of A-modules which
contains all finite A-modules, so that the property "M 2 F" is stably generic.
If M 2 F, we say that M is stably finite. By definition of F, we have
Proposition 1.8 If P is a stably generic property which holds for all finite
A-modules, then P holds for all modules in F.
We combine these definitions with the earlier results on the Steenrod algebra
to get a few simple corollaries; first, we need this proposition.
Proposition 1.9 If M is finite and N is any A-module, then for s < 0,
{M; N}s;tA= 0.
6
Proof: We show that if M is a finite module and N is any module, then
Hom A(M; N) = 0; this is certainly sufficient. For every x 2 M, there is an
n such that if r > n then Sqrx = 0. On the other hand, since N ,! P for
some projective P , then for every y 2 N and for every n, there is an r > n
such that Sqry 6= 0. Therefore there can be no non-trivial A-maps from M
to N. 2
This gives an immediate generalization of the vanishing line theorem (1.4).
Proposition 1.10 If M and N are stably finite with M = M, then
{M; N}s;tA= 0 if t < ms - c, for some c depending on |M|, m, and N.
Using stable module language, Theorem 1.4 can be "realized" at the module
level; for an A-module M, let ||M|| be the stable connectivity of M, defined
by ||M|| = |{M; Fp}0;*A|. (This is also the connectivity of a module stably
equivalent to M which contains no free summands_see [6].)
Proposition 1.11 (22.6 in [6 ], for p = 2) For each m there is an integer
e so that for any bounded below module M, we have M = M if and
only if ||sM|| |M| - e + sm for all s 0.
Proof: First, assume M = M. By Theorem 1.4, for all s 1 we have
Exts;tA(M; Fp) = {sM ; Fp}0;tA= 0 when t < sm-e+|M|, for some e = e(m).
Therefore, ||sM|| |M| - e + sm.
Conversely, assume that H(M; x) 6= 0 for some x with slope(x) < m.
Then H(2sM; x) = 2s.slope(x)H(M; x). So for any e, for s 0, we have
||2sM|| |H(2sM; x)| = |H(M; x)| + 2s . slope(x) < |M| - e + 2sm. 2
2 Constructing self-maps
In this section we prove Theorem A, generalized to stably finite modules
(Theorem 2.2). First, we need to describe some particular self-maps. Let
7
A(n) be the sub-Hopf algebra of A generated (as an algebra) by {Pts: s + t
n + 1} for p = 2, and by {Pts: s + t n} [ {Qt : t n} for p odd. Given
a differential x in A(n), let C(x) be the subalgebra of A(n) generated by
x (either Fp[x]=xp for x = Pts, or E[x] for x = Qt). Then Ext**C(x)(Fp ; Fp)
contains as a subalgebra a polynomial algebra on a generator v(x) in bidegree
(2; 2 . slope(x)) (see [7], for example). We need this result, due to Lin at the
prime 2 and to Wilkerson in the general case.
Lemma 2.1 ([5 ],[11 ]) Let x be a differential in A(n). Then we can lift some
power of v(x) through the restriction map Exts;tA(n)(Fp ; Fp) ! Exts;tC(x)(Fp ;*
* Fp).
Now we come to our main result, that modules in F have self-maps parallel to
their vanishing lines. Part (a) is a slight generalization of a result of Hopki*
*ns
and Smith ([4]).
Theorem 2.2 Fix a differential x in A with slope(x) = m. Assume that
M ' M is stably finite with H(M; x) 6= 0 (i.e., M 6= M).
(a) Then for some k there is a map v : kM ! km M which is non-
nilpotent in {M; M}*;*A; H(v; x) is an isomorphism, and H(v; y) = 0
when y 6= x.
v -2km 2k v -km k v
(b) Let K = lim-(. .-.--! M---! M---! M). Then K
is bounded below, H(K; y) = 0 when y 6= x, and the map K ! M
induces an isomorphism in H(-; x). We denote K by M.
'
(c) M is loop-periodic of period k, via v : -km kM---! M:
(d) For stably finite N, we have {M ; N}*;*A~=v-1{M; N}*;*A.
Proof: First we construct the element v 2 {M; M}k;kmA. By Proposition 1.10,
{M; M}*;*Ahas a vanishing line of slope m. If n is large enough so that
x 2 A(n), we have an isomorphism in a band parallel to the vanishing line:
{M; M}s;tA~={M; M}s;tA(n)
8
for t < ms - k(n), where k(n) is some constant depending on n (and on m
and M). Indeed, as n ! 1 we have k(n) ! 1, so that for n large enough,
the line t = ms is in this band.
If x 2 A(n), then by Lemma 2.1 we can construct a non-nilpotent element
w 2 {Fp ; Fp}k;kmA(n).
Now choose n 0. Let C(x) be the subalgebra of A(n) generated by x.
Consider this diagram:
ae s;t s;t
{M; M}s;tA ---! {M; M}A(n)x ---! {M; M}C(x)x
??-M ??-M
{Fp ; Fp}s;tA(n)---! {Fp ; Fp}s;tC(x)
Since H(M; x) 6= 0, then {Fp ; Fp}*;*C(x)---!{M; M}*;*C(x)is injective; the di-
agram clearly commutes, so the element w 2 {Fp ; Fp}k;kmA(n)maps to a non-
nilpotent element v 2 {M; M}k;kmA(n). Since the restriction map ae is an isomor-
phism in that bidegree (and in all bidegrees (j; jm)), we have a non-nilpotent
element v 2 {M; M}k;kmA.
We want our self-map v to satisfy certain homology properties, so we may
have to use a power of v: we claim that we can choose j so that vj induces
an isomorphism in x-homology, and H(vj; y) = 0 when y 6= x. The first of
these is easy_multiplication by v induces an isomorphism on {M; M}*;*C(x),
so H(v; x) must be an isomorphism (so any choice of j will work). Now
choose j so that vj lies below the line with slope m + 1 and intercept given
by the minimum degree of a (vector-space) map from H(M; y) to itself, for
all y 6= x. (By genericity, (max deg H(M; y)-min deg H(M; y)) is uniformly
bounded, so this minimum exists.) Then for all y with slope(y) > slope(x),
the restriction of vj to {M; M}*;*C(y)will be zero; hence H(vj; y) = 0. This
finishes (a).
Let v 2 {M; M}k;kmAbe the map just constructed. Let K = lim-(. . .!
-2km 2kM ! -km kM ! M). By part (a) and Proposition 1.2 we have
(
H(M; x) if y = x
H(K; y) =
0 if y 6= x
9
There is obviously a map from K to M which is an isomorphism on H(-; x);
thus to finish part (b) we only need to show that K is bounded below. This
follows from Proposition 1.11: there exists an e independent of M and a
choice of rM so that |-rm rM| |M| - e for all r; therefore the tower is
uniformly bounded by |M| - e, so the inverse limit is bounded below. This
finishes part (b).
Applying v to the tower that defines K clearly induces a stable equivalence
of the inverse limits, proving (c).
Part (d) follows from another vanishing line argument_for any stably finite
module N, there is a vanishing line in {M; N}*;*Aof slope m; we have an
isomorphism between {M; N}s;tAand {M ; N}s;tAbelow a line of slope at
least m + 1. By part (c), we know that {M ; N}*;*Ais v-periodic, and this
isomorphism says that {M; N}*;*Ais v-periodic below a line of slope at least
m + 1. Hence inverting v extends the periodicity to the whole plane, yielding
the result. 2
3 Margolis' killing construction
Our goal here is to prove the following theorem:
Theorem 3.1 (21.1 in [6 ], for p = 2) Given any bounded below A-module
M and any integer m 1,
(a) there exists a bounded below module N = N and a map f : M !
N so that H(f; x) is an isomorphism if slope(x) m;
(b) there exists a bounded below module L = L<1; m - 1> and a map g :
L ! M so that H(g; x) is an isomorphism if slope(x) < m.
Given a module M, if we have N and f : M ! N as in part (a) of the
theorem, we say that N is of type M, and write N = M.
Similarly we write L = M<1; m - 1> if we have L and g as in part (b). Given
10
a module M and two integers k m, we can also construct a module of type
M, as
M ! M M
or
M M<1; m> ! M;
where the maps are as in the theorem. We also write M for M. One
can check that for any integers 0 < k < m 1, the modules M and
M are unique up to stable equivalence.
Proof of Theorem 3.1: (a) First of all, it suffices to prove the theorem in
the case M = Fp: if we have Fp ! Fp, then given any module M we
have
M = M Fp ! M Fp:
By Proposition 1.1, M Fp is of type , and the map is an
isomorphism on H(-; y) when slope(y) m.
So we need to construct Fp. Let {x0; x1; : :;:xj} be the differentials
with slope less than m; let si = slope(xi), and assume that s0 < s1 < . .<.sj.
Lemma 3.2 For 0 i j + 1, there are stably finite modules Mi, with
M0 = Fp and Mi = Mi, so that there are short exact sequences
fi
0---! -kisikiMi---! Mi---! Mi+1---! 0;
for some integers ki > 0.
Proof: This is a simple consequence of Theorem 2.2_given Mi = Mi,
we construct a self-map fi 2 {Mi ; Mi}ki;kisiA; then we have fi : -kisikiMi !
Mi. Since kiMiis a sub-module of a projective module, we may arrange that
fiis injective by replacing Miby a stably equivalent module. Let Mi+1be the
cokernel of fi. We know that H(fi; xi) is an isomorphism and H(fi; y) = 0 for
y 6= xi; thus by the long exact sequences in homology, H(Mi+1; xi) = 0 and
H(Mi+1; y) 6= 0 when slope(y) > si. The lemma follows by induction. 2
__
Of course,_Mj+1 depends on the (j + 1)-tuple k = (k0; k1; : :;:kj); we write
Mj+1(k) to distinguish them. Assume that at the ith stage one has self-maps
11
f; g 2 {Mi ; Mi}*;*Aof bidegrees (k; sik) and (`; si`) respectively, and let Mi*
*+1;k
and Mi+1;`denote their cokernels. If k < `, then one can check that there is
a map Mi+1;`!_Mi+1;k; hence increasing the ki's gives us an inverse system
of the Mj+1(k)'s. We claim that
__
Fp = lim-Mj+1(k):
Let N denote the inverse limit; we need to check that the effect of Fp ! N
in homology is right, and that the N is bounded below.
For homology, we give the argument for H(-; y) where y2 = 0; the other
case follows similarly. Using the long exact sequence in homology and_the
fact that H(fi; y) = 0, at the ith stage in the construction of Mj+1(k), we
have
0 ! H(Mi; y) ! H(Mi+1; y) ! -slope(y)H(-sikikiMi; y) ! 0:
If slope(y) < m, then by Proposition 1.2 we see that
__
H(lim Mj+1(k); y) = 0:
If slope(y) m, then by induction_we have a vector space isomorphism
between the y-homology of Mj+1(k) and an exterior algebra:
__ Oj
H(Mj+1(k); y) ~= E[zi];
i=0
with |zi| = ki(slope(y) - si) - slope(y). We have inclusions
H(Fp; y) = H(M0; y) ,! H(M1; y) ,! . .,.! H(Mj+1; y);
so by Proposition 1.2 we have
__
H(Fp; y) ~=H(lim Mj+1(k); y):
Thus the homology of the inverse limit is what it should be.
__
By applying_Proposition 1.11 to ki-1Mi(k), we can construct an_inclusion_
-sikikiMi(k)_,! P , with P projective_and |P | -e(si)-si+|Mi(k)|; thus
|Mi+1(k)| |P | -e(si) - si+ |Mi(k)|. By induction, then, we have
__ X
|Mj+1(k)| - (e(si) + si);
i
12
__
for any k; therefore N is bounded below. This finishes part (a).
(b) Again, it suffices to construct Fp<1; m - 1>. By part (a) we have f :
Fp ! Fp. By replacing Fp with a (bounded below) stably equivalent
module, we may assume that f is surjective. Let L be the kernel of f; then
it is easy to check that L is a module of type Fp<1; m - 1>. 2
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Massachusetts Institute of Technology
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