NILPOTENCE FOR MODULES OVER
THE MOD 2 STEENROD ALGEBRA I
JOHN H. PALMIERI
Abstract.Let B be a sub-Hopf algebra of the mod 2 Steenrod algebra, and *
*let
M be a finite dimensional B-module. We prove that one can detect nilpote*
*nce
of elements in Ext**B(M; M) by restricting to elementary sub-Hopf algebr*
*as,
and we prove a similar result at odd primes.
1.Introduction
In [4], Devinatz, Hopkins, and Smith proved the nilpotence theorem, a remark-
able result which provides algebraic means for detecting nilpotence in the coll*
*ection
of homotopy classes of self-maps of any finite spectrum. This theorem has many
important consequences, and so has opened up new approaches to studying homo-
topy theory (see [19, 10, 7], for example). See [20] for a thorough discussion *
*of this
material; [5] also gives an overview of the nilpotence theorem and related resu*
*lts.
Let A be the mod 2 Steenrod algebra, and let M be a finite A-module. In
this paper we show that there is an analogous result, Theorem 1.1, for detecting
nilpotence in Ext**A(M; M). We hope that this leads to structure theorems for t*
*he
category of finite A-modules, comparable to those for finite spectra in [8]. We
begin to develop this material in a sequel [17]. Also, the nilpotence theorem f*
*or A-
modules extends the strong parallel between results in stable homotopy theory a*
*nd
results for A-modules, as described in [11] and [16]. Some of these earlier A-m*
*odule
results have been used to prove results in homotopy theory via the Adams spectr*
*al
sequence, as in [10] and [18]; we hope that one can do likewise with Theorem 1.*
*1.
In order to state our main theorem, we need a few definitions. Given a Hopf
algebra B over a field k of characteristic p, an elementary sub-Hopf algebra E *
* B
is a bicommutative sub-Hopf algebra such that ep = 0 for all e 2 IE. (Here, IE
denotes the augmentation ideal of E, IE = ker(" : E ! k).) Of course, given any
Hopf algebra inclusion C : C ! B and B-modules L and M, we have a restriction
map *C : Ext**B(L; M) ! Ext**C(L; M). Note that if is a (coassociative) B-
coalgebra (so that there are B-module maps : ! and " : ! k making
certain diagrams commute), then Ext**B(; k) is an (associative) algebra, via the
map *. We prove the following result.
Theorem 1.1. Let B be a sub-Hopf algebra of the mod 2 Steenrod algebra A.
(a) Let be a bounded below coassociative B-coalgebra; fix z 2 Ext**B(; F2).
Then z is nilpotent if and only if *E(z) is nilpotent for all elementary*
* sub-
Hopf algebras E of B.
1
2 JOHN H. PALMIERI
(b) Let M be a finite dimensional B-module; fix z 2 Ext**B(M; M). Then z is
nilpotent under Yoneda composition if and only if *E(z) is nilpotent for*
* all
elementary sub-Hopf algebras E of B.
(c) Let L and M be B-modules with L bounded below and M finite dimen-
sional; fix z 2 Ext**B(L; M). Then zn 2 Ext**B(Ln ; Mn ) is zero for n
sufficiently large if and only if (*E(z))nE is zero for nE sufficiently*
* large,
for all elementary sub-Hopf algebras E of B.
We also prove a similar result about detecting nilpotence for sub-Hopf algebr*
*as
of the Steenrod reduced powers at an odd prime; see Section 4.
In [21] Wilkerson proved the special case of (b) where B is finite dimensiona*
*l and
M = F2; Lin [9] proved this for a more general collection of sub-Hopf algebras *
*(those
with finite "profile" function_see Section 2), again only for M = F2. Hopkins a*
*nd
the author [6] generalized this to arbitrary sub-Hopf algebras of A, but still *
*for
M = F2. These are analogous to proving Nishida's theorem [15] for various B's.
We view Theorem 1.1 as analogous to the nilpotence theorem of [4]; it is also
directly analogous to a result of Carlson [3] regarding detection of nilpotent *
*elements
in Ext*kG(M; M) for G a finite group, k a field of characteristic p > 0, and M *
*a kG-
module. This opens up other lines of questions; for example, what other results*
* for
group cohomology also hold for the Steenrod algebra (or for other Hopf algebras*
*)?
What sorts of structure theorems for modules over the Steenrod algebra carry ov*
*er
to give interesting results in group theory? Are there interesting parallels be*
*tween
kG-modules and (finite) spectra? Wilkerson [21] has discussed some aspects of t*
*he
first question.
Our other main results are Corollary 2.7 and Theorem 3.1; we would have to
introduce too much notation to state them precisely here, so we only give rough
descriptions. Let B be a finite dimensional graded connected cocommutative Hopf
algebra over a finite field k. Corollary 2.7 is the generalization of Wilkerso*
*n's
result [21, Corollary 5.6] about detection of nilpotent elements in Ext**B(k; k*
*) to the
case Ext**B(M; M) for M any B-module. As far as Theorem 3.1 goes, if one views
restriction to the elementary sub-Hopf algebras of B in Theorem 1.1 as analogou*
*s to
applying Morava K-theories, then Theorem 3.1 gives an analog of the BP -version
(or the MU-version) of the nilpotence theorem; namely, it gives a single sub-Ho*
*pf
algebra D of B so that restriction to D detects nilpotence.
Our proof is a combination of the work about Hopf algebra cohomology in [9]
and [21] with the proof of Carlson's theorem as presented in [2]; we also need a
variant on an argument in [6]. In Section 2 we give some background about the
Steenrod algebra, its sub-Hopf algebras, and their cohomology; we also prove our
main lemma, Lemma 2.4. In Section 3 we prove the main theorem. In Section 4
we discuss the odd prime case; we summarize which of the results still hold, and
we indicate the changes necessary in the proofs.
I would like to thank the topologists at the University of Minnesota for their
hospitality while this research was being conducted. Also, the notion of "quas*
*i-
elementary Hopf algebra" in Section 4 is due to Haynes Miller and Clarence Wilk-
erson; I would like to thank them for pointing it out to me.
NILPOTENCE OVER THE STEENROD ALGEBRA I 3
2. Preliminaries and the main lemma
In this section we recall some notions about the mod 2 Steenrod algebra, and *
*we
prove our main lemma, Lemma 2.4. [13] and [11] are basic references for results*
* on
the Steenrod algebra.
Recall that the mod 2 Steenrod algebra A is a Hopf algebra; its dual is isomo*
*r-
phic, as an algebra, to F2[1; 2; : :]:.sLet the Milnor basis be the dual to the*
* mono-
mial basis; as usual, we set Pts= (2t)*. Given a sequence (n1; n2; : :):where e*
*achnni
is either a non-negative integer or 1, consider the subspace (F2[1; 2; : :]:=(2*
*i ))i*
of A. Under certain conditions on the sequence of ni's, this is a sub-Hopf alge*
*bra
of the Steenrod algebra_see [1] or [11], for example. Call the corresponding su*
*b-
Hopf algebra A(n1; n2; : :):; the sequence (n1; n2; : :):is the profile for thi*
*s sub-Hopf
algebra. This is the main result we need:
(a) If (n1; n2; : :):is a non-decreasing sequence of elements of {0; 1; 2; :*
* :}:[
{1}, then it is the profile function of a normal sub-Hopf algebra of A_s*
*ee
[11].
Here are some examples:
(b) Let D = A(1; 2; 3; : :):(Lin [9] called this Ad).
(c) Fix r 1. Let D0= A(0;_0;_:-:;:0z___"; r; r + 1; : :):.
r-1
(d) Fix m 0. Let E(m) = A(0;_0;_:-:;:0z___"; m+1; m+1; m+1; : :):. Then E(m)
m
is a sub-Hopf algebra of A, and (as an algebra) it is an exterior algebr*
*a (see
[9], for example). Indeed, the E(m)'s are the maximal elementary sub-Hopf
algebras of A.
(e) Fix m 0 and r > m. Let
E(m)0= A(0;_0;_:-:;:0z___"; m_+_1;_:_:;:m-+z1_____"; r; r + 1; r +*
* 2; : :)::
m r-m-1
Finally, we have one remark.
(f)By convention, we let n0 = 0. This is important in order to have the full
generality for Lemmas 2.2, 2.3 and 2.4.
Definition 2.1.Denote by hj;ithe element [2ij] in the cobar construction for A *
*(and
similarly for any sub-Hopf algebra B of A with profile function satisfying nj >*
* i).
If this element is a permanent cycle, then let hj;ialso denote the corresponding
element in Ext**A(F2 ; F2) (or Ext**B(F2 ; F2)).
For example, if E = A(n1; n2; : :):is an elementary sub-Hopf algebra of A (so
that there is some t so that ni= 0 for all i t, and ni t + 1 for all i > t), t*
*hen
Ext**E(F2 ; F2) ~=F2[hj;i: j 1; nj > i 0]:
Note that hj;iis in bidegree (1; 2i(2j- 1)).
Lemma 2.2. Let B = A(n1; n2; : :):be a sub-Hopf algebra of the Steenrod algebra
with n1 n2 . ...Let j 1 be an integer for which nj-1 < nj; then hj;qis a
nonzero element in Ext**B(F2 ; F2), for all q with nj-1 q < nj if nj is finite*
*, and
for all q with nj-1 q if nj = 1.
4 JOHN H. PALMIERI
Proof.This is a simple computation with the cobar construction. __|_|
Lemma 2.3. If B and j are as in Lemma 2.2 with nj < 1, then there is a Hopf
algebra extension
B0! B ! E[Pjnj-1];
where B0has the same profile sequence as B except that its jth coordinate is nj*
*-1,
rather than nj.
Proof.This is also straightforward; one only needs to know that B0 is a normal
sub-Hopf algebra of B. __|_|
The following is our key lemma, analogous to [21, 5.2] and [2, Vol. II, 5.2.1*
*].
Lemma 2.4. Let B = A(n1; n2; : :):be a sub-Hopf algebra of the Steenrod algebra
with n1 n2 . .,.and assume that there is an integer j 1 so that nj-1 < nj <
1. Let B0 = A(m1; m2; : :):be the sub-Hopf algebra defined by setting mi = ni if
i 6= j, and by setting mj = nj- 1.
(a) Let be a coassociative B-coalgebra, with structure maps : ! and
" : ! F2. Assume that z 2 Ext**B(; F2) is an element of positive homo-
logical degree which restricts to zero in Ext**B0(; F2); then z = "*(hj;*
*nj-1)z0
for some z0 2 Ext**B(; F2).
(b) Let M be a B-module. Assume that z 2 Ext**B(M; M) is an element of
positive homological degree which restricts to zero in Ext**B0(M; M); th*
*en
z = (hj;nj-1. 1M )z0 for some z0 2 Ext**B(M; M).
Proof.We prove this just as in [2]. We prove part (b); (a) is done similarly. S*
*et
x = Pjnj-1. We have a Hopf algebra extension
(1) B0! B ! E[x]:
Consider the following short exact sequence of E[x]-modules (and hence of B-
modules):
0 ! |x|F2 ! E[x] ! F2 ! 0:
Tensoring with M gives a short exact sequence
0
(2) 0 ! |x|M ! E[x] M --j--!M ! 0:
Corresponding to these we have hj;nj-12 Ext1;|x|B(F2 ; F2) and hj;nj-1. 1M 2
Ext1;|x|B(M; M).
Note that the composite
0)*
Ext **B(M; M)-(j---!Ext**B(E[x] M ; M) ~=Ext**B0(M; M)
is just the restriction map_see Section 2.8, Volume I, of [2], for instance. So*
* since
the restriction of z is zero, then we have (j0)*(z) = 0.
From the short exact sequence (2) we get a long exact sequence in Ext:
0)*
. .!.Extn-1B(M; M) --ffi--!ExtnB(M; M)(j----!ExtnB(E[x] M ; M) ! . .:.
Here ffi is Yoneda composition with hj;nj-1. 1M . So since (j0)*(z) = 0, we see*
* that
z = z0 O (hj;nj-1. 1M ) for some z0 2 Ext**B(M; M). The element hj;nj-1. 1M is
central in Ext**B(M; M), so this finishes the proof. __|_|
NILPOTENCE OVER THE STEENROD ALGEBRA I 5
Remark 2.5.Lemma 2.4 holds if we work with chain complexes of modules instead
of modules. For example, if C is a finite chain complex of finite A-modules and*
* if
z 2 Ext**B(C; C) restricts to zero in Ext**B0(C; C), then z = (hj;nj-1. 1C )z0 *
*for some
z0 2 Ext**B(C; C). See also Remarks 2.10 and 3.4.
We want to generalize Wilkerson's work in [21] on detecting nilpotence; we re*
*fer
the reader to that paper for more information. In particular, we use Wilkerson's
notation for Steenrod operations acting on Hopf algebra cohomology. Let B be a
finite dimensional graded connected cocommutative Hopf algebra over a finite fi*
*eld
k; let p be the characteristic of k. We make the following definition; we give *
*the
version for p = 2, since Wilkerson gives the odd prime version.
Q
Definition 2.6 ([21], DefinitionT5.3).he fundamental class of B is uB = v,
0 1
where the product ranges over all nonzero v 2 ker(fSq) \ ExtB(k; k).
Note that Lemma 2.4 can be stated more generally for finite dimensional Hopf
algebras over k, as in [2] or [21]; one replaces the Steenrod algebra condition*
*s by
conditions that ensure the existence of a Hopf algebra extension as in (1). Nam*
*ely,
0 1
given a nonzero v 2 ker(fSq)\Ext B(k; k), there is a normal Hopf algebra extens*
*ion
(1), where the polynomial generator of Ext*E[x](k; k) hits v under the map in E*
*xt
induced by B ! E[x]. There is also an odd prime version. See [21, Section 5] for
more details.
This is a generalization of [21, Corollary 5.6].
Corollary 2.7.Assume that every sub-Hopf algebra C of B is either elementary
or has uC nilpotent.
(a) Let be a coassociative B-coalgebra; fix z 2 Exts;tB(; k). Then z is nil*
*po-
tent if and only if *E(z) is nilpotent for all elementary sub-Hopf algeb*
*ras
E of B.
(b) Let M be a B-module; fix z 2 Exts;tB(M; M). Then z is nilpotent under
Yoneda composition if and only if *E(z) is nilpotent for all elementary *
*sub-
Hopf algebras E of B.
Proof.Use Wilkerson's argument for the proof of [21, Corollary 5.6], replacing *
*his
Lemma 5.2 with (the appropriate restatement of) Lemma 2.4. __|_|
In the next section we will need the next lemma and its corollary. One can pr*
*ove
a better result, a "relative vanishing line" theorem, but we don't need anything
so precise. By the way, these are not new results; we do not know to whom they
should be attributed, though.
Lemma 2.8. Let C : C ,! B be an inclusion of graded connected cocommutative
Hopf algebras over a field k. Let n be the connectivity of the cokernel of C (v*
*iewed
as a map of k-vector spaces). Let L and M be B-modules with L bounded below
and M bounded above, and let c be the connectivity of Hom *k(L; M). Then the map
*C: Exts;tB(L; M) ! Exts;tC(L; M)
is an isomorphism for all (s; t) with t - s < c + n.
6 JOHN H. PALMIERI
Proof.Assume that L is zero below dimension ` and M is zero above dimension
m, so that Hom tk(L; M) is zero for t < ` - m. Since B is connected, then there*
* is
a projective resolution
. .!.P2 ! P1 ! P0 ! L ! 0
for L over B so that the bottom dimension of Ps is at least s + `, for each s *
*0. B
is free over C by a result in [14], so we can view P* ! L as a projective resol*
*ution
for L over C, as well. Under the conditions on C ,! B, we see that the map
Hom tB(Ps; M) ! Hom tC(Ps; M)
is an isomorphism for all t with t < s + n + ` - m. This proves the result. __*
*|_|
Corollary 2.9.Let C B be sub-Hopf algebras of A with non-decreasing profile
functions (m1; m2; : :):and (n1; n2; : :):, respectively. Suppose that for some*
* integer
r 1, we have mi = ni for all i < r, and mr < nr; let L and M be B-modules
with L bounded below and M bounded above. Given a bidegree (s; t), if either r *
*is
sufficiently large or mr is sufficiently large, then the restriction map
Exts;tB(L; M) ! Exts;tC(L; M)
is an isomorphism. (Here, "sufficiently large" depends only on (s; t) and the c*
*on-
nectivity of Hom *F2(L; M).)
Remark 2.10.As with Lemma 2.4, both Lemma 2.8 and Corollary 2.9 hold for
chain complexes of modules, in place of modules. More precisely, if D and F are
finite chain complexes of finite B-modules, then there is a number c depending *
*on
D and F , so that if C ,! B is an inclusion of Hopf algebras with cokernel that*
* is
"(n - 1)-connected," then the restriction map
Exts;tB(D; F ) ! Exts;tC(D; F )
is an isomorphism for all (s; t) with t-s < c+n. To see this, note that the sta*
*tement,
"There is a number c : : :" is generic with respect to D and F , and since the
statement holds for all finite A-modules, it must hold for all finite chain com*
*plexes
of finite A-modules. (See [17] for a discussion of chain complexes, genericity,*
* and
related topics. See also Remarks 2.5 and 3.4.)
Finally, we will need the following results of Lin (see also [21, Theorem 6.4*
*]).
Theorem 2.11 (Theorem 3.1 in Part I of [9]).Let B = A(n1; n2; : :):be a
sub-Hopf algebra of A so that for some k, ni is finite for all i < k. Suppose
hj; and hk; are elements in Ext**B(F2 ; F2), with j k and j . Then the class
hj;hk; is nilpotent in Ext**B(F2 ; F2).
Corollary 2.12 (Corollary 3.2 in Part I of [9]).Let B = A(n1; n2; : :):be a
sub-Hopf algebra of A so that for some k, ni is finite for all i < k. If hk; is*
* an
element in Ext**B(F2 ; F2) with k , then hk; is nilpotent.
NILPOTENCE OVER THE STEENROD ALGEBRA I 7
3. Proof of Theorem 1.1
We prove Theorem 1.1 as well as the following result. For a Hopf algebra B
and B-modules L and M, given an element z 2 Ext**B(L; M), we say that z is
tensor-nilpotent if zn = 0 in Ext**B(Ln ; Mn ), for all n sufficiently large.
Theorem 3.1. Let B be a sub-Hopf algebra of the mod 2 Steenrod algebra A; let
D be the sub-Hopf algebra of A with profile function (1; 2; 3; : :):.
(a) Let be a bounded below coassociative B-coalgebra; fix z 2 Ext**B(; F2).
Then z is nilpotent if and only if *D\B(z) is nilpotent.
(b) Let M be a finite dimensional B-module; fix z 2 Ext**B(M; M). Then z is
nilpotent if and only if *D\B(z) is nilpotent.
(c) Let L and M be B-modules with L bounded below and M finite dimensional;
fix z 2 Ext**B(L; M). Then z is tensor-nilpotent if and only if *D\B(z) *
*is
tensor-nilpotent.
We view Theorem 1.1 as being analogous to the Morava K-theory version of the
nilpotence theorem, [8, Theorem 3]; so Theorem 3.1 corresponds to the BP -versi*
*on
or the MU-version, [4, Theorem 1]. (See also [5] or [20].) By the way, Lin [9]
conjectured part (b) of Theorem 3.1 for M = F2. We will prove Theorem 3.1 as
one of the steps in proving Theorem 1.1.
Proofs of Theorems 1.1 and 3.1.Suppose z is in the appropriate Ext group (either
Ext**B(; F2), Ext**B(M; M), or Ext**B(L; M)). It is clear that if z is nilpoten*
*t in the
appropriate sense, then so is *E(z) for any sub-Hopf algebra E of B. (For the f*
*irst
two cases, for instance, *Eis an algebra map.)
So assume that *E(z) is nilpotent for every elementary sub-Hopf algebra E of *
*B.
In particular, the restriction of z to every maximal elementary sub-Hopf algebr*
*a is
nilpotent. By Corollary 2.9, the restriction of z is zero for all but finitely *
*many of
these; so by raising z to a large enough power, we may assume that *E(z) = 0 for
all E.
If z is in homological degree zero, then the restriction maps are inclusions;*
* hence
if some restriction of z is nilpotent, then so is z. Most of the remaining work*
* is in
proving Theorem 1.1(a) for z of positive homological degree. We do this for the*
* full
Steenrod algebra A; the case for a sub-Hopf algebra follows similarly. There a*
*re
two steps in the proof of (a): first we prove that if z 2 Exts;tA(; F2) is a cl*
*ass with
s > 0 which restricts to zero over all elementary sub-Hopf algebras of A, then *
*z is
nilpotent when restricted to D. Then we prove Theorem 3.1(a), and this finishes
the proof. (The first step is a generalization of one of Lin's main theorems in*
* [9].)
Following that, we prove parts (b) and (c) of both theorems.
3.1. z is nilpotent over D. Let be a bounded below coassociative A-coalgebra;
let z 2 Exts;tA(; F2) be an element with s > 0. Let DR = A(0;_0;_:-:;:0z___"; R*
*; R+1; R+
R-1
2; : :):. Now we apply Corollary 2.9 to F2 ,! DR _since is bounded below, then
for R large enough, we can make sure that Exts;tDR(; F2) ! Exts;tF2(; F2) is an
isomorphism. But since s > 0, then Exts;tF2(; F2) = 0; hence for R large enough*
*, z
restricts to zero over DR .
8 JOHN H. PALMIERI
We assume that z restricts to zero over every elementary sub-Hopf algebra; in
particular, z restricts to zero over E(0), E(1), E(2), : :.:Using Corollary 2.9*
* again,
we see that if rm is a large enough integer for each m, then z restricts to zer*
*o over
the sub-Hopf algebra A(0;_0;_:-:;:0z___"; m_+_1;_:_:;:m-+z1_____"; rm ; rm + 1;*
* rm + 2; : :):. In
m rm -m-1
fact, since the isomorphism in Corollary 2.9 depends only on the connectivity of
the cokernel of the Hopf algebra inclusion, we may choose rm to be the same for
all m, say all equal to r0.
Let r = max(R; r0). Define sub-Hopf algebras
Dr = A(0;_0;_:-:;:0z___"; r; r + 1; r + 2; : :):;
r-1
E(m)0= A(0;_0;_:-:;:0z___"; m_+_1;_:_:;:m-+z1_____"; r; r + 1; r + 2;*
* : :):;
m r-m-1
for m < r. Then z restricts to zero over Dr and all the (E(m)0)'s; we want to s*
*how
that *D(z) is nilpotent.
We use induction to build D out of Dr and the (E(m)0)'s. Here is the precise
statement: we claim that if B = A(n1; n2; : :;:nr-1; r; r + 1; : :):is a sub-H*
*opf
algebra of D with n1 n2 . . .nr-1, then *B(z) is nilpotent. We prove this
by induction on n1 + n2 + . .+.nr-1. Let NB denote this sum. (Note that since
B D, then ni i for each i < r.)
If NB = 0, then we have B = Dr and so *B(z) = 0. For the inductive step, we
need the following observation.
Sublemma 3.2. Suppose B = A(n1; n2; : :;:nr-1; r; r + 1; : :):is a sub-Hopf al-
gebra of D with n1 n2 . . .nr-1, so that B not a sub-Hopf algebra of any
E(m)0. Let j = min{i : ni6= 0}; let k = min{i : ni j + 1}. Then j k < r.
Proof.This is immediate. __|_|
Assume that *C(z) is nilpotent for all C as above with NC N -1. By replacing
z with some power, we may assume that *C(z) = 0 for all such C. Let B denote
a sub-Hopf algebra of D as above, with NB = N. If B is a sub-Hopf algebra of
E(m)0 for some m, then *B(z) = 0 by assumption. Otherwise we have j and k
as in the sublemma; by Lemma 2.2, hj;nj-1and hk;nk-1are nonzero elements of
Ext1B(F2 ; F2), and by Theorem 2.11, the product hj;nj-1hk;nk-1is nilpotent. So
apply induction and Lemma 2.4 twice to get z = hj;nj-1z0= hk;nk-1z00.
This completes the inductive step. When NB = r(r-1)_2, then B = D. __|_|
3.2. z is nilpotent over A. Let z 2 Exts;tA(; F2) be an element with s > 0, and
assume that *E(z) = 0 for all elementary sub-Hopf algebras E of A. We know from
the previous subsection that *D(z) is nilpotent; we may assume that *D(z) = 0, *
*by
replacing z with an appropriate power. We want to show that z is nilpotent. (In
other words, we are proving Theorem 3.1(a) here.)
For each integer q 1, let Bq = A(1; 2; : :;:q - 1; q; 1; 1; : :):, and let B*
*0 = A.
We will show by downward induction on q that z is nilpotent when restricted to
Bq, for each q. By Corollary 2.9, we can find an integer q large enough so that*
* z
NILPOTENCE OVER THE STEENROD ALGEBRA I 9
restricts to zero over Bq; this starts the induction. Now we use a variant on o*
*ne of
the arguments used in [6]. For each q we have a normal extension of Hopf algebr*
*as
(3) Bq ----! Bq-1 ----! Bq-1==Bq;
where (as algebras) we have Bq-1==Bq ~=E[Pqu: u q]. For k q, let Gk be
the Bq-1==Bq-module defined by Gk = E[Pqu: k u q], and let Gq-1 = F2; of
course, Gk is also a Bq-1-module for each k q - 1. For any Bq-1-module M,
Ext**Bq-1( M ; F2) is a module over Ext**Bq-1(; F2), via the composite
Ext**Bq-1(; F2) Ext**Bq-1( M ;-F2)---!Ext**Bq-1( M ; F2)
??
y(1M )*
Ext**Bq-1( M ; F2):
Let "zdenote the restriction of z to Ext**Bq-1(; F2). We have the following lem*
*ma.
Lemma 3.3. For k sufficiently large, we have "z-1Ext**Bq-1( Gk ; F2) = 0.
Proof.We have the following Cartan-Eilenberg spectral sequence associated to the
extension (3):
Ep;q2~=ExtpBq-1==Bq(Gk ; ExtqBq(; F2)) ) Extp+qBq-1(Gk ; F2):
Precisely as in the proof of [6, Lemma 2.2], we use a vanishing plane argument *
*to
show that some power of "zannihilates every element of the abutment. __|_|
Now we finish the proof of the theorem. For each k q, there is a short exact
sequence
k(2q-1)
(4) 0 ! 2 Gk-1 ! Gk ! Gk-1 ! 0:
We tensor with and apply Ext**Bq-1(- ; F2); the result is a long exact sequence
of Ext**Bq-1(; F2)-modules, in which the boundary homomorphism is composition
with hq;k 1Gk-1 . Inverting "zis exact, so if we choose k sufficiently large, *
*as in
Lemma 3.3, then "z-1Ext**Bq-1( Gk ; F2) = 0; and so we have an isomorphism
hq;k 1Gk-1 : "z-1Ext**Bq-1( Gk-1 ; F2) ! "z-1Ext**Bq-1( Gk-1 ; F2):
Now we use Corollary 2.12: since k q, then hq;kis nilpotent; hence
"z-1Ext**Bq-1( Gk-1 ; F2) = 0:
By downward induction on k, we see that "z-1Ext**Bq-1(; F2) = 0, which means
that the restriction of z to Bq-1 is nilpotent. Replacing z with an appropriate
power allows us to assume that z restricts to zero over Bq-1. This completes the
inductive step. __|_|
10 JOHN H. PALMIERI
3.3. Proofs of Theorems 1.1(b-c) and 3.1(b-c). These proofs are essentially
the same as the corresponding arguments for spectra; both parts follow from (a)
easily. Given a B-module M, let M* denote the dual of M, with B-module structure
given as usual.
Proof of Theorem 1.1(b).If M is a finite dimensional B-module, then M M* is
a coassociative B-coalgebra_the coproduct map
M M* ! (M M*) (M M*)
is adjoint to the identity on M M* M, and the counit map M M* ! F2 is
adjoint to the identity on M*. Also, as algebras we have
Ext**B(M; M) ~=Ext**B(M M* ; F2):
So apply part (a). __|_|
Proof of Theorem 1.1(c).We have
Ext**B(L; M) ~= Ext**B(L M* ; F2);
z $ ^z:
Now, *E(z) = 0 for all E if and only if *E(^z) = 0 for all E, and zn = 0 if an*
*d only if
^zn = 0. So it suffices to prove the corollary for an element z of Ext**B(L M**
* ; F2);
i.e., we may assume that M = F2. Furthermore, by suspending L we may assume
that L is zero below dimension 0 (z 2 Exts;tB(L; F2) is tensor-nilpotent if and*
* only
if z 2 Exts;t+nB(nL ; F2) is).
So assume that z 2 Ext**B(L; F2). Let be the tensor coalgebra on L: let
M
= Ln
n0
(where L0 = F2), with the evident coproduct. Then is a coassociative B-
coalgebra, and since L is zero in negative dimensions, is bounded below. Also,
tensor powers of z correspond to powers of the corresponding element _zin the
algebra Ext**B(; F2), and *E(z) = 0 if and only if *E(_z) = 0. So apply part (a*
*). __|_|
Proof of Theorem 3.1(b-c).Imitate the proof of Theorem 1.1(b-c). __|_|
This finishes the proofs of Theorems 1.1 and 3.1. __|_|
Remark 3.4.This proof relied on certain facts about Ext**B(F2 ; F2), for B a su*
*b-
Hopf algebra of A_2.11 and 2.12_as well as some basic homological algebra ap-
plied to Ext**B(L; M) for B-modules L and M_2.4 and 2.9. Hence the whole proof
carries through for chain complexes of A-modules, in place of A-modules_see Re-
marks 2.5 and 2.10. For example, if C is a finite chain complex of finite A-mod*
*ules
and z is an element of Ext**A(C; C), then z is nilpotent if and only if *E(z) i*
*s nilpo-
tent for every elementary sub-Hopf algebra E of A. We will need the nilpotence
theorem in this generality in the sequel [17].
NILPOTENCE OVER THE STEENROD ALGEBRA I 11
4.The odd prime case
In this section we discuss our results for the algebra of reduced powers. Let*
* p
be an odd prime, let A denote the mod p Steenrod algebra, and let P = A=AfiA.
We prove the odd prime analog of Theorem 3.1 for sub-Hopf algebras of P , and we
prove a weaker version of Theorem 1.1.
P is a Hopf algebra, and its dual is isomorphic to Fp[1; 2; : :]:. One has the
same description of sub-Hopf algebras of P as for the mod 2 Steenrod algebra; we
write P (n1; n2; : :):for the sub-Hopf algebra with profile as indicated. Many*
* of
the results from Section 2 hold without change. There are a few exceptions: if
E = P (n1; n2; : :):is an elementary sub-Hopf algebra of P , then
Ext**E(Fp ; Fp) ~=E[hj;i: j 1; nj > i 0] Fp[fi(hj;i) : j 1; nj > i 0];
where fi denotes the Bockstein acting on Hopf algebra cohomology. So we have
to modify Lemma 2.4 to reflect this; the conclusion of the lemma will be that
z2 = "*(fi(hj;nj-1))z0 for some z0 (see [2] for the changes necessary in the pr*
*oof).
Also, we would like to have analogs of Theorem 2.11 and Corollary 2.12; we ha*
*ve
one for the latter, but not the former. Wilkerson [21, 6.3] gives an example in*
* which
the odd primary version of Theorem 2.11 fails. We have the following, though.
Proposition 4.1 (Proposition 4.1 in [12]).Let B = P (n1; n2; : :):be a sub-
Hopf algebra of P so that for some k, ni is finite for all i < k. If hk; is an
element in Ext**B(Fp ; Fp) with k , then fi(hk; ) is nilpotent.
Here is our first result for odd primes.
Theorem 4.2. Theorem 3.1 holds for sub-Hopf algebras of P .
Proof.The proof is almost identical to that given above. Let D[x] denote the
algebra Fp[x]=(xp); then with Bq and Bq-1 as in Section 3, we have Bq-1==Bq ~=
D[Pqu: u q]. We set Gk = D[Pqu: k u q], so that Gk-1 ~=Gk=(Pqk); we also
set "Gk-1~=Gk=((Pqk)p-1). Now, instead of the short exact sequence (4), we have
two short exact sequences:
k|
0 ! |PqG"k-1! Gk ! Gk-1 ! 0
k|
0 ! (p-1)|PqGk-1 ! Gk ! "Gk-1! 0:
We get two long exact sequences in Ext**Bq-1==Bq( - ; Fp), and both boundary
homomorphisms are isomorphisms once "zis inverted. Hence so is the composite of
the boundary homomorphisms, and this is just fi(hk;q) 1Gk-1 . Now, as above,
fi(hk;q) is nilpotent. This finishes the proof. __|_|
In light of the failure of Theorem 2.11, we cannot expect the precise analog
of Theorem 1.1 to hold; indeed, the same counterexample shows that the analog
does not hold. Here is an alternative. Say that a sub-Hopf algebra E of P is
quasi-elementary if for some r 1 it is a sub-Hopf algebra of
Q(r) = P (0;_0;_:-:;:0z___"; 1; r + 1; r + 1; r + 1; : :)::
r-1
12 JOHN H. PALMIERI
Theorem 4.3. Suppose B is a sub-Hopf algebra of P and is a bounded below
coassociative B-coalgebra. One can detect nilpotence in Ext**B(; Fp) by restric*
*ting
to quasi-elementary sub-Hopf algebras.
Proof.The only step we need is the following version of Theorem 2.11 (see also
Sublemma 3.2). This is essentially [21, Theorem 6.4], modified to work at odd
primes. As one might guess, for suitable r 0 we set
D = P (1; 2; 3; : :):;
Q(m)0 = P (0;_0;_:-:;:0z___"; 1; m_+_1;_:_:;:m-+z1_____"; r; r + 1; r*
* + 2; : :)::
m-1 r-m-1
Lemma 4.4. Suppose B = P (n1; n2; : :;:nr-1; r; r + 1; : :):is a sub-Hopf alge*
*bra
of D with n1 n2 . . .nr-1, so that B is not a sub-Hopf algebra of any Q(m)0.
Let j = min{i : ni6= 0}. If nj = 1, then let k = min{i : ni j + 2}; otherwise, *
*let
k = min{i : ni j + 1}. Then i k < r, and furthermore, the class
fi(hj;nj-1)fi(hk;nk-1) 2 Ext**B(Fp ; Fp)
is nilpotent.
Proof.Such a Hopf algebra B satisfies the conditions of [21, Lemma 6.2] with r *
*= j
and n = j + k. Hence we have a non-trivial relation hk;jhj;0= 0 in Ext**B(Fp ; *
*Fp);
now apply Steenrod operations. See [21, Theorem 6.4] for details. (The key here*
* is
that either hj;1or hk;j+1is nonzero in Ext**B(Fp ; Fp).) __|_|
This finishes the proof of Theorem 4.3. __|_|
Corollary 4.5.If L and M are B-modules with L bounded below and M finite,
then one can detect nilpotence in Ext**B(M; M) and tensor-nilpotence in Ext**B(*
*L; M)
by restricting to quasi-elementary sub-Hopf algebras.
Remark 4.6. (a)Theorem 4.3 and Corollary 4.5 hold more generally for chain
complexes of B-modules_see Remark 3.4.
(b) The quasi-elementaryQsub-HopfQalgebras Q of P are defined so that no pro*
*d-
uct v fiP"0u is zero, where v ranges over any finite subset of nonze*
*ro
elements in Ext1;oddQ(Fp ; Fp), and u ranges over any finite subset of n*
*onzero
elements in ker"P0\ Ext1;evenQ(Fp ; Fp)_see [21, Definition 5.3]. We exp*
*ect
that if we used this Ext condition as a definition for quasi-elementary,*
* then
the analog of Theorem 4.3 for the full mod p Steenrod algebra A would ho*
*ld.
It would be nice to have a characterization of these sub-Hopf algebras in
terms of profile functions.
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University of Minnesota
Current address: Department of Mathematics, University of Wisconsin, Madison,
WI 53706
E-mail: palmieri@math.umn.edu