Some quotient Hopf algebras of the
dual Steenrod algebra
J. H. Palmieri
20 November 2002
1 Introduction
Fix a prime number p, and let A be the following familiar Hopf algebra: as an
algebra,
A = Fp[,1, ,2, ,3, . .].,
and the coproduct is determined by
Xn i
(,n) = ,pn-i ,i, (1.1)
i=0
where ,0 = 1. There is also an antipode - see Lemma 3.6 - but it does not play
a central role here. A is graded by putting ,n in degree 2n - 1 when p = 2,
and in degree 2pn - 2 when p is odd. This makes A into a graded connected
commutative Hopf algebra.
When p = 2, the graded dual of A is the mod 2 Steenrod algebra; when p
is odd, the graded dual of A is the algebra of reduced power operations, which
is a quotient of the mod p Steenrod algebra. We abuse the language and refer
to A as the dual Steenrod algebra, regardless of the prime. We write A* for its
graded dual.
Homotopy theorists would like to understand the cohomology of A, which can
be defined as Ext*,*A*(Fp, Fp), where Ext is computed in the category of graded
A*-modules, or equivalently as Ext*,*A(Fp, Fp), where Ext is computed in the
category of graded A-comodules. (In places where it makes a difference, we will
use the comodule definition). Note that in this setting, Ext is bigraded: the f*
*irst
grading is the usual homological one, and the second is induced by the grading
on the (co)module category. The Frobenius map a 7! ap on A is a Hopf algebra
map, and so it induces an algebra map on cohomology. We denote this map by
Pf0 when p is unspecified or odd, and by fSq0when p = 2 (see Proposition 5.1
for an explanation of this notation). Since the Frobenius does not preserve the
grading, neither does fP0 - it works like this:
Pf0 : Exts,tA(Fp, Fp) ! Exts,ptA(Fp, Fp).
1
We would like to understand how the map fP0interacts with the multiplicative
structure of Ext*A(Fp, Fp). This paper has two goals: to advertise some questio*
*ns
and conjectures related to this interaction, and to study some Hopf algebras
related to A that arise when studying fP0 and the Frobenius.
We start by pointing out that
fP0: Ext0,0A(Fp, Fp) ! Ext0,0A(Fp, Fp)
is the identity map. Regarding Ext in positive dimensions, one can deduce the
following from the main result in [Pal99].
Theorem 1.2. Let p = 2 and fix z 2 Exts,tA(F2, F2). If s is positive, then there
0
is an n so that (fSq)n(z) is nilpotent.
This fails if A is replaced by the sub-Hopf algebra F2[,1] - the cohomology *
*of
0
this is F2[h10, h11, h12, . .]., with fSq(h1,n) = h1,n+1- but holds if A is rep*
*laced
by any quotient Hopf algebra. See the appendix for the theorem's proof. One
might ask whether there is a simpler proof of this, one which would hold at any
prime.
Question 1.3. (a)Is the analogous result true when p is odd?
(b)Can one prove Theorem 1.2 without appealing to the F -isomorphism of
[Pal99, Theorem 1.2]?
It is also natural to ask, given z as in Theorem 1.2, how large must n be to
0 0
make (fSq)n(z) nilpotent? fSq is an algebra map, so if z is nilpotent, then so
0 0
is fSq(z). For every known non-nilpotent element z (except for z = 1), fSq(z)
is nilpotent as well. This leads us to make the following conjecture. All of our
evidence is valid only for the prime 2, but we brashly state it for all primes.
Conjecture 1.4. Fix a prime p and an element z 2 Exts,tA(Fp, Fp) with s > 0.
Then fP0(z) is nilpotent.
This is rather specific to A: Theorem 1.2 fails for Fp[,1], and the conjectu*
*re
implies the theorem. Thus the conjecture does not hold for sub-Hopf algebras
of A. It also fails for quotient Hopf algebras of A: for example, the cohomology
0
of F2[,2]=(,42) is isomorphic to F2[h20, h21], with fSq(h20) = h21.
One can interpret many of these questions in terms of the öd uble" A of
A: let A be the sub-Hopf algebra of A generated by the pth powers of the
generators:
A = Fp[,p1, ,p2, ,p3, . .]..
Then A and A are isomorphic via the Frobenius map x 7! xp, and so their
Ext algebras are isomorphic. We write OE for the isomorphism:
~= s,pt
OE : Exts,tA(Fp, Fp) -! Ext A (Fp, Fp).
2
Since A is a sub-Hopf algebra of A, there is also an induced map which pre-
serves the grading:
æ : Exts,t(AFp, Fp) ! Exts,tA(Fp, Fp).
It is not hard to see that for any z 2 Exts,tA(Fp, Fp),
æ(OE(z)) = fP0(z).
So, for example, Conjecture 1.4 is equivalent to the statement that every posit*
*ive-
dimensional element in the image of æ is nilpotent.
One can also reformulate these questions, conjectures, and results as follow*
*s.
Let bAbe the dual complete Steenrod algebra: Ab is the Hopf algebra obtained
from A by adjoining all pkth roots of each indeterminate ,n:
bA= Fp[,pkn: n 1, k 2 Z],
with the apparent relations. The coproduct on A induces one on bA:
k Xn pi+k pk
(,pn) = ,n-i ,i ,
i=0
for all n 1 and all k 2 Z, where ,0 = 1 as usual. The antipode for A also
induces one for bA, which we will not need. The dual bA*of bAis the complete
Steenrod algebra, which was introduced by Arnon [Arn94 , Arn00] and has also
been studied by Llerena and Hu'ng [LH96 ].
In other words, one obtains bAfrom A by inverting the Frobenius map. As
a result, one can show - see Proposition 5.3 - that the cohomology of Ab is
isomorphic to the cohomology of A with the operation fP0 inverted.
Thus Theorem 1.2 says that every positive dimensional element in Ext*bA(F2, *
*F2)
is nilpotent, and Question 1.3(a) asks whether this is true at odd primes. bA
seems as though it should be easier to understand than A. Hence we have the
following natural question.
Question 1.5. What is the cohomology of bA?
In order to approach all of these questions, we try something simpler: we
constructed bAby formally inverting the Frobenius and fP0, thus making them
isomorphisms with infinite order. We will construct Hopf algebras in which they
are invertible with finite order.
Thus we study the following quotient Hopf algebras of A: fix n 1 and
define Hn by n
Hn = A=(,pi - ,i: i 1).
This is a version of A in which the Frobenius has order n. In this paper, we
show that the topological linear dual of Hn is a group algebra and we identify
3
the group. We use this observation to show that Ext*,*Hn(Fp, Fp) consists en-
tirely of nilpotent elements. We also try to relate these Ext calculations to t*
*he
cohomology of the complete Steenrod algebra.
A few words of warning may be warranted for any topologists who may be
reading this. First, Hn does not inherit the grading from A, since the ideal is*
* not
homogeneous. Many topologists are used to dualizing quotient Hopf algebras of
A to get sub-Hopf algebras of the Steenrod algebra, but since Hn is not graded,
its dual sits inside the ungraded dual of A, which is distinctly different from*
* the
usual Steenrod algebra. After all, since A is finite-dimensional in each gradin*
*g,
it is isomorphic to its double graded dual; since it is infinite-dimensional as*
* a
whole, it is not isomorphic to its double ungraded dual.
For example, let p = 2 and n = 1. If one dualizes with respect to the
monomial basis of A, then ,i1is dual to an element called Sqi. Hence when one
dualizes H1, since ,1 is identifiedPwith ,i1for all i, the element dual to ,1 in
(H1)* deserves to be called j 1Sqj. This element is non-nilpotent - compare
this to the graded dual of A, in which such infinite sums do not occur and in
which everyPelement in the augmentation ideal is nilpotent. The non-nilpotence
of x = j 1Sqj is reflected by the coproduct in H1: (,2) contains a term
,21 ,1 = ,1 ,1, and hence x2 6= 0 - it is detected by ,2. (,4) has a term
,2 ,2, so the iterated coproduct of ,4 has a term ,1 ,1 ,1 ,1, and hence
x4n6= 0. For each n, (,2n) has a term ,2n-1 ,2n-1, and hence inductively
x2 6= 0 for all n.
In other words, the Hopf algebras Hn and their duals may challenge the
intuition of any readers (or authors) who are used to working with the Steenrod
algebra.
Here is an outline of the paper: in Section 2, we show that Hn is dual to
a group algebra, and we identify the group - it is the group of strict auto-
morphisms of the additive formal group law over Fpn. We accomplish this by
imitating work of Ravenel and others on the Morava stabilizer groups. We also
note that this automorphism group is a torsion-free pro-p group. In Section 3,
we collect some observations about Hn and the associated automorphism group;
for example, we point out that the group is not p-adic analytic. In Section 4, *
*we
use the fact that the group is torsion-free to show that its cohomology consists
entirely of nilpotent elements. In Section 5, we fill in some of the details of
the ideas in this introduction: the relationship between the Frobenius on A and
the Steenrod operation Pf0, and also the completed Steenrod algebra and its
cohomology. There is also an appendix which contains the proof of Theorem 1.2.
Convention. Throughout, an unadorned tensor product means tensor prod-
uct over Fp.
Acknowledgments. Much of the work on the Hopf algebras Hn was inspired
by a brief conversation with Mike Hopkins. I have had many helpful discussions
with Ethan Devinatz and Hal Sadofsky.
4
2 Hn is dual to a group algebra
Let Hn be as above. Let Bnjbe the sub-Hopf algebra of Hn generated by ,i
with 1 i j. Let Hnjbe the quotient of Hn by Bnj. Let F = Fpn. That is,
for fixed n 1 and j 0,
n
Hn = Fp[,1, ,2, . .].=(,pi - ,i: i 1),
n
Hnj= Fp[,j+1, ,j+2, . .].=(,pi - ,i: i j + 1),
n
Bnj= Fp[,1, ,2, . .,.,j]=(,pi - ,i: j i 1),
F = Fpn.
Note that Hn0= Hn.
In this section, we show that after tensoring with F, each of these Hopf
algebras is dual to a group algebra, we identify the groups, and we observe that
the group associated to Hnjis torsion-free for each n and j.
First we should verify that Hnjis a quotient Hopf algebra of A for each n
and j. Given this, it is clear from the formula for the coproduct (1.1)that Bnj
is a sub-Hopf algebra of Hnj.
Lemma 2.1. Fix n 1 and j 0. Then
n
Inj= (,1, . .,.,j; ,pi - ,i: i j + 1)
is a Hopf ideal of A. Thus Hnjis a quotient Hopf algebra of A.
Proof. We have defined I = Injas an ideal in the algebra A, so we need to show
that it interacts well with the coproduct. That is, for each x 2 I, we need to
show that (x) 2 I A + A I. It suffices to check this fornthe generators of
the ideal; it is clear when x = ,m with m j, so let x = ,pm-,m , where m > j.
We compute:
n Xm i pn+i pn pi j
(,pm - ,m )= ,m-i ,i - ,m-i ,i
i=0
n pn
= (,pm - ,m ) 1 + 1 (,m - ,m )
m-1Xi n+i i n+i n j
+ (,pm-i- ,pm-i) ,i+ ,pm-i (,pi - ,i).
i=1
Thus (,pnm- ,m ) is in I A + A I, as desired. |___|
As in [Rav86 , Section 6.2], we can discuss the öt pological linear dualö f
the Hopf algebras Hnj: since Hn, for instance, is the direct limit of the finit*
*e-
dimensional sub-Hopf algebras Bnj, then the topological linear dual (Hn)* is
defined to be (Hn)* = lim-j(Bnj)*, with the topology induced by the inverse
limit.
The first part of the proof of [Rav86 , Theorem 6.2.3] applies in our setting
to give the following.
5
Proposition 2.2. Fix n 1, j 0, and ` 2 {1, 2, . .}.[ {1}, and let H be
the following sub-Hopf algebra of Hnj F:
n
H = F[,j+1, . .,.j+`]=(,pi - ,i: j + ` i j + 1).
Then the topological linear dual H* of H is a group algebra. |___|
For example, this applies when H = Bnj F or H = Hnj F. By "the
topological linear dualö f a finite-dimensional Hopf algebra like Bnj, we mean
its ordinary vector space dual with the discrete topology.
We would like to identify the groups involved. Let Gn, Gnj, and Pjnbe the
groups so that
F[Gn] ~=(Hn F)*,
F[Gnj] ~=(Hnj F)*,
F[Pjn] ~=(Bnj F)*.
Of course, Gn0= Gn since Hn0= Hn. Also, note that Pjnis a finite p-group and
Gnjis a pro-p group: Hnjis the direct limit of the finite-dimensional sub-Hopf
algebras n
Fp[,j+1, . .,.,j+`]=(,pi - ,i),
and this direct system becomes an inverse system of group algebras, induced by
an inverse system of groups, after tensoring up with F and dualizing.
Proposition 2.3. Fix n 1 and j 0.
(a)Gn is the group of strict automorphisms of the additive formal groupPlaw
over F. That is, Gn is the group of power series of the form x+ i 1aixpi
with ai2 F, under composition.
(b)Gnjis the subgroup of Gn consisting of such power series with ai = 0 for
1 i j.
(c)Pjnis the quotient group Gn=Gnj. That is, Pjnis the group of power series
of the form in (a), with xpj+1= 0.
Proof. This follows from the well-known observation that the dual Steenrod
algebra A corepresents the strict automorphisms of the additive formal group
law for Fp-algebras - see [Rav86 , p. 378], for example. That is, A is the mod p
group scheme for this strict automorphism group.
In more detail: we prove (c) first. Bnjis dual to the group algebra of a
finite group, and_the group_is precisely the group of points in the group scheme
defined by Bnj Fp, where Fp is the algebraic closure of Fp:
__ __
Pjn= Hom _Fp-alg(Bnj Fp, Fp).
By imitating the analysis for the dual Steenrod algebra, one can see that this *
*is
the group of power series of the form
2 pj
x + a1xp + a2xp + . .+.ajx
6
__ pn
under composition,_subject to xpj+1 = 0, where ai 2 Fp and_ai_ = ai. In n
particular, ai 2 Fp is the image of ,i. The elements a of Fp satisfying ap = a
form a subfield isomorphic to F, and this completes the proof of part (c).
To prove (a), take limits: since Hn is the direct limit of the finite-dimens*
*ional
sub-Hopf algebras Bnj, Hn represents the group scheme
__ n __
Hom _Fp-alg(lim-!Bnj Fp, -) = lim-Hom_Fp-alg(Bj Fp, -).
j j
__
Gn is the group of Fp-points in this group scheme, which is as described in_(a).
The proof of part (b) is similar. |__|
We end this section with the following result.
Proposition 2.4. Fix n 1 and j 0. Then Gnjis torsion-free.
P i
Proof. Fix ff 2 Gnjdifferent from the identity element, so ff = x + i>jaixp
with at least one ai nonzero. Indeed, suppose that am is thepfirstmnonzero
coefficient. Then we claim that in ffp,ithe coefficient of xp is the first no*
*nzero
term. Given this, the elements {ffp |i 0} are distinct, and so ff is not a to*
*rsion
element. i
To verify the claim, we need to know the coefficient of xp in ffp. Since the
product in Gnjis determined by the coproduct in Hnj, to find any coefficient in
ffp, we use the pth iterated coproduct in Hnj, which is defined in Notation 2.5
and a formula for which is given in Lemma 2.6. The pth iterated coproduct on
,pm has a term
(p-1)m p2m pm
,pm_______._._.,m_-z__,m____,m___",
p factors
which produces the nonzero term
(p-1)m+p(p-2)m+...+p2m+pm +1
apm
in the coefficient of xppm. Since am is the first nonzero coefficient in ff, th*
*en only
terms in the iterated coproduct having no ,ifactors with 0 < i < m are relevant.
So all of the other relevant terms in the pth iterated coproduct of ,pm , and a*
*ll of
the relevant terms in the pth iterated coproduct of ,k for k < pm, will have at
least one tensor factor equal to 1, and hence will occur with some multiplicity.
It is easy to check that the multiplicity is divisible by p - see Lemma_2.7_- a*
*nd
hence is zero in F. |__|
Notation 2.5. Given a coassociative coalgebra C with coproduct , we let
2 = , and m = ( 1 m-2 ) O m-1 ; we call m the "mth iterated
coproduct". This is indexed so that m (x) lies in C m .
The following is immediate from the formula for the coproduct in Hnj.
7
Lemma 2.6. For any m 1, the mth iterated coproduct in Hnjof ,` is
X pi1+...+im-1 pi1+i2 pi1
m (,`) = ,im . . .,i3 ,i2 ,i1.
i1+...+imi=`
k=0 orik>j
Lemma 2.7. Given an ordered list (b1, . .,.bm ) and a number p m, then the
number of ways of adding p - m 1's to the list, while preserving the ordering_of
the bi's, is pm. |__|
The proof is a simple combinatorial argument.
Thus given a pth iterated coproduct with at p - m 1's in it, it occurs with
multiplicity pm. If p is prime and 0 < m < p, this multiplicity is congruent to
zero mod p.
3 Examples and observations
In this section, we collect some miscellaneous observations about the Hopf al-
gebras Hnjand the corresponding groups Gnj.
Example 3.1 (H1). Let n = 1. In this case, the Hopf algebra H1 is bi-
commutative. Avinoam Mann pointed out (private communication) that the
correspondingPabelian group G1 is isomorphic to the group of power series of
the form 1 + aixi with multiplication as the group operation. According to
[Cam00 , p. 218], this group is a free abelian pro-p group of infinite rank. In
particular, it is not p-adic analytic, since such groups must have finite rank.
Since G1 is a subgroup of Gn for every n, no Gn is p-adic analytic.
One might also note that for any n, the group Gn is a subgroup ofPthe
Nottingham group N = N (F), which consists of all power series x + j 2bjxj
with bj 2 F, under composition. Such an observation is almost a triviality, sin*
*ce
every finite p-group, and every countably-based pro p-group, can be embedded
as a subgroup of N ; the group Gn is embedded in a particularly nice way, though
- it is an "index subgroup". See [Cam00 , Theorems 10 and 11, and pp. 217-8]
for more on the Nottingham group.
Example 3.2 (Pj1and G1). The group Pjncorresponding to the Hopf algebra
Bnjis a finite p-group of order pnj. P1n is abelian for all n; indeed, P1n is
isomorphic to the additive group of F = Fpn: P1n~= (Cp)n. (We write Cd for
the cyclic group of order d.) For each j, Pj1is abelian of order pj, and one can
show that Y
Pj1~= Cffp(j+1_i),
i j, p-i
where ffp(r) is the smallest power of p greater than or equal to r:
ffp(r) = mini{pi: pi r} = pdlogp(r)e.
8
In this decomposition, the cyclic summand of order ffp(j+1_i) is generated by t*
*he
power series 1 + xi. Thus in the inverse system
. .!.Pj1+2! Pj1+1! Pj1! . . .
defining G1, the cyclic summands at the jth stage are the images from the
later stages of cyclic summands of larger and larger orders. Of course this must
happen, since G1 is torsion-free; in this particular case, though, we can obser*
*ve
it directly.
Example 3.3 (A filtration). Note that Bnjis a sub-Hopf algebra of Bnj+1, and
n
the (conormal) quotient is Fp[,j]=(,pj - ,j) with ,j primitive. This quotient is
isomorphic to Bn1. Dually, for each j 1 there is a group extension
1 ! (Cp)n ! Pjn+1! Pjn! 1.
Gn is the inverse limit of the groups Pjnwith respect to these surjections. So
the associated graded consists of a copy of P1n= (Cp)n = F in each degree.
Equivalently, since Gnjis the kernel of the map Gn ! Pjn, Gnjis normal in
Gn. Indeed, each Gnjis normal in Gnkwhenever j k. This gives a filtration of
Gn:
Gn = Gn0 Gn1 Gn2 . ...
Again, the associated graded is isomorphic to F in each degree.
We also point out that whenever n divides m, Fpnis a subfield of Fpm, so Gn
is a subgroup of Gm . For the same reason, the colimit of the groups_Gn is the
group of strict automorphisms of the additive formal group over Fp, which we
denote by G1 . One can define G1j analogously to Gnj: G1j consists of power
series in G1 with the first j coefficients equal to zero. Summarizing, we have
the following.
Proposition 3.4. (a)Fix j 0 and k, n 1. Then Gnjis a subgroup of Gknj.
Dually, Hnjis a quotient of Hknj: the map sending ,i 2 Hknjto ,i 2 Hnj
is a surjective Hopf algebra map.
(b)Fix j 0. Then with respect to the inclusions in part (a),
lim-!Gnj~=G1j. __
n |__|
As far as dualizing part (b), see Proposition 5.4 for a partial result.
Note that it is not necessary to tensor up to Fpkncoefficients to get the
quotient map Hknji Hnj. We also point out that these subgroups are not
normal:
Proposition 3.5. Fix n 1. Gn is not normal in Gkn if k 2. Thus Hn is
not a conormal quotient of Hkn if k 2; that is, Hkn Hn Fp and Fp Hn Hkn
are different as sub-vector spaces of Hkn.
9
In order to prove this, we need to know how to invert elements of Gkn. We
recall the formula for the antipode in A from [Mil58]. Let Part(n) denote the s*
*et
of ordered partitions of n (so, for example, (3, 1) and (1, 3) are distinct ele*
*ments
of Part(4)). Given an ordered partition ff 2 Part(n), write `(ff) for the length
of ff; for i `(ff), let ff(i) be the ith term of ff, and let oe(i) be the sum*
* of the
first i - 1 terms of ff.
Lemma 3.6 (Lemma 10 in [Mil58 ]). The conjugate of ,n in A is equal to
X `(ff)Ypoe(i)
Ø(,n) = (-1)`(ff) ,ff(i).
ff2Part(n) i=1
The same formula applies in Hnjand Bnjfor all n and j.
In particular,2thensummand-corresponding1to the partition (1, 1, . .,.1) is
(-1)n,1+p+p1+...+p .
Proof of Proposition 3.5.Fix n 1 and k 2. To see that Gn is not normal in
Gkn, conjugate ff = x + xp 2 G1 Gn by fi = x + b1xp for some b1 2 Fpkn\ Fpn.
Using Lemma 3.6 and the observation following it, one can see that
1X i-1i
fi-1 = x + (-1)ib1+p+...+p1xp .
i=1
So
fffi-1= fi-1 + (fi-1)p
1X i-1 2 i-1 i
= x + (-b1 + 1)xp + (-1)i(b1+p+...+p1 - bp+p1+...+p )xp .
i=2
So one can compute fifffi-1:
fifffi-1= (fffi-1) + b1(fffi-1)p
i 2 3 4 5 j
= x + xp + (-bp1+ b1) xp - b1xp + b1+p1xp - b2+p1xp + . ...
We are assuming that b1 62 Fpn, so bp16= b1. If the coefficient -bp1+ b1 of xp2*
* is
in Fpn, then the coefficient (-bp1+ b1)b1 of xp3 is not. So fifffi-1 is_not_in *
*Gn,
although ff is. |__|
4 The cohomological variety of Hn is trivial
Let Hn be as above. In this section, we examine the cohomology of Hn. By öc -
homology," we mean Ext*Hn(Fp, Fp), where Ext is computed in the category of
Hn-comodules. The main result is that every positive dimensional cohomology
class is nilpotent.
10
Since after tensoring up with Fpn, the topological linear dual of Hn is the
group algebra of the profinite group Gn, we have the following result, which
is analogous to [Rav86 , Corollary 6.2.4]. First note that any Hn-comodule M
may be viewed as an (Hn)*-module via the composite.
(Hn)* M ! (Hn)* Hn M ! M
This makes M Fpninto a discrete Gn-module.
Proposition 4.1. There is an isomorphism
Ext*Hn(Fp, Fp) Fpn~=H*c(Gn, Fpn),
where H*cdenotes continuous group cohomology.
(There should be a similar result for cohomology with coefficients in any
Hn-comodule M, but we are only interested in the case when M is the trivial
comodule.)
Proof. Since Gn is the inverse limit of the finite groups Pjn, H*c(Gn, Fpn) may
be defined as follows - see [Ser02, I.2.2, Proposition 8]:
H*c(Gn, Fpn) = lim-!H*(Pjn, Fpn).
j
Since Pjnis finite and its group algebra is dual to Bnj, we have an isomorphism
H*(Pjn, -) ~=Ext*Bnj Fpn(Fpn, -). The result follows from the observation that
Hn = lim-!Bnj. |___|
We also need the following. Here, means k.
Lemma 4.2. Let C be a coaugmented coalgebra over a field k, and let F be a
field extension of k. Then there is an isomorphism of algebras Ext*C F(F, F) ~=
Ext *C(k, k) F .
Proof. Take an injective resolution of k as a C-comodule, and tensor with F . __
The result is an injective resolution of k F = F as a C F -comodule. |_*
*_|
Here is the main result of this section.
Theorem 4.3. Every positive-dimensional element in Ext*Hn(Fp, Fp) is nilpo-
tent.
Proof. As above, let F = Fpn. Hn F is dual to the group algebra F[Gn],
where Gn is the group defined before Proposition 2.3. By Proposition 4.1,
Ext *Hn F(F, F) ~= H*c(Gn; F). Proposition 2.4 says that Gn is torsion-free,
and so [Qui71, Proposition 13.4] applies, to say that every positive-dimensional
element in Ext*Hn F(F, F) is nilpotent. Hence by Lemma 4.2, the same is true
for Ext*Hn(Fp, Fp). |___|
11
Given the description of the group G1 in Example 3.1, we expect these Ext
algebras to be infinitely generated. This contrasts with the situation for Mora*
*va
stablizer groups, some of which are Poincar'e duality groups, and all of which
have reasonably well-behaved cohomology: [Rav86 , Theorem 6.2.10] says that
the cohomology of Sn is always finitely generated, and either satisfies Poincar*
*'e
duality or is periodic.
Question 4.4. When n = 1, the corresponding group is a free abelian pro-p
group, and so every element in its cohomology has nilpotence height p. Is there
a uniform nilpotence height for the cohomology of Hn when n 2?
5 On the cohomology of the complete Steenrod
algebra
As we discussed in the introduction, the results in this paper arose in part as
an attempt to study the cohomology of the complete Steenrod algebra. In this
section, we explore this algebra, its cohomology, and the relationship to Hn.
We start by recalling from [May70 ] the relationship between the Frobenius
map on a commutative Hopf algebra and the Steenrod operation Pf0 on Ext.
As usual, Ext is computed in the category of comodules.
Let B be a graded commutative Hopf algebra over Fp. By May's work
[May70 ], one has Steenrod operations acting on Ext*B(Fp, Fp). In this paper, we
0
are only interested in the operation called Pf0 (also called fSq when p = 2),
which is an operation
Pf0 : Exts,tB(Fp, Fp) ! Exts,ptB(Fp, Fp).
Proposition 5.1 (Proposition 11 in [May70 ]). Let B be a graded connected
commutative Hopf algebra over the field Fp. The Frobenius map OE : b 7! bp is
a Hopf algebra map, and the induced map OE* on Ext is the Steenrod operation
Pf0. |___|
We defined the dual complete Steenrod algebra bAin the introduction:
bA= Fp[,pkn: n 1, k 2 Z],
with the coproduct inherited from A. Arnon [Arn94 , Arn00] defined the com-
plete Steenrod algebra, and we note that it is dual to bA. (This is also the as*
*sertion
in [LH96 , Definition 3.6].)
Proposition 5.2. The dual bA*of bAis the complete Steenrod algebra.
Proof. The dual complete Steenrod algebra bAmay be defined as a direct limit
- see the proof of Proposition 5.3, for instance. The complete Steenrod algebra
Ab*is defined in [Arn00 , p. 185] as an inverse limit which is dual to this. *
* |___|
12
Ab inherits a grading from A: ,pknis in degree 2pk(pn - 1) when p is odd,
2k(2n - 1) when p = 2. Thus bAis a Z[1_p]-graded Hopf algebra. It is not finite-
dimensional in each grading, though, so while the monomial basis is a pleasant
one for bA, there is no nice basis for its dual. Furthermore, the dual is not g*
*raded
anymore, strictly speaking. In the language of [LH96 , Definition 1.1], Ab*is a
äl rge" Z[1_p]-graded algebra - additively, it is not isomorphic to the direct *
*sum
of its homogeneous pieces, but rather is a (proper) subgroup of the product of
those pieces.
As far as we can tell, the cohomology of the complete Steenrod algebra has
not been studied in any published paper.
Proposition 5.3. The cohomology of bAis isomorphic to the cohomology of A
with the operation fP0 inverted:
Ext *bA(Fp, Fp) ~=(Pf0)-1 Ext*A(Fp, Fp).
Proof. bAmay be defined as the direct limit with respect to the Frobenius map
OE:
bA= lim(A OE-!A OE-!.)...
-!
Coalgebra cohomology commutes with direct limits, so take Ext of this directed
system:
i OE OE j
Ext*bA(Fp, Fp) = lim-!Ext*A(Fp, Fp) -!*Ext *A(Fp, Fp) -!*... .
Proposition 5.1 says that OE* = fP0, and this finishes the proof. |*
*___|
As mentioned in the introduction, much of the work in this paper was mo-
tivated, directly or indirectly, by Conjecture 1.4, which is equivalent to the
statement that every positive degree element in Ext*bA(Fp, Fp) is nilpotent. Th*
*e-
orem 1.2 says that this is true when p = 2.
For example, let p = 2. There is a family of cohomology classes {h1,k: k 2
Z} in the cohomology of Ab, represented in the cobar complex by [,2k1]. The
classes with k 0 lift to classes of the same name in Ext*A(F2, F2), and in the
cohomology of A, h1,0is non-nilpotent - its powers produce the spike in the zero
stem at the E2-term of the Adams spectral sequence - while h1,kis nilpotent
when k 1. In particular, h41,k= 0 for all k 1. These classes are connected
0 0
by the algebra map fSq: fSq(h1,k) = h1,k+1, so if h1,kis nilpotent, so is h1,k+*
*1.
0
In Ext*bA(F2, F2), fSq has been inverted, so the relation h41,1= 0 implies that
h41,k= 0 for all k 2 Z.
The multiplicative structure in the cohomology of the odd primary Steen-
rod algebra is less well-understood. For example, there are classes h1,k 2
k
Ext 1A(Fp, Fp), for k 0, represented in the cobar complex by [,p1]. These
classes are odd-dimensional, so by graded-commutativity of Ext, they square to
zero. There are related classes b1,k2 Ext2, though, which can be obtained by
13
applying the Bockstein operation to h1,k; equivalently, b1,kis represented in t*
*he
cobar complex by
p-1X1` ' k k
_ p [,ip1|,(n-i)p1].
i=1p i
One can show that b1,0is non-nilpotent, in analogy with h1,0at the prime 2.
The nilpotence (or lack thereof) of b1,kfor k > 0 has not been determined
in general; a result of Nakamura [Nak75 , Proposition 1.1(c)] shows that these
classes are all nilpotent when p = 3, and this is the best known result in this
direction.
Given Conjecture 1.4, one could conclude that b1,kis nilpotent in Ext*bA(Fp,*
* Fp),
and hence b1,kis nilpotent in Ext*A(Fp, Fp) for sufficiently large k. Then [Pal*
*01,
Lemma B.3.3] would imply that b1,kis nilpotent for all k 1. In general, a
positive solution to Conjecture 1.4 would be a helpful step in trying to prove
odd primary analogues of the nilpotence and F -isomorphism results in [Pal99].
See [Pal01, Conjecture 5.4.1] and the surrounding discussion.
We end this section by exploring the connection between the dual complete
Steenrod algebra Ab and the Hopf algebras Hn, with an eye toward Conjec-
ture 1.4. The philosophy is that, while bAis defined by formally inverting the
Frobenius OE, Hn is defined by forcing OEn to be the identity. At the level of *
*co-
homology, fP0 is invertible of infinite order in Ext*bA(Fp, Fp), and it is inve*
*rtible
of order n in Ext*Hn(Fp, Fp). So one can hope that knowing that every positive-
dimensional element in Ext*Hn(Fp, Fp) is nilpotent would be relevant to studying
nilpotence of classes in Ext*bA(Fp, Fp).
More formally, we start by noting that one can generalize the definition of
Ab; in keeping with the definition and notation for Hnjfrom Section 2, for any
j 0 we let
bAj= Fp[,pkn: n j + 1, k 2 Z].
Thus Ab0= Ab, and Abjis a conormal quotient Hopf algebra of Abkwhenever
j k. Note that for each n and j, one can view Hnjas being a quotient of bAj:
n+k pk
Hnj= bAj=(,pi - ,i : i 1, k 2 Z).
Also, as noted in Proposition 3.4, Hnjis a quotient of Hmj whenever n divides
m. Thus for fixed j, the Hopf algebras Hnjform an inverse system.
Proposition 5.4. Fix j 0. Abjembeds in the inverse limit of the Hopf algebras
Hnj. This embedding is not an isomorphism.
Proof. The quotient maps bAji Hnjinduce maps from bAjto the inverse limit,
and one can see that the kernel of the map to the limit is zero.
Let H1j be the inverse limit. To see that bAjis not isomorphic to H1j, we
note that there are elements in H1j of unipotence height p: elements x different
from 1 satisfying xp = x. To construct these elements, it suffices to construct*
* a
family of such elements, one in each Hnj, mapping to each other in the inverse
n-1 __
system. The relevant elements are ,pi , for each i > j. |__|
14
Question 5.5. (a)What does this imply about the cohomology of bA?
(b)What is the cokernel of the map bAj,! lim-Hnj?
(c)How are H1j and G1j (from Proposition 3.4) related? How are bAjand
G1j related?
A Proof of Theorem 1.2
We restate the theorem.
Theorem A.1. Let p = 2 and fix z 2 Exts,tA(F2, F2). If s is positive, then there
0
is an n so that (fSq)n(z) is nilpotent.
Proof. This follows from the main result of [Pal99].
In detail, for each pair of integers (S, T ) with 0 S < T , let RT,S be the
polynomial algebra
RT,S= F2[hts : 0 s S < T t].
Let R be the inverse limit of the RT,S under the apparent maps (map each
polynomial generator to the generator of the same name if present, and to zero
otherwise). Then
R = F2[hts : s < t]=(htshvu : u t).
Also, Rn,n-1is the cohomology of this quotient Hopf algebra of A:
n
E(n) = A=(,1, . .,.,n-1, ,2i : i n),
with the polynomial generator htscorresponding to the cobar element [,2st]. So
there is a restriction map
æn : Ext*A(F2, F2) ! Rn,n-1.
These can be assembled to give a map
æ : Ext*A(F2, F2) ! R,
and theorems 1.3 and 1.4 of [Pal99] say that up to nilpotence, æ is an isomor-
phism between the cohomology of A and a certain subring of the codomain. In
particular, æ is a monomorphism mod nilpotence.
Since æ is the inverse limit of the maps æn, and since each æn is a restrict*
*ion
0
map, æ commutes with the action of fSq. Hence there is an induced map
0-1 * 0-1
æ : (fSq) ExtA(F2, F2) ! (fSq) R,
which is a monomorphism mod nilpotence. But the codomain is zero - the
Frobenius map acts nilpotently on each element of the Hopf algebra E(n).
*
* 0
Equivalently, at the level of cohomology, there is a multiplicative action of f*
*Sq
0 0
on R, defined by fSq(hts) = ht,s+1if s + 1 < t, but fSq(ht,t-1) = 0. Thus for
0 __
each element y 2 R, (fSq)n(y) = 0 for n sufficiently large. |__|
15
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