Commutative Morava homology Hopf algebras
Hal Sadofsky and W. Stephen Wilson
Abstract.We give the Dieudonne module theory for Z=2(pn --1)graded bi-
commutative Hopf algebras over Fp. These objects arise as the Morava K-
theory of homotopy associative, homotopy commutative H-spaces.
1. Introduction
In [DG70 , V], Demazure and Gabriel classify commutative unipotent alge-
braic groups over a perfect field of characteristic p in terms of Dieudonne mod*
*ules.
Making appropriate translations of terminology, this classifies a category of H*
*opf
algebras in terms of Dieudonne-modules.
Bousfield, in an appendix to [Bou96b ], proves many of the results of inter*
*est
in an accessible way, adapting and clarifying [DG70 , V] for his specific purp*
*oses.
In particular, he gives much of the structure for Z=(2)-graded bicommutative Ho*
*pf
algebras over Fp. He needs these results for his work with mod p K-theory. We
are interested in similar results for Morava K-theory. The Morava K-theory of
homotopy commutative H-spaces gives rise to bicommutative Z=2(pn - 1)-graded
Hopf algebras satisfying certain conditions. Having a Dieudonne module theory f*
*or
these Hopf algebras gives us a great deal of control over their structure. When*
* the
grading is forgotten then Bousfield's results can be applied, so our contributi*
*on is
really to unravel the gradings in our case and make appropriate definitions so *
*that
Bousfield's proofs go through with little or no modification.
Specifically, we want to study the category C(n) of commutative Morava Hopf
algebras. These are bicommutative, biassociative, Hopf algebras over Fp which a*
*re
graded over Z=2(pn - 1) (n > 0) and have an exhaustive primitive filtration. Th*
*is
last condition, an exhaustive primitive filtration, means that each element maps
trivially under some iterate of the reduced coproduct (into the tensor product *
*of
the cokernel of coaugmentation ideal). (Bousfield calls this property "irreduci*
*ble,"
a term we wish to reserve for a Hopf algebra with no proper non-trivial sub-Hopf
algebras). Bousfield's work analyzes these Hopf algebras quite thoroughly by r*
*e-
ducing the Z=(2)-graded case to the ungraded case by splitting off the sub-alge*
*bra
generated by odd-dimensional primitives. The study of these Hopf algebras in
[HRW98 ] concentrates on important consequences of the grading. Our goal here
is to put the grading into Bousfield's results.
____________
1991 Mathematics Subject Classification. Primary 16W30, 55N20.
The first author was partially supported by the National Science Foundation.
1
2 HAL SADOFSKY AND W. STEPHEN WILSON
[HRW98 ] shows C(n) splits (for odd primes) as the product of two categori*
*es;
OC(n), whose objects are Hopf algebras generated by primitives in odd degrees,
and hence primitively generated exterior algebras, and EC(n), whose objects are*
* all
concentrated in even degrees. (Note that [Bou96b ] proves the same result for *
*the
Z=(2)-graded case, and his proof also holds without alteration for C(n).) So, o*
*ur
real category of study will be EC(n).
In [Bou96a ], Bousfield shows the Z=(2)-graded case is an abelian category,
but again, his proof works just as well for C(n). Our contribution is to defin*
*e a
functor to Dieudonne modules, to which we again appeal to [Bou96b ] with small
alterations.
We define a Morava-Dieudonne module to be a Z=(pn-1)-graded abelian group
M* with endomorphisms F : Mt ! Mpt, and V : Mt ! Mt=p(note that in
Z=(pn - 1) every t=p exists) such that:
(i)F V = p = V F ;
(ii)for each x 2 M, there exists a q 1 with V q(x) = 0.
We denote the category of Morava-Dieudonne modules by MD(n)*. Note the
definition implies the modules are all p-torsion, i.e. pq(x) = (F V )q(x) = F q*
*V q(x) =
0 where we have used (i) and (ii).
Our main theorem is:
Theorem 1.1. There is a functor m* : EC(n) ! MD (n)* which is an equiva-
lence of abelian categories.
Let Z=(n) act on Z=(pn - 1) by k * j = jpk. The orbits are given by fl =
{j; jp; jp2; jp3; : :;:jpn = j}. We can see with no work that MD (n)* splits a*
*s a
product of categories MD fl(n)* where all elements are in the degrees contained
in fl. This translates into a similar theorem, a major result of [HRW98 ], ab*
*out
EC(n).
Corollary 1.2 ([HRW98 ], Theorem 1.13).As categories:
Y
EC (n) ' EC(n)fl
fl
where EC(n)flconsists of Hopf algebras in EC(n) with all primitives in degrees *
*2fl.
This result was proven in [HRW98 ] so it could be combined with results of
[RW80 ] to show that there is only the trivial map from K(n)*(K(Z=(pi); s)) to
K(n)*(K(Z=(pj); t)) when s 6= t.
The motivation for us is Morava K-theory, K(n)*(-), which is a generalized
homology theory with a K"unneth isomorphism. The coefficient ring is K(n)* '
Fp[vn; v-1n] where the degree_of_vn is 2(pn - 1). By setting vn = 1 we get a
Z=2(pn - 1)-graded theory,_K(n)_*(-). If a space X is connected and the loop
space of an H-space then K(n) *(X) is in C(n) for odd primes (there are some
complications with commutativity for p = 2).
In [RW80 ] the Morava K-theory of all (abelian) Eilenberg-Mac Lane spaces
was computed. They all, with the exception of K(Z; 1), the circle, lie in EC(n*
*).
Let I = (i1; : :;:iq), 0 i1 < i2 < . .<.iq < n. When i1 > 0 define s(I) =
(i1-1; i2-1; : :;:iq-1) and when iq < n-1 define s-1(I) = (i1+1; i2+1; : :;:iq+*
*1).
We can just read off the Morava-Dieudonne module stucture from [RW80 ].
COMMUTATIVE MORAVA HOMOLOGY HOPF ALGEBRAS 3
_____
Theorem 1.3. The Morava-Dieudonne module, m*(K(n) *(K(Z=(pj); q))), is
a free Z=(pj) module on generators aI = a(i1)a(i2): :a:(iq), I = (i1; : :;:iq),*
* 0
i1 < i2 < . .<.iq < n in degree pi1+ pi2+ . .+.piqwith
8
< as(I) if i1 > 0
V (aI) = :
(-1)q-1pa(i2-1)a(i3-1): :a:(n-1)if i1 = 0
and 8
< pas-1(I) if iq < n - 1
F (aI) = :
(-1)q-1a(0)a(i1+1): :a:(iq-1+1)if iq =:n - 1
We have, from [RW80 , Corollary 13.1], a short exact sequence of Hopf algeb*
*ras:
_____ i _____ i+j _____ j
Fp ! K(n)*(K(Z=(p ); q)) ! K(n)*(K(Z=(p ); q)) ! K(n)*(K(Z=(p ); q)) ! Fp:
This translates into the most obvious exact sequence in the Morava-Dieudonne
modules. For the surjection, [RW80 , Proposition 11.4] is enough and for the
injection we need that and [RW80 , Theorem 11.1(c)].
Since
_____ _____ _____
lim!K(n)*(K(Z=(pi); q)) ' K(n)*(K(Q=Z(p); q)) ' K(n)*(K(Z; q + 1))
the above map shows the Morava-Dieudonne module for this is a bunch of copies
of Q=Z(p)and gives both F and V .
The Morava-Dieudonne module description of the Morava K-theory of Eilenberg-
Mac Lane spaces is much simpler than the Hopf algebra description of [RW80 ].
This is because it is no longer necessary to know the order of truncation of th*
*e al-
gebra generators to give a precise description (as it is for Hopf algebras). Th*
*e order
of F on generators can be calculated from the description of the Morava-Dieudon*
*ne
module structure, but it is not necessary to know it to give the description. T*
*his
is a definite improvement because the orders are probably wrong in [RW80 ].
Note that because Morava K-theory has a K"unneth isomorphism the Morava-
Dieudonne module of the Morava K-theory of a finite product of Eilenberg-Mac La*
*ne
spaces is just the direct product of the respective Morava-Dieudonne modules.
We define an irreducible Morava Hopf algebra as one with no proper non-trivi*
*al
sub-Hopf algebras. Since we are working with an abelian category, it is clear t*
*hat
any finite Morava Hopf algebra can be filtered so that the quotients are irredu*
*cible.
The remaining interesting problem is to identify these finite irreducibles. Thi*
*s is
easiest to do in the category of Morava-Dieudonne modules. We need only work in
the category EC(n)flbecause of Corollary 1.2. Let |fl| be the order of the orbi*
*t fl.
Theorem 1.4. Every finite irreducible Morava-Dieudonne module can be fil-
tered so the quotients are irreducible Morava-Dieudonne modules. The finite irr*
*e-
ducible Morava-Dieudonne modules in ECfl(n) are:
(i)An Fp (in any degree in fl) with F and V trivial.
(ii)A q dimensional vector space over Fp in each degree in fl with V and f(F *
*|fl|)
acting trivially (F is an isomorphism) where f is a primitive polynomial *
*over
Fp (of degree q).
These irreducibles correspond to the following Morava Hopf algebras. The fir*
*st
is the truncated polynomial algebra P (x)=(xp), where x is primitive. The second
is the more complicated P (x)=(f(F |fl|)(x)) where again x is primitive. This *
*last
4 HAL SADOFSKY AND W. STEPHEN WILSON
relation just tells us how to write xpq|fl|in terms of xpi|fl|for i < q. The pr*
*imitive
nature of the polynomial prevents us from finding sub-Hopf algebras of this.
This theory can be generalized to a work over a perfect field of characteris*
*tic
p, see [Rav75 ]. If we do this over the agebraic closure of Fp, then the irredu*
*cibles
are much simplified. In fact, none of the more exotic irreducible Hopf algebras*
* have
been spotted in nature, raising the question of whether they can arise as part *
*of
the Morava K-theory of a space or not.
There is an additional theorem worth pointing out. Under very mild condition*
*s,
see [Bou96b , Appendix B, Theorem B.4], Sweedler proves what we would call a
dual Borel theorem (about the coalgebra structure) in [Swe67a ] and [Swe67b ].
We remark that our classification of Hopf algebras constructs Dieudonne mod-
ules by, as usual, considering maps of some variant of the Witt ring (see secti*
*on 2
for the variant suited to the problem in this paper) into the Hopf algebra under
consideration. There is another context familiar to algebraic topologists where*
* ob-
jects called Dieudonne modules arise. This is in the classification of formal g*
*roups
over a field k of characteristic p. We would like to point out the similarities*
* of these
situations as the authors have not seen this made explicit elsewhere.
Recall that bicommutative Hopf k-algebras are abelian group objects in the
category of commutative k-algebras. The Witt algebra (or as in this paper, some
variant of it) is a particular bicommutative Hopf k-algebra that serves as a ge*
*nerator
for this category. Our classification comes about by taking maps of Hopf algebr*
*as
from the Witt algebra to some other Hopf algebra and considering this group as a
"Diedonne module" - a module over the ring of self maps of the Witt Hopf algebr*
*a.
Commutative formal groups over k are, on the other hand, abelian group ob-
jects in the category of formal schemes over k. There is a special commutative
formal group (infinite dimensional) called the Witt vectors. (The Witt algebra *
*is
essentially the representing ring of the Witt vectors in the category of k-alge*
*bras.)
To classify formal groups [Laz75 ], we use the "curves" functor which is maps of
formal groups from the Witt vectors (as a formal group) to the formal group und*
*er
consideration. This gives us a "Diedonne module" which is a module over the self
maps of the Witt vectors. Because the Witt vectors are represented by the Witt
Hopf algebra, the Diedonne module is also a module over the self maps of the Wi*
*tt
Hopf algebra.
Background. A Dieudonne module theory has long been developed for graded
Hopf algebras, [Sch70 ]. The authors had been talking about this project for so*
*me
time when the second author was decimated by the chairmanship of his depart-
ment. By the time he had recovered, the two papers, [Bou96b ] and [Bou96a ],
were available and the project was easy to complete by co-opting the proofs from
Bousfield, [Bou96b ]. Pete Bousfield politely but firmly declined the authors'*
* in-
vitation to be a coauthor. The authors regret his decision and make no claim to
originality. We feel the theorems should be made available however. Following a
suggestion of Igor Kriz's we were developing the general Dieudonne module theory
for Hopf rings only to find that Paul Goerss had already done it. We thank the
referee for his helpful suggestions.
COMMUTATIVE MORAVA HOMOLOGY HOPF ALGEBRAS 5
2. Differences from Bousfield
At the suggestion of the referee we assume the basic facts about the element*
*ary
Hopf algebras which we use without a detailed review. Some basic and specific
references are [Wit36 ], [Ser67 ], [Kra72 ], [HRW98 ], and [Bou96bn ]. n-1
There is a bicommutative Hopf algebra W = Z[x0; x1; : :]:with xp0+ pxp1 +
. .+.pn-1xpn-1+ pnxn primitive. We can grade W by setting |x0| = 2t and |xi| =
pi2t. The primitives are homogeneous, so we get a graded Hopf algebra which we
call Wt. From it we get Hopf algebras graded over Z=2(pn - 1) by defining
W (s; t) Fp[x0; x1; : :;:xs] Wt Fp:
Bicommutative Hopf algebras over perfect fields of characteristic p come with t*
*wo
self maps, the Frobenius F and the Verschiebung, V . The Frobenius is the pthpo*
*wer
map, i.e. F (x) = xp, which is a map of Hopf algebras since we are in character*
*istic
p, but not of graded Hopf algebras. The Verschiebung can be defined as the dual
of the Frobenius map on the dual Hopf algebra, as in [HRW98 ], or in terms of
the coproduct, as in [Bou96b ]. It gives a map of Hopf algebras, but not a map
of graded Hopf algebras since |V (x)| = |x|=p. We have V (xi) = xi-1 in W (s; t*
*).
When x is primitive, V (x) = 0, so V is 0 on primitively generated bicommutative
Hopf algebras. Clearly F V = V F since V is a map of algebras. Hom EC(n)(A; B) *
*is
an abelian group with addition of f and g defined using the coproduct on A and
the product on B:
A -! A A fg-!B B -! B:
It is standard that F V = p in the group Hom EC(n)(A; B).
The proof of Theorem 1.1 is the same as Bousfield's, [Bou96b ], and at the
suggestion of the referee has not been included here. We must include the defin*
*ition
of the functor though because here we do differ from Bousfield.
There is a surjective map W (s; t=p) ! W (s - 1; t) which takes the primitive
generator in degree 2t=p to zero.
We use the sequence:
W (0; t) W (1; t=p) . . .W (s; t=ps) W (s + 1; t=ps+1) . . .
to define
mt: EC(n) -! MD(n)t
by
mt(A) = lim!Hom(W (s; t=ps); A):
Remark 2.1. To get the equivalence of EC(n)fland MD (n)flwe need only
restrict our attention to the t in fl. In particular, when fl = {0} we are prec*
*isely
in the ungraded case of [Bou96b ] except with a slightly modified definition o*
*f the
functor between them.
To complete our definition of m* : EC(n) ! MD(n)* we need to define the
action of F and V on m*(A). We require a minor departure from the standard
definition in order to preserve grading. In our case we use the fact that the i*
*mage
of F on W (s; t) is canonically isomorphic to W (s; pt), i.e. the composite
W (s; t) '-!W (s; pt) ,! W (s; t)
that sends xi7! xi7! xpiis F . The second map here preserves degree, so is in o*
*ur
category.
6 HAL SADOFSKY AND W. STEPHEN WILSON
We get a map of sequences
W(0; t) W(1; t=p) . . . W(s; t=ps) W(s + 1; t=ps+1). . .
" " " "
W(0; pt) W(1; t) . . . W(s; t=ps-1) W(s + 1; t=ps). . .
which induces the map F
mt(A) = lim!Hom(W (s; t=ps); A) -! mpt(A) = lim!Hom(W (s; pt=ps); A):
We now need the map V . The image of V acting on W (s; t) is the sub-Hopf algeb*
*ra
W (s - 1; t). There is a canonical map of W (s; t=p) which surjects to this ima*
*ge. In
particular, this gives us maps
W (s; t=p) ! W (s - 1; t) W (s; t):
The corresponding map of sequences
W(0; t) W(1; t=p) . . . W(s; t=ps) W(s + 1; t=ps+1). . .
" " " "
W(0; t=p) W(1; t=p2) . . . W(s; t=ps+1) W(s + 1; t=ps+2). . .
induces V
mt(A) = lim!Hom(W(s; t=ps); A) -! mt=p(A) = lim!Hom(W(s; (t=p)=ps); A):
3. Irreducibles
Proof of Theorem 1.4. We construct the filtration of an arbitrary finite
Morava-Dieudonne module and see what we get. Assuming that M* is finite, we
look at the kernel of V on it. This is always a nontrivial sub-Morava-Dieudonne
module because V iterated enough times must be zero. Of course this kernel has
trivial V action so we have reduced our problem to filtering a Morava-Dieudonne
module with V = 0. This is a Fp vector space because p = F V = 0.
If F has a kernel then the kernel is a sub-Morava-Dieudonne module with
trivial F and V action. Any element of this generates a sub-module of type (i) *
*of
the Theorem. We have now reduced our problem to the case where V acts trivially
and F has no kernel, i.e. where F is an isomorphism of finite dimensional vector
spaces. Thus the dimensions of Mt for each t 2 fl must be the same.
Pick any x 2 M*. Because M* is finite dimensional there must be some minimal
u such that F u(x) can be written in terms of F i(x) for i < u. We first claim *
*that
u is a multiple of |fl|. If not, then F u(x) is not in the same degree as x and*
* the
above relation does not include x = F 0(x). Since F is an isomorphism, we can
divide the relation by some power of F until x is in the relation. This contrad*
*icts
the minimality of u.
Our relation can now be written as f(F |fl|)(x) = 0 because elements in
dimensions other than that of x cannot be part of the relation. Factor f
into primitive polynomials f1f2: :f:k. The element y = f2: :f:k(F |fl|)(x) *
*is
in M|x|. It has the property that f1(F |fl|)(y) = 0 and the vector*
* space
{y; F |fl|(y); F 2|fl|(y); F 3|fl|(y); : :;:F (q-1)|fl|(y)} (where q is the deg*
*ree of f1) is an
irreducible sub-vector space of M|x|under the action of F |fl|. We now take the
sub-Morava-Dieudonne module generated by y. This completes our proof and gives
our modules of type (ii).
COMMUTATIVE MORAVA HOMOLOGY HOPF ALGEBRAS 7
Note that we have just reproven the fundamental structure theorem for finite
modules over the P.I.D. Fp[z] (where z = F |fl|) and we could have just referre*
*d_to
it instead. |__|
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University of Oregon, Eugene, OR, 97403
E-mail address: sadofsky@math.uoregon.edu
Johns Hopkins University, Baltimore, Maryland 21218
E-mail address: wsw@math.jhu.edu