INJECTIVE COMODULES AND LANDWEBER EXACT
HOMOLOGY THEORIES
MARK HOVEY
Abstract.We classify the indecomposable injective E(n)*E(n)-comodules,
where E(n) is the Johnson-Wilson homology theory. They are suspensions of
the Jn,r= E(n)*(MrE(r))*where 0 r n, with the endomorphism ring of
Jn,rbeing [E(r)[E(r), where [E(r)denotes the completion of E(r).
Introduction
Perhaps the most important homology theory in algebraic topology is complex
bordism MU and the many theories derived from it. In particular, if p is a prim*
*e,
and we localize at p as we do throughout this paper, then MU splits as a coprod*
*uct
of suspended copies of a spectrum known as BP , with BP*(*) ~=Z(p)[v1, v2, . .]*
*.,
where vi has degree 2(pi- 1). Derived from BP are the many Landweber ex-
act homology theories [Lan76], such as the Johnson-Wilson theory E(n), with
E(n)*(*) ~=Z(p)[v1, . .,.vn, v-1n] and Morava E-theory En (see [DH95 ]).
As is well known [Rav86 , Theorem 2.2.8], for all of these homology theories
h, the homology groups h*X form a graded comodule over the graded Hopf al-
gebroid (h*, h*h). It therefore behooves us to learn as much as possible about
the category of h*h-comodules. In particular, products are not exact in the cat*
*e-
gory of h*h-comodules, and their derived functors form the E2-term of a spectral
sequence [Hov05 ] converging to the h*-homology of a product of spectra. This
spectral sequence offers one possible approach to the chromatic splitting conje*
*cture
of Hopkins [Hov95 ].
The author has thus been engaged in attempting to understand as much of the
structure of the category of h*h-comodules as possible, following in the footst*
*eps
of Peter Landweber, whose papers [Lan76], [Lan73], and the less well-known but
excellent [Lan79] are the basis for our understanding of BP*BP -comodules. In
particular, the author and Strickland, in [HS05a ] and [HS05b ], have shown that
analogues of Landweber's theorems hold in the category of E*E-comodules, where
E denotes, as it will throughout the paper, a Landweber exact homology theory of
height n such as E(n) or En. Recall that this means that there is a ring homomo*
*r-
phism BP* -!E* so that the sequence (p, v1, . .,.vn) is regular in E* and vn is*
* a
unit in E*=(p, v1, . .,.vn-1).
The present paper is devoted to the study of injective BP*BP -comodules and
E*E-comodules. Injective comodules have not been studied before, except briefly
in [HS05b ], because one can almost always use relatively injective comodules to
compute the derived functors of interest in topology. This includes the Ext gro*
*ups
ExtE*E(E*, E*X) and even the derived functors of product.
____________
Date: March 27, 2006.
1
2 MARK HOVEY
However, we will show in this paper that the injective comodules have a very *
*rigid
structure, in analogy to the Matlis theory of injective modules over a Noetheri*
*an
commutative ring (described in [Lam99 , Section 3I] ). There are only n+1 diffe*
*rent
isomorphism classes of indecomposable injective E*E-comodules, where E is, as
always, a Landweber exact homology theory of height n. They are the injective
hulls Jr of E*=Ir, where 0 r n and Ir = (p, v1, . .,.vr-1). Furthermore, if*
* E*
is evenly graded, Jr ~=E*(MrE(r)) up to suspension, where Mr is the fiber of the
map of Bousfield localizations*Lr -!Lr-1. Also, the endomorphism ring of Jr as
an E*E-comodule is [E(r)[E(r), where [E(r)= LK(r)E(r) is the completion of E(r)
at Ir. *
Unfortunately, a good algebraic description of [E(r)[E(r)is not known. We know
that E*rEr is the twisted completed group ring Er*[[Sr]] on the large Morava st*
*abi-
lizer group (this is an old result of Hopkins and Ravenel;*see [Bak89 ] or [Hov*
*04b ] for
proofs of it). It therefore seems likely that [E(r)[E(r)is very closely related*
* to the
stabilizer group Sr. If we accept this, then we are seeing all of the stabilize*
*r groups
Sr for 0 r n in the category of E*E-comodules. Since the chromatic splitting
conjecture is in some sense about how the apparently unrelated stabilizer groups
actually are related to each other, seeing all of the stabilizer groups togethe*
*r like
this is a good sign and might be useful. We can also find a decomposable inject*
*ive
comodule whose endomorphism ring is E*rEr.
As a corollary of our work, we rediscover the folklore isomorphisms
[E(n)*(X) ~=Hom E(n)*(E(n)*(MnX), -n E(n)*=I1n)
and
E*nX ~=Hom En*(En*(MnX), -n En*=I1n).
As far as the author knows, these isomorphisms have not been written down befor*
*e,
though they were certainly known to Hopkins, Greenlees, Sadofsky, and others.
Throughout this paper p will be a fixed prime integer, all spectra will be lo*
*calized
at p, and n > 0 will be a fixed positive integer. The symbol E will denote a
Noetherian Landweber exact homology theory of height n, like E(n) or En. The
symbol (A, ) will denote a flat Hopf algebroid. *
The author thanks Andy Baker for sharing his insights into [E(r)[E(r).
1. BP*BP -comodules and E*E-comodules
The purpose of this section is to remind the reader of some of the results on*
* the
structure of BP*BP -comodules, E*E-comodules, and the relation between them.
Recall that the Hopf algebroid (A, ) is called an Adams Hopf algebroid
when is the colimit of a filtered system of comodules i, where each iis fin*
*itely
generated and projective over A. As explained in [Hov04a , Section 1.4] (though
originally due to Hopkins), both (BP*, BP*BP ) and (E*, E*E) are Adams Hopf
algebroids. When (A, ) is an Adams Hopf algebroid, the category of -comodules
is a Grothendieck category [Ste75, Chapter V]. This means that it is an abelian
category in which filtered colimits areLexact, and that there is a family of ge*
*ner-
ators {Pi}, in the sense that -comod( Pi, -) is a faithful functor. The most
natural collection of generators consists of the dualizable comodules; these ar*
*e the
comodules that are finitely generated and projective over A. These are studied *
*and
proved to be generators in Sections 1.3 and 1.4 of [Hov04a ]. Note that BP* and*
* its
INJECTIVE COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 3
suspensions do not generate the category of BP*BP -comodules, and we know of
no naturally defined proper subcollection of the dualizable comodules which does
so.
As is well-known, the category of -comodules is a closed symmetric monoidal
category [Hov04a , Section 1.3]. The tensor product is denoted M ^ N; it is iso*
*mor-
phic as an A-module to the tensor product M A N of left A-modules (a different
notation is used because the usual tensor product symbol is reserved for the te*
*nsor
product of A-bimodules, which occurs frequently in the theory). The internal Hom
object is denoted F (M, N); there is a natural map F (M, N) -! Hom A(M, N) of
A-modules that is an isomorphism when M is finitely presented over A.
There are enough injectives in any Grothendieck category. One can see this
directly for -comodules by noting that A J is an injective -comodule whenev*
*er
J is an injective A-module. Note that this is a bimodule tensor product, and all
the -coaction is concentrated on the tensor factor. Moreover, if A is Noethe*
*rian,
as it is for (A, ) = (E*, E*E), the category of -comodules is locally Noether*
*ian
as well. This means that subcomodules of the generators satisfy the ascending
chain condition; this is obvious because dualizable comodules are finitely gene*
*rated
over A. In any locally Noetherian Grothendieck category, direct sums of injecti*
*ves
are injective and every injective is a direct sum of indecomposable injectives *
*in an
essentially unique way (see [Ste75, Sections V.4, V.5]).
There is a functor * from BP*BP -comodules to E*E-comodules defined by
*M = E* BP* M. This functor is much studied in [HS05a ] and [HS05b ]. The
functor * is exact and has a right adjoint *. The composite * * is naturally
equivalent to the identity, and the composite * * is the localization functor *
*Ln
with respect to the hereditary torsion theory of vn-torsion comodules. In parti*
*cu-
lar, * defines an equivalence of categories between E*E-comodules and Ln-local
BP*BP -comodules. This result is valid even if E* is not Noetherian, so the res*
*ults
in this paper about injective E*E-comodules also apply to the case when E* is
not Noetherian. It is proved in [HS05b , Section 2] that * and *, preserve i*
*n-
jectives, filtered colimits, and arbitrary direct sums. In fact, if TnM denotes*
* the
subcomodule of vn-torsion elements in M, then Tn also preserves injectives, and
LnJ = J=TnJ for injective BP*BP -comodules J.
2. Indecomposable injectives
We have remarked above that every injective E*E-comodule is a direct sum
of indecomposable injectives. Matlis has a well-known theory of indecomposable
injectives over Noetherian rings (see, for example,[Lam99 , Section 3I]), and we
will mimic his theory for indecomposable injective E*E-comodules and BP*BP -
comodules. The object of this section is to classify all the indecomposable inj*
*ective
E*E-comodules.
We first point out that indecomposable injectives are relevant even for BP*BP*
* -
comodules.
Proposition 2.1. Suppose J is an injective BP*BP -comodule for which there
exists an n such that J has no vn-torsion. Then J is a direct sum of indecompos*
*able
injectives in an essentially unique way.
Before proving this proposition, we point out the following lemma.
4 MARK HOVEY
Lemma 2.2. If N is an E*E-comodule, then N is indecomposable if and only if
*N is indecomposable.
Proof.In general, an object M in an abelian category C is indecomposable if and
only if the ring C(M, M) has no nontrivial idempotents. Since * is fully faith*
*ful,
the result follows.
Proof of Proposition 2.1.Since J has no vn-torsion and J is injective, J is Ln-
local. Hence J = * *J. Now *JLis an injective object in the category of
E*E-comodules, and hence *J ~= Jff, where each Jffis an indecomposable
injective E*E-comodule. Since * preserves direct sums, we get
M
J ~= *Jff.
Since * preserves injectives, and also indecomposables by Lemma 2.2, we see th*
*at
J is a direct sum of indecomposable injectives. The uniqueness of this direct s*
*um
decomposition follows from the Krull-Remak-Schmidt-Azumaya theorem as in Sec-
tion V.5 of [Ste75], using the fact that the endomorphism ring of an indecompos*
*able
injective is always local [Ste75, Proposition V.5.1].
The next thing to do is to enumerate all the indecomposable injectives. Recall
that the indecomposable injective modules over a commutative Noetherian ring
R are the injective hulls of the R=p, where p is a prime ideal in R. The ideals
Ir = (p, v1, . .,.vr-1) are the prime invariant ideals in E(n)* for 0 r n a*
*nd
in BP* for 0 r 1 (see [HS05a , Theorem 5.6] for the E(n)* case). For an
arbitrary Landweber exact theory of height n, it is possible that the Ir are not
actually prime, but they remain the "categorically prime" invariant ideals in E*
**,
as explained in [HS05a , Theorem 5.6]. Hence we let Jn,rfor 0 r n denote the
injective hull of E*=Ir in the category of E*E-comodules, and let Jr for 0 r *
* 1
denote the injective hull of BP*=Ir in the category of BP*BP -comodules.
Lemma 2.3. The injective comodules Jn,rfor 0 r n and Jr for 0 r 1
are indecomposable.
Proof.According to [Ste75, Proposition V.2.8], it suffices to show that E*=Ir a*
*nd
BP*=Ir are coirreducible, which means that any two nontrivial subcomodules
M, N have nontrivial intersection. This is obvious for BP*=I1 = Fp. For the oth*
*er
cases, we use the fact that every nontrivial BP*BP -comodule or E*E-comodule has
a nonzero primitive [HS05a , Theorem 5.1]. For example, assume 0 < r < n. Then
the primitives in E*=Ir are isomorphic to Fp[vr] by [HS05a , Theorem 5.2]. Hence
M contains all sufficiently high powers of vr, as does N, and so M \ N 6= 0. The
other cases are similar, using the computations of the primitives in BP*=Ir [Ra*
*v86 ,
Theorem 4.3.2] and in E*=Ir [HS05a , Theorem 5.2].
We also note that Jr has no vr-torsion in view of [HS05b , Theorem 2.7]. (We
will see below that Jn,ralso has no vr-torsion).
Theorem 2.4. An E*E-comodule J is an indecomposable injective comodule if and
only if J ~= tJn,rfor some 0 r n and some t. Similarly, a BP*BP -comodule
with no vn-torsion is an indecomposable injective comodule if and only if J ~= *
*tJr
for some 0 r n and some t.
There are probably many different indecomposable injective BP*BP -comodules
that are vn-torsion for all n.
INJECTIVE COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 5
Proof.Suppose J is an indecomposable injective E*E-comodule. Then J has a
nonzero subcomodule that is finitely generated over E* (see [Hov04a , Proposi-
tion 1.4.1]). By the filtration theorem [HS05a , Theorem 5.7], any nontrivial f*
*initely
generated E*E-comodule contains an isomorphic copy of tE*=Ir for some t and
some r with 0 r n. Since J is injective, we see that tJr J; since J is a*
*lso
indecomposable, tJr ~=J.
Similarly, suppose J is an indecomposable injective BP*BP -comodule with no
vn-torsion. It is once again true that J contains a nonzero subcomodule that is
finitely generated over BP*. The new ingredient is [Lan79, Corollary 7], which
implies that, since J has no vn-torsion, its finitely generated subcomodules are
in fact finitely presented over BP*. Therefore we can use the usual Landweber
filtration theorem [Lan76] to find a subcomodule of the form tBP*=Ir for 0 r
n. As before, this implies that J ~= tJr.
Note that the obvious inclusions
BP*=In -! tBP*=(p1 , . .,.v1n-1) -!v-1n tBP*=(p1 , . .,.v1n-1)
are essential extensions, so these important comodules (defined more precisely *
*after
Theorem 3.1) have the same injective hull Jn. Here t = |v1| + . .+.|vn-1|, and *
*the
inclusion sends 1 2 BP*=In to 1=pv1. .v.n-1.
It is also useful to know that * and * map these indecomposable injectives *
*as
one would expect.
Lemma 2.5. We have *Jr ~=Jn,rand *Jn,r~=Jr for 0 r n.
Proof.Recall that both * and * preserve injectives by [HS05b , Corollary 2.5]*
*, and
* preserves indecomposables by Lemma 2.2. Hence *Jn,ris an indecomposable
injective, necessarily without vn-torsion. On the other hand, since
*BP*=Ir = E*=Ir Jn,r,
we conclude that
Ln(BP*=Ir) *Jn,r.
But Ln(BP*=Ir) is either BP*=Ir itself if r < n, or v-1nBP*=In if r = n [HS05a ,
Lemma 5.3]. In either case we see that Jr *Jn,r. Equality must hold since
*Jn,ris an indecomposable injective.
It then follows that *Jr = * *Jn,r~=Jn,r, as required.
This lemma implies that Jn,rand Jr share much of the same structure. For
example, we have the following corollary.
Corollary 2.6. Suppose 0 r n. Then Jr and Jn,rare Ir-torsion and vr-
periodic.
Proof.For Jr, this follows from Proposition 2.2, Proposition 2.6, and Theorem 2*
*.7
of [HS05b ]. It is clear that if M is Ir-torsion, so is *M, and so Jn,ris Ir-t*
*orsion.
In particular, since vr is a primitive modulo Ir, this means that v-1rJn,ris a *
*well-
defined comodule. The kernel K of Jn,r-! v-1rJn,ris the vr-torsion in Jr and K
intersects E*=Ir trivially. Since Jn,ris an essential extension of Jn,r, we con*
*clude
that Jn,rhas no vr-torsion. This means that v-1rJn,ris an essential extension of
Jn,r, so v-1rJn,r= Jn,r.
6 MARK HOVEY
3. Structure of indecomposable injectives
We would like to know more of the structure of the indecomposable injectives
Jr and Jn,r, in analogy to Matlis' theory described in [Lam99 , Section 3I].
The object of this section is to prove the following theorem.
Theorem 3.1. If E = E(n), then there is an isomorphism of comodules
Jn,n~=E(n)*E(n) E(n)* tE(n)*=I1n,
where t = |v1| + . .+.|vn-1|.
Note that in this theorem, we are not free to use any Noetherian Landweber
exact homology theory of height n, but must use E(n) itself. In particular, we
cannot use Morava E-theory.
In this theorem E(n)*=I1n is thought of as an E(n)*-module, not as a comodule,
and it is the usual construction used in algebraic topology. That is, we induct*
*ively
define E(n)*=I1r via the short exact sequence
0 -!E(n)*=I1r -!v-1rE(n)*=I1r -!E(n)*=I1r+1-!0.
Additive generators of E(n)*=I1n are given by
____~vrn____
pi0vi11. .v.in-1n-1
where ~ 2 Fp, r 2 Z and i0, i1, . .,.in-1 all positive integers. The action of
p, v1, . .,.vn is the obvious one, with a product being 0 if it ever removes any
of p, v1,. . . ,vn-1 from the denominator. For example, E(n)*=In is the submodu*
*le
of tE(n)*=I1n generated by 1=pv1. .v.n-1.
We can make a similar construction to form En*=I1n, where
En* = W Fq[[u1, . .,.un-1]][u, u-1]
is the coefficient ring of Morava E-theory, where q = pn, W Fq is the Witt vect*
*ors
of the Galois field Fq, the ui have degree 0, andru has degree -2. There is a r*
*ing
homomorphismnE(n)* -! En* that takes vr to uru1-p for 1 r < n and vn to
u1-p . Hence Ir = (p, u1, . .,.ur-1) as an ideal of En*. The elements
_____~ur_____,
pi0ui11. .u.in-1n-1
where ~ 2 Fq, r 2 Z and i0, i1, . .,.in-1 are all positive integers, are additi*
*ve
generators for En*=I1n.
It is important to note that E(n)*=I1nis not the increasing union of the E(n)*
**=Irn
for n > 1. Indeed, E(n)*=I2nhas distinct elements p and v1 that are both killed*
* by
In, whereas the only elements of E(n)*=I1n killed by In are ffvrn=pv1. .v.n-1for
ff 2 Fp and r 2 Z. Thus no shifted copy of E(n)*=I2ncan sit inside E(n)*=I1n for
n > 1.
Theorem 3.1 has the following corollary.
Corollary 3.2. We have
Jr ~=BP*BP BP* tE(r)*=I1r
and, for any Noetherian Landweber exact E of height n,
Jn,r~=E*BP BP* tE(r)*=I1r
for 0 r n. Here t = |v1| + . .+.|vr-1|.
INJECTIVE COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 7
In this corollary, E is, as always, a Noetherian Landweber exact homology the*
*ory
of height n.
Proof.The corollary follows from the fact that
*(E*E E* M) = BP*BP BP* M
(see Lemma 2.4 of [HS05a ]), and Lemma 2.5.
We will prove Theorem 3.1 by first using Matlis theory to show that tE(n)*=I*
*1n
is the injective hull of E(n)*=In in the category of E(n)*-modules. This shows *
*that
E(n)*E(n) E(n)* tE(n)*=I1n is an injective comodule. We then show that both
of the maps in the composition
E(n)*=In i-!E(n)*E(n) E(n)*E(n)*=In 1-j-!E(n)*E(n) E(n)* tE(n)*=I1n
are essential extensions. Here i is the unit of the adjunction between the forg*
*etful
functor and the extended comodule functor, with i(a) = jL(a) 1, and j is the
embedding of the E(n)*-module E(n)*=In into its injective hull.
Proposition 3.3. The injective hull of E(n)*=In in the category of E(n)*-modules
is tE(n)*=I1n, where t = |v1|+. .+.|vn-1|. Similarly, the injective hull of En*
**=In
in the category of En*-modules is En*=I1n.
Proof.The proof is a computation using Matlis theory, modeled on [Lam99 , Sec-
tion 3J]. We first note that Q=Z(p)[vn, v-1n] is an injective object in the cat*
*egory of
graded Z(p)[vn, v-1n]-modules and graded homomorphisms (not necessarily of degr*
*ee
0). To see this, one can just use the same proof as the proof that shows Q=Z(p)
is an injective Z(p)-module. This proof works for discrete valuation rings such*
* as
W Fq as well, so that (W Fq Q)=Q[u, u-1] is an injective object in the catego*
*ry
of graded W Fq[u, u-1]-modules and graded homomorphisms. The rest of the proof
for En* is the same as the proof for E(n)* given below.
Now let M = Hom *Z(p)[vn,v-1n](E(n)*, Q=Z(p)[vn, v-1n]); M is an injective gr*
*aded
E(n)*-module by the generalization of [Lam99 , Lemma 3.5] to the graded case. We
define the element
_____vrn____= _______1________
pi0vi11. .v.in-1n-1pi0vi11. .v.in-1n-1v-rn
of M to be the element that takes the monomial
vi11. .v.in-1n-1
to vrn=pi02 Q=Z(p)[vn, v-1n], and takes the complementary Z(p)[vn, v-1n]-summand
of E(n)* to 0. Then the submodule of M generated by these elements is E(n)*=I1n.
An arbitrary element of M can be written as an infinite sum of these elements.
By Proposition 3.88 of [Lam99 ] (again, in the graded case), the In-torsion in
M is still an injective E(n)*-module. We claim that the In-torsion in M is just
E(n)*=I1n. Indeed, it is clear that each of the elements
_____vrn____
pi0vi11. .v.in-1n-1
is In-torsion. On the other hand, if f 2 M is In-torsion, then Iknf = 0 for lar*
*ge k, so
f(Ikn) = 0. Thus f kills all but finitely many monomials pi0vi11vi22. .v.in-1n-*
*1in E(n)*.
It follows that f is a finite sum of our generating elements, so is in E(n)*=I1*
*n.
8 MARK HOVEY
Now it is clear that -tE(n)*=In -!E(n)*=I1n, where 1 goes to 1=pv1. .v.n-1,
is an essential extension, completing the proof.
We need a simple test for essential extensions of comodules.
Lemma 3.4. Suppose (A, ) is a flat Hopf algebroid for which every -comodule
has a primitive. Then an extension M -!N of -comodules is essential if and only
if Ax \ M 6= 0 for all primitives x 2 N.
Note that every E*E-comodule has a primitive by [HS05a , Theorem 5.1].
Proof.If N0 is an arbitrary nonzero subcomodule of N, it must contain a primiti*
*ve
x of N. The subcomodule Ax generated by x is then inside N0. The result follows
easily.
Lemma 3.5. If (A, ) is a flat Hopf algebroid and M is an A-module, the primiti*
*ves
in the extended comodule A M are the elements 1 m.
Proof.Note that x is a primitive if and only if x = f(1) for some comodule map
f :A -! A M. But then adjointness implies f is induced by a map g :A -!M
of A-modules. This means that f(a) = jL(a) g(a) for all a 2 A. In particular
x = f(1) = 1 g(1). The converse is clear.
Proposition 3.6. The map
E(n)*=In i-!E(n)*E(n) E(n)*E(n)*=In
defined by i(a) = jL(a) 1, is an essential extension of E(n)*E(n)-comodules.
We note that this proposition is also true for Morava E-theory En, but the pr*
*oof
requires more care.
__
Proof.The primitives_in E(n)*E(n) E(n)*E(n)*=In are the elements 1 ~vknfor
~ 2 Z(p)(so ~ is the reduction of ~ in Fp) and k 2 Z by Lemma 3.5. But we have
__k k
1 ~vn = jR (~)jR (vn) 1.
Of course jR (~) = ~, and also jR (vn) vn (mod In). Hence
__k k
1 ~vn = jL(~vn) 1.
Lemma 3.4 now completes the proof.
Proposition 3.7. The map
E(n)*E(n) E(n)*E(n)*=In 1-j-!E(n)*E(n) E(n)* tE(n)*=I1n
is an essential extension of E(n)*E(n)-comodules. Here t = |v1| + . .+.|vn-1|.
Note that this proposition completes the proof of Theorem 3.1. It is this pro*
*po-
sition that we believe to be false for Morava E-theory En.
Proof.In view of Lemma 3.4 and Lemma 3.5, it suffices to show that there is an
a 2 E(n)* such that
k ~vk
jL(a) ____~vn_____ii= 1 _____n____,
pi0v1 . .v.n-1n-11 pv1. .v.n-1
ignoring suspensions. But then it is clear that we should take
a = pi0-1vi1-11. .v.in-1-1n-1.
INJECTIVE COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 9
There is a slight subtlety, since jL(vj) 6= jR (vj). We get around this by goin*
*g in
order, from p to v1 to v2, et cetera, using the fact that vj is primitive modul*
*o Ij.
More precisely, for the inductive step, we have
k
xjL(vij-1j) ________vn_________ii
pv1. .v.j-1vj . .v.njn-1-1
k
= xjR (vij-1j) ________vn_________ii
pv1. .v.j-1vj . .v.njn-1-1
k
= x ________vn_________ii,
pv1. .v.jvj+1 . .v.nj+1n-1-1
as required.
4. Indecomposable injectives and duality
In this section, we study the dualities in the stable homotopy category that *
*arise
from the indecomposable injectives studied in this paper.
Fix a p-local spectrum X, and integers n r, and consider the functor that
takes a p-local spectrum Y to the abelian group Hom E*E(E*(X ^ Y ), Jn,r), where
E is a Landweber exact homology theory of height n as usual. Since Jn,ris an
injective E*E-comodule, this functor is exact and so is a cohomology theory of *
*Y .
Hence it is representable by some spectrum n,rX, for which we have the natural
isomorphism
[Y, n,rX] ~=Hom E*E(E*(X ^ Y ), Jn,r).
Using the injective BP*BP -comodule Jr, we can also define rX with the property
that
[Y, rX] ~=Hom BP*BP(BP*(X ^ Y ), Jr).
We have the following basic results about these duality functors.
Proposition 4.1. For a p-local spectrum X, define n,rX and rX as above. We
denote n,rS0 and rS0 by n,rand r. Then:
(1) n,rX ~= rX for all n r, and so n,ris independent of the choice of
E.
(2) [Y, rX] ~=Hom E(r)*(E(r)*(X ^ Y ), tE(r)*=I1r), where t = |v1| + . .+.
|vn-1|.
(3) rX ~=F (X, r).
(4) r is a BP -injective spectrum with BP* r ~=Jr, and the isomorphism
[X, r] ~=Hom BP*BP(BP*X, Jr)
is induced by taking BP -homology. If E* is concentrated in even dimen-
sions, then r is also an E-injective spectrum for all n r, with E* r *
*~=
Jn,r, and the isomorphism
[X, r] ~=Hom E*E(E*X, Jn,r)
is induced by taking E-homology.
(5) rX is Lr-local. In particular, rX = F (LrX, r).
(6) rX = 0 if and only if LrX = 0.
10 MARK HOVEY
Proof.For part (1), we have
[Y, n,rX] = Hom E(n)*E(n)(E(n)*(X ^ Y ), Jn,r)
~=Hom BP*BP(BP*(X ^ Y ), *Jn,r)
~=Hom BP*BP (BP*(X ^ Y ), Jr) = [Y, rX],
using the adjointness between * and * and the isomorphism *Jn,r= Jr of
Lemma 2.5.
Part (2) follows from part (1) and the isomorphisms
[Y, r,rX] = Hom E(r)*E(r)(E(r)*(X ^ Y ), Jr,r)
~=Hom E(r)*(E(r)*(X ^ Y ), E(r)*=I1r),
the last of which follows from Theorem 3.1.
Part (3) is an exercise in adjointness, and part (4) follows from [Dev97 , Pr*
*opo-
sition 1.3 and Theorem 1.5].
Part (5) is clear, since if Y is Lr-acyclic, then E(r)*(X ^ Y ) = 0 and so
[Y, rX] = 0.
For part (6), suppose LrX 6= 0. Then E(r)*X 6= 0, and so E(r)*X contains
a subcomodule M isomorphic to rE(r)*=Ij for some 0 j r and some r
by [HS05a , Theorem 5.7]. There is obviously a nonzero map (of some degree)
M -! E(r)*=I1r of E(r)*-modules, obtained by modding out by Ir and including.
Since E(r)*=I1r is injective by Proposition 3.3, this map extends to a nonzero
map (of some degree) E(r)*X -! E(r)*=I1r. This map corresponds to a nonzero
homotopy class in rX, and so rX is nonzero.
We also point out that, as usual, there is a natural map X -! 2rX obtained by
taking the image of the identity under the isomorphisms
[ rX, rX] ~=Hom E(r)*(E(r)*(X ^ rX), tE(r)*=I1r) ~=[X, 2rX].
The spectrum r has in fact been studied before in stable homotopy theory.
Recall that Ln denotes Bousfield localization with respect to E(n), or any Land*
*we-
ber exact theory of height n. There is a natural map LnX -! Ln-1X, and the
homotopy fiber of this map is traditionally denoted MnX.
Theorem 4.2. Fix an integer r 1. Then r ~= t+rMrE(r), where t = |v1| +
. .+.|vr-1|. In particular, if E* is an evenly graded Landweber exact homology
theory of height n r, then
Jn,r~= t+rE*(MrE(r)).
Proof.Recall from [Rav84 , Theorem 6.1] that ss*MrBP ~= -rv-1r(BP*=I1r). In
view of the localization theorem [Rav92 , Theorem 7.5.2], we see that
BP*MrS0 ~= -rv-1r(BP*=I1r).
By Landweber exactness, we conclude that
E(r)*MrS0 ~= -rE(r)*=I1r.
But MrE(r) ~=E(r) ^ MrS0 because of the smash product theorem [Rav92 , Theo-
rem 7.5.6], and the flatness of E(r)*E(r) implies that
E(r)*(E(r) ^ X) ~=E(r)*E(r) E(r)*E(r)*X
(see Lemma 2.2.7 of [Rav86 ]). Some diagram chasing is necessary to show that
this isomorphism is an isomorphism of comodules, where the right side is given *
*the
INJECTIVE COMODULES AND LANDWEBER EXACT HOMOLOGY THEORIES 11
extended comodule structure, so that all the coaction is on the E(r)*E(r) tensor
factor. Theorem 3.1 then implies that
E(r)*(MrE(r)) ~= -t-rJr,r
as E(r)*E(r)-comodules. This isomorphism then defines a map
MrE(r) -! -t-r r
that is an isomorphism in E(r)-homology. Since both MrE(r) and r are E(r)-
local, the theorem follows.
This theorem has the following corollary, apparently not written down before
but known to several mathematicians, including Hopkins, Greenlees, and Sadofsky.
Corollary 4.3. Let [E(n)denote LK(n)E(n), the completion of E(n) at In. Then
[E(n)*(X) ~=Hom E(n)*(E(n)*(MnX), -n E(n)*=I1n).
On the simplest level, we can think of this corollary as analogous to the fac*
*t that
Hom Z(p)(Q=Z(p), Q=Z(p)) ~=Z(p). As pointed out by Greenlees, one can look at t*
*his
corollary as a reflection of the fact that completion (represented by [E(n)) sh*
*ould
be maps out of local cohomology (represented by MnE(n)). This is the viewpoint
of [GM95 , Section 4].
Proof.Theorem 6.19 of [HS99 ] tells us that
[E(n)*(X) ~=[LK(n)X, LK(n)E(n)]* ~=[MnX, MnE(n)]*.
But Theorem 4.2 implies that
[MnX, MnE(n)]* ~=[ t+nMnX, n]*
~=Hom E(n)*(E(n)*( t+nMnX), tE(n)*=I1n)
~=Hom E(n)*(E(n)*MnX, -n E(n)*=I1n).
We can also determine the endomorphism rings of the indecomposable injectives
Jr and Jn,r.
Corollary 4.4. The endomorphism rings EndBP*BP (Jr) and EndE(n)*E(n)(Jn,r)
*
for 0 r n are isomorphic to [E(r)[E(r).
Proof.Both of these endomorphism rings are isomorphic to [ r, r]* in view of
Proposition 4.1(4). But Theorem 4.2 implies that
[ r, r]* ~=[MrE(r), MrE(r)]*,
*
which is in turn isomorphic to [E(r)[E(r)by [HS99 , Theorem 6.19].
It is unfortunate*that these endomorphism rings of indecomposable injectives
turn out to be [E(r)[E(r)instead of E*rEr, where Er is Morava E-theory.The ring
E*rEr of operations in Morava E-theory is the twisted completed group ring Er*[*
*[ ]],
where is the semidirect product of the automorphism group of the height r Hon*
*da
formal group law over Fpr with the Galois group of Fpr. This is an old result of
Hopkins and Ravenel; see [Bak95 ] or [Hov04b ] for a proof. Since Er is a*fini*
*te
free module over [E(r), there is probably some Galois theory relating [E(r)[E(r*
*)to
Er*[[ ]], but the author does not know any details.
12 MARK HOVEY
We can, however, make a decomposable injective comodule whose endomorphism
ring is E*rEr ~=Er*[[ ]]. Indeed, we can simply take the comodule
J0r,r= Er*Er Er*Er*=I1r.
This is an Er*Er-comodule that is injective because Er*=I1r is an injective Er*-
module 3.3. Under the equivalence of categories between E(r)*E(r)-comodules
and Er*Er-comodule of [HS05a ], J0r,rcorresponds to the direct sum of r(pr - 1)
copies of Jr,r, with r copies in every even dimension from 2 to 2(pr- 1); this *
*is the
E(r)*E(r)-comodule
E(r)*E(r) E(r)*Er*=I1r.
There is a spectrum 0rwith
[X, 0r] ~=Hom Er*Er(Er*X, J0r,r) ~=Hom Er*(Er*X, Er*=I1r).
The analogue to Theorem 4.2 tells us that 0r~= rMrEr. We then get isomor-
phisms
E*nX ~=Hom En*(En*(MnX), -n En*=I1n),
again known before by Hopkins, Greenlees, Sadofsky, and others, and
EndEr*Er(J0r,r) ~=E*rEr ~=Er*[[ ]].
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Department of Mathematics, Wesleyan University, Middletown, CT 06459
E-mail address: hovey@member.ams.org