LOCALIZATION AND COMPLETION THEOREMS FOR
MU-MODULE SPECTRA
J.P.C. GREENLEES AND J.P. MAY
Abstract. Let G be a finite extension of a torus. Working with highly
structured ring and module spectra, let M be any module over MU; ex-
amples include all of the standard homotopical MU-modules, such as the
Brown-Peterson and Morava K-theory spectra. We shall prove localization
and completion theorems for the computation of M
∗
(BG) and M
∗
(BG). The
G-spectrum M U
G
that represents stabilized equivariant complex cobordism is
an algebra over the equivariant sphere spectrum S
G
, and there is an M U
G
-
module M
G
whose underlying M U-module is M . This allows the use of topo-
logical analogues of constructions in commutative algebra. The computation
of M
∗
(BG) and M
∗
(BG) is expressed in terms of spectral sequences whose
respective E
2
terms are computable in terms of local cohomology and local
homology groups that are constructed from the coefficient ring MU
G
∗
and its
module M
G
∗
. The central feature of the proof is a new norm map in equivariant
stable homotopy theory, the construction of which involves the new concept of
a global I
∗
-functor with smash product.
Contents
1. Introduction and statements of results 1
2. The strategy of proof 6
3. Constructing sufficiently large finitely generated ideals 9
4. The idea and properties of norm maps 11
5. Global I
∗
-functors with smash product 13
6. The passage to spectra 16
7. Wreath products and the definition of the norm map 17
8. The proof of the double coset formula 20
9. The norm map on sums and its double coset formula 22
10. The Thom classes of Thom spectra 25
11. The proof of Lemma 3.4 26
References 27
1. Introduction and statements of results
Completion theorems relate the nonequivariant cohomology of classifying spaces
to algebraic completions of associated equivariant cohomology theories. They are at
the heart of equivariant stable homotopy theory and its nonequivariant applications.
The authors are grateful to Gustavo Comeza˜na, Jim McClure, and Neil Strickland for helpful
conversations. The first author thanks the University of Chicago for its hospitality and the Nuffield
Foundation for its support. The second author acknowledges support from the NSF..
1
2 J.P.C. GREENLEES AND J.P. MAY
The most celebrated result of this kind is the Atiyah-Segal completion theorem
[1]. For any compact Lie group G, it computes K(BG) as the completion of the
representation ring R(G) at its augmentation ideal. A more recent such result is the
Segal conjecture [3]. For any finite group G, it computes the cohomotopy π
∗
(BG)
as the completion of the equivariant cohomotopy π
∗
G
at the augmentation ideal of
the Burnside ring A(G). Unlike the Atiyah-Segal completion theorem, in which the
representation ring is under good algebraic control, the Segal conjecture relates two
sequences of groups that are largely unknown and difficult to compute.
Shortly after the Atiyah-Segal completion theorem appeared, Landweber [31] and
others raised the problem of whether an analog might hold for complex cobordism.
It was seen almost immediately that the appropriate equivariant form of complex
cobordism to consider was the stabilized version, MU
∗
G
, introduced by tom Dieck
[8]. Shortly after the question was raised, L¨offler [36] sketched a proof of the
following result. A complete argument has been given by Comeza˜na and May [6].
Theorem 1.1 (L¨offler). If G is a compact Abelian Lie group, then
(MU
∗
G
) ˆ
J
G
∼
=
MU
∗
(BG),
where J
G
is the augmentation ideal of M U
∗
G
.
Here M U
∗
(BG) is completely understoo d [30, 35, 36], and the result is not
difficult because the Euler classes of the irreducible complex representations of G,
which of course are all 1-dimensional, are under good control. There has been no
further progress in over twenty years. In fact, in his 1992 survey of equivariant
stable homotopy theory [4], Carlsson stated the problem as follows:
“Formulate a conjecture about MU
∗
(BG), for G a finite group.”
Landweb er [31] had noted that the problem of studying MU
∗
(BG) seemed to be
even harder than the problem of studying MU
∗
(BG).
In [15], the first author introduced a new approach to the Atiyah-Segal com-
pletion theorem (for finite groups), in which he deduced it from what we now
understand to be a kind of localization theorem giving a computation of K
∗
(BG)
in terms of local cohomology. When such a localization theorem holds in homol-
ogy, it is a considerably stronger result than the implied completion theorem in
cohomology. For example, the localization theorem for stable homotopy theory is
false, although the completion theorem for stable cohomotopy is true. We refer
the reader to [21, §§6-8] and [22] for a general discussion of localization theorems
in equivariant homology and completion theorems in equivariant cohomology. We
shall here prove theorems of this kind for stabilized equivariant complex cobordism.
Our results were announced in [9], and an outline of the proofs has appeared in
[23].
To make sense of the approach of [15], one must work in a sufficiently precise
context of highly structured ring and module spectra that one can mimic construc-
tions in commutative algebra topologically. The theory developed by Elmendorf,
Kriz, Mandell, and the second author [11] provides these essential foundations.
That paper was written nonequivariantly but, as stated in a metatheorem in its
intro duction and explained in more detail in [12], all of its theory applies verbatim
to G-sp ectra for any compact Lie group G; see also [10, 21, 13]. In the language
of [11], stabilized equivariant cobordism is represented by a commutative algebra
MU
G
over the equivariant sphere G-spectrum S
G
. The underlying nonequivari-
ant S-algebra of MU
G
is MU. In earlier language, this means that M U
G
is an
LOCALIZATION AND COMPLETION THEOREMS FOR MU -MODULE SPECTRA 3
E
∞
ring G-spectrum with underlying nonequivariant E
∞
ring spectrum MU. We
understand S
G
-algebras to be commutative from now on.
A considerable virtue of the kind of localization theorem that we have in mind
is that, when it applies to an S
G
-algebra R
G
with underlying nonequivariant S-
algebra R, it automatically implies localization and completion theorems for the
computation of M
∗
(BG) and M
∗
(BG) for the underlying R-module M of any split
R
G
-module M
G
. (The notion of a split G-spectrum is defined and discussed in [34,
II.84], [19, §0], and [21, §3].) This is an especially happy feature of our work since
MU
G
is split and every MU-module M is the underlying nonequivariant spectrum
of a certain split MU
G
-module M
G
= MU
G
∧
MU
M [38]. Therefore, by [11, V§4],
our work applies to all of the standard MU-modules that are constructed from
MU by quotienting out the ideal generated by a regular sequence of elements of
MU
∗
and localizing by inverting some other elements. In particular, it applies
to the Brown-Peterson spectra BP , the Morava K-theory spectra K(n), and the
Johnson-Wilson spectra E(n). There is a long and extensive history of explicit
calculations of groups M
∗
(BG) and M
∗
(BG) in special cases. Some of the relevant
authors are: Landweber; Johnson, Wilson, and Yan; Tezuka and Yagita; Bahri,
Bendersky, Davis, and Gilkey; Hopkins, Kuhn, and Ravenel; Hunton; and Kriz.
See [31, 30, 32, 2] for MU, [28, 29, 40, 41, 42, 43] for BP , and [26, 27, 25] for K(n).
Our theorem gives a general conceptual framework into which all such computations
must fit.
As we shall make precise shortly, the theorem shows that these nonequivari-
ant homology and cohomology groups are isomorphic to the equivariant homotopy
groups of certain homotopical J
G
-power torsion MU
G
-modules Γ
J
G
(M
G
) and ho-
motopical completion MU
G
-modules (M
G
)
∧
J
G
, where J
G
is the augmentation ideal
of MU
G
∗
. There result spectral sequences for the computation of these homotopy
groups in terms of “local cohomology groups” and “local homology groups” that
can be computed from knowledge of the ring
MU
G
∗
and its module
M
G
∗
. Thus the
theorem establishes a close connection between the geometrically defined equivari-
ant cobordism groups and the homology and cohomology of classifying spaces with
coefficients in MU-modules.
This is entirely satisfactory on a conceptual level. However, like the Segal con-
jecture, our theorem relates two sequences of groups that are largely unknown and
difficult to compute. Thus, on the computational level, it merely points the direc-
tion towards further study. Explicit computations will require better understanding
of MU
G
∗
than is now available. We recall an old and probably false conjecture.
Conjecture 1.2. MU
G
∗
is MU
∗
-free on generators of even degree.
The conjecture is true when G is Abelian, as was announced by L¨offler [35, 36]
and proven in detail by Comeza˜na [5]. Little is known for non-Abelian groups.
Since our work is based on the importation of techniques of commutative algebra
into equivariant stable homotopy theory, we briefly recall the relevant algebraic
constructions; see [20] for details and discussion. Let R be a graded commutative
ring and let I = (α
1
, . . . , α
n
) be a finitely generated ideal in R. Define K
•
(I) to be
the tensor product of the graded co chain complexes
K
•
(α
i
) = (R → R[1/α
i
]),
4 J.P.C. GREENLEES AND J.P. MAY
where R and R[1/α
i
] lie in homological degrees 0 and 1. Up to quasi-isomorphism,
K
•
(I) depends only on the radical of I. For a graded R-module M , define
H
s,t
I
(R; M ) = H
s,t
(K
•
(I) ⊗ M ),
where s indicates the homological degree and t the internal grading. Such “local
cohomology groups” were first defined by Grothendieck [24]. It is easy to see that
H
0
I
(R; M ) is the submodule
Γ
I
(M) = {m ∈ M |I
N
m = 0 for some N}
of I-power torsion elements of M. If R is Noetherian it is not hard to prove directly
that the functor H
∗
I
(R; −) is effaceable and hence that lo cal cohomology calculates
the right derived functors of Γ
I
(−) [24]. It is clear that the local cohomology
groups vanish above degree n, but in the Noetherian case Grothendieck’s vanishing
theorem shows the powerful fact that they are zero ab ove the Krull dimension of
R. One key fact that we shall use is that if β ∈ I then H
∗
I
(R; M )[1/β] = 0; this
is a restatement of the easily proven fact that K
•
(I)[1/β] is exact [20, 1.1]. We
abbreviate
H
∗
I
(R) = H
∗
I
(R; R).
These algebraic local cohomology groups are relevant to topological homology
groups.
There are dual “local homology groups” which, to the b est of our knowledge,
were first introduced in [17, 18]. Replacing K
•
(I) by a quasi-isomorphic R-free
chain complex K
0
•
(I), define
H
I
s,t
(R; M ) = H
s,t
(Hom(K
0
•
(R), M )).
There is a tri-graded universal coefficient spectral sequence that converges to these
groups; ignoring the internal grading t, which is unchanged by the differentials, it
converges in total degree s = −(p + q) and satisfies
E
p,q
2
= Ext
p
R
(H
−q
I
(R), M ) and d
r
: E
p,q
r
→ E
p+r,q−r +1
r
.
There is a natural epimorphism H
I
0
(R; M ) → Mˆ
I
whose kernel is a certain lim
1
group. It is not hard to check from the definition that if R is Noetherian and M
is free or finitely generated then H
I
0
(R; M )
∼
=
M
∧
I
, and one may also prove that
in these cases the higher local homology groups are zero. It follows that, at least
when R is No etherian, H
I
∗
(R; M ) calculates the left derived functors of the (not
necessarily right exact) I-adic completion functor. These algebraic local homology
groups are relevant to topological cohomology groups.
Now, returning to topology, let R
G
be an S
G
-algebra and M
G
be an R
G
-module;
we always understand algebras and modules in the highly structured sense of [11].
We understand G-spectra to be indexed on a complete G-universe U, which im-
plies that our equivariant homology and cohomology theories are RO(G)-graded.
However, we restrict attention to integer degrees except where explicitly stated oth-
erwise. We write E
G
n
= E
−n
G
for the nth homotopy group π
G
n
(E) = [S
G
, E
G
]
G
n
of a
G-spectrum E
G
.
For α ∈ R
G
k
, let R
G
[1/α] be the telescope of iterates
R
G
→ Σ
−k
R
G
→ Σ
−2k
R
G
→ · · ·
of multiplication by α and let K(α) be the fiber of the canonical map R
G
→
R
G
[1/α]. For a finitely generated ideal I = (α
1
, . . . , α
n
) in R
G
∗
, let K(I) be
LOCALIZATION AND COMPLETION THEOREMS FOR MU -MODULE SPECTRA 5
the smash product over R
G
of the R
G
-modules K(α
i
). Up to equivalence of R
G
-
modules, K(I) depends only on the radical of I. Define
Γ
I
(M
G
) = K(I) ∧
R
G
M
G
and
(M
G
)
∧
I
= F
R
G
(K(I), M
G
).
There is a spectral sequence converging to Γ
I
(M
G
)
G
∗
(in total degree p+ q), with
E
2
p,q
= H
−p,−q
I
(R
G
∗
; M
G
∗
) and d
r
: E
r
p,q
→ E
r
p−r,q+r−1
,
and there is a spectral sequence converging to ((M
G
)
∧
I
)
∗
G
(in total degree p + q)
with
E
p,q
2
= H
I
−p,−q
(R
∗
G
; M
∗
G
) and d
r
: E
p,q
r
→ E
p+r,q−r+1
r
.
Now take I to be a finitely generated ideal contained in the augmentation ideal
J
G
= Ker(R
G
∗
→ R
∗
). Note that the ring R
G
∗
need not be Noetherian and the
augmentation ideal need not be finitely generated. In particular, MU
G
∗
is not
Noetherian and its augmentation ideal is not finitely generated, even when G is
finite.
Since R[1/α] is nonequivariantly contractible for α ∈ J
G
, the canonical map
K(I) → R
G
is an equivalence of underlying spectra and so induces an equivalence
upon smashing with EG
+
, where EG
+
is the union of EG and a G-fixed disjoint
basepoint. Inverting this equivalence and using the projection EG
+
−→ S
0
, we
obtain a canonical map of R
G
-modules
κ : EG
+
∧ R
G
→ K(I).
For an R
G
-module M
G
, κ induces maps of R
G
-modules
EG
+
∧ M
G
' EG
+
∧ R
G
∧
R
G
M
G
→ K(I) ∧
R
G
M
G
= Γ
I
(M
G
)
and
(M
G
)
∧
I
= F
R
G
(K(I), M
G
) → F
R
G
(EG
+
∧ R
G
, M
G
) ' F (EG
+
, M
G
),
both of which will be equivalences for all R
G
-modules M
G
if κ is an equivalence.
We can now state our completion theorem for modules over M U
G
.
Theorem 1.3. Let G be finite or a finite extension of a torus. Then, for any
sufficiently large finitely generated ideal I ⊂ J
G
, κ : EG
+
∧ MU
G
→ K(I) is an
equivalence. Therefore,
EG
+
∧ M
G
→ Γ
I
(M
G
) and (M
G
)
∧
I
→ F (EG
+
, M
G
)
are equivalences for any MU
G
-module M
G
.
It is reasonable to define K(J
G
) to be K(I) for any sufficiently large I and to
define Γ
J
G
(M
G
) and (M
G
)
∧
J
G
similarly. The theorem implies that these MU
G
-
modules are independent of the choice of I.
Our main interest is in finite groups. However, the fact that the result holds for
a finite extension of a torus and therefore for the normalizer of a maximal torus in
an arbitrary compact Lie group suggests the following generalization. There should
be an appropriate transfer argument, but we have not succeeded in finding one.
Conjecture 1.4. The theorem remains true for any compact Lie group G.