VARIETIES AND LOCAL COHOMOLOGY FOR CHROMATIC GROUP
COHOMOLOGY RINGS.
J. P. C. GREENLEES AND N. P. STRICKLAND
Abstract.Following Quillen [20, 21], we use the methods of algebraic geo*
*metry to study
the ring E*(BG) where E is a suitable complete periodic complex oriented*
* theory and
G is a finite group: we describe its variety in terms of the formal grou*
*p associated to
E, and the category of Abelian psubgroups of G. This also gives informa*
*tion about the
associated homology of BG.
For example if E is the complete 2periodic version of the JohnsonWil*
*son theory E(n)
the irreducible components of the variety of the quotient E*(BG)=Ik by t*
*he invariant
prime ideal Ik = (p; v1; : :;:vk1) correspond to conjugacy classes of A*
*belian psubgroups
of rank n  k. Furthermore, if we invert vk the decomposition of the va*
*riety into irre
ducible pieces corresponding to minimal primes becomes a decomposition i*
*nto connected
components, corresponding to the fact that the ring splits as a product.
Contents
1. Informal introduction *
* 2
2. Description of results. *
* 4
3. Schemes and varieties *
* 9
4. Admissible cohomology theories *
* 14
5. The proof of Theorem 2.4 *
*17
6. Multiple level structures *
* 22
7. The geometric Frobenius map *
*28
8. Thickenings *
*29
9. Pure strata *
*35
10. The Ehomology of BG. 39
11. Some examples. 41
Appendix A. Proof of the theorem of HopkinsKuhnRavenel *
*43
Appendix B. The Evens norm map 44
Appendix C. Varieties and reduction mod Ik. *
* 45
References *
*47
___________
The authors thank the University of Chicago for its hospitality during Autumn*
* 1994 when this work
was begun, and the Transpennine Topology Triangle for providing the opportunity*
* to meet later. The first
author also thanks the Nuffield Foundation for its support.
1
2 J. P. C. GREENLEES AND N. P. STRICKLAND
1.Informal introduction
The article investigates the interface between equivariant topology and the c*
*hromatic
approach to stable homotopy theory, by giving a geometric description of the ri*
*ng E*(BG)
for a certain class of complete periodic complex oriented theories E*(.) and fi*
*nite groups G.
The archetypal examples of such theories are the complete, 2periodic versions *
*E = En of
the JohnsonWilson theories E(n) with coefficients E(n)* = Z(p)[v1; v2; : :;:vn*
*1; vn; v1n]
for some prime p, and we restrict attention to this case in the introduction. *
*To obtain
a general description, we follow the example of Quillen [20, 21], and concentra*
*te on the
geometric properties of the ring E*(BG). We use Quillen's descent argument to *
*reduce
to the study of E*(BA) for Abelian psubgroups A. The Abelian case is translate*
*d into
the theory of formal groups, and the resulting questions studied by extending t*
*he second
author's theory of multiple level structures [23]. The work is also strongly in*
*fluenced by
that of HopkinsKuhnRavenel [14], discussed in [13].
The description of the ring E*(BG) allows us to understand the first author's*
* local
cohomology approach to the homology E*(BG) [6] from the chromatic point of view*
*, and
sets the results of [10, 11] in a wider context. The slogan is that equivariant*
* topology is
trivial over pure chromatic strata, and thus the important geometry concerns th*
*e way the
strata are attached to each other.
Quillen's method involves considering the cohomology E*(EG xG Z) of the Borel*
* con
struction for a finite Gcomplex Z, even if one is only concerned with the spec*
*ial case
Z = *. Our results apply to all such Z, but it may help the reader if we spend *
*the rest
of the introduction summarising the highlights when Z = *. We explain the geom*
*etric
language in Section 3 below, but the outline should be clear without precise de*
*finitions.
The essential distinction is between schemes, which encode all the ring theoret*
*ic informa
tion, and varieties, which ignore nilpotents and only capture a crude picture o*
*f the ring.
Nonetheless many important features, such as the dimension, connected component*
*s and
irreducible components are visible at the level of varieties. Readers comfortab*
*le with the
formal framework may wish to skip directly to the next section, where our resul*
*ts are
stated precisely and in appropriate generality.
Following Morava's chromatic philosophy we concentrate on the invariant prime*
* ideals
Ik = (p; u1; : :;:uk1) of the complete local ring E0. Indeed, if we let X = sp*
*f(E0) denote
its formal scheme, the geometric counterpart of the filtration
0 = I0 I1 I2 . . .In;
is the filtration
X = X0 X1 X2 . . .Xn;
where Xk = spf(E0=Ik) is the formal subscheme defined by Ik. Evidently Xk is a *
*formal
affine space of dimension n  k. One of the themes will be that we can underst*
*and
phenomena over X by restricting to the subschemes Xk, and that they will be esp*
*ecially
simple over the pure strata X0k= Xk \ Xk1. The notational convention that a su*
*bscript k
denotes restriction to the kth chromatic stratum and that a dash denotes restri*
*ction to a
pure stratum will remain in force throughout the paper.
CHROMATIC GROUP COHOMOLOGY RINGS 3
Now consider the formal scheme X(G) = spf(E0(BG)), which is finite over X; we*
* shall
give a description of the primary features of its underlying variety. Let Xk(G)*
* denote the
restriction of X(G) to the part over Xk, and note that this is spf(E0(BG)=Ik). *
* On the
other hand, we are often interested in the rings (E=Ik)0(BG); for example if k *
*= n it is
simply the Morava Ktheory K(n)0(BG) made 2periodic. If, as often happens, E0(*
*BG)
is free over E0 then E0(BG)=Ik is actually equal to (E=Ik)0(BG); we shall see t*
*hat in any
case their varieties agree.
Next we describe the irreducible components of Xk(G), and how they behave ove*
*r pure
strata. In fact there is a decomposition
[
Xk(G) = Yk(G; A)
(A)
into irreducible components all of which have dimension n  k, where the indexi*
*ng set is
the set of conjugacy classes of Abelian psubgroups A of rank n  k. Furthermo*
*re, over
the pure stratum X0kthis decomposition becomes a disjoint union
a
X0k(G) = Yk0(G; A):
Indeed, there is a single formal scheme Levelk(A*) for each Abelian pgroup of *
*rank
n  k so that at the level of varieties, Yk0(G; A) = Level0k(A*)=WG(A) where W*
*G(A) =
NG(A)=A is the Weyl group of A in G. At one extreme, spf(E0(BG)) has one irredu*
*cible
component for each conjugacy class of Abelian psubgroups of rank n whilst at *
*the
other, spf(K(n)0(BG)) is itself irreducible. The schemes Levelk(A*) classify `m*
*ultiple level
structures' (see Section 6) for the formal group G of E; they are formal spectr*
*a of complete
regular local rings Dk(A) of dimension n  k.
Translating geometric statements back into ring theory, our decompositions st*
*ate that
the minimal primes of E0(BG)=Ik are in bijective correspondence with the conjug*
*acy
classes of Abelian psubgroups of rank n  k. Moreover, if we invert uk the ri*
*ng splits
up to F isomorphism as a product of rings of invariants of wellunderstood rin*
*gs:
Y
u1kE0(BG)=Ik Fiso!D0k(A)WG(A):
(A)
Here WG(A) = NG(A)=A, but the action factors through NG(A)=CG(A), so that in the
Abelian case the action is trivial. In the Abelian case G = A, we identify u1k*
*E0(BA)=Ik
exactly, and it decomposes in the same way:
Y
u1kE0(BA)=Ik = D0k(A; B):
B
The pieces correspond in the sense that D0k(A; B)red= D0k(B), and D0k(A; B) is *
*A=Bk
times larger than D0k(B). In particular D0k(A; A) is reduced and D0k(A; A) = D0*
*k(A).
The reader may also wish to consider applications to E*(BG) in Sections 10 an*
*d 11
before confronting the general results.
4 J. P. C. GREENLEES AND N. P. STRICKLAND
2. Description of results.
In this section we give precise statements of our main results.
Let p be a prime. We shall consider the following class of cohomology theorie*
*s. In this
paper, E*(Y ) will always denote the unreduced cohomology of a space Y , and th*
*e reduced
cohomology of a based space will be indicated by a tilde.
Definition 2.1. Let pCP1 :CP 1 !CP 1 be the map classifying the p'th tensor po*
*wer of
the canonical line bundle. A plocal commutative ring spectrum E is admissible *
*if
(a) E0 is a complete local Noetherian ring.
(b) E1 = 0.
(c) E2 contains a unit.
(d) pCP1 induces a nonzero selfmap of eE0(CP 1).
These conditions will be discussed further in Section 4. Examples include the*
* cohomology
theory obtained from [E(n)by adjoining a (pn1)'st root_of vn, or various spect*
*ra obtained
from this by tensoring with the Witt ring of Fpn or Fp, or killing some generat*
*ors (provided
that p > 2). For suitable versions of elliptic cohomology, the spectrum LK(2)El*
*lwill be a
finite product of admissible ring spectra. The version of integral Morava Kthe*
*ory used by
Igor Kriz in [16] is also admissible. It can be shown that every admissible rin*
*g spectrum
E is K(n)local for some n, called the height of E (this is the same as the hei*
*ght of the
formal group law associated to E).
We will specify one example more precisely. Let W be the Witt ring of Fpn, an*
*d consider
the following graded ring:
E* = W [[u1; : :;:un1]][u; u1]:
The generators uk have degree 0, and u has degree 2. We take u0 = p and un = 1*
* and
k1
uk = 0 for k > n. There is a map BP *!E* sending vk to up uk. Using this, we*
* define
a functor from spectra to E*modules by
E*(X) = E* BP* BP*(X):
The BP *module E* is Landweber exact, so this functor is a homology theory, wh*
*ich we
shall call Morava Etheory. It is represented by an admissible ring spectrum.
Convention 2.2. Throughout the whole of this paper the following notation is f*
*ixed.
o E is an admissible cohomology theory of height n,
o G is a finite group, and
o Z is a finite Gcomplex.
Our main results will give a crude picture of the ring
E*G(Z) = E*(EG xG Z):
In particular, when Z is a point this is E*(BG), as discussed in the introducti*
*on. Our ap
proach is influenced heavily by the work of Quillen [20, 21] and HopkinsKuhnR*
*avenel [13,
14].
CHROMATIC GROUP COHOMOLOGY RINGS 5
Because of axiom (c), the ring E*Ghas period two. Our methods cannot see E1G(*
*Z), and
anyway it is zero in many cases, so we focus attention on E0G(Z), which is an u*
*ngraded
commutative ring. We shall prove in Corollary 5.4 that it is a finitely generat*
*ed module
over E0, and thus a complete semilocal Noetherian ring.
It will be convenient to give our results in geometric language. We will set *
*up a suitable
technical context in Section 3. For the moment, we just define the following af*
*fine objects:
a scheme is a covariant representable functor from Noetherian rings to sets, an*
*d a variety
is a functor from algebraically closed fields to sets that is represented by a *
*Noetherian
ring. We will define formal schemes in Section 3. We write X(Z; G) for the form*
*al scheme
spf(E0G(Z)) represented by E0G(Z), or for its underlying scheme spec(E0G(Z)), o*
*r for its
underlying variety. We omit Z from the notation when it is a point, so X(G) me*
*ans
X(*; G) = spec(E0(BG)).
We also write X for spf(E0), and G for spf(E0(CP 1)), which turns out to be a*
* formal
group. We will construct a "chromatic" filtration of X by closed formal subsche*
*mes:
X = X0 X1 : : :Xn:
In terms more familiar to topologists, we have Xk = spec(E0=Ik), where Ik is th*
*e ideal
k1
generated by the coefficients of x; xp; : :;:xp in the pseries of a suitable*
* formal group
law over E0. We also consider
X0k= Xk \ Xk+1 = spec(u1kE0=Ik) = k'th pure stratum of the filtration,
which are schemes, but not formal schemes. One of the themes will be that we c*
*an
understand phenomena over X by restricting to the subschemes Xk, and that they *
*will be
especially simple over the pure strata X0k. We adopt the permanent notational c*
*onvention
that a subscript k denotes restriction to the k'th chromatic stratum and that a*
* dash denotes
restriction to a pure stratum. Thus
Xk(Z; G) = X(Z; G) xX Xk
(which is a formal scheme) and
X0k(Z; G) = X(Z; G) xX X0k
(which is an ordinary scheme).
The idea is to describe X(Z; G) by working up from X(A), where A is Abelian. *
*This
can be described completely in terms of the formal group G. Suitably interpret*
*ed (see
Proposition 4.10), the answer is that
X(A) = Hom (A*; G);
where A* is the character group Hom (A; S1). We define
Hom k(A*; G) = Hom (A*; G) xX Xk = spf(E0(BA)=Ik) = Xk(A);
and similarly for Hom 0k(A*; G).
If G were a group in the category of sets (rather than the category of formal*
* schemes),
we could argue as follows: any map OE: A* ! G factors uniquely as the projecti*
*on to a
quotient of A*, followed by a monomorphism. The quotients of A* are precisely t*
*he groups
6 J. P. C. GREENLEES AND N. P. STRICKLAND
`
B* where B A. We would thus have a decomposition Hom (A*; G) = BA Mon (B*; *
*G).
As the category of formal schemes is more complicated than the category of sets*
*, this does
not quite work out as expected. The right substitute for the notion of a monomo*
*rphism
A* ! G turns out to be a levelA* structure on G. This concept is essentially*
* due to
Drinfel'd [2], and is studied in detail in the present context in [23]. When wo*
*rking over the
subscheme Xk, it is useful to refine this notion and use pkfold level structur*
*e instead; the
relevant theory is developed in Section 6, where we prove the following theorem.
Theorem 2.3. There are formal schemes Levelk(A*; G) = spf(Dk(A)) (for 0 k < *
*n)
with the following properties.
1. Levelk(A*; G) is a closed subscheme of Hom k(A*; G).
2. Levelk(A*; G) is flat of degree Ak Mon (A*; (Qp=Zp )nk) over Xk. In p*
*articular, it
is empty if rankp(A) > n  k.
3. The evident map
Levelk(B*; G) ae Hom k(B*; G) ae Hom k(A*; G)
is a closed embedding.
4. The evident map of formal schemes
a
Levelk(B*; G) !Hom k(A*; G)
BA
induces an isomorphism of varieties
a
Level0k(B*; G) !Hom 0k(A*; G):
BA
This will be proved as Theorem 6.4.
We now consider a general finite group G. Let A denote the category whose obj*
*ects are
the Abelian psubgroups of G, with morphisms
A(A; B) = Map G(G=A; G=B) ' {g 2 G  Ag B}=B:
We have a functor
Aopx A ! Formal Schemes
given by (A; B) 7! ss0(ZAR) x Hom (B*; G), where ZA is the subspace of Afixed *
*points. We
denote its coend by A ss0(ZA ) x Hom (A*; G) (see [18, Chapter IX] for the the*
*ory of ends
and coends).
Theorem 2.4. There is a natural map of formal schemes
Z A
ss0(ZA ) x Hom (A*; G) !X(Z; G);
which induces an isomorphism of varieties. This gives a natural map of formal s*
*chemes
Z A
ss0(ZA ) x Hom k(A*; G) !Xk(Z; G)
CHROMATIC GROUP COHOMOLOGY RINGS 7
and a natural map of ordinary schemes
Z A
ss0(ZA ) x Hom 0k(A*; G) !X0k(Z; G);
both of which induce isomorphisms of varieties.
This will be proved as Theorem 5.10.
We also show that the last of the above coends, where we have restricted to a*
* pure
chromatic stratum, can be greatly simplified. Recall that Levelk(A*; G) is emp*
*ty unless
rankp(A) n  k.
Theorem 2.5. There is a natural map of formal schemes
!
a
ss0(ZA ) x Levelk(A*; G)=G !Xk(Z; G)
A2A
such that the induced map of schemes
!
a
ss0(ZA ) x Level0k(A*;=G)G !X0k(Z; G)
A2A
is an isomorphism on the underlying varieties.
This will be proved as Theorem 9.2.
In the case k = 0 our result is not the best available. The main theorem of [*
*14] can be
reformulated as follows:
Theorem 2.6 (HopkinsKuhnRavenel). There is a natural isomorphism of schemes
!
a
X(ZA ) xX Level00(A*; G)=G !X00(Z; G);
A2A
where X(ZA ) = spf(E0(ZA )). Moreover, it is easy to see that X(ZA ) has the sa*
*me variety
as ss0(ZA ) x X.
We outline a proof of this in Appendix A; although it does not differ in an e*
*ssential way
from that of [14] we feel the difference of language and inaccessibility of [14*
*] make this
worthwhile.
One can deduce from Theorem 2.5 that`the underlying variety of X0k(Z; G) spli*
*ts as a
disjoint union of pieces indexed by ( A ss0(ZA ))=G where the coproduct runs o*
*ver Abelian
subgroups of prank n  k, and that the irreducible components of Xk(Z; G) are*
* indexed
by the same set. In particular, when Z = * the set of pieces correspond to conj*
*ugacy classes
of Abelian subgroups of prank n  k. For some applications, we need to know t*
*hat the
scheme itself splits. Moreover, we would like a splitting that is valid after c*
*ompleting at Ik,
rather than just reducing modulo Ik. In other words, we would like to study the*
* scheme
bX0k(Z; G) = spec((u1kE0G(Z))^):
Ik
8 J. P. C. GREENLEES AND N. P. STRICKLAND
For this we need some further notation. Let A G be Abelian`and z an element of*
* ss0(ZA ).
The pair (A; z) defines a point of the finite Gset A ss0(ZA ). We write NG(A*
*; z) for its
stabiliser and [A; z] for its orbit, and put WG(A; z) = NG(A; z)=A.
We will prove the following theorem.
Theorem 2.7. There is a closed subscheme bYk0(Z; G; A; z) bX0k(Z; G) dependi*
*ng only on
the orbit of (A; z), such that
a
bX0k(Z; G) = bYk0(Z; G; A; z);
[A;z]
where the coproduct is indexed by orbits. Moreover, there is a map of schemes
Level0k(A*; G)=WG(A; z) !Xk xX bYk0(Z; G; A; z);
which is an isomorphism on the underlying varieties.
This will be proved as Theorem 9.4. `
In the case Z = * we write bYk0(G; A) = bYk0(*; G; A; *), so that Xb0k(G) = *
* (A)bYk0(G; A).
Using the material developed in Section 8, we can make this somewhat more expli*
*cit when
G is Abelian. In particular, we have the following theorem.
Theorem 2.8. If A is a finite Abelian pgroup and B A then bYk0(A; B) is fla*
*t over bX0k,
with degree Ak Mon (B*; (Qp=Zp )nk).
This will be proved as Theorem 9.5.
We believe that we can prove the following result, but we have not yet worked*
* out all
the details.
Conjecture 2.9. In good cases, the map ss0LK(k)F (BG+; E) !ss0F (BG+; LK(k)E)*
* is the
projection OXb0k(G)i ObYk0(G;1). It is probably enough for E to be Landweber ex*
*act and E*G
to be free over E* and concentrated in even degrees.
One application of these results is to the study of EG*(EG) = E*BG, in the sp*
*irit
of [10, 11], and equally to EG*(EG x Z). For this discussion, we shall assume (*
*as in the
conjecture) that E is Landweber exact and that E*Gis free over E* and concentra*
*ted in even
degrees. This is known to be true for a large class of groups. In the cases whe*
*re E*(BG)
is known, it is rather complicated, containing many pieces like E*=(p1 ; : :;:u*
*1n1), often in
odd degrees. Usually it can be calculated using local cohomology at the augment*
*ation ideal
J = ker(E0Gi E0); in fact, if G is a pgroup or E admits an equivariant E1 str*
*ucture,
there is a spectral sequence
E*2= H*J(E0G) =) EG*(EG) = E*(BG):
The groups H*J(M) occuring here are the derived functors of the functor
JM = H0J(M) = {m 2 M  JN m = 0 forN 0}:
Our description of the variety suggests we should consider the Cousin complex a*
*ssociated
to the chromatic filtration of X(G), which has the form
Co(E0G) = (p1E0G! u11E0G=p1 ! . ..!u1n1E0G=I1n1!E0G=I1n):
CHROMATIC GROUP COHOMOLOGY RINGS 9
^
Note that the k'th term is a module over (u1kE0G)Ik= OXb0k(G). There is a spec*
*tral sequence
Es;t1= HsHtJCo(E0G) =) Hs+tJ(E0G):
We will show using our splitting of bX0k(G) that this is concentrated on the li*
*ne t = 0, giving
the case Z = * of the following calculation.
Theorem 2.10. Assuming that E is Landweber exact and E*G(Z) is a free module *
*over
E* concentrated in even degrees, there is a natural isomorphism
H*J(E0G(Z)) = H*JCo(E0G(Z)):
This will be proved as Theorem 10.1. Geometrically, the point is that
a
Xb0k(G) = bYk0(G; A);
(A)
and only the component indexed by A = 1 meets V (J), and this lies entirely ove*
*r V (J) in
the sense that
V (J) xX(G)Xb0k(G) = bYk0(G; 1);
and the local cohomology of this one factor is concentrated in degree zero.
We should point out that the complex JCo(E0G) is still highly nontrivial. Fo*
*r example
if G is Abelian and of rank r then J may be generated as a radical ideal by r e*
*lements,
and so the local cohomology must vanish above degree r. If r < n it is far from*
* obvious
that the above complex is exact in degrees greater than r.
We conclude this introduction with a brief outline of the various sections of*
* the paper.
Section 3 contains some preliminary material about schemes and varieties, and S*
*ection 4
develops the basic theory of admissible ring spectra. In Section 5, we prove Th*
*eorem 2.4,
following Quillen's proof of the parallel result in ordinary cohomology. In Sec*
*tions 6 to 8,
we develop the theory of multiple level structures. In Section 9, we combine t*
*his with
Theorem 2.4 to prove Theorem 2.5 and Theorem 2.7. The applications to the calcu*
*lation
of E*(BG) are developed in Section 10. In Appendix B, we outline an alternative*
* approach
to some of our results (under some quite restrictive hypotheses) using a varian*
*t of the Evens
norm map.
3. Schemes and varieties
In this section we set up the technical framework for our use of schemes and *
*varieties.
All rings will be assumed to be commutative and plocal.
First, recall that the Jacobson radical JA of a ring A is the intersection of*
* its maximal
ideals, and that a ring is said to be semilocal if it has only finitely many ma*
*ximal ideals. A
semilocal ring is said to be complete if it is complete with respect to its Jac*
*obson radical.
The complete semilocal Noetherian rings are precisely the finite products of co*
*mplete local
10 J. P. C. GREENLEES AND N. P. STRICKLAND
Noetherian rings. We define three categories:
Rb= { complete Noetherian semilocal rings}
R = { Noetherian rings}
K = { algebraically closed fields}:
In bR, we consider only maps f :A ! B such that f(JA) JB ; equivalently, f mu*
*st be
continuous with respect to the Jadic topologies on A and B. In the other two c*
*ases, we
consider all ring maps. Of course, any ring map between fields is injective.
Definition 3.1. If A 2 bR, we define a functor spf(A): bR!Sets by
spf(A)(B) = bR(A; B):
We define a formal scheme to be a functor X :bR! Setssuch that X ' spf(A) for *
*some
A.
If X is a formal scheme, we write OX for the set of natural maps from X to th*
*e forgetful
functor. These form a ring, and an isomorphism X ' spf(A) gives an isomorphism *
*OX ' A
by Yoneda's lemma. Thus, the functors A 7! spf(A) and X 7! OX give an equivale*
*nce
between bRopand the category of formal schemes. We prefer to think about the f*
*unctor
category because many examples arise in topology where the functor is easier to*
* describe
than the representing ring. See [23] for some basic theory of formal schemes in*
* this sense,
and extensive discussion of some examples. See [24] for more examples and topo*
*logical
motivation, in a rather more general technical framework. We will make heavy us*
*e of the
functorial viewpoint in Sections 6 to 8 of the present work.
We next define schemes and varieties.
Definition 3.2. If A 2 R, we define a functor spec(A): R !Sets by
spec(A)(B) = R(A; B):
We define a scheme to be a functor X :R ! Setssuch that X ' spec(A) for some A*
*. If
X is a scheme, we write OX for the ring of natural maps from X to the forgetful*
* functor.
Definition 3.3. Let var(A): K ! Setsbe the restriction of spec(A) to K. We de*
*fine a
variety to be a functor X :K ! Sets such that X ' var(A) for some A. Thus, ev*
*ery
scheme X has an underlying variety; this will often also be called X, but we wi*
*ll write
Xvarwhere it is necessary to emphasise the distinction.
We say that a map f :X ! Y of schemes is a V isomorphism if the induced map
Xvar!Yvaris an isomorphism, and that a map A !B of rings is a V isomorphism *
*if the
induced map spec(A) spec(B) is so.
This turns out to be closely related (see Proposition 3.8) to the following m*
*ore elementary
condition. We say that a map u: A ! B of Fpalgebras is an F isomorphism if *
*every
element of the kernel is nilpotent, and for each b 2 B there is an integer k 0*
* such that
k
bp 2 u(A).
CHROMATIC GROUP COHOMOLOGY RINGS 11
We next define a number of useful properties of (formal) schemes and maps bet*
*ween
them. We refer the reader to [19] (for example) for the basic facts about these*
* concepts.
Definition 3.4. 1.A (formal) scheme X is connected if it is not the disjoint *
*union of
two nonempty (formal) schemes, equivalently if OX is not the product of tw*
*o nonzero
rings.
2. A (formal) scheme X is reducedpif_OX has no nonzero nilpotents. The reduce*
*d part
Xredof a scheme X is spec(OX = 0).
3. A connected formal scheme X is smooth if OX is a regular local ring.
4. A closed subscheme of a scheme X is a scheme of the form V (I) = spec(OX =*
*I) for
some ideal I OX , and similarly for formal schemes. We say that X is an i*
*nfinitesimal
thickening of Y if Y = V (I) for some nilpotent ideal I OX .
5. A map f :W ! X of (formal) schemes is finite if the corresponding map OX *
*! OW
makes OW into a finitely generated module over OX . The degree of f is th*
*e smallest
possible number of generators. When working over a fixed base scheme X, we*
* write
deg[W ] for the degree of the given map W ! X.
6. A map f :X ! Y of (formal) schemes is dominant if the corresponding map O*
*Y ! OX
has nilpotent kernel. If f is finite and dominant, then f induces a surje*
*ctive map
Xvar!Yvar.
7. A map f :X ! Y of (formal) schemes is flat if it makes OX into a flat mo*
*dule over
OY , and similarly for faithful flatness. If Y is a connected formal sch*
*eme and f is
finite and flat then OX is a free module over OY .
It is more usual to define spec(A) to be the space of prime ideals of A, but *
*this is
not convenient for our purposes. It turns out that a V isomorphism of schemes*
* gives a
continuous bijection of prime ideal spaces, which is a homeomorphism if the map*
* is finite;
as we shall not use this, we leave the proof to the reader.
We next state various results about V isomorphisms, mostly proven in [21, Ap*
*pendix
B].
*
* __
Proposition 3.5. An infinitesimal thickening is a V isomorphism. *
* __
Proposition 3.6. Every V isomorphism is dominant.
Proof.Let f :X ! Y be a V isomorphism. Given a prime ideal p OY , we can emb*
*ed
the integral domain OY =p in an algebraically closed field K. As f is a V isom*
*orphism, we
can find a map u: OX ! K such that u O f* is the composite OY i OY =p ae K. It*
* follows
that ker(f*) p. As the intersection of all primes in OY is the ideal of nilpot*
*ents, we_see
that ker(f*) is nilpotent, as required. *
* __
Proposition 3.7 (Quillen).Let B be a ring, M a Bmodule, d: B ! M a derivation,
and A the kernel of d (which is a subring of B). If B is a finitely generated A*
*module then
the inclusion A !B is a V isomorphism.
*
* __
Proof.This is Corollary B.6 of [21]. *
* __
Proposition 3.8 (Quillen).A finite map of schemes over spec(Fp) is a V isomorp*
*hism if
and only if it is an F isomorphism.
12 J. P. C. GREENLEES AND N. P. STRICKLAND
*
* __
Proof.This combines Propositions B.8 and B.9 of [21]. *
* __
We next consider finite limits and colimits of (formal) schemes. We will oft*
*en apply
the following results to (co)ends, which can be converted to (co)limits as desc*
*ribed in [18,
Chapter IX]. Whenever we write something like lim Wi, we implicitly refer to a *
*finite
i
diagram. We first make the following convention:
Convention 3.9. When we talk about (co)limits of varieties, we mean (co)limits*
* com
puted in the category of all functors from K to sets. It is easy to see that a*
* limit of
varieties is a variety, but a colimit of varieties need not be.
Most of the time, we will work over a fixed base scheme X. We write CX for th*
*e category
of schemes W equipped with a finite map W ! X, and (V; W ) = CX (V; W ) for th*
*e set
of maps of schemes V ! W that commute with the given maps to X.
Proposition 3.10. The category CX has finite limits and colimits, which are co*
*nverted by
the functor W 7! OW into colimits and limits in the category of OX algebras. *
*In particular,
the products are given by V xX W = spec(OV OX OW ) and the coproducts are given*
* by
V q W = spec(OV x OW ). For any ring A 2 R we have
(limWi)(A) = lim(Wi(A));
 
i i
where the limit on the right is computed in the category of sets over X(A). Fo*
*r any
algebraically closed field K 2 K, we have
(limWi)(K) = lim(Wi(K)):
! !
i i
Proof.It is not hard to see that CX is equivalent to the opposite of the catego*
*ry of finite OX 
algebras. Given this, the only nontrivial point is the last sentence, which is *
*Proposition_B.1
of [21]. *
* __
Proposition 3.11. If X is a formal scheme, then CX is equivalent to the categ*
*ory of
formal schemes W equipped with a finite map to X.
Proof.This just means that every finite OX algebra is a complete semilocal Noe*
*therian
ring, and that all maps of such algebras are continuous with respect to the Jac*
*obson __
radicals. This is a standard piece of commutative algebra. *
* __
Definition 3.12. Let {Wi} be a finite diagram of schemes, equipped with compati*
*ble
maps Wi! W . We say that W is a V colimit of the diagram if the induced map l*
*im Wi!
*
*! i
W is a V isomorphism.
Proposition 3.13. Any map of schemes X0! X gives rise to a functor W 7! W xX *
*X0
from CX to CX0; explicitly, W xX X0= spec(OW OX OX0). This has the following pr*
*operties:
(limWi) xX X0= lim(WixX X0):
 
i i
CHROMATIC GROUP COHOMOLOGY RINGS 13
The canonical map
lim(WixX X0) !(limWi) xX X0
! !
i i
is a V isomorphism, and a genuine isomorphism if the colimit is a coproduct. *
*We also
have
(W xX X0)var= WvarxXvarX0var;
so that the pullback functor preserves V isomorphisms and V colimits.
Proof.The only nontrivial point is the V isomorphism. For any algebraically cl*
*osed field
K, it is easy to see that (WixX X0)(K) = Wi(K) xX(K) X0(K) and
((limWi) xX X0)(K) = (limWi)(K) xX(K) X0(K):
! !
i i
Proposition 3.10 gives (lim Wi)(K) = lim Wi(K) and (lim X0xX Wi)(K) = lim(X0xX
!i ! i !i ! i *
* __
Wi)(K), and the claim follows. *
* __
Remark 3.14. In the light of Proposition 3.8, this has the following conseque*
*nce. If
we have a Noetherian Fpalgebra A and a finite diagram of finite Aalgebras Ai,*
* and a
Noetherian Aalgebra B, then the natural map B A lim Ai ! lim B A Ai is an F 
i  i
isomorphism. It would be nice to have a more direct proof of this fact.
We conclude by showing that a splitting of varieties gives rise to a splittin*
*g of the
underlying schemes.
Proposition 3.15. Suppose that we have a finite`collection of finite maps of s*
*chemes
fi:Xi ! Y , such that the combined map f : iXi !`Y is a V isomorphism. Th*
*en
there are closed subschemes Yi Y such that Y = iYi and fi factors through *
*a finite
V isomorphism Xi! Yi.
Proof.Write B = OY and Ai= OXi and Ii=Tker(B !Ai). Because f is a V isomorphi*
*sm
and thus dominant, we see that J = iIi is nilpotent. We claim that if i 6= j*
* then
Ii+Ij = B. If not, we can find a map B=(Ii+Ij) !K for some algebraically close*
*d field K.
As the map B=Ii! Aiis finite and injective, we can factor the map B=Iii B=Ii+I*
*j !K
through Ai. Similarly, we can factor it through Aj, so the map B !B=Ii+`Ij !K*
* defines
a point of Xi(K) \ Xj(K), which contradicts the assumption that Y (K) = iXi(K*
*).
Q We can now apply the Chinese Remainder Theorem to see that the natural map B=*
*J !
iB=Ii is an isomorphism.TThis means that there are elements ei 2 B such that *
*ei = 1
(mod Ii) and ei2 j6=iIj, so that e2i ei lies in the nilpotent ideal J. By t*
*he Idempotent
Lifting Theorem,Pwe can modify the ei moduloQJ in a unique way to ensure that e*
*2i= ei,
and then iei = 1. This means that B = iCi as rings where Ci = Bei. It is ea*
*sy to
check that the map B ! Ai factors through a V isomorphism Ci ! Ai, so we can*
* put_
Yi= spec(Ci). *
*__
14 J. P. C. GREENLEES AND N. P. STRICKLAND
4. Admissible cohomology theories
In this section we prove some basic facts about admissible cohomology theorie*
*s. For ease
of reference, we repeat the definition.
Definition 4.1. Let pCP1 :CP 1 !CP 1 be the map classifying the p'th tensor po*
*wer of
the canonical line bundle. A plocal commutative ring spectrum E is admissible *
*if
(a) E0 is a complete local Noetherian ring.
(b) E1 = 0.
(c) E2 contains a unit.
(d) pCP1 induces a nonzero selfmap of eE0(CP 1).
It follows from (b) and (c) that E* is concentrated in even degrees, so the A*
*tiyah
Hirzebruch spectral sequence
H*(CP 1; E*) =) E*CP 1
collapses. This means that E*CP 1 = E*[[x]]for suitable x 2 Ee*CP 1, in other *
*words,
E is complexorientable. Because of (c), we can choose such an x in degree zero*
*, so that
E0(CP 1) = E0[[x]]. In particular, this is a complete Noetherian local ring. By*
* a similar
argument, we see that
E0(CP 1 x CP 1) = E0(CP 1)b E0E0(CP 1) = E0[[x 1; 1 x]]:
For convenience, we shall assume that we have chosen an element x as above once*
* and for
all, but as far as possible we shall state our results in a form which is indep*
*endent of this
choice.
We now define formal schemes X = spf(E0) and G = spf(E0(CP 1)). Because CP 1 *
*is
a commutative Hspace, we get a multiplication map
: G xX G = spf(E0(CP 1 x CP 1)) !spf(E0(CP 1)) = G:
This makes G into a group object in the category of formal schemes over X, or i*
*n other
words, a formal group scheme over X. The basic theory of formal groups from thi*
*s point
of view is developed in [23]. Condition (d) has a natural reformulation in term*
*s of G. As
G is an Abelian group object, its endomorphisms form an Abelian group under add*
*ition,
so it makes sense to consider
pkG= pk:1G = spf(E0(pkCP1)): G !G:
Condition (d) says that pG is not the zero homomorphism. We can of course trans*
*late back
to the more traditional language: there is a unique formal group law F (s; t) o*
*ver OX such
that
*(x) = F (x 1; 1 x) 2 E0(CP 1 x CP 1) = E0[[x 1; 1 x]]:
In terms of this formal group law, condition (d) simply says that [p]F(x) 6= 0.
We will often consider the following subgroups of G.
CHROMATIC GROUP COHOMOLOGY RINGS 15
Definition 4.2.
G(k) = ker(pkG:G !G) = spfE0[[x]]=[pk](x):
The next two results are wellknown. Proofs are given under more restrictive *
*hypotheses
in [5], but they generalise readily.
Proposition 4.3. There is an integer k 0 (called the strict height of G) such*
* that [p](x)
k 0
has the form f(xp ) with f (0) 6= 0.n There is an integer n 0 (called the heig*
*ht of G) __
such that [p](x) has the form g(xp ) modulo m[[x]], with g0(0) 2 (E0=m)x. *
* __
*
* __
Proposition 4.4. The maps pkG:G !G and G(k) !X are flat of degree pnk. *
* __
Note that the strict height of G may change if we pull back along a finite ma*
*p Y ! X,
but the height does not. Note also that [p](x) has Weierstrass degree pn, where*
* n is the
height.
k
Definition 4.5. For 0 k n, let uk be the coefficient of xp in [p](x). Note th*
*at u0 = p,
that uk 2 m for 0 < k < n, and that un 62 m. Write
Ik= (u0; : :;:uk1) E0
Xk = spf(E0=Ik) X
X0k= spec(u1kE0=Ik)
bX0k= spec((u1kE0)^):
Ik
One can check quite directly that these objects do not depend on the choice of *
*coordinate
x. Moreover, Ik is also the ideal generated by the coefficients of xiin [p](x) *
*for 0 < i < pk.
Thus G has strict height k over Xk.
Some of our results will only be valid if E is Landweber exact. We recall the*
* definition,
adapted to take account of our other assumptions.
Definition 4.6. An admissible cohomology theory E is Landweber exact if the the*
* se
quence (u0; u1; : :;:un1) is regular in E0.
The main example is Morava Etheory: all other examples are very closely rela*
*ted to it.
Before proceeding to the next definition, we must clear up a possible ambigui*
*ty. Given
a space Y , there are two possible meanings for u1kE0(Y ). On the one hand, we*
* can form
the module E0(Y ) and then invert uk algebraically; we call the result u1k(E0(*
*Y )). On the
other hand, multiplication by uk gives a selfmap of the spectrum E, whose tele*
*scope is a
spectrum u1kE, so we can form (u1kE)0(Y ). These two objects can be very diff*
*erent. For
example, let E be Morava Etheory and let k = 0. Then u1kE is a rational spect*
*rum, so
(u1kE)0(BG) = u1kE0, but u1k(E0(BG)) is much more complicated, as described *
*in [14].
Convention 4.7. The symbol u1kE0(Y ) will always mean u1k(E0(Y )), as discus*
*sed in
the previous paragraph.
16 J. P. C. GREENLEES AND N. P. STRICKLAND
Definition 4.8. Let G be a finite group, and Z a Gspace. We write E*G(Z) = E*(*
*EG xG
Z), and
X(Z; G) = spf(E0G(Z))
Xk(Z; G) = spf(E0G(Z)=Ik) = X(Z; G) xX Xk
X0k(Z; G)= spec(u1kE0G(Z)=Ik) = X(Z; G) xX X0k
bX0k(Z; G)= spec((u1kE0G(Z))^)
Ik
Remark 4.9. It is often the case that there is an admissible ring spectrum F *
*and a ring
map E !F inducing an isomorphism ss*(E)=Ik ' ss*(F ) (so F could reasonably be*
* called
E=Ik). If so, the map E0G(Z)=Ik !FG0(Z) need not be an isomorphism, but it wil*
*l induce
an isomorphism of varieties. This can be proved by applying Theorem 2.5 to both*
* E and F .
There is a much more direct proof using Bockstein spectral sequences in any con*
*text where
these spectral sequences exist, but one needs either the technology of highly s*
*tructured ring
spectra [3, Chapter V] or the technology of generalised Moore spectra [15] to s*
*et them up.
We give this argument in Appendix C.
Proposition 4.10. Let A be a finite Abelian group. Then there is a scheme Hom *
*(A; G)
over X such that
(W; Hom (A; G)) = Hom (A; (W; G))
for all schemes W 2 CX . Moreover, Hom (A; G) is finite and flat over X, wit*
*h degree
A(p)n.
Proof.If A has order prime to p then Hom (A; (W; G)) = 0, so we may assume that*
* A is
a pgroup, say
r1M
A = = Z=pdk:
k=0
Q
We then have Hom (A; G) = kG(dk). This means that
OHom(A;G)= OX [[x0; : :;:xr1]]=([pd0](x0); : :;:[pdr1](xr1));
Q *
* __
by Proposition 4.4 this is a free module of rank ipndi= An over E0 = OX . *
* __
Convention 4.11. We write OE: A !G when we mean there is a homomorphism OE: A*
* !
(W; G) for some scheme W over X that is to be understood from the context.
Proposition 4.12. If A is a finite Abelian group, then there is a natural isom*
*orphism
X(A) = Hom (A*; G);
where A* = Hom (A; S1) is the character group of A. In particular, X(A) is fini*
*te over X.
Proof.We may assume that A is a pgroup. By construction, maps X(A) !Hom (A*;*
* G)
over X biject with homomorphisms OE: A* ! (X(A); G), and (X(A); G) is the set *
*of
continuous E0algebra maps E0(CP 1) ! E0(BA). By regarding CP 1 as BS1, we get*
* a
CHROMATIC GROUP COHOMOLOGY RINGS 17
canonical map OE: A* !(X(A); G) defined by OE(ff) = E0(Bff), and thus a canoni*
*cal map
X(A) !Hom (A*; G). One can check that there is a natural commutative square
X(A B) _____________wHom ((A B)*; G)
 
 
  
u 
X(A) xX X(B) ______Homw(A*; G) xX Hom (B*; G):
Next, recall the wellknown cofibration
pk
BZ=pk !CP 1 !(CP 1)L :
Using the Thom isomorphism, we get a long exact sequence
[pk](x)
. .. E0BZ=pk OX [[x]] OX [[x]]. .:.
As [pk](x) is not a zerodivisor in OX [[x]], we see that
E0(BZ=pk) = OX [[x]]=[pk](x) = OG(k)= OHom((Z=pk)*;G):
One can check that this is the same as the map considered previously. Similarly*
*, we see
that E1(BZ=pk) = 0. As E0(BZ=pk) is free of finite rank over E0, we have a K"u*
*nneth
isomorphism
E0(B(Z=pk) x Z) = E0(BZ=pk) E0 E0(Z):
It follows easily that X(A B) = X(A) xX X(B), and that our map X(A) !Hom (A**
*; G)__
is an isomorphism for all A. *
* __
5. The proof of Theorem 2.4
In this section, we prove Theorem 2.4. The method is essentially due to Quill*
*en, but we
adapt some of his ideas to our particular technical context, and we present the*
* argument
in the language of equivariant topology.
We begin by outlining the argument. Consider the space T of complete flags in*
* a faithful
complex representation of G; we show that
(a) T has Abelian isotropy groups.
(b) There is an equaliser diagram
E0G(Z) !E0G(Z x T ) E0G(Z x T x T ):
(c) E0G(Z) is a finitely generated module over E0.
Next, we try to extract information from the filtration of Z by Gskeleta, or*
* equivalently
from the equivariant AtiyahHirzebruch spectral sequence. The conclusion is tha*
*t E0G(Z)
has the same variety as the ring H0G(Z; E_), where H0G(.; E_) denotes ordinary *
*cohomology
in the sense of Bredon, for the coefficient system E_: G=H 7! E*G(G=H) = E*H.
18 J. P. C. GREENLEES AND N. P. STRICKLAND
We next consider the category A of Abelian psubgroups of G, and the functor
Z
EA (Z) = Map (ss0(ZA ); E0A)
A2A
considered by Quillen (he denotes it AG(Z) and writes the end as a limit). We s*
*how that
this is the same as H0G(Z; E_) if Z has Abelian isotropy groups. By combining t*
*his with (a),
(b) and (c) above, we shall see that
Z A
X(Z; G) = ss0(ZA ) x X(A)
as varieties.
We now start work on the proof. Choose a faithful complex representation V of*
* G, of
dimension d, say. Let T = Flag(V ) be the space of complete flags of subspaces *
*of V , so a
point of V is a chain
V_= (0 = V0 < V1 < : :<:Vd = V )
where dim(Vk) = k for all k. As T is a compact Gmanifold, it can be made into *
*a finite
GCW complex.
Lemma 5.1. Every point in T has Abelian isotropy group.
Proof.If HL G stabilises a flag 0 = V0 < V1 < . .<.Vd = V then we have an isomo*
*r
phismQV ' iVi=Vi1of Hrepresentations. Since V is faithful, H embeds in the *
*group __
iAut (Vi=Vi1), and since Vi=Vi1is one dimensional, Aut(Vi=Vi1) is Abelian.*
* __
We will need the following wellknown calculation.
Proposition 5.2. Let Y be a space, and W a complex vector bundle of dimension *
*d over
Y . Let ss :Flag(W ) ! Y be the associated flag bundle, so a point of ss1y i*
*s a complete
flag in the vector space Wy. There is an evident filtration
0 = W0 < W1 < : :<:Wd = ss*W
of bundles over Flag(W ). We let xk 2 E0(Flag(W )) denote the Euler class of Wk*
*=Wk1,
and write oek for the k'th symmetric function in the variables xi. We also let *
*ck 2 E0(Y )
denote the k'th Chern class of W . There is a natural isomorphism
E*(Flag(W )) = E*(Y )[[x1; : :;:xd]]=(oek  ck  0 k d):
Moreover,Qthis is a free module over E*(Y ) of rank d!, spanned by the monomial*
*s xff=
d ffi *
* __
i=1xi for which 0 ffi< i for all i. *
* __
Corollary 5.3. For any finite Gcomplex Z, there is an equaliser diagram
E*G(Z) !E*G(Z x T ) E*G(Z x T x T ):
Proof.We use an explicit calculation to replace Quillen's use of faithfully fla*
*t descent; this
avoids a discussion of K"unneth isomorphisms.
Let S be the set of multiindices ff = (ff1; : :;:ffd) such that 0 ffi < i. *
*By applying
Proposition 5.2 to the bundle V ! EG xG (Z x V ) ! EG xG Z, whose associated *
*flag
CHROMATIC GROUP COHOMOLOGY RINGS 19
bundle is EG xG (Z x T ), we find that E*G(Z x T ) is the free module E*G{xff *
*ff 2 S}. On
replacing Z by Z x T we find that
E*G(Z x T x T ) = E*G(Z x T ){yfi fi 2 S} = E*G(Z){xffyfi ff; fi 2 S*
*}:
The two projections Z xT xT Z xT induce maps sending xffto xffand yffrespectiv*
*ely. __
It follows immediately that the given diagram is an equaliser. *
* __
Corollary 5.4. E*G(Z) is a finitely generated module over E* (and thus a comple*
*te semilo
cal Noetherian ring).
Proof.First suppose that Z has Abelian isotropy groups, so it is built from cel*
*ls of the
form Bk x G=A with A Abelian. We know from Proposition 4.12 that E*G(G=A) = E*A*
*is
finitely generated over E*. An evident induction on the number of cells shows t*
*hat E*G(Z)
is also finitely generated.
Even if Z does not have Abelian isotropy, the product Z x T does, so E*G(Z x *
*T ) is
finitely generated. We know from Proposition 5.2 that E*G(Z) is a submodule of *
*E*G(Z xT_),
so it is also finitely generated. *
* __
Remark 5.5. The finite generation does not hold without some completeness hyp*
*othesis
on E. For example it fails for uncompleted Ktheory.
We now look at E0G(Z) through the eyes of the equivariant AtiyahHirzebruch s*
*pectral
sequence. This is the spectral sequence arising from the filtration of Z by ske*
*leta, and it
has
Es;t1= Es+tG(Z(s); Z(s1)):
If we let E_tdenote the coefficient system G=A 7! EtG(G=A) = Et(BA), then we ha*
*ve
Es;t2= HsG(Z; E_t):
We need to know this spectral sequence is multiplicative from E2 onwards: this *
*follows
by using the fact that the diagonal map Z ! Z x Z is skeletal if Z x Z is give*
*n the
product filtration. An alternative is to use the Postnikov filtration of E to o*
*btain a spectral
sequence: it is proved in [8, Appendix B] that this agrees with the first from *
*the E2 term
onwards.
Lemma 5.6. The edge homomorphism
i: E0G(Z) !H0G(Z; E_0)
of the AtiyahHirzebruch spectral sequence is a V isomorphism.
Proof.Let d be the dimension of Z, so that Es;*1= 0 for s > d. We first show t*
*hat the
kernel of i consists of nilpotents. Indeed, if i(x) = 0 then x has positive fil*
*tration, so xd+1
has filtration at least d + 1 and thus is zero. Thus, E0G(Z) has the same vari*
*ety as its
image under i, which is precisely E0;01= E0;0d+1. Next, observe that E0;0r+1is*
* the kernel of
a derivation from the ring E00rto the module Er+1;rr, and that everything in s*
*ight is a
finitely generated module over E0 (by Corollary 5.4). By Proposition 3.7, we se*
*e that E0;0r+1_
has the same variety as E0;0r, and the claim follows. *
* __
20 J. P. C. GREENLEES AND N. P. STRICKLAND
We now appeal to an explicit description of H0G(Z; E_), which is valid when Z*
* has Abelian
isotropy. Recall that A is the category whose objects are the Abelian psubgrou*
*ps of G,
with morphisms
A(A; B) = Map G(G=A; G=B) ' {g 2 G  Ag B}=B:
We have two contravariant functors from A to sets, given by A 7! ss0(ZA ) and A*
* 7! E0A=
E0G(G=A). We define
Z
EA (Z) = NatA2A (ss0(ZA ); E0A) = Map (ss0(ZA ); E0A):
A
Lemma 5.7. If H G is Abelian (but not necessarily a pgroup) then EA (G=H) =*
* E0H.
Proof.We can write H = B x C where B is a pgroup and C is a p0group. Note th*
*at
the action of C on E0Bby conjugation is trivial because H is commutative. Since*
* B is a
retract of H, E0Bis a retract of E0H; by a transfer argument E0H= E0B. Note tha*
*t if A is
a pgroup, a coset g(B x C) is Afixed only if the corresponding coset gB is A*
*fixed, so
(G=(B x C))A = (G=B)A=C, so that ss0((G=H)A) = A(A; H) = A(A; B)=C. By Yoneda's
lemma, we have NatA(A(A; B); E0A) = E0B, so NatA(A(A; B)=C; E0A) = (E0B)C = E0B*
*. Thus_
EA (G=H) is just E0B= E0H. *
* __
Lemma 5.8. There is a natural map H0G(Z; E_) !EA (Z), which is an isomorphis*
*m if all
isotropy groups of points of Z are Abelian.
Proof.From the definitions one sees that there are natural maps
H0G(Z; E_) !H0A(ZA ; E_) !H0(ZA ; E0A) = Map (ss0(ZA ); E0A);
and that these assemble to give a natural map
ffZ :H0G(Z; E_) !EA (Z):
Now suppose that Z is a union of two subcomplexes Z0 and Z1. We have a MayerVi*
*etoris
sequence
0 !H0G(Z; E_) !H0G(Z0; E_) H0G(Z1; E_) !H0G(Z0 \ Z1; E_):
Similarly, for each A we have ZA = ZA0[ ZA1, so there is a coequaliser of sets
ss0(ZA0\ ZA1) ss0(ZA0) q ss0(ZA1) !ss0(ZA ):
By applying Map (; E0A), we get an equaliser diagram. It is easy to see that *
*equalisers
commute with ends, so we get a left exact sequence
0 !EA (Z) !EA (Z0) EA (Z1) !EA (Z0 \ Z1):
By comparing these two sequences, we see that ffZ is an isomorphism provided th*
*at ffZ0,
ffZ1 and ffZ0\Z1are isomorphisms. As Z has Abelian isotropy groups, it is built*
* from cells
of the form Bm x G=H, with H Abelian. By induction on the number of cells, it i*
*s enough
to check that ffZ is an isomorphism when Z = Sm1 x G=H or Z = Bm x G=H. This
reduces easily to a check that ff: H0G(G=H; E_) ! EA (G=H) is an isomorphism. *
*The left__
hand side is just E0H, which is the same as the right hand side by Lemma 5.7. *
* __
CHROMATIC GROUP COHOMOLOGY RINGS 21
Lemma 5.9. For any Z and T , the evident diagram
EA (Z) !EA (Z x T ) EA (Z x T x T )
is an equaliser.
Proof.Suppose that A 2 A. Because ss0((Z x T )A) = ss0(ZA ) x ss0(T A) and so *
*on, it is
clear that the diagram
ss0(ZA ) ss0((Z x T )A) ss0((Z x T x T )A)
is a coequaliser. By applying Map (; E0A) we therefore get an equaliser. As *
*equalisers_
commute with ends, the claim follows. *
* __
We now restate and prove Theorem 2.4.
Theorem 5.10. There is a natural map of formal schemes
Z A
ss0(ZA ) x Hom (A*; G) !X(Z; G);
which gives an isomorphism of the underlying varieties. Thus, there are also V *
*isomorphisms
Z A
ss0(ZA ) x Hom k(A*; G) !Xk(Z; G)
and
Z A
ss0(ZA ) x Hom 0k(A*; G) !X0k(Z; G):
Proof.As previously, we let T denote the space of flags in a faithful complex r*
*epresentation
of G. By Lemma 5.1, we know that T has only Abelian isotropy groups. It follows*
* that
the same is true of Z x T and Z x T x T . Consider the diagram
EG(Z) v____________EG(Zwx T ) ________wEG(Z_xwT x T )
  
  
i0 i1 i2
  
u u u
H0G(Z; E_)__________H0G(Zwx T ; E_)_____H0G(Zwx_Twx T ; E_)
  
  
j0 j1 j2
  
u u u
EA (Z)v____________EAw(Z x T )________wEA_(Zwx T x T )
The first and third rows are exact by Corollary 5.3 and Lemma 5.9 respectively*
*. The maps
i0, i1 and i2 are V isomorphisms by Lemma 5.6. The maps j1 and j2 are isomorph*
*isms by
Lemma 5.8. Everything in sight is a finitely generated module over the Noether*
*ian ring
E0, by Corollary 5.4. We now apply the functor var() to get a diagram of varie*
*ties with
the arrows reversed. The first and third lines are coequalisers by Proposition*
* 3.10, and
all vertical maps except for var(j0) are known to be isomorphisms. It follows i*
*mmediately
22 J. P. C. GREENLEES AND N. P. STRICKLAND
that var(j0) is also an isomorphism, so that X(Z; G) = var(EA (Z)). We can rewr*
*ite the
end that defines EA (Z) as the inverse limit of a finite diagram of rings that *
*are finitely
generated as modules over E0. By Proposition 3.10, this becomes colimit diagram*
* when
we apply var(). This shows that
Z
(5.0.1) X(Z; G)var= var Map (ss0(ZA ); E0A)=
A
Z A Z A
var(Map (ss0(ZA ); E0A)) = ss0(ZA ) x *
*X(A):
Using Proposition 4.10, we can rewrite this as
Z A
X(Z; G)var= ss0(ZA ) x Hom (A*; G);
*
* __
as claimed. The other two claims follow at once using Proposition 3.13. *
* __
6.Multiple level structures
In this section, we study the notion of a multiple level structure on a forma*
*l group. This
theory is closely related to questions studied in [23], and we will assume some*
* familiarity
with that paper. For the purposes of the next three sections, we do not need t*
*o start
with an admissible cohomology theory. We merely assume that we have a formal g*
*roup
G of finite height n over a connected formal scheme X. We also fix an integer *
*k with
0 k n, and assume that G has strict height at least k. Thus, to apply this se*
*ction to
the topological questions studied in previous sections, we need to replace X by*
* Xk and G
by G xX Xk.
We now define the main object of interest.
Definition 6.1. Let k 0 be an integer, A a finite Abelian pgroup, and Y a sch*
*eme over
X. We say that a homomorphism OE: A* ! (Y; G) is a pkfold levelA* structure*
* if we
have an inequality of divisors
pk[OEA*(1)] G(1) 2 (Y; Div(G)):
Here as usual A*(1) = ker(p: A* ! A*) = (A=p)* and G(1) = ker(p: G ! G). It *
*is
equivalent to require that [p](x) be divisible by
Y k
(x  x(OE(ff)))p
ff2A*(1)
in OY [[x]].
>From a purely algebraic point of view, there is no reason to consider maps A*
** ! G
rather than maps A !G, but it is convenient for the topological applications.
The case k = 0 just gives level structures as studied in [23]. We shall usual*
*ly ignore this
case.
The following result follows easily from [23, Proposition 16].
CHROMATIC GROUP COHOMOLOGY RINGS 23
Proposition 6.2. The functor from schemes over X to sets defined by
Y 7! { pkfold levelA* structures on G over}Y
is represented by a scheme Levelk(A*; G) over X. Moreover, Levelk(A*; G) is a *
*closed_
subscheme of Hom (A*; G) and thus is finite over X. *
* __
The next introduce notation for a group that will occur in many places.
Definition 6.3. We write k = (Qp=Zp )nk.
The main result is as follows.
Theorem 6.4. The scheme Levelk(A*; G) is flat over X, of degree Ak Mon (A**
*; k). If
(G; X) is of universal type (see Definition 6.5) then Levelk(A*; G) is smooth. *
* For each
B A, there is a natural closed inclusion Levelk(B*; G) ! Hom (A*; G). Togethe*
*r these
maps give a map
a
Levelk(B*; G) !Hom (A*; G);
B
and the resulting map
a
Level0k(B*; G) !Hom 0(A*; G)
B
*
* __
is an infinitesimal thickening (and thus a V isomorphism). *
* __
We shall prove this theorem at the end of this section, except that we will o*
*nly obtain
an inequality
deg[Levelk(A*; G)] Ak Mon (A*; k):
The reverse inequality will be proved as Corollary 8.4.
For the rest of this section, A will be a finite Abelian pgroup, and B will *
*be a subgroup
of A. There is an evident restriction map ss :A* i B*; the kernel is the annihi*
*lator of B,
which we write as BO. We can choose a presentation
A = :
We shall order the generators so that 0 < d0 : : :dr1. We write ffi for the e*
*lement of
A* that is dual to ai in the evident sense, so that A* = . Gi*
*ven a scheme
Y over X and a map OE: A* !(Y; G), we write xff= x(OE(ff)) and xi= xffi= x(OE(*
*ffi)).
Let OE be a pkfold levelA* structure on G over Y . Over the special fibre o*
*f Y we have
r+k pn
OE = 0 so xp divides [p](x), which is a unit multiple of x . It follows tha*
*t r n  k
(unless Y is empty). Thus, Levelk(A*; G) = ; if rankp(A) > n  k.
Many questions about multiple level structures become simpler if the base sch*
*eme X
has good properties. Fortunately, we can often assume this, as we now explain.
Definition 6.5. We shall say that the pair (G; X) is of universal type if the s*
*equence of
elements (uk; : :;:un1) is regular on OX and generates the maximal ideal (and *
*thus X is
a smooth scheme).
24 J. P. C. GREENLEES AND N. P. STRICKLAND
If k > 0 and (G; X) has universal type then OX = K[[uk; : :;:un1]]for some f*
*ield K
of characteristic p, and if k = 0 then OX = R[[u1; : :;:un1]]for some discrete*
* valuation
ring R whose maximal ideal is generated by p. In either case, OX is a unique fa*
*ctorisation
domain and has many other good properties. We can often assume that X has unive*
*rsal
type, and deduce the general case by base change using the following propositio*
*n. We will
give this argument in detail in Corollary 6.11, and subsequently leave similar *
*arguments
to the reader.
Proposition 6.6. Let G be a formal group of height n and strict height at leas*
*t k over a
formal scheme X. Then there are pullback diagrams of formal groups
G u_____H_ ______wK
  
  
  
u u u
X u_____W_f______wVg
where f is faithfully flat and (K; V ) has universal type.
Proof.Let m < OX be the maximal ideal, and let K be the perfect closure of OX =*
*m. By
EGA 0III10:3:1 [12] there is a complete Noetherian local ring OW and a faithfu*
*lly flat map
OX ! OW such that the associated extension of residue fields is the inclusion*
* OX =m ae K.
We define H = G xX W . By the LubinTate deformation theory (see [17], or [23, *
*Section
6] for an account in geometric language) there is a smooth scheme U, a formal g*
*roup L
over U, and a pullback diagram
H ______Lw
 
 
 
u u
W ______Uw
Moreover, we have OU = R[[w1; : :;:wn1]]for some complete discrete valuation *
*ring R
i
with R=p = K, where wi is the coefficient of yp in the [p](y) for some coordina*
*te y on L.
It follows that
V = spf(OU=(p; w1; : :;:wk1)) = spf(K[[wk; : :;:wn1]])
is the closed subscheme where L has strict height at least k. As H has strict h*
*eight at most
k, we see that the map W ! U must factor through V . We thus get a diagram as *
*claimed_
with K = L xU V . *
*__
Definition 6.7. We use the following notation.
D = OX
R(A) = OHom(A*;G)= D[[x0; : :;:xr1]]=([pd0](x0); : :;:[pdr1](xr1))
D(A) = OLevelk(A*;G):
CHROMATIC GROUP COHOMOLOGY RINGS 25
We also write D0= u1kD, R0(A) = u1kR(A) and so on, and X0= spec(D0). Note tha*
*t D0
is not a complete local ring, so our theorems about functors represented by var*
*ious rings
cannot be applied directly to D0.
Proposition 6.8. If OE: A* ! (X; G) is such that for all 0 6= ff 2 A*(1) the *
*element
xff2 OX is not a zerodivisor, then OE is a pkfold levelA* structure. In pa*
*rticular, this
holds if OX is an integral domain and OE is injective.
Proof.We may assume that A* = A*(1) (which just simplifies notation). Let g(x)*
* be
theQWeierstrass polynomial that is a unit multiple in OX [[x]]of [p](x), and wr*
*ite f(x) =
pk
ff(x  xff) . We need to prove that f divides g in OX [x]. Because divisibil*
*ity of monic
polynomials is aQclosed condition, it is enough to proveQthis over any extensio*
*n ring of OX .
Note that = ff6=fi(xff xfi) is a unit multiple of ff6=fixfffi(because *
*x  y is a
unit multiple of x F y), so is not a zerodivisor, so the natural map OX ! *
*1OX is
injective. If ff 6= fi then xxffand xxfigenerate the unit ideal in 1OX [x], *
*so the same
k pk
is true of (x  xff)p and (x  xfi) . It follows from the Chinese Remainder T*
*heorem that
k
the intersection of the ideals ((x  xff)p ) is the same as their product. It i*
*s thus enough
k *
* pk
to check that (x  xff)p divides f(x) in OX [x], or equivalently that (x F xf*
*f) divides
k
[p](x) in OX [[x]]. As G has strict height at least k, we know that xp divide*
*s [p](x) and
k
thus that (x F xff)p divides [p](x F xff). However, [p](x F xff) = [p](x) *
*F [p](xff) =
k *
* __
[p](x) F xpff= [p](x), so (x F xff)p divides [p](x) as required. *
* __
The following result is a partial converse to the above.
Proposition 6.9. Let G be a formal group of strict height precisely k over an *
*integral
domain OX . Then a map OE: A* !(X; G) is a pkfold level structure if and only*
* if it is
injective.
k+1 k
Proof.The hypothesis on G is that xp does not divide [p](x). If OE is a p *
*fold level
k
structure and 0 6= ff 2 A*(1) then xp (x  xff) divides [p](x) so we must have *
*xff6= 0. It
follows that ker(OE) \ A*(1) = 0 and thus that OE is injective. The converse is*
* covered_by
the previous proposition. *
* __
Proposition 6.10. If X is of universal type then the scheme Levelk(A*; G) is s*
*mooth and
is flat over X.
Proof.Choose m so large that pm A = 0, and let g(x) be the Weierstrass polynomi*
*al that
is a unit multiple of [pm ](x), so g is monic and has degree pnm . Let L be th*
*e splitting
field of g over the field of fractions of OX , and R the subring of L generated*
* by OX and
the roots of g. These roots form a group A0under formal addition. It is easy to*
* see that
they all have the same multiplicity, and that the multiplicity of zero is pmk; *
*it follows
that A0 = p(nk)m. A similar argument shows that the subgroup A0(j) has order*
* p(nk)j
for 0 j m, and it follows from the structure theory of finite Abelian groups *
*that
A0 ' (Z=pm )nk. Write Y = spf(R), so A0 is naturally a subgroup of (Y; G). *
*Because
rankp(A*) n  k and pm A* = 0, we can choose a monomorphism OE: A* !A0 (Y; G).
By Proposition 6.8, this is a pkfold level structure (because R is an integral*
* domain). The
26 J. P. C. GREENLEES AND N. P. STRICKLAND
evident map Y ! X (which is finite and dominant) thus factors through Levelk(A*
**; G). It
follows that Levelk(A*; G) !X is dominant, so that dim(Levelk(A*; G)) dim(X) *
*= nk.Q
Note that Levelk(A*; G) is by construction a closed subscheme of Hom (A*; G) *
*= iG(di).
It follows that the maximal ideal in D(A) is generated by x0; : :;:xr1; uk; : *
*:;:un1, and
also that Levelk(A*; G)_is finite over X. Let S be the set {x0; : :;:xr1; uk+r*
*; : :;:un1},
so S = n  k. Write D (A) = D(A)=(S). Over this ring we knowQthat OE(ffi) = 0*
* for all
*
* k pr+k
i and thus that OE(ff) = 0 for all ff 2 A*. It follows that ff2A*(1)(x  xff)*
*p = x . We
*
* __
also know that this divides [p](x), so that_G has strict height at least r + k *
*over D (A).
This implies that uk = : :=:ur+k1= 0_in_D (A). The rest of the u's vanish by d*
*efinition.
It follows that the maximal ideal in D (A) is zero, or equivalently, that S gen*
*erates the
maximal ideal in D(A). As S is the same as the Krull dimension of D(A), we se*
*e that
D(A) is a regular local ring, or in other words that Levelk(A*; G) is a smooth *
*scheme. As
a finite dominant map of smooth schemes is flat, we see that Levelk(A*; G) is f*
*lat_over
X. *
*__
Corollary 6.11. Even when X is not universal, the map Levelk(A*; G) !X is fla*
*t.
Proof.Choose a diagram as in Proposition 6.6. By looking at the represented fun*
*ctors, we
see easily that
Levelk(A*; H) = Levelk(A*; G) xX W = Levelk(A*; K) xV W:
As (K; V ) has universal type, we see that Levelk(A*; K) is flat over V . As f*
*latness
is preserved by pullbacks, we see that Levelk(A*; H) is flat over W . Thus, th*
*e map
Levelk(A*; G) ! X becomes flat after pullback along the faithfully flat map W *
* !X; __
this means that it is itself flat, by standard properties of flatness. *
* __
We next prove a bound on the degree of Levelk(A*; G).
L r1
Lemma 6.12. If A = i=0Z=pdiwith di> 0 then
r1Y
 Mon (A*; k) = p(nk)(di1)(pnk  pi) = A*=A*(1)nk Mon (A*(1); *
*k):
i=0
It follows that
r1Y
Ak Mon (A*; k) = pn(di1)(pn  pi+k):
i=0
Proof.Write fii = pdi1ffi 2 A*(1). Let be a map A* ! k. If is not injec*
*tive, say
(ff) = 0 with ff 6= 0, then piff 2 ker( ) \ A*(1) for some i. Thus, is injec*
*tive if and
only if A*(1)is injective, if and only if we have
(fii) 62 < (fi0); : :;: (fii1)>
for all i. Suppose that we have chosen (ffj) for j < i; there are then pnk  *
*pi possible
choices for (fii) = pdi1 (ffi). As k(di1) = p(nk)(di1), there are p(nk)*
*(di1)(pnkpi)
CHROMATIC GROUP COHOMOLOGY RINGS 27
possible choices for (ffi). It follows that
Y
 Mon (A*; k) = p(nk)(di1)(pnk  pi)
i
*
* __
as claimed. The rest is trivial. *
* __
Proposition 6.13. deg[Levelk(A*; G)] Ak Mon (A*; k).
Proof.In view of Lemma 6.12, it is enough to show that xihas degree at most pn(*
*di1)(pn
pi+k) over OX [x0; : :;:xi1].QBecause OE is a pkfold level structure, we see *
*that [pdi1](xi)
k
is a root of f(x) = [p](x)= fi(x  xfi)p , where fi runs over . *
* As f(x) has
Weierstrass degree pn  pi+k and [pdi1](x) has Weierstrass degree pn(di1), we*
* see that
xk is a root of a power series f([pdi1](x)) of Weierstrass degree pn(di1)(pn *
* pi+k)_as
required. *
* __
Proof of Theorem 6.4.We proved as Corollary 6.11 that Levelk(A*; G) is flat ove*
*r X,
and as Proposition 6.13 that the degree is at most Ak Mon (A*; k). The pro*
*of of
the other inequality is postponed to Corollary 8.4. We proved as Proposition 6*
*.10 that
Levelk(A*;`G) is smooth in the universal case. All that is left is to define a*
*nd study the
map B Levelk(B*; G) !Hom (A*; G).
By the usual argument, we may assume that (G; X) is of universal type, so tha*
*t D is a
regular local ring.
For any B A, it is easy to see that the epimorphism ss :A* !B* gives a clos*
*ed embed
ding Hom (B*; G) ae Hom (A*; G), or equivalently a surjective map R(A) i R(B). *
*By com
posing this with the closed embedding Levelk(B*; G) ae Hom (B*; G), we get a cl*
*osed em
bedding Levelk(B*; G) ae Hom (A*; G), corresponding to a surjective map R(A) i *
*D(B).
Write pB for the kernel; this is prime because D(B) is a regular local ring and*
* thus an in
tegral domain. Because D(B) is flat (and thus free) over D, it is also clear th*
*at pB \ D = 0
and in particular uk 62 pB . We also write p0B= pB R0(A), so that R0(A)=p0B= D0*
*(B).
Now let p be any prime ideal in R(A) such that uk 62 p (these biject with pri*
*me ideals
OE
in R0(A), of course). For some B A, the map A* ! (R(A)=p; G) can be factored *
*as
the projection ss :A* i B* followed by a monomorphism :B* ae (R(A)=p; G), and
Proposition 6.9 tells us that is a pkfold levelB* structure. It follows tha*
*t pB p, so
that the intersection of the p0B's is the samepas_the intersection of all prime*
*s in R0(A). It
is wellknown that this intersection is just 0= {x 2 R0(A)  x is nilpotent}.
We next consider pB + pC, where B and C are distinct subgroups of A, say C 6 *
*B.
There is an element fl 2 BO\ CO, so xfl2 pB . On the other hand, there is an in*
*teger j such
that pjfl has exact order p modulo CO and thus [pj](xfl) divides uk mod pC by t*
*he argument
of Proposition 6.9. As xfldivides [pj](xfl), we deduce that uk = 0 (mod pB + p*
*C), and thus
that p0B+ p0C= R0(A). The Chinese remainder theorem now tells us that
p __ Y Y
R0(A)= 0 = R0(A)=p0B= D0(B):
BA BA
28 J. P. C. GREENLEES AND N. P. STRICKLAND
In geometric terms, this means that the map
a
Level0k(B*; G) !Hom 0(A*; G)
BA
*
* __
is an infinitesimal thickening, as claimed. *
* __
7. The geometric Frobenius map
We next discuss an interesting reformulation of the definition of a multiple *
*level structure.
The concepts involved will be useful later. Assuming that k > 0, we have p = 0*
* in OY
for all schemes Y under consideration, so the map x 7! xp is a ring endomorphis*
*m of OY ,
giving rise to a map FY :Y ! Y of schemes, called the geometric Frobenius. For*
* any d 0,
we let G be the pullback of G along the map FXkd:X ! X. The map FGk:G !G c*
*overs
FXkand thus induces a morphism f :G !G<1> of formal groups over X. A coordinat*
*e x
kd
on G gives rise to a coordinate y on G suchPthat y(fd(a)) = x(a)p for any p*
*oint a of
G. If the formal group law arising from x is ijcijxi0xj1, then the formal gro*
*up law arising
P pkd j
from y is ijcij xi0x1.
Because G has strict height at least k, we see that pG factors through f, say*
* pG = q O f
for some map q :G<1> !G. The kernel of q is a subgroup divisor K < G<1> of deg*
*ree pnk.
k
More explicitly, we have [p](x) = h(xp ) for some power series h over OX , and *
*q is given
by x(q(b)) = h(y(b)) for any point b of G<1>. The subgroup K is just spf(OX [[y*
*]]=h(y)).
Proposition 7.1. A homomorphism OE: A* ! (Y; G) is a pkfold level structure *
*if and
only if we have an inequality of divisors
[fOEA*(1)] K 2 (Y; Div(G<1>)):
Proof.Let g :H ! K be any isogeny of formal groups. Let D and D0 be divisors *
*on K.
Because g is faithfully flat, it is easy to see that D D0if and only if the pu*
*llbacks satisfy
g*D g*D0. It is also easy to see that for any point a of H, the divisor g*[g(*
*a)] is the
translate by a of the divisor ker(g). In the case g = f we have ker(f) = pk[0]*
*, so the
relevant translate is just pk[a]. It follows that
X
f*[fOEA*(1)] = f*[fOE(ff)] = pk[OEA*(1)]:
ff2A*(1)
On the other hand, we have
f*K = f* ker(q) = ker(q O f) = ker(pG) = G(1):
It follows that [fOEA*(1)] K if and only if pk[OEA*(1)] G(1), which means pre*
*cisely that_
OE is a pkfold level structure. *
* __
We can pull the maps p = pG, q and f back along the map FXek:X ! X to obtain*
* maps
which we still call f, q and p. These fit into an infinite diagram whose top le*
*ft corner looks
CHROMATIC GROUP COHOMOLOGY RINGS 29
like this:
f f
G _______G<1>w______wG<2>
  
 q ae  q ae 
p  ae p ae p 
 ae  ae 
  
uaeaeAEf uaaeeAEf u
G<1> ______G<2>w______wG<3>
  
 q ae  q ae 
p  ae p ae p 
 ae  ae 
  
uaeaeAEf uaaeeAEf u
G<2> ______G<3>w______wG<4>
(Half of the triangles commute because p = q O f; the squares commute because *
*f is a
homomorphism of formal groups; because all maps are isogenies and thus epimorph*
*isms, it
follows that the remaining triangles commute.) It follows from this that we hav*
*e a filtration
ker(fd) ker(fd1 O p) : : :ker(f O pd1) ker(pd) = G(1):
On the other hand, ker(fdiO pi) is the pullback along piGof the divisor ker(fd*
*i) = p(di)k,
which is just p(di)kG(i). Thus, the above filtration has the form
pdk[0] p(d1)kG(1) : : :pkG(d  1) G(d):
We can thus define divisors
Di= p(di)kG(i)  p(di+1)G(i  1);
P d
and we find that G(d) = i=0Di.
Algebraically, this decomposition takes the form of the following lemma, whos*
*e proof we
leave to the reader.
k
Lemma 7.2. We can factor [p](x) as xp g(x) for some power series g of Weierst*
*rass
degree pn  pk. We write g0(x) = x and gi(x) = g([pi1](x)) for i > 0. We then *
*have
Yd
k(di)
[pd](x) = gi(x)p ;
i=0
k(di) *
* __
and the divisor Di is just spf(OX [[x]]=gi(x)p ). *
* __
8.Thickenings
In this section we "thicken up" the subschemes Levelk(B*; G) of`Hom (A*; G) t*
*o get
schemes Y (A; B) with nice properties such that Hom 0(A*; G) = B Y 0(A; B) as*
* schemes
(where Y 0(A; B) means Y (A; B) xX X0kas usual).
Definition 8.1. Let A be a finite Abelian pgroup, B a subgroup of A, and ss :A*
** i B*
the canonical map. Let W be a scheme over X, and OE a homomorphism A* !(W; G).
We say that OE is a pkfold level(A*; B*) structure if
30 J. P. C. GREENLEES AND N. P. STRICKLAND
(I) If C* is a subgroup of ker(ss) with pdC* = 0 (such as BO(d)) then the comp*
*osite
OE fd
C* !G ! G
is zero.
(II)If C* is a subgroup of A* with pdC* = 0 and ker(ss) \ C* = pC* then there *
*is a pkfold
level structure :ssC* !(W; G) such that the following diagram comm*
*utes.
OE
C* _________wG

 
 
  d1
ss f
 
 
uu u
ssC* ______G
The following result follows easily from [23, Proposition 16].
Proposition 8.2. The functor from schemes over X to sets defined by
Y 7! { pkfold level(A*; B*) structures on G over}Y
is represented by a scheme Y (A; B) over X. This is a closed subscheme of Hom *
*(A*;_G)_
and thus is finite over X. *
* __
The main result is as follows.
Theorem 8.3. (1)Y (A; B) is flat over X, of degree Ak Hom (B*; k).
(2) Y (A; B) is an infinitesimal`thickening of Levelk(B*; G).
(3) Hom 0(A*; G) = BY 0(A; B).
(4) Y (A; A) = Levelk(A*; G).
(5) Y 0(A; B) \ Hom (B*; G) = Levelk(B*; G).
Corollary 8.4. By parts (1) and (4), we see that
deg[Levelk(A*; G)] = Ak Mon (A*; k):
Part (5) will be proved separately as Proposition 8.12. The proof of the rest*
* will rely on
the following three results.
Proposition 8.5. deg[Y (A; B)] Ak Mon (B*; k).
Proposition 8.6. There are closed subschemes Z(A; B) Hom (A*; G) such`that Z(*
*A; B)
is an infinitesimal thickening of Levelk(B*; G), and Hom 0(A*; G) = B Z0(A; B*
*). More
over, uk is not a zerodivisor on OZ(A;B). (Warning: unlike almost all other co*
*nstructions
we use, the formation of Z(A; B) is not compatible with base change.)
Proposition 8.7. If Z(A; B) is as in Proposition 8.6, then Z(A; B) Y (A; B).
We also record some notation
CHROMATIC GROUP COHOMOLOGY RINGS 31
Definition 8.8.
D = OX
R(A) = OHom(A*;G)
D(B) = OLevelk(B*;G)
D(A; B) = OY (A;B)
E(A; B) = OZ(A;B):
We now prove Theorem 8.3 assuming the above propositions.
Proof of Theorem 8.3.Write dB = Ak Mon (B*; k)`and d = An = Ak Hom (A**
*;Pk).
Note that the obvious decomposition Hom (A*; k) = B Mon (B*; k) gives d = B*
* dB ,
and recall from Proposition 4.10 that R(A) is a free module of rank d over D. P*
*roposi
tion 8.5 gives us an epimorphism tB :DdBQi D(A; B) of Dmodules, and putting th*
*ese
together gives an epimorphism t: Dd i B D(A; B). The closed inclusions Z(A; B*
*) ae
rA sA
Y (A; B)Qae Hom (A*; G)Qgive epimorphisms R(A) ! D(A; B) ! E(A; B) and thus m*
*aps
R(A) r! B D(A; B) s! B E(A; B) with s surjective.
We now invert uk everywhere, and denote this by adding a dash. We may assume *
*that
we are in the universal case, so that uk acts injectively on freeQDmodules.
By Proposition 8.6, we know that s0r0is an isomorphism, so BE0(A; B) is fre*
*e of rank
d over D0. This means that s0t0is an epimorphism between free modules of the sa*
*me rank
d over D0; by a general fact about Noetherian rings, we conclude that it is an *
*isomorphism.
As s0and t0are epimorphisms and s0t0is an isomorphism, we conclude that s0and t*
*0are
isomorphisms. As s0r0is an isomorphism, we now seeQthat r0is an isomorphism.
As t0is iso, we see that the kernel of t: Dd i BD(A; B) is a uktorsion mod*
*ule, but
uk is regular on Dd, so t is injective. It is also surjective, so each tB is an*
* isomorphism,
so D(A; B) is a free module over D of rank dB . This proves (1). It also show*
*s that uk
is regular on D(A; B); as s0B:D(A; B) !E(A; B) is an isomorphism, the same arg*
*ument
shows that sB is an isomorphism, so Y (A; B) = Z(A; B). Given this, part (2) o*
*f the
theorem follows from Proposition 8.6.
Part (3) of the theorem says that r0 is an isomorphism, which we have already*
* seen.
Part (4) holds by inspection of the definitions. As mentioned previously, part *
*(5) will_be
proved separately as Proposition 8.12. *
* __
The proof of Proposition 8.5 relies on three lemmas.
Lemma 8.9. Let B be a subgroup of A, so that (A=B)* A*. Then the natural map
r :Hom (A*; G) !Hom ((A=B)*; G)
is flat of degree Bn.
Proof.Write d = deg[Hom (A*; G) !Hom ((A=B)*; G)]. By definition, d is the sa*
*me as the
degree of the restriction of r over the special fibre of Hom ((A=B)*; G). The s*
*pecial fibre is
contained in the closed subscheme of Hom ((A=B)*; G) where the universal homomo*
*rphism
(A=B)* !G vanishes, which is a copy of X. The inverse image of this copy of X *
*under
32 J. P. C. GREENLEES AND N. P. STRICKLAND
r is just Hom (B*; G). Thus, d = deg[Hom (B*; G)] = Bn. We can choose gener*
*ators
x1; : :;:xd giving a surjective map f :OdHom((A=B)*;G)!OHom(A*;G). We also kn*
*ow that
OHom((A=B)*;G)and OHom(A*;G)are free over OX , with ranks e = A=Bn and ed = *
*An
respectively. Thus, f is an epimorphism between free modules of the same finite*
* rank over
the Noetherian ring OX , so it must be an isomorphism. Thus, r is a flat map o*
*f degree_
d = Bn, as claimed. *
* __
Lemma 8.10. There is a pullback diagram of the following form:
Y (A; B)________wYw(A=pB; B=pB)
v v
 
 
 
u u
Hom (A*; G)______Homw((A=pB)*;wG)
Proof.Note that (A=pB)* is the preimage under ss :A* ! B* of B*(1). Thus, in *
*condi
tions (I) and (II) of Definition 8.1, the subgroup C* is necessarily contained *
*in (A=pB)*.
This means that a map OE: A* !G is a pkfold level(A*; B*) structure if and o*
*nly if the __
restriction of OE to (A=pB)* is a pkfold level((A=pB)*; (B=pB)*) structure, a*
*s required. __
Lemma 8.11. If pB = 0 then we can choose a presentation of A* of the form
A* = ;
where ss(flj) = 0 for all j, and {ss(fi1); : :;:ss(fis)} is a basis for B*. We *
*may also assume
that 0 < d1 : : :ds.
Proof.Choose a presentation
A* = ;
where 0 < c0 : : :cr1. Write
S = {i  ss(ffi) 2 }:
If i 2 S, then we can choose an element ff0iwhich is congruent to ffimod , such
that ss(ff0i) = 0. Because we have cj ciwhen j < i, we see that ff0ihas order *
*precisely pci.
We relabel {ffi  i 62 S} as {fi1; : :;:fis} and {ff0i i 2 S} as {fl1; : :;:fl*
*t}. It is_easy to see
that these elements give a presentation of the required type. *
* __
We can now prove that deg[Y (A; B)] Ak Mon (B*; k).
Proof of Proposition 8.5.For the moment we assume that pB = 0, and we choose a *
*pre
sentation of A as in Lemma 8.11. Write A*j= A*, and B*j= ss(A*j) '*
* A*j=p.
We also write xi = x(OE(fii)) 2 OY (A;B)and yj = x(OE(flj)), and note that thes*
*e generate
OY (A;B)as an algebra over OX . We let Ri be the subalgebra generated by {x1; :*
* :;:xi}.
For each fi 2 B*jwe choose a lift "fi2 A*j, and we write
Y
hj(x) = (x  x(OE(f"i))):
fi2B*j
CHROMATIC GROUP COHOMOLOGY RINGS 33
We now apply condition (II) of Definition 8.1 to the group C* = B*j, giving a d*
*iagram
A*j ________wB*jwss

 
 
OE 
 
 
u u
G ______Gd 1
f j
k pk(dj1)*
* pkdj
in which is a pkfold levelB*jstructure. This means that (hj(x)p ) =*
* hj(x)
k(dj1)
divides [p](x)p . It follows that
k(dj1) pkdj
kj(x) = [p](x)p =hj1(x)
is a power series of Weierstrass degree pkdj(pnk  pj1) over Rj1, and that k*
*j(xj) = 0.
AsQRj is generated by xj over Rj1 we see that the degree of Rs over R0 = OX is*
* at most
s kdj nk j1
j=1p (p  p ).
Next, we apply condition (I) of Definition 8.1 to the subgroup spanned by flj*
* to see that
kej
fejOE(flj) = 0 and thus that ypj = 0. It follows that the degree of OY (A;B)ov*
*er OX is at
most
Yt Ys
pkei pkdj(pnk  pj1):
i=1 j=1
A comparison with Lemma 6.12 shows that this is just Ak Mon (B*; k), as req*
*uired.
We now remove the assumption that pB = 0. We can still apply the above argume*
*nt to
see that
deg [Y (A=pB; B=pB)] A=pBk Mon ((B=pB)*; k):
On the other hand, Lemma 8.10 and Lemma 8.9 give
deg[Y (A; B) !Y (A=pB; B=pB)] = deg[Hom (A*; G) !Hom ((A=pB)*; G)] = pB*
*n:
It follows that
deg[Y (A; B)] pBnA=pBk Mon ((B=pB)*; k):
Using (B=pB)* = B*(1) and (pB)* = B*=B*(1) and Lemma 6.12, we can rewrite this *
*as
AkB*=B*(1)nk Mon (B*(1); k) = Ak Mon (B*; k);
*
* __
as required. *
* __
We next prove, as promised, that Z(A; B) Y (A; B).
Proof of Proposition 8.7.Let OE be the tautological map A* ! (Z(A; B); G). We*
* need
to prove that this satisfies conditions (I) and (II) of Definition 8.1. We may *
*assume that
(G; X) is of universal type, and we reuse the notation of Definition 8.8.
34 J. P. C. GREENLEES AND N. P. STRICKLAND
We first address condition (I). Suppose that ff 2 A* satisfies pdff = 0 and s*
*s(ff) = 0;
kd
it will be enough to prove that ap = 0, where a = x(OE(ff)). If we restrict to*
* the closed
subscheme Levelk(B*; G) then OE factors through ss so a becomes zero, in other *
*words a is
in the kernel of the map E(A; B) ! D(B). As Z(A; B) is an infinitesimal thicke*
*ning of
Levelk(B*; G), this means that a is nilpotent. Using the factorisation in Lemm*
*a 7.2, we
also see that
Yd
k(di)
gi(a)p = 0:
i=0
Suppose that i > 0. In D(B) we have gi(a) = gi(0) = uk, so gi(a) becomes a unit*
* in D0(B).
As ker(E0(A; B) i D0(B)) is nilpotent, it follows that gi(a) is a unit in E0(A;*
* B). Thus,
kd 0
only the i = 0 term in the above product is not a unit, so ap = 0 in E (A; B).*
* As E(A; B)
kd
has no uktorsion, we conclude that ap = 0 in E(A; B), as required.
We next prove condition (II). Let C* be a subgroup of A* such that pdC* = 0 a*
*nd
ker(ss) \ C* = pC*. Suppose that ff 2 C* \ pC*, and write a = x(OE(ff)). As bef*
*ore, we have
Yd
k(di)
gi(a)p = 0:
i=0
Because ss(ff) has exact order p and OE becomes a pkfold levelB* structure ov*
*er D(B),
k pk
we see that xp (x  a) divides [p](x) in D(B)[[x]]and thus that g(a) becomes *
*zero in
D(B). This means that g0(a) = a divides the constant term g(0) = uk in D(B), so*
* that
g0(a) becomes a unit in D0(B). It also means that in D(B) we have [p](a) = 0 an*
*d thus
gi(a) = g([pi1](a)) = g(0) = uk for i > 1, so that gi(a) is also a unit in D0(*
*B). Arguing as
k(d1)
before, we see that g(a)p = 0 in E(A; B).
k(d1)
As g(a) divides [p](a), we conclude that [p](a)p = 0, which implies that *
*the map
OE fd1
pC* ae C* !(Z(A; B); G) ! (Z(A; B); G)
is zero. This means that there is a map :ss(C*) = C*=pC* !(Z(A; B); G*
*) such
that fd1OE = ss. In the previous paragraph we saw that a was a unit in E0(A; *
*B) and thus
not a zerodivisor in E(A; B). It follows by Proposition 6.9 that is a pkfol*
*d levelssC*_
structure. *
* __
Proposition 8.12. Y 0(A; B) \ Hom (B*; G) = Level0k(B*; G).
Proof.It is clear that Levelk(B*; G) is a closed subscheme of W = Y (A; B)\Hom *
* (B*; G)
Hom (A*; G), so that Level0k(B*; G) W 0= Y 0(A; B) \ Hom (B*; G). For the conv*
*erse, let
OV be the image of OHom(B*;G)in OW0, or equivalently the quotient of OW by the*
* ideal
of elements annihilated by a power of uk. Note that V is a closed subscheme of *
*W with
V 0= W 0, so it is enough to show that V Levelk(B*; G). Over W , the map OE: A*
** !G
factors through a map :B* ! G; we need to show that this becomes a level str*
*ucture
when we restrict over V .
CHROMATIC GROUP COHOMOLOGY RINGS 35
Let fi be a nonzero element of B*(1), and let ff be a preimage of fi in A*, o*
*f order pd say.
kd pk(d*
*1)
By the definition of Y (A; B) we see that (x  x(OE(ff)))p divides [p](x) *
* over Y (A; B).
Thus, over W we see that x( (fi)) = x(OE(ff)) divides a power of uk. It follows*
* that over V_,
the element x( (fi)) is not a zerodivisor. The proposition follows by Proposit*
*ion 6.8. __
9. Pure strata
In this section, we return to the framework of Sections 4 and 5, so we again *
*have an
admissible ring spectrum E, a finite group G, a finite Gcomplex Z, and a categ*
*ory A
whose objects are the Abelian psubgroups of G. We define a new functor L from*
* A to
schemes as follows. Each object A 2 A is sent to the scheme
a
MA = Levelk(B*; G):
BA
Consider a morphism u: B ! A in A, so u is really a map of Gsets from G=B to *
*G=A,
sending B to gA say. As this is a map of Gsets we must have Bg A. The map
Mu: MB ! MA is defined to send each component Levelk(C*; G) of MB to the compo*
*nent
Levelk((Cg)*; G) of MA by the obvious map induced by g. It is easy to see that*
* this is
welldefined.
Another point of view is to consider the category A0, whose objects are the s*
*ame as
the objects of A, and whose maps are the isomorphisms in A. The assignment A 7!
Levelk(A*; G) gives a functor from A0to schemes, and M is the left Kan extensio*
*n of this
functor along the inclusion A0! A.
Lemma 9.1. There is an isomorphism of formal schemes
Z A2A a ! a !
ss0(ZA ) x Levelk(B*; G)= ss0(ZA ) x Levelk(A*; G)=G;
BA A2A
and similarly with Levelk(A*; G) replaced by Level0k(A*; G).
Proof.Both statements are formalities: we prove the first statement for definit*
*eness.`
For brevity, we write PA = ss0(ZA ) and LA = Levelk(A*; G). By definition, ( *
* A PA x
LA)=G is the initial example of a scheme Y with maps kA :PA x LA ! Y such that
kAg = kA O (Pg x Lg1): PAg x LAg !Y:
` R A
Thus, to construct a map k0:( PA x LA)=G ! PA x MA, we need to construct m*
*aps
RA
k0A:PA x LA ! PA x MA with k0Ag= k0AO (Pg x Lg1).
RA
Similarly, the coend PA xLA is the initial example of a scheme Y equipped *
*with maps
jA :PA x MA ! Y such that for each morphism u: B ! A in A, the following dia*
*gram
36 J. P. C. GREENLEES AND N. P. STRICKLAND
commutes.
1xMu
PA x MB ______PAwx MA
 
 
Pux1  jA
 
u u
PB x MB _________Ywj
B
Each such morphism u: B ! A is given by an element g 2 G such that Bg A;
the element g is not welldefined, but its coset gB is welldefined.R By lookin*
*g at the
components of the schemes in the above diagram, we see that A PA x MA is the i*
*nitial
example of a scheme Y with maps jA;B: PA xLB ! Y whenever B A, such that when*
*ever
Cg Bg A we have a commutative diagram as follows.
1xLg
PA x LC ______PAwx LCg
 
 
Pux1 jA;Cg
 
u u
PB x LC _________wYj
B;C
RA `
Thus, to construct`a map j0: PA x MA ! ( PA x LA)=G, we need to construc*
*t maps
j0A;B:PA x LB ! ( PA x LA)=G making diagrams like the above commute.
We define
Z A
k0A= jA;A:PA x LA ! PA x MA
and
kB a
j0A;B= (PA x LB Pux1!PB x LB ! ( PA x LA)=G):
A
(Here Pu is the evident morphism PA ! PB induced by the inclusion B !A.) We l*
*eave it
to the reader to check that these maps have the necessary compatibilities to in*
*duce maps
a k0 Z A j0 a
( PA x LA)=G ! PA x MA ! ( PA x LA)=G;
A A
*
* __
and that these maps are mutually inverse. *
* __
Theorem 9.2. There is a natural map of formal schemes
!
a
ss0(ZA ) x Levelk(A*; G)=G !Xk(Z; G);
A
which induces a V isomorphism
!
a
ss0(ZA ) x Level0k(A*; G)=G !X0k(Z; G):
A
CHROMATIC GROUP COHOMOLOGY RINGS 37
Proof.When B A 2 A, we have an inclusion
Levelk(B*; G) !Hom k(B*; G) !Hom k(A*; G):
These can be assembled to give a map
a
MA = Levelk(B*; G) !Hom k(A*; G);
BA
and one can check that this is natural for maps A !A0in A. We therefore get an*
* induced
map of coends
! Z Z
a A A
f : ss0(ZA ) x Levelk(A*; G)=G = ss0(ZA ) x MA ! ss0(ZA ) x Hom k(A*
**; G):
A
We compose this with the map
Z A
g : ss0(ZA ) x Hom k(A*; G) !Xk(Z; G)
provided by Theorem 2.4 to get the map claimed in the Proposition.
General nonsense provides a map
! !
a a
ss0(ZA ) x Level0k(A*; G)=G !X0xX ss0(ZA ) x Levelk(A*; G)=G;
A A
which is a V isomorphism by Proposition 3.13. By composing this with the pullb*
*ack of f
and the isomorphism of Lemma 9.1, we get a map
! Z
a A
f0: ss0(ZA ) x Level0k(A*; G)=G ! ss0(ZA ) x Hom 0k(A*; G):
A
We know by Theorem 6.4 that the maps X0kxX MA ! Hom 0k(A*; G) are V isomorphi*
*sms,
and it follows using Proposition 3.13 that f0 is a V isomorphism. We also kno*
*w from
Theorem 2.4 that g0 = 1X0kxX g is a V isomorphism, so g0f0 is a V isomorphism*
*, as_
required. *
* __
Remark 9.3. One way to think about this result is that it guarantees that the*
* intersection
of any two irreducible components of Xk(Z; G) lies over an infinitesimal thicke*
*ning of Xk+1.
There is another way that one might hope to prove this, at least in the case wh*
*ere E is
Morava Etheory. As the components are distinct and have Krull dimension n  k*
*, the
image of the intersection in X will have dimension strictly less than n  k. Be*
*cause of the
way that the components arise from Abelian subgroups, this image will also be i*
*nvariant
under the action of the Morava stabiliser group. It is widely believed that the*
* only prime
ideals of E0 that are invariant under this action are the ideals Ik, and that t*
*his is a simple
consequence of the classification of invariant primes in MU*. We suspect that t*
*he first of
these beliefs is true, but the second seems to be false. We also suspect that *
*the Ik are
the only invariant radical ideals, or equivalently that they are the only prime*
*s that are
invariant under an open subgroup of the stabiliser group. If we assume this, we*
* see easily
38 J. P. C. GREENLEES AND N. P. STRICKLAND
that the reduced part of the image of the intersection of two irreducible compo*
*nents is Xj
for some j > k.
We now prove Theorem 2.7. We repeat the statement for ease of reference.
Theorem 9.4. Let A G be Abelian`and z an element of ss0(ZA ). The pair (A; z*
*) defines
a point of the finite Gset A ss0(ZA ). We write NG(A; z) for its stabiliser *
*and [A; z] for
its orbit, and put WG(A; z) = NG(A; z)=A.
There is a closed subscheme bYk0(Z; G; A; z) Xb0k(Z; G) depending only on th*
*e orbit of
(A; z), such that
a
Xb0k(Z; G) = bYk0(Z; G; A; z)
[A;z]
(where the coproduct is indexed by orbits). Moreover, there is a map of schemes
Level0k(A*; G)=WG(A; z) !Xk xX bYk0(Z; G; A; z);
which is an isomorphism on the underlying varieties.
Proof.It is not hard to see that
! 0 1
a a
ss0(ZA ) x Level0k(A*; G)=G= @ Level0k(A*; G)A=G
A (A;z)
a
= Level0k(A*; G)=WG(A; z):
[A;z]
This scheme maps by a V isomorphism to X0k(Z; G) = spec(u1kE0G(Z)=Ik), which *
*can be
infinitesimally thickened to get the scheme spec(u1kE0G(Z)=Imk) for any m > 0.*
* Applying
Proposition 3.15,`we get an induced splitting of the ring u1kE0G(Z)=Imkinto pi*
*eces indexed
by the orbits in A ss0(ZA ). Moreover, the piece indexed by (A; z) is V iso*
*morphic to
Level0k(A*; G). All this is canonical and thus compatible as m varies, and by *
*taking_the
inverse limit we get the theorem. *
* __
We can be a little more explicit in the Abelian case. We write bYk0(A; B) for*
* bYk0(*; A; B; *).
Theorem 9.5. If A is an Abelian pgroup and B A then ObY(0A;B)is a free modu*
*le of
rank Ak Mon (B*; k) over OXb0k.
`
Proof.The decomposition bX0k(A) = BYb0k(A; B) gives a decomposition
a
Hom 0k(A*; G) = X0k(A) = bX0k(A) xX Xk = bYk0(A; B) xX Xk:
B
It is not hard to see that this must coincide with the splitting in Theorem 8.3*
*, so that
ObYk0(A;B)=Ik = OY 0(A;B). We also know that OY 0(A;B)can be generated over u*
*1kE0=Ik =
OXb0k=Ik by dB elements, where dB = Ak Mon (B*; k). It follows easily tha*
*t we can
CHROMATIC GROUP COHOMOLOGY RINGS 39
choose an epimorphism tB :OdBbX0i ObY(0A;B). The direct sum of all these is an *
*epimorphism
P k k
OdbX0i OXb0(A)where d = B dB = An. In other words, it is an epimorphism bet*
*ween
k k
free modules of the same finite rank over the Noetherian ring OXb0k, hence an i*
*somorphism.
*
* __
This implies that tB is an isomorphism, as required. *
* __
10.The Ehomology of BG.
In this section we use our results as input to the calculation of E*(BG) by e*
*quivariant
means. We have been much concerned with the cohomology theory E*G(Z) = E*(EGxG *
*Z),
and now we consider the calculation of EG*(EGxZ) = E*(EGxG Z) for finite Gcomp*
*lexes
Z, particularly if Z = * when we obtain EG*(EG) = E*(BG). It is necessary to re*
*mind the
reader that EG*(Z) is usually not the homology of the Borel construction unless*
* Z is free.
For this section we assume that
(1) E is Landweber exact, and
(2) E*G(Z) is a free module over E*, concentrated in even degrees.
Condition (2) is known to hold for Z = * for a large class of groups.
Let J be the augmentation ideal of E0G, so J = ker(E0G i E0). We will ap*
*ply
Theorem 2.7 to the study of the local cohomology H*J(E0G(Z)) and the local Tate*
* co
homology Hb*J(E0G(Z)), which is the same as the Cech cohomology H*J(E0G(Z)) bec*
*ause
E0G(Z) is complete at J. In good cases, these will be relevant to the calculat*
*ion of
EG*(EG x Z) = E*(EG xG Z) and the ETate cohomology groups t(E)*G(Z) of [8]; in
deed, we expect to have spectral sequences
E**2= H*J(E*G(Z)) =) EG*(EG x Z) = E*(EG xG Z)
and
E**2= H*J(E*G(Z)) =) t(E)*G(Z):
These spectral sequences can be constructed by elementary means if G is a pgro*
*up [11, 7],
but for a general finite group we rely on the theory of highly structured ring *
*spectra [9, 7].
The particular way that the algebra is reduced to the ungraded ring E0Gis discu*
*ssed in
detail in [11].
This approach is a continuation of the progression starting with [10, 11], al*
*though the
answer is necessarily less explicit in the higher dimensional cases.
Recall that for a finitely generated ideal J = (ff1; : :;:ffd) of E0Gwe may d*
*efine the flat
stable Koszul complex
Ko(J) = (A !A[1=ff1]) . . .(A !A[1=ffd]);
this is independent of the generators up to quasiisomorphism and the local coh*
*omology
of a module M can be defined by
H*J(M) = H*(Ko(J) M):
Note that
H0J(M) = J(M) := {x 2 M  JN x = 0 for N sufficiently large};
40 J. P. C. GREENLEES AND N. P. STRICKLAND
geometrically, J(M) is the module of sections of the sheaf fM with support in V*
* (J).
Since E0Gis a Noetherian ring, a result of Grothendieck states that local cohom*
*ology
calculates the right derived functors of the left exact functor J. Note that i*
*n general
the local cohomology of a ring depends on the nilpotents, so that it is not a f*
*unctor of
the variety var(R). However the Jlocal cohomological dimension only depends o*
*n the
variety, and in particular HiJ(R) = 0 for i > 0 if and only if J is nilpotent, *
*or equivalently
var(R) = V (J) = var(R=J).
For any module M over E0Gwe may form the Cousin complex corresponding to the
chromatic filtration
1 1 1 1 1 1 1 1 1
Co(M) = p M ! u1 M=p ! u2 M=p ; u1 ! . ..!un1M=In1 !M=In ;
where the terms are defined recursively by the exact sequences
M=I1k ae u1kM=I1k i M=I1k+1:
If p; u1; : :;:un1 is an Mregular sequence the complex is acyclic and H*(Co(M*
*)) = M.
Our main result is as follows.
Theorem 10.1. Assuming that E is Landweber exact and E*G(Z) is a free module *
*over
E* concentrated in even degrees, there is a natural isomorphism
H*J(E0G(Z)) = H*JCo(E0G(Z)):
*
* __
Proof.This follows immediately from Lemma 10.2 and Proposition 10.3. *
* __
Lemma 10.2. Let M be a module over E0G. If p; u1; : :;:un1 is a regular sequ*
*ence for M
then there is a spectral sequence
H*H*J(Co(M)) =) H*J(M):
Proof.Consider the double complex
T o;o= Co(M) Ko(J):
If p; u1; : :;:un1 is a regular sequence for M then if we take Cousin cohomolo*
*gy of T o;owe
obtain M Ko(J) whose Koszul cohomology is H*J(M); this is therefore the cohomo*
*logy
of the total complex. The spectral sequence is the one for calculating the coho*
*mology_of_
the total complex obtained by taking Koszul homology first. *
* __
For any module M over E0G, it is easy to see that Ck(M) = u1kM=I1k admits a *
*unique
^ Q
structure as a module over (u1kE0G=Ik)Ik= OXb0k(G)= (A)ObYk0(G;A)extending i*
*ts structure
as an E0Gmodule. It follows that Ck(M) has a canonical splitting as a direct s*
*um of pieces
CkA(M), where CkA(M) is a module over ObYk0(G;A). In particular, we have a pie*
*ce Ck1(M)
corresponding to the trivial subgroup A = 1.
Proposition 10.3. For any module M over E0Gwe have
JCk(M) = H0JCk(M) = Ck1(M)
and HiJ(Ck(M)) = 0 for i > 0.
CHROMATIC GROUP COHOMOLOGY RINGS 41
Proof.Write V = V (J). Note that the maps 1 ae G i 1 induce`an isomorphism V *
* '
X, and thus Vk0= X0k= Level0k(1; G). Recall that X0k(G) = (A)Yk0(G; A), wher*
*e the
map Level0k(A*; G) !Yk0(G; A) factors through a V isomorphism Level0k(A*; G)=*
*WG(A) !
Yk0(G; A). In particular, we have Vk0 Yk0(G; 1).
The splitting gives an idempotent e 2 u1kE0G=Ik such that e 7! 0 in OYk0(G;1*
*)and e 7! 1
in OYk0(G;A)for all A 6= 1. For large enough N we have a = uNke 2 E0G=Ik. As Vk*
*0 Yk0(G; 1),
we see that a becomes zero in u1kE0=Ik, but E0=Ik has no uktorsion so a becom*
*es zero
in E0=Ik. We can thus choose a lift b 2 E0Gof a such that b becomes zero in E0,*
* in other
words b 2 J. When A 6= 1, it is clear that b becomes a unit in OYk0(G;A)and thu*
*s also in
ObYk0(G;A).
Now let M be an E0Gmodule. If A 6= 1 then b acts as an isomorphism on CkA(M)*
* and
b 2 J; by standard properties of local cohomology, we see that H*J(CkA(M)) = 0.
Next, recall that Vk0! Yk0(G; A) is a V isomorphism and thus dominant; this*
* means
that the image of J in OYk0(G;A)= ObYk0(G;A)=Ik is nilpotent. It follows that f*
*or each m 1
the image of J in ObYk0(G;A)=Imk is nilpotent. As every element of Ck1(M) is a*
*nnihilated
by Imk for some m, we see that it is also annihilated by a power of J. The def*
*inition of
local cohomology now tell us that H0J(Ck1(M)) = Ck1(M), and since J preserves i*
*njectives,_
HiJ(Ck1(M)) = 0 for i > 0. *
* __
The geometric content of the proof is that the components bYk0(G; A) do not m*
*eet V (J)
unless A = 1 and that bYk0(G; 1) is essentially a colimit of infinitesimal thic*
*kenings of V (J).
We may argue rather similarly for Cech cohomology, and obtain a chromatic rou*
*te to
calculating it, and hence to calculating the Tate cohomology t(E)*G(Z) via the *
*local Tate
spectral sequence. Note that there is a fibre sequence Ko(J) !A !Co (J), wher*
*e Co(J)
is the usual flat complex for calculating Cech cohomology; therefore we may arg*
*ue exactly
as in 10.2 to obtain a spectral sequence for calculating the Cech cohomology. F*
*urthermore
in case H1J(M) = 0 we find H0J(M) = M=JM, so that by 10.3 we obtain the followi*
*ng in
which J(M) := M=JM.
Corollary 10.4. Provided that p; u1; : :;:un1 is a regular sequence for E0G(Z*
*), the Cech
cohomology groups H*J(E0G(Z)) are the cohomology groups of the complex JCo(E0G(*
*Z))._
*
*__
11. Some examples.
Let us consider some special cases. We have pcomplete Ktheory with n = 1, w*
*here the
sequence reads
J(p1R(G)^p) !J(R(G)^p=p1 )
By character theory we see that Jpower torsion in the zero'th term consists of*
* multiples
of the regular representation, and if G is a pgroup Jn (p) for some n, so all*
* of the first
term is Jpower torsion. The sequence is thus
Z^p[1=p] !R(G)=p1 ;
42 J. P. C. GREENLEES AND N. P. STRICKLAND
we recover the pcomplete version of the calculation [6] H0J(R(G)) = Z and H1J(*
*R(G)) =
(R(G)=Z) Z=p1 . Geometrically, there are components Y0(1), and Y0(A) when A i*
*s a
cyclic psubgroup. The statement that the local cohomology of p1R(G)^pis conce*
*ntrated
in degree 0, corresponds to the fact that Y00(1) does not meet Y00(A) if A 6= 1*
*, or that
Y0(1) only meets other components over the point (p) 2 Spec(Z^p). The statemen*
*t that
there is a single component Y1(1) mod p suggests that the components Y0(A) meet*
* Y0(1)
`tangentially' in an infinitesimal neighbourhood of p, and the rank of the answ*
*er suggests
that Y0(A) contributes with multiplicity equal to the number of generators of t*
*he group
A. In this case one may make a more precise integral analysis using Segal's des*
*cription of
Spec(R(G)).
The results of [11] concern the admissible theory
En;i= En=(p; u1; : :;:ui1; ui+1; : :;:un1);
and give a precise calculation for an elementary Abelian group of rank r, fitti*
*ng into the
following picture. Again the local cohomology is given by the complex
J(v1iE0n;i(BG)) !J(E0n;i(BG)=v1i):
If i = n1 these terms correspond to the nonzero terms of the complex of JCo(E0*
*n;i(BG))
in degrees n  1 and n. If G is elementary Abelian of order pr, the (n  1)'st*
* term is a
free v1iE0module of rank pir, and the nth term is a direct sum of pnr copies *
*of E0=v1i.
This again suggests that X0(G) has components Y0(1) and Y0(A) for A a cyclic p*
*subgroup,
and that Y0(1) only meets the other components over (vi). The difference is tha*
*t it now
has multiplicity pir. Again, in an infinitesimal neighbourhood of (vi), there *
*is a single
component Yi(1), but now with multiplicity pnr. This suggests that the interse*
*ction of
Y0(A) with Y0(1) is `tangential', but now the multiplicity contributed by each *
*group element
of order p is (pnr pir)=(pn  1).
Now consider the case of an Abelian pgroup G = A, and E = En. For each m 1 *
*there
are rings D0k(A; B; m), where D0k(A; B; 1) = D0k(A; B), and there is a splitting
Y
u1kE0(BA)=Imk~= D0k(A; B; m):
B
Furthermore, D0k(A; B; m) is a free u1kE0=Imkmodule of rank Ak Mon (B; (Z=*
*p1 )nk),
independent of m. Thus we find that H*J(E*(BA)) is the cohomology of a `chroma*
*tic
complex with multiplicities'
A0[p1E*] !A1[u11E*=p1 ] !A2[u12E*=p1 ; u11] !. . .
. ..!An1[u1n1M=I1n1] !An[E*=*
*I1n]:
This then suggests the general picture that each J(u1kE0(BG)=I1k) is a direct *
*sum of
lk(G) copies of u1kE0=I1k. However, the differential
J(u1kE0(BG)=I1k) !J(u1k+1E0(BG)=I1k+1)
CHROMATIC GROUP COHOMOLOGY RINGS 43
is not given by just projecting onto lk+1(G) factors and then composing with th*
*e natural
map
u1kE0=I1k !u1k+1E0=I1k+1
in each one, contrary to what one might guess from the onedimensional case. To*
* see this,
consider the case where G is elementary Abelian. The naive guess would give E0 *
*in degree
0, and a direct sum of pkr p(k1)rcopies of E0=I1k in degree k 1. This is no*
*nzero
for 0 k n. However, the local cohomology vanishes above dimension r if J can*
* be
generated as a radical ideal by r elements (for example if G is Abelian of rank*
* r); this gives
a contradiction.
Appendix A. Proof of the theorem of HopkinsKuhnRavenel
In this appendix we prove Theorem 2.6. The proof we give here is essentially*
* that of
HopkinsKuhnRavenel; we record it partly because of notational differences, an*
*d partly
because [14] is not easily accessible.
Proof of Theorem 2.6.First we construct a natural map
! G
Y
:p1E*G(Z) ! p1E*(ZA ) E0 D(A) :
A
Indeed, it suffices to construct the components
A : E*G(Z) !E*(ZA ) E0 D(A)
in a suitably natural way, since G acts trivially on the domain. For this we u*
*se the
restriction maps:
E*G(Z) !E*A(Z) !E*A(ZA ) = E*(ZA x BA) = E*(ZA ) E0 E0(BA)
together with the natural map E0(BA) !D(A). It remains to show that is an iso*
*mor
phism for all finite Gcomplexes Z.
The domain is evidently a cohomology theory, and the codomain is too, because*
* D(A) is
flat and the group order is inverted before we take invariants. Thus, if is an*
* isomorphism
for Z = G=B whenever B is an Abelian subgroup, it is an isomorphism whenever Z *
*has
finite Abelian isotropy groups. The general case follows since both sides give*
* equaliser
diagrams when applied to Z x T x T Z x T ! Z. For the domain this is Corollar*
*y 5.3.
For the codomain, it suffices to show that for each A when we apply E*(.) to th*
*e diagram
ZA xT AxT A ZA xT A! ZA we obtain an equaliser. Indeed, the subsequent operati*
*ons
preserve equalisers: D(A) is flat, localisation and products are exact, and pas*
*sage to G
invariants is exact since the group order is invertible. However T A is the sp*
*ace of A
invariant complete flags in V , and this is a disjoint union of products of the*
* flag spaces
of the Aisotypical parts of V . The result therefore again follows from the no*
*nequivariant
instance of Corollary 5.3.
44 J. P. C. GREENLEES AND N. P. STRICKLAND
It remains to show is an isomorphism if Z = G=B, which we do by direct calcu*
*lation
and the theory of level structures. A simple argument reduces us to the case wh*
*ere B is a
pgroup. In this case the codomain of is
!G
Y
p1 Hom (A(A; B); D(A))
A
in each even degree. By Lemma 9.1 and the Yoneda Lemma, this coincides with
Z Y Y
Hom (A(A; B); p1D(C)) = p1 D(C):
A2A CA CB
By Theorem 6.4, the natural map
Y
p1EB = p1OHom(B*;G)!p1 D(C)
CB
is surjective (with nilpotent kernel). Both sides arePfree modules over p1E. T*
*he domain
has rank  Hom (B*; 0), and the codomain has rank CB  Mon (C*; 0), which *
*is the
same. An epimorphism of freeQmodules of the same rank over a Noetherian ring i*
*s an_
isomorphism, so p1EB = p1 CB D(C), as required. *
* __
Appendix B. The Evens norm map
In this section we outline an alternative approach to Theorem 2.5, parallel t*
*o that of
Evens [4] for the case of ordinary cohomology. This has the advantage that it c*
*onstructs
some useful elements of E0Gexplicitly. However, we need to assume that E has th*
*e structure
of an H1 ring spectrum [1]. There are essentially only two admissible theories*
* for which
there is a published construction of an H1 structure (rational cohomology and *
*padic K
theory). However, it is widely believed that Morava Etheory admits even an E1 *
*structure.
The key ideas needed to prove this are due to Mike Hopkins and Haynes Miller, b*
*ut the
foundations needed to support them have turned out to be unexpectedly hard to c*
*onstruct.
We understand that an account will appear in a paper currently being written by*
* Paul
Goerss and Mike Hopkins.
For this appendix we assume that E has an H1 structure. For simplicity, we *
*also
assume that k > 0. The H1 structure allows us to construct an Evens norm map E*
*0H!
E0Gwhenever H G, which should be thought of as a multiplicative analogue of the
transfer. The action of G on G=H together with a choice of coset representative*
*s G=H =
{x1H; : :;:xdH} gives rise to a homomorphism G ! d o H, which is canonical up *
*to
conjugacy. If u 2 E0H= [BH+; E] then we define
Dd(u) d
norm GH(u) = (BG+ ! B(d o H) = Dd(BH+) ! DdE ! E);
where Dd is the d'th extended power construction, and d is provided by the H1 s*
*tructure.
This norm map has the following properties:
1. It is natural for isomorphisms of groups, and thus behaves in the obvious *
*way under
conjugation.
CHROMATIC GROUP COHOMOLOGY RINGS 45
2. It is multiplicative: norm GH(uv) = normGH(u) normGH(v).
3. There is a double coset formula: For any two subgroups H and K of G we have
Y g
resGKnormGH(u) = norm KK\gHresHK\gHcg(u):
KgH2K\G=H
Definition B.1. For any Abelian psubgroup A of G we write O(A) 2 E0Afor the Eu
ler class of the reduced regular representation of A. Under the identification*
* X(A) =
Hom (A*;QG), this becomes the natural transformation Hom (A*; G) ! Forget give*
*n by
OE 7 ! ff2A*\0x(OE(ff)). This is a unit over Level0k(A*; G) and nilpotent ov*
*er Level0k(B*; G)
when B < A. It is also invariant under Aut(A). We write WGA = pab where p doe*
*s not
divide b, and we note that g(x) = (1 + x)1=b 1 is a power series over Zp. We d*
*efine
zA = g(norm GA(1 + O(A))  1):
Using the above properties of the norm map, we see that
(
0 if B contains no conjugate of A
resGB(zA) = a
O(A)p ifB = A:
If A and B are nonconjugate Abelian psubgroups, with B A say, then B can*
*not
contain a conjugate of A. Thus, zA is zero over the image of Level0k(B*; G) in *
*X0k(G), and
invertible over the image of Level0k(A*; G), so these images are disjoint.
Consider the natural map
f :u1kE0G=Ik = OX0k(G)!D0k(A)WGA = OLevel0k(A*;G)=WGA:
a
We know that O(A)p lies in the image of f and that it is invertible. The invers*
*e is integral
over the image of f (because everything is finite over X0k).a After multiplying*
* a monica
equation of integral dependence byaa suitable power of O(A)p , we find that O(A*
*)p also
lies in the image of f, say O(A)p = f((A)). Now suppose that y 2 Dk(A)WGA , *
*and
choose an element "y2 E0Alifting y. One can check that
a
f(g(norm GA(1 + O(A)"y))(A)) = yp :
Thus, f is F surjective in the evident sense. A little more work in this direc*
*tion recovers
Theorem 2.5, at least in the case Z = *. For general Z, we can observe that F =*
* F (Z+; E)
is again an H1 ring spectrum, with FG0= E0G(Z).
Appendix C. Varieties and reduction mod Ik.
In this appendix we give the elementary argument that E*(BG)=Ik and (E=Ik)*(B*
*G)
have isomorphic varieties, assuming that Bockstein spectral sequences with appr*
*opriate
multiplicative properties exist, and we sketch a construction under certain cir*
*cumstances.
Proposition C.1. For any finite group G there is an Fisomorphism
E*(BG)=Ik !(E=Ik)*(BG):
46 J. P. C. GREENLEES AND N. P. STRICKLAND
Proof.We shall in fact prove the corresponding result for any space X with K(n)*
**(X)
finitely generated over K(n)*; by a result of Ravenel [22] or by 5.4 above, thi*
*s applies to
X = BG.
By a Bockstein_spectral sequence argument given in [11] or by 5.4 above, sinc*
*e E is
* __* __
complete, E (X) is finitely generated over E for E = E=Ik for any k n. We *
*argue
by induction on k, supposing that we have constructed_an Fisomorphism E*(X)=Ik*
* !
(E=Ik)*(X), using_the_identity_if_k = 0. Now let E = E=Ik_and v = uk, and consi*
*der the
* __ *
cofibre sequence E v!E !E =v; this gives a ring map E (X)=v !(E =v) (X). Com*
*bining
this with the Fisomorphism already constructed, we obtain
E*(X)=Ik+1 = (E*(X)=Ik)=v !{E=Ik}*(X)=v !{E=(Ik + (v))}*(X) = {E=Ik+1}*(X):
The fact that the first map is an Fisomorphism is an amusing exercise.
__
Lemma C.2. If :A !B is an Fisomorphism and a 2 A then :A=a !B=(a) is al*
*so_
an Fisomorphism. *
*__
It therefore suffices to deal with the principal case.
__* __*
Lemma C.3.__ If E (X) is finitely generated over the characteristic p ring E *
*then for any
* __ *
v the map E (X)=v !(E =v) (X) is an Fisomorphism.
Proof.Of course the map is injective_by_construction, so it suffices to show th*
*at there is
t
an integer_t such that for any y 2 (E =v)*(X) the power yp is the reduction of *
*some class
*
x 2 E (X). This is a job for the Bockstein spectral sequence arising from the d*
*iagram
__ ________v________v________v______v
E [[^ Ew [[^ Ew [[^ Ew .w. .
[  [  [ 
ae ae ae
fi[  fi[  fi[ 
__u __u __u
E =v E =v E =v
__ __*
by applying_[X; .]*. Here fi is the connecting homomorphism (E =v)*(X) ! E (*
*1X).
*
Since E (X) is Noetherian it has bounded vtorsion in the sense that there is a*
*n integer
t so that if vt+1y_= 0 then also vty = 0. Now by definition, if dtz = 0 we have*
* fiz = vtzt
* t+1 t
for some zt2 E (X), and since 0 = vfiz_= v zt it follows that 0 = v zt= fiz, *
*and hence
* *
* t
z is the reduction of an element of E (X). It thus suffices to show that any *
*p th power
t __ *
yp 2 (E =v) (X) is a tcycle. But the spectral sequence is multiplicative by as*
*sumption,
t *
* __
and so for any scycle z, ds+1zp = 0, thus yp is a tcycle by induction. *
* __
*
* __
*
*__
If we assume E is an Salgebra in the sense of [3] then the tower can be cons*
*tructed
in the category of highly structured Emodules, since v comes from the coeffici*
*ents of E.
This gives the first property we required. The second property is that the diff*
*erentials are
CHROMATIC GROUP COHOMOLOGY RINGS 47
__
derivations. For this we first need to know that E =v is an Ealgebra up to hom*
*otopy, and
that the diagram
__ __ __
E =v ^E E =v_______wE =v
 
ifi^1j 
1^fi  fi
 
__ __ __u __ __u
E ^E E =v _ E=v ^E E E
 
 
'  ae
 
__ __u __ u
E =v _ E =v ______Ew=vr
commutes. This follows by elementary obstruction theory as in Chapter V of [3]*
* provided
E* is concentrated in degrees divisible by 4, and_Ik is generated by a regular *
*sequence (and
also under slightly weaker hypotheses). Indeed, E is the Esmash product of the*
* spectra_
E=u for the elements u in the regular sequence, so it suffices to deal with the*
* case E = E.
Here it is easy to be explicit since E=v ^E E=v = E=v _ E=v, and the product i*
*s the
identity on the first summand and zero on the second. The clockwise composite *
*is zero
fi
on the second factor, and E=v ! E ! E=v on the first. Consider the anticlock*
*wise
contribution from the second factor: the components into each of the factors at*
* the bottom
left are 1 and 1, and thus cancel. On the first factor, one of the terms is zer*
*o and the
fi
other is E=v ! E !E=v, depending on the isomorphisms used.
References
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School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH, UK.
Email address: j.greenlees@sheffield.ac.uk
Trinity College, Cambridge CB2 1TQ, UK.
Email address: n.strickland@pmms.cam.ac.uk