CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY
N. P. STRICKLAND
Let G be a finite group, and let E* be a generalised cohomology theory, subje*
*ct to certain
technical conditions ("admissibility" in the sense of [4]). Our aim in this pap*
*er is to define and
study a certain ring C(E; G) that is in a precise sense the best possible appro*
*ximation to E0BG
that can be built using only knowledge of the complex representation_theory of *
*G. There is a
natural map C(E; G) -! E0BG, whose image is the subring C(E; G) E0BG generated*
* over
E0 by all Chern classes of all complex representations. There is ample preceden*
*t for considering
this subring in the parallel_case of ordinary cohomology; see for example [13, *
*14, 3]. However,
although the generators of C(E; G) come from representation theory, the same ca*
*nnot be said
for the relations; one purpose of our construction is to remedy this. We also a*
*lso develop a kind
of generalised character theory which gives good information about Q C(E; G). *
* In the few
cases that we have been able to analyse completely, either Q C(E; G) 6= Q E0B*
*G for easy
character-theoretic reasons, or we have C(E; G) = E0BG.
Rather than working directly with rings, we will study the formal schemes X(G*
*) = spf(E0BG)
and XCh(G) = spf(C(E; G)); note that the map C(E; G) -!E0BG corresponds to a ma*
*p X(G) -!
XCh(G). See [4, 9, 8] for foundational material on formal schemes. Suitably int*
*erpreted, our main
definition is that XCh(G) is the scheme of homomorphisms from the -semiring R+(*
*G) of complex
representations of G to the -semiring scheme of divisors on the formal group G *
*associated to E.
We start by fixing some conventions in Section 1. We then recall the basic t*
*heory of -
semirings (Section 2), set up the parallel theory of -semiring schemes, and def*
*ine the -semiring
scheme of divisors (Section 3). We then recall the definition of Adams operatio*
*ns and study their
basic properties (Section 4). Using this we give a precise definition of XCh(G)*
* and an implicit
presentation of C(E; G) by generators and relations (Section 5). In Section 6, *
*we work out the case
of the symmetric group 3 at the prime 3, and show that X(3) = XCh(3). In Sectio*
*n 7 we show
that X(G) = XCh(G) when G is Abelian, and in Section 8 we show that the same is*
* true when E is
the p-adic completion of complex K-theory and G is a p-group. We then use Adams*
* operations to
reduce certain questions to the Sylow subgroup of G (Section 9) and to prove th*
*at XCh(G) is finite
over X = spf(E0) (Section 10). In Section 11, we recall the Hopkins-Kuhn-Ravene*
*l generalised
character theory, which relates QE0BG to the set (G) of conjugacy classes of ho*
*momorphisms
Znp-! G. We give a parallel (but less precise) relationship between Q C(E; G)*
* and the set
Ch(G) of -semiring homomorphisms R+(G) -!N[(Qp=Zp)n]. These descriptions are re*
*lated by
a map : (G) -! Ch(G). In Section 12, we compare (G) and Ch(G) with two other se*
*ts
that are sometimes easier to understand. We next return to examples: in Section*
* 14 we show that
X(4) = XCh(4) at the prime 2, and in Section 15 we study (G) and Ch(G) when G i*
*s an
extraspecial group at an odd prime. We then show that a certain approach which *
*appears more
precise actually captures no more information (Section 16). We conclude in Sect*
*ion 17 by proving
a result in representation theory that was used in Section 9.
1. Notation and conventions
Fix a prime p. Throughout this paper, E will denote a p-local generalised coh*
*omology theory
with an associative and unital product. We write Ek for Ek(point), so E* is a Z*
*-graded ring and
E0 is an ungraded ring. We assume that E has the following properties:
1.E0 is a commutative complete local Noetherian ring, with maximal ideal m sa*
*y.
2.Ek = 0 whenever k is odd.
3.E-2 contains a unit (so EkX ' Ek-2X for all X).
4.Either p > 2 and E is commutative, or p = 2 and E is quasi-commutative, whi*
*ch means that
there is a natural derivation Q: EkX -!Ek+1X and an element v 2 E-2 such th*
*at 2v = 0
and ab - (-1)|a||b|ba = vQ(a)Q(b) for all a; b 2 E*X.
___________
Date: June 16, 1999.
1
2 N. P. STRICKLAND
There is one more condition, which needs some background explanation. Note tha*
*t the quasi-
commutativity condition means that whenever E1X = 0, the ring E0X is commutativ*
*e (in the
usual ungraded sense.) In particular, E* = E*(point) is commutative. A collap*
*sing Atiyah-
Hirzebruch spectral sequence argument shows that
E*CP 1 ' E*b H*CP 1 = E*b Z[[y]] = E*[[y]];
where y 2 eE2CP 1; it follows that E is complex-orientable. We can multiply y b*
*y a unit in E-2
to get an element x 2 eE0CP 1such that E*CP 1 = E*[[x]]. We fix such an element*
* x once and
for all (although we will state our results in a form independent of this choic*
*e as far as possible).
This gives rise in the usual way to a formal group law F over E0.
5.In addition to the above properties, we will assume that the reduction of F*
* modulo the
maximal ideal of E0 has height n < 1.
In the language of [4], our conditions say that E is a K(n)-local admissible *
*theory. The
only difference is that previously we insisted that E should be commutative rat*
*her than quasi-
commutative; the reader can easily check that everything in [4] goes through in*
* the quasi-
commutative case.
We will describe our results in the language of formal schemes. Most of the f*
*ormal schemes that
we consider have the form spf(R), where R is a complete local Noetherian E0-alg*
*ebra. For these
the foundational setting discussed in [9] is satisfactory: one can regard the c*
*ategory of formal
schemes as the opposite of the category of complete semilocal NoetherianQrings *
*and continuous
homomorphisms. We also make some use of formal schemes such as spf( k2ZE0[[c1;*
* c2; : :]:]); a
set of foundations covering these is developed in [8]. The older category of fo*
*rmal schemes embeds
as a full subcategory of the newer one.
Definition 1.1.We let X be the formal scheme spf(E0), and write G for the forma*
*l group
spf(E0CP 1) over X. Note that our element x 2 eE0CP 1can be regarded as a coord*
*inate on G,
with the property that
x(a + b) = F (x(a); x(b)) = x(a) +F x(b):
Remark 1.2. Many of our constructions work with an arbitary formal group G over*
* a formal
scheme X; it is not usually necessary to assume that G comes from a cohomology *
*theory, although
that is the case of most interest for us.
We will let G denote a finite group. We write e = e(G) for the exponent of G*
*, in other
words the least common multiple of the orders of the elements. We factor this *
*in the form
e = pve0= pv(G)e0(G), where e06= 0 (mod p). We also choose a Sylow p-subgroup P*
* G.
2.-(semi)rings
We will use the following definition:
Definition 2.1.A semiring is a set R equipped with the following structure.
o A commutative and associative addition law with neutral element (written as*
* 0); we do not
assume that there are additive inverses.
o A commutative and associative multiplication law with neutral element 1, wh*
*ich distributes
over addition.
A -semiring is a semiring RPequipped with Maps k:R -!R for k 0 satisfying 0(*
*x) = 1
and 1(x) = x and k(x + y) = k=i+ji(x)j(y).
A -ring is a -semiring which has additive inverses.
The initial -semiring is N and the initial -ring is Z; in both cases we have
k(n) = (nk)= n(n - 1) : :(:n - k + 1)=k!:
Definition 2.2.An N-augmented -semiring is a -semiring R equipped with a homomo*
*rphism
dim:R -!N of -semirings. A Z-augmented -ring is a -ring R equipped with a homom*
*orphism
dim:R -!Z of -rings.
Example 2.3.Let R+(G) be the semiring of isomorphism classes of complex represe*
*ntations of
G. It is well-known that this is a -semiring with operations k given by exterio*
*r powers. There
is an augmentation dim:R+(G) -!N sending each representation to its dimension.
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 3
Example 2.4.Let A be anPAbelian group, and let N[A] be the group semiring of A,*
* in other words
the set of expressions` ana[a] with na 2 N and na = 0 for all but finitely man*
*y a. Equivalently,
we have N[A] = n An=n. This has a canonical structure as a -semiring, with
X
k([a1] + : :+:[an]) = [ai1+ : :+:aik];
I
where the sum on the right runs over all lists I = (i1; : :;:ik) such that 1 i*
*1 < : :<:ik n.
There is an augmentation dim:N[A] -!N defined by
dim([a1] + : :+:[an]) = n:
If A is finite and A* = Hom (A; S1) then N[A] = R+(A*) as -semirings.
Example 2.5.For any space X, let Vect+(X) denote the semiring of isomorphism cl*
*asses of
complex vector bundles over X. This is a -semiring with operations as for R+(G*
*). We will
always allow vector bundles to have different dimensions over different compone*
*nts of the base, so
we do not have a natural map dim:Vect+(X) -!Z. We write Vect+d(X) for the set o*
*f isomorphism
classes of bundles all of whose fibres have dimension d, and we put Pic(X) = Ve*
*ct1(X) ' H2(X).
This is an Abelian group, and there is an evident map N[Pic(X)] -! Vect+(X). I*
*n the case
X = BG, there is a well-known homomorphism R+(G) -! Vect+(BG) sending a represe*
*ntation
V to the bundle V xG EG.
Remark 2.6. In the important examples of -(semi)rings, some extra identities ho*
*ld that relate
the elements ij(x) and i(xy) to the elements k(x) and l(y). For many purposes i*
*t would be
preferable to take these identities as part of the definition of a -(semi)ring.*
* However, it turns out
that this would make no difference for us and the identities are complicated (p*
*articularly in the
semiring case) so we omit them. In Section 16 we will discuss an approach which*
* is apparently
even more precise, and show that it actually gives no more information than our*
* approach using
-semirings without extra identities.
Remark 2.7. Let R+ be a -semiring, and let R be its Grothendieck completion, or*
* in other
words the group completion of R+ considered as a monoid under addition. It is w*
*ell-known that
this can be made into a -ring in a canonical way, and that any homomorphism fro*
*m R+ to a
-ring factors uniquely through R. Moreover, if R+ is augmented over N then R is*
* augmented
over Z.
Example 2.8.The Grothendieck completion of R+(G) is of course the ring R(G) of *
*virtual
representations of G, and the completion of N[A] is the group ring Z[A]. We wri*
*te Vect(X) for
the Grothendieck completion of Vect+(X). It is well-known that the complex K-th*
*eory K0(X)
is a Z-augmented -semiring and that there is a natural map Vect(X) -! K0(X) whi*
*ch is an
isomorphism whenever X is compact Hausdorff.
Remark 2.9. We will occasionally use the notation Z[A]+ = N[A] and Z[A]+d= Ad=d*
* N[A].
3. -(semi)ring schemes
The theory of -semirings is an instance of universal algebra: it is defined i*
*n terms of operations
! :Rk -!R with k = 0; 1 or 2, and identities between operations derived from th*
*ese. It is thus
formal to define the notion of a -semiring object in any category C with finite*
* products: such a
thing is an object R 2 C equipped with maps
0; 1:-1!R
+; x: R2-!R
k:R -! R for allk 2 N
making the evident diagrams commute. (Here the object 1 2 C is the terminal obj*
*ect.) Similar
remarks apply to -rings.
Next, suppose that C has arbitrary coproducts such that the natural map
a a a
Xix Yj -! Xix Xj
i;j i j
`
is always an isomorphism. We then have a product-preserving functor S 7! S_:= *
*s2S1 from sets
to C, so N_is a -semiring object in C. Similarly, Z_is a -ring object in C.
4 N. P. STRICKLAND
__
Example 3.1.Take C = hT_, the homotopy category of unbased CW-complexes. We ha*
*ve a
functor`Vect+(-) from hT opto the category of`-semirings, which is represented_*
*by the space
dBU(d). It follows_by_Yoneda's lemma that dBU(d) is a -semiring in hT . Simi*
*larly, the
functor K0(-) from hT opto the category of -rings is represented by the -ring s*
*pace Z x BU. `
Note that in this context the object N_is just the discrete space N and similar*
*ly for Z_, so d BU(d)
is augmented over N_and Z x BU is augmented over Z_.
Now let X be a formal scheme, and consider the category bXXof category of for*
*mal schemes
over X in the sense of [8]. For simplicity we will assume that X is solid, whi*
*ch means that
X = spf(OX ) for some formal ring OX . Let A be the category of discrete OX -al*
*gebras, and let
F be the category of functors from A to sets. The category bXXcan be regarded a*
*s a subcategory
of F (compare [8, Remark 2.1.5]), and the inclusion bXX-! F preserves products.
We will refer to -(semi)ring objects in XbXas -(semi)ring schemes (suppressin*
*g the words
"formal" and "over X" for brevity).
Let G be an ordinary formal group over X, in other words a commutative group *
*object in bXX
that is isomorphic in bXXto bA1X= spf(OX [[x]]). We can then define the schemes
Div+d(G)= Gd=d
a
Div+(G) = Div+d(G)
d2N
Div0(G)= lim-!Div+d(G)
d
Div(G) = Z_x Div0(G)
Divd(G)= {d} x Div0(G) Div(G):
More detailed definitions are given in [8, Section 5], where it is also explain*
*ed how these formal
schemes relate to the theory of divisors on G. In [8, Proposition 6.2.7] it is *
*observed that
1.Div+(G) is the free commutative monoid object in C generated by G.
2.Div(G) is the free commutative group object in C generated by G.
3.Div0(G) is the free commutative monoid object generated by G considered as *
*a based object
in C, which is the same as the free commutative group object generated by G*
* considered as
a based object in C.
Moreover, all these universal properties are stable under base change: if X0 is*
* a formal scheme
over X then Div+(G) xX X0 is the free commutative monoid in bXX0generated by G *
*xX X0 and
so on.
RecallQthat OG = OX [[x]] and thus OGd = OX [[x1; : :;:xd]]. IfQck denotes th*
*e coefficient of td-k
in i(t - xi) then ODiv+d(G)= OX [[c1; : :;:cd]] and ODiv+(G)= d0 OX [[c1; :*
* :;:cd]]. There are
also isomorphisms
ODiv0(G)= OX [[c1; c2; : :]:]
Y
ODiv(G)= OX [[c1; c2; : :]:]:
d2Z
Using these, one sees that Div+d(G) is a closed subscheme of Divd(G), and Div+(*
*G) is a closed
subscheme of Div(G).
If E is an even periodic ring spectrum, X = spec(ss0E) and G = spf(E0CP 1) th*
*en there are
natural isomorphisms
spf(E0BU(d))= Div+d(G)
spf(E0BU) = Div0(G)
spf(E0(Z x BU))= Div(G):
This is just a translation of well-known calculations; details are given in [8,*
* Section 8].
Proposition 3.2.Let G be an ordinary formal group over a scheme X. Then Div+(G)*
* has a
natural structure as a -semiring scheme, and Div(G) has a natural structure as *
*a -ring scheme.
Moreover, there is a canonical homomorphism dim:Div(G) -!Z_of -ring schemes, wh*
*ich sends
Divd(G) to d.
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 5
Proof.Recall that F is the category of functors from discrete OX -algebras to s*
*ets. Define R+; R 2
F by R+(A) = N[G(A)] and R(A) = Z[G(A)]. It is clear that R+ is a -semiring obj*
*ect in F,
and R is a -ring object.
There is an evident inclusion j :G = Div+1(G) -! Div+(G). As Div+(G) is a co*
*mmutative
monoid scheme, the set Div+(G)(A) is a commutative monoid for all A 2 A. As R+(*
*A) is the free
commutative monoid generated by the set G(A), there is a unique homomorphism OE*
*+ :R+(A) -!
Div+(G)(A) extending j. These maps are natural in A so we get a`map OE+ :R+ -!*
* Div+(G)
in F. If we interpret the colimits in bXXthen we have Div+(G) = dGd=d; this t*
*ranslates to
the statement that Div+(G) is the initial example of a formal scheme in XbXequi*
*pped with a
map R+ -! Div+(G) in F. By similar arguments, we find that Div(G) is the initia*
*l example of
a formal scheme over X with a map OE: R -! Div(G) in F. Moreover, one can chec*
*k that the
schemes Div+(G)k and Div(G)k enjoy the evident analogous universal properties f*
*or all k 0.
It now follows that there is a unique map x: Div+(G) x Div+(G) -! Div+(G) mak*
*ing the
following diagram commute:
x
R+ x R+ ____________R+w
| |
OE+xO|E+ |OE+
| |
|u |u
Div+(G) x Div+(G)_____Div+(G):wx
Similarly, all the other structure maps for the -semiring structure on R+ induc*
*e operations on
Div+(G), and one checks easily that this makes Div+(G) into a -semiring scheme.*
* A similar __
argument works for Div(G). It is clear that there is a map dim:Div(G) -!Z_as de*
*scribed. |__|
The above -semiring structure can be made more explicit as follows. Let cd;k2*
* OX [[x1; : :;:xd]]
be defined by
Yd Xd
(t - xi) = cd;ixd-i:
i=1 i=0
Let pd;e;k(cd;1; : :;:cd;d; c0e;1; : :;:c0e;e) be defined by
Yd Ye deX
(t - (xi+F x0j)) = pd;e;ktde-k:
i=1j=1 k=0
Suppose d; r 2 N and put N = (dr). Let qd;r;k(cN;1; : :;:cN;N) be defined by
Y FX NX
(t - xij) = qd;r;ktN-k ;
I j k=0
where the sum on the left runs over all lists I = (i1; : :;:ir) such that 1 i1*
* < : :<:ir d. Then
the multiplication map
x: Div+d(G) x Div+e(G) -!Div+de(G)
corresponds to the map
OX [[cde;1; : :;:cde;de]] -!OX [[cd;1; : :;:cd;d; c01;e; : :;:c*
*0e;e]]
(of formal OX -algebras) sending cde;kto pd;e;k. Similarly, the map correspondi*
*ng to r: Div+d(G) -!
Div+N(G) sends cN;kto qd;r;k.
4. Adams operations
We now recall the theory of Adams operations in -semirings; for a more detail*
*ed exposition
see [6], for example.
Let R be a -ring. For any a 2 R we can form the power series
X
t(a) = k(a)(-t)k 2 R[[t]]:
k0
This is equal to 1 mod t and thus is invertible in R[[t]]. It is easy to check *
*that t(0) = 1 and
t(a + b) = t(a)t(b).
6 N. P. STRICKLAND
We next define
t(a) = -t-t(a)-1d-t(a)=dt 2 R[[t]];
and let k(a) be the coefficient of tk in t(a). This defines an additive map *
*k: R -!R, called
the k'th Adams operation.
Now consider the case R = Z[A] for some Abelian group A. It is not hard to se*
*e that
X X X
k( ni[ai]) = ni[ai]k = ni[kai];
i i i
so k is just the map induced by the homomorphism k:1A :A -!A. Thus, if A is ac*
*tually a Z(p)-
module or a Zp-module, then there is a natural way to define k: Z[A] -! Z[A] f*
*or all k 2 Z(p)
or k 2 Zpas appropriate. Moreover, we see that k is a ring homomorphism which *
*preserves the
semiring Z[A]+ and the subsets Z[A]+d, and that k j = kjand kj = j k.
Now consider the -ring scheme Div(G). As our original definition of k is nat*
*ural, we evidently
get morphisms
k: Div(G) -!Div(G)
of schemes. It is well-known that G is actually a Zp-module scheme, or in terms*
* of our coordinate,
that one can define the series [k]F(x) in a sensible way for all k 2 Zp. This m*
*eans that each ring
Z[G(A)] admits Adams operations k for all k 2 Zp, with properties as above. Th*
*e argument of
Proposition 3.2 shows that
o We can define operations k on Div(G) for all k 2 Zp, extending the definit*
*ion given previ-
ously.
o These maps are maps of ring schemes, induced by the maps k :G -!G.
o We have j k = jkfor all j; k 2 Zp, and kj = j k for all k 2 Zpand j 2 N.
o The map k preserves Divd(G), Div+(G) and Div+d(G) = Gd=d for all d.
Lemma 4.1. For any discrete OX -algebra A, the group G(A) is a p-torsion group.
Proof.Our coordinate x gives an isomorphism x: G(A) -!bA1(A) = Nil(A) (the set *
*of nilpotents
in A). As p lies in the maximal ideal of E0 = OX and A is a discrete OX -algebr*
*a, we see that
pr = 0 2 A for somesr, and thus [pr](x) is divisiblesby x2 in A[[x]]. It follo*
*ws that [prs](x) is
divisible by x2 . For any a 2 G(A) we have x(a)2 = 0 for large s, so x(prsa) = *
*0 for_large_s, so
pm a = 0 for large m as required. *
* |__|
Lemma 4.2. Let A be a discrete OX -algebra, and suppose we have a divisor D 2 D*
*iv(G)(A).
Then k(D) = dim(D)[0] whenever the p-adic valuation vp(k) is sufficiently larg*
*e.
Proof.First suppose that D 2 Div+d(G)(A). We can thenPchoose a faithfully flat *
*map A -!A0such
that the image of D in Div+d(G)(A0) has the form di=1[ai]. The map Div+d(G)(A*
*)m-!Div+d(G)(A0)PP
is automatically injective, so it suffices to show that for large m we have p *
*( i[ai]) = i[pm ai] =
d[0], which is immediate from the previous lemma.
Now suppose that D 2 Divd(G)(A). We can then write D in the form D0- e[0] fo*
*r some
D02 Div+d+e(G)(A) and we reduce easily to the previous case.
Finally, consider a general divisor D 2 Divd(G)(A), which need not have const*
*antQdimension.
Instead, we have a splitting A = A1 x : :x:Ar giving a bijection Div(G)(A) = *
*iDiv(G)(Ai)
under which D becomes an r-tuple (D1; : :;:Dr) with Di 2 Div+di(G)(Ai) forQsome*
* integers di.
This means that dim(D) becomes (d1; : :;:dr) under the bijection Z_(A) = iZ_(*
*Ai). The cases
considered previously imply that
kD = ( kD1; : :;: kDr) = (d1[0]; : :;:dr[0]) = dim(D)[0]
when vp(k) 0, as required. *
* |___|
Now consider instead the -ring R(G). In this case we can define Adams operati*
*ons k for
k 2 N, and it is well-known that in terms of characters we have
O kV(g) = OV (gk):
As a virtual representation is determined by its character and dim(V ) = OV (1)*
*, it follows easily
that k is a degree-preserving map of -rings and that j k = jk. Moreover, if *
*e is the exponent
of G (in other words, least common multiple of the orders of the elements) then*
* k depends only
on the congruence class of k modulo e. If k is coprime to e (or equivalently, t*
*o |G|), it follows that
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 7
k j = 1 for some j, so k is an isomorphism. In this case the map g 7! gk is a*
* bijection, and
it follows easily that k preserves the usual inner product on R(G). A virtual *
*representation V
is an irreducible honest representation iff OV (1) > 0 and = 1, and it *
*follows that k sends
irreducibles to irreducibles and thus sends R+d(G) to R+d(G). (Compare [7, Exer*
*cise 9.4].)
However, if k is not coprime to e then k need not preserve R+(G). For exampl*
*e, take G = 3,
let ffl be the nontrivial one-dimensional representation, and let ae be the irr*
*educible two-dimensional
representation. We then have 2(ae) = ae + 1 - ffl 62 R+(G).
Lemma 4.3. Let pv be the p-part of the exponent of G. Then for any homomorphism*
*vf :R(G) -!
Div(G)(A) of -rings and any V 2 Rd(G) we have f(V ) 2 Divd(G) and p f(V ) = d[*
*0].
Proof.Let the exponent of G be e = pve0, where e0is0coprimevto p. The map e0:D*
*iv(G) -!
Div(G) is an isomorphism and fixes d[0], and e p = e so it suffices to show *
*that ef(V ) = d[0].
To see this note that ef(V ) = f( eV ) and O eV(g) = OV (ge) = OV (1) = d for *
*all g, so eV
is the trivial representation of rank d. As f is a ring map, we have f( eV ) = *
*f(d) = d[0], as
required. k k
This implies that p f(V ) = d[0] for k 0 but Lemma 4.2 says that p f(V ) =*
* dim(f(V ))[0]_
for k 0, so dim(f(V )) = d, so f(V ) 2 Divd(G) as claimed. *
* |__|
5. Chern approximations
Definition 5.1.Let G be a finite group, and let A be a discrete OX -algebra. We*
* define a functor
XCh(G) from discrete OX -algebras to sets by
XCh(G)(A) = { homomorphisms R+(G) -!Div+(G)(A) of -seimirings}:
We write C(E; G) for the ring OXCh(G)of natural transformations from XCh(G) to *
*the forgetful
functor A1. We also put X(G) = spf(E0BG). We refer to C(E; G) as the Chern appr*
*oximation
to E0BG, and to XCh(G) as the Chern approximation to X(G).
Remark 5.2. We say that a homomorphism f :R(G) -! Div(G)(A) of -rings is positi*
*ve if
f(R+(G)) Div+(G)(A). It is clear from Remark 2.7 that XCh(G)(A) bijects natura*
*lly with the
set of positive homomorphisms R(G) -!Div(G)(A), and we will implicitly use this*
* identification
where convenient. We also see from Lemma 4.3 that positive homomorphisms satisf*
*y f(R+d(G))
Div+d(G)(A).
Proposition 5.3.The functor XCh(G) is a formal scheme over X. The ring C(E; G) *
*= OXCh(G)
is a quotient of a formal power series ring in finitely many variables over OX *
*(and thus is a
complete Noetherian local ring).
Proof.Let V1; : :;:Vh be the irreducible representations of G, and let d1; : :;*
*:dh be their degrees.
We assume that these are ordered so that V1 is the trivial representation of ra*
*nk one. There are
then natural numbers mijkand lrijfor r 0 and 1 i; j; k h such that
M
Vi Vj ' mijk:Vk
Mk
rVi ' lrij:Vj
j
(Here m:W means the direct sum of m copies of W .)
To give a homomorphism f :R+(G) -!Div+(G)(A) is the same as to give divisors *
*Di= f(Vi) 2
Div+di(G) for i = 1; : :;:h such that
X
DiDj = mijkDk
ijk
X
rDi = lrijDj
j
Q h
This exhibits XCh(G)(A) as the equaliser of a pair of maps from i=1Divdi(G) to
Y Y
Divdidj(G) x Div(di)(G):
i;j r;i r
In particular, this is a pair of maps between formal schemes over X, so the equ*
*aliser is a formal
scheme over X.
8 N. P. STRICKLAND
More explicitly, we have XCh(G) = spf(C(E; G)), where C(E; G) is defined as f*
*ollows. We start
with OX and adjoin powerPseries variables cikfor i = 1; : :;:h and k = 1; : :;:*
*di, and put ci0= 1.
We then put fi(t) = dik=0ciktdi-kand impose the relations obtained by equatin*
*g coefficients in
the following identities between polynomials:
didjX Y
pdi;dj;a(ci*; cj*)tdidj-a= fk(t)mijk
a=0 k
(dir)X
di)-aY lr
qdi;r;a(ci*)t( r= fj(t) ij
a=0 j
The resulting quotient ring is C(E; G). *
* |___|
__
We next explain how to compare XCh(G) to X(G). Let G be the category whose ob*
*jects are
Lie groups, and whose morphisms are the conjugacy classes of continuous homomor*
*phisms. We
then have a natural map
a __ B __ a spf(E0(-))
R+(G) = G (G; U(d)) -! hT (BG; BU(d)) -------! bXX(X(G); Div+(G)):
d d
By taking adjoints, we obtain a map X(G) -!Map (R+(G); Div+(G)), and one checks*
* easily that
this actually lands in the subscheme XCh(G) Map(R+(G); Div+(G)) of -semiring h*
*omomor-
phisms. We thus have a natural map
G :X(G) -!XCh(G):
In terms of our explicit description of C(E; G), the map *:C(E; G) -!E0BG sends*
* cikto the
k'th Chern class of the representation Vi.
It is natural to ask whether a homomorphism f :R(G) -!Div(G)(A) of -rings is *
*automatically
positive. We next show that we always have f(R+1(G)) Div+1(G) ' G, but the cor*
*responding
claim for d > 1 seems to be false.
Proposition 5.4.If D 2 Div1(G)(A) and k(D) = 0 for all k > 1 then D 2 Div+1(G)(*
*A).
Proof.We can write D = E - e[0] for some e 0 and E 2 Div+e+1(G)(A). Put D0= e+*
*1E 2
Div+1(G). We have E = D + e[0] so
X
D0= i(D)j(e[0]) = 1(D)e(e[0]) = D;
i+j=e+1
so D 2 Div+1(G) as claimed. *
* |___|
Corollary 5.5.If L 2 R+1(G) and f :R(G) -! Div(G)(A) is a map of -rings then f(*
*L) 2
Div+1(G).
Proof.Clearly kL = 0 for k > 1 so kf(L) = 0 for k > 1, and f(L) 2 Div1(G)(A) by*
* Lemma_4.3
so f(L) 2 Div+1(G) by the proposition. *
* |__|
Proposition 5.6.For suitable formal groups G and rings A, there exist divisors *
*D 2 Div2(G)(A)
such that kD = 0 for k > 2 but D 62 Div+2(G).
Proof.We will assume that OX = F2, so p = 2. Suppose that a; b 2 G(A) and 2a = *
*2b = 0. Put
c = a + b so 2a = 2b = 2c = a + b + c = 0, and put E = [a] + [b] + [c] and D = *
*E - [0]. Then
2E = [a+b]+[b+c]+[c+a] = [c]+[a]+[b] = E and 3E = [0] = 1 so t(E) = 1+tE +t2E +*
*t3 =
(1 + tD + t2)(1 + t), so t(D) = 1 + tD + t2. Thus kD = 0 for k > 2. If D is in *
*Div+2(G)(A) we
must have x(a)x(b)x(c) = c3(E) = c3(D + [0]) = 0. Note also that x(c) = x(a - b*
*) = x(a) -F x(b),
which is a unit multiple of x(a)-x(b), so the condition is equivalent tonx(a)2x*
*(b)n= x(a)x(b)2. The
universal example for A is OX [[y; z]]=([2](y); [2](z)) = F2[y; z]=(y2 ; z2 ) (*
*where y = x(a);_z = x(b)).
Clearly in this case we have y2z 6= yz2 so D 62 Div+2(G). *
* |__|
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 9
6. The group 3
In this section we work through the case where G = 3and E is the 2-periodic v*
*ersion of Morava
K-theory at the prime 3 with height 2. Many constructions discussed here will b*
*e generalised later.
Recall that the coefficient ring is E* = F3[u; u-1], where |u| = -2.
We have a coordinate x on G such that
x(-a)= [-1](x(a)) = -x(a)
x(3a)= [3](x(a)) = x(a)9
x(a + b)= x(a) +F x(b) = x(a) + x(b) (mod x(a)3x(b)3)
for all a; b 2 G. (The first equation isPtrue because the formal group law F as*
*sociated to E has an
integral lift whose logarithm logF(x) = kx9k=3k satisfies logF(-x) = - logF(x*
*). The second is
well-known, and the third follows from [9, Lemma 80].)
Define y; z :Div+2(G) -!A1 by y([a] + [b]) = x(a)x(b) and
z([a] + [b]) = x(2([a] + [b])) = x(a + b) = x(a) +F x(b)
(which is a unit multiple of x(a) + x(b)). One checks that that ODiv+2(G)= F3[*
*[y; z]]. If we let
Z = SDiv+2(G) be the scheme of divisors D 2 Div+2(G) such that 2(D) = [0] then *
*it follows that
OZ = F3[[y; z]]=z = F3[[y]]. There is an evident map ffi :G -! Z defined by ffi*
*(b) = [b] + [-b], and
y(ffi(b)) = x(b)x(-b) = -x(b)2 so the map ffi*:F3[[y]] -! F3[[x]] sends y to -x*
*2. In particular, we
see that ffi is finite and faithfully flat, with degree two.
Next, note that
ffi(b)2 = [2b] + [-2b] + 2[0] = 2(ffi(b)) + 2[0];
as ffi is faithfully flat, it follows that D2 = 2D + 2[0] for any D 2 Z.
Let Y be the scheme of divisors D 2 Z such that 2(D) = D. To analyse this, n*
*ote that
x(2b) = x(-b + 3b) = [-1](x(b)) +F [3](x(b)) = -x(b) + x(b)9 (mod x(b)1*
*2);
so
-x(2b)2 = -x(b)2- x(b)10 (mod x(b)12);
or in other words
y( 2ffi(b)) = y(ffi(b)) - y(ffi(b))5 (mod y(ffi(b))6):
As ffi is faithfully flat, we deduce that
y( 2(D)) = y(D) - y(D)5 (mod y(D)6)
for all D 2 Z. It follows that ( 2)*y - y is a unit multiple of y5 in F3[[y]] a*
*nd thus that OY =
F3[y]=y5.
The character table of G = 3 is
________________
|____|1_|ffl_o|e_3|
|_1__|1_|1__|2__|
|_1:21||-1_|_0__|
|__3_1||_1__|-1_|
From this we see that
R(G) = Z[ffl; oe]=(ffl2- 1; ffloe - oe; oe2 - oe - 1 - ffl):
The only interesting -operation is that 2(oe) = ffl.
Let f :R+(G) -! Div+(G)(A) be a -semiring homomorphism, in other words a poin*
*t of
XCh(G). As Div+1(G) ' G, there is a unique point a 2 G such that f(ffl) = [a]. *
*We also write
D = f(oe) 2 Div+2(G). As f is a map of -semirings, these satisfy
[2a]= [a]2 = f(ffl2) = f(1) = [0]
[a]D= f(ffloe) = f(oe) = D
D2 = f(oe2) = f(oe + 1 + ffl) = D + [0] + [a]
2D = f(2oe) = f(ffl) = [a]:
10 N. P. STRICKLAND
As we work mod 3, the map 2: G -!G is an isomorphism so the first equation give*
*s a = 0, so the
second equation is automatic and the last equation says that D 2 Z. Thus, the t*
*hird equation
becomes
2D + 2[0] = D2 = D + 2[0]:
The semiring Div+(G) embeds in the ring Div(G) so we can cancel to see that 2(*
*D) = D, so
D 2 Y . We can thus define a map O: XCh(G) -! Y by O(f) = f(oe). One can check *
*that the
whole argument is reversible, so O is an isomorphism and
C(E; G) = OXCh(G)' OY = F3[y]=y5:
We also have a short exact sequence C3 -! G -! C2 leading to an Atiyah-Hirzeb*
*ruch-Serre
spectral sequence
Hp(C2; EqBC3) =) Ep+qBG:
We have E*BC3 = F3[u1 ][x]=x9 (where |u| = 2 and |x| = 0), and C2 acts on this *
*by u 7! u and
x 7! [-1](x) = -x. Because C2 has order coprime to 3 we see that the spectral s*
*equence collapses
to an isomorphism
E*BG = (E*BC3)C2 = E*[y]=y5;
where y = -x2. After some comparison of definitions we see that the map G :X(G)*
* -!XCh(G)
is an isomorphism.
7.The Abelian case
Theorem 7.1.If G is Abelian then G :X(G) -!XCh(G) is an isomorphism.
Proof.Put G* = Hom (G; S1), so R+(G) = N[G*], and let A be an OX -algebra. If f*
* :N[G*] -!
Div+(G)(A) is a -semiring homomorphism, then f induces a group homomorphism f0:*
*G* =
R+1(G) -! G(A) = Div+1(G)(A). Conversely, given a group homomorphism f0:G* -!G(*
*A) we
get a map R+(G) = N[G*] -! N[G(A)] of -semirings. We can compose this with the*
* map
N[G(A)] -! Div+(G)(A) in the proof of Proposition 3.2 to get a map R+(G) -! Div*
*+(G)(A),
or in other words a point of XCh(G)(A). One checks that these constructions gi*
*ve a bijection
Hom(G*; G(A)) ' XCh(G)(A), or equivalently an isomorphism Hom (G*; G) ' XCh(G).*
* There is
also an isomorphism X(G) ' Hom (G*; G) (see [4, Proposition 2.9]), so we have a*
*n isomorphism
X(G) ' XCh(G). A straightforward comparison of definitions shows that this is_*
*the same as
G. |__|
8. The height one case
In this section we choose a prime p and let E be the p-complete complex K-the*
*ory spectrum.
We thus have X = spf(Zp), so discrete OX -algebras are just p-torsion rings. We*
* also have G = bGm,
so
G(A) = {u 2 Ax | 1 - u is nilpotent}:
Theorem 8.1.If E is the p-adic complex K-theory spectrum and G is a p-group the*
*n the map
G :X(G) -!XCh(G) is an isomorphism.
The rest of this section constitutes the proof. We fix a p-group G and write *
* = G for brevity.
The first ingredient is the Atiyah-Segal completion theorem. In the case of a*
* p-group, this says
that
OX(G)= E0BG = Zp R(G):
We know that Div(bGm) is the free ring scheme generated by the group scheme b*
*Gm. Recall that
the affine line A1 is just the forgetful functor from p-torsion rings to sets. *
*This is a ring scheme in
a natural way, and it contains bGmas a subgroup of its group of units. We thus *
*have a ring map
:Div(bGm) -!A1
P P
extending the inclusion of bGmin A1. If D = ini[ui] 2 Div(bGm)(A) then (D) = *
* iniui 2
A1(A) = A.
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 11
Next recall that Div+d(bGm) =_bGdm=d,_so to describe a function f :Div+d(bGm)*
* -!A1 it suffices
to give the symmetric function f:bGdm-!A1 such that
Xd __
f( [ui]) = f(u1; : :;:ud):
i=1
Let oej:Ad -!A1 be the j'th elementary symmetric function and define
X
cj( [ui])= oej(1 - u1; : :;:1 - ud)
Xi
aj( [ui])= oej(u1; : :;:ud)
iX X
a0j( [ui])= aj( [ui]) - dj:
i i
Recall that ODiv+d(bGm)is the set of all maps Div+d(bGm) -! A1, so cj, aj and a*
*0jcan be viewed
as elements of this ring. The function u 7! 1 - u is a coordinate on the formal*
* group bGmso a
well-known argument gives an isomorphism
ODiv+d(bGm)= Zp[[c1; : :;:cd]]:
It is not hard to deduce that
ODiv+d(bGm)= Zp[[a01; : :;:a0d]];
which is the completion of the ring Zp[a1; : :;:ad] at the ideal generated by t*
*he elements a0i=
ai- (di). Note also that a1(D) = (D) for D 2 Div+d(bGm).
Lemma 8.2. For any divisor D 2 Div+d(bGm) we have aj(D) = (j(D)).
P
Proof.We may assume that D = i[ui] forQsome elements u1; : :;:udP2 bGm. For *
*any I
{1; : :;:d} with |I| = j we put uI = i2Iui. We then have jD = I[uI] and thus
X
j(D) = uI = oej(u1; : :;:ud) = aj(D):
I
|___|
Let V be a complex vector bundle over a space Z with associated projective bu*
*ndleLP V , and
let D(V ) = spf(E0P V ) be the corresponding divisor on G over spf(E0Z). If V =*
* di=1Li for
some complex line bundles Lithen eachPLican be regarded as an element of E0Z = *
*K0(Z; Zp),
and one sees easily that D(V ) = i[Li] so aj(D(V )) = oej(L1; : :;:Ld) = j(V *
*). By the splitting
principle we see that
j(D(V )) = aj(D(V )) = j(V )
even when V does not split as a sum of line bundles.
Now consider the case Z = BG and suppose that V comes from a representation o*
*f G, which
we also call V . Suppose we have a point x 2 X(G)(A) for some p-torsion ring A,*
* corresponding
to a ring map ^x:E0BG -!A. One can now check from the definitions that (x)(V ) *
*= ^x*(D(V )),
so f((x)(V )) = ^x(f(D(V ))) for any f 2 ODiv+d(bGm). In particular, we have
aj((x)(V )) = ^x(aj(D(V ))) = ^x(j(V )):
We can now construct the map i :XCh(G) -!X(G) that will turn out to be invers*
*e to . Let
A be a p-torsion ring. A positive homomorphism f 2 XCh(G)(A) gives rise to a ho*
*momorphism
A O f :R(G) -! A1(A) = A, which factors canonically through Zp R(G) = E0BG = O*
*X(G)
because A is a p-torsion ring. This gives a continuous homomorphism OX(G) -!A, *
*or in other
words a point of X(G)(A), which we call i(f). This construction gives a natural*
* map i :XCh(G) -!
X(G), as required.
Suppose we start with a point x 2 X(G)(A), corresponding to a ring map ^x:Zp *
*R(G) =
E0BG -!A. We need to check that i(x) = x, or equivalently that A((x)(V )) = ^x(*
*V ) for all
V 2 Zp R(G), and it will suffice to do this for all honest representations V 2 *
*R+d(G) for all d.
In that context we have = a1 so
A((x)(V )) = a1((x)(V )) = ^x(1(V )) = ^x(V )
12 N. P. STRICKLAND
as required. Thus i = 1X(G).
Suppose instead that we start with a point f 2 XCh(G)(A), in other words a po*
*sitive homomor-
phism f :R+(G) -!Div+(bGm)(A). We need to check that ((i(f)))(V ) = f(V ) 2 Div*
*+d(bGm)(A)
for all V 2 R+d(G). To see this, note that
aj((i(f))(V ))= di(f)(j(V ))
= (f(j(V )))
= (j(f(V )))
= aj(f(V )):
It follows that a0j((i(f))(V )) = a0j(f(V )) and the functions a0jgenerate ODiv*
*+d(bGm)so (i(f))(V ) =
f(V ), as required. This shows that i = 1XCh(G), so is an isomorphism as claim*
*ed.
9.Reduction to the Sylow subgroup
By a well-known transfer argument, if P is a Sylow p-subgroup of G then the r*
*estriction map
E0BG -!E0BP is injective, and similar methods give some control over the image.*
* In this section
we develop some analogous results for the approximation C(E; G) to E0BG. Let I *
*be the kernel
of the restriction map R(G) -!R(P ) (which is independent of the choice of P ).*
* Note that R(G)=I
is isomorphic to the image of the restriction map, so it is a free Abelian grou*
*p of finite rank and
it inherits a -ring structure.
Proposition 9.1.Any -ring homomorphism f :R(G) -!Div(G)(A) factors through R(G)*
*=I.
Proof.As usual, we let the exponent of G be e = pve0, where e0is coprime to p. *
*If V 2 I and g 2 G
has p-power order then g is conjugate to an element of P and thus0OV (g) = 0. I*
*f g is0an arbitrary
element of g then0the order of0g divides pve0so the order of ge divides pv, so *
*OV (ge0) = 0. This
proves that e (V ) = 0, so e (f(V )) = 0 in Div(G)(A). However, the action of*
* e on Div(G)(A)
is induced by the action of e0on G, which is invertible because e0is coprime to*
* p._This_implies
that f(V ) = 0 as claimed. *
* |__|
Corollary 9.2.If |G| is prime to p then XCh(G) = X(G) = X. *
* |___|
Corollary 9.3.XCh(G) is a closed subscheme of the scheme of -ring maps R(G)=I_-*
*!Div(G).
|__|
Remark 9.4. The disadvantage of this point of view is that the positivity condi*
*tions f(R+(G))
Div+(G) become less visible when we work with R(G)=I. However, the situation si*
*mplifies again
if we assume that P is normal in G. In that case, it is known that the map R+(G*
*) -!R+(P )G is
surjective; this was first proved by Gallagher [2], and we will give an alterna*
*tive proof in Section 17.
The sums of G-orbits of irreducibles in R+(P ) give a canonical system of gener*
*ators for R+(P )G,
which we can lift to get representations ae1;P: :;:aetof G say. Let dibe the de*
*gree ofPaei. If ae 2 R+d(G)
then resGP(ae)P2 R+d(P )G so resGP(ae) ' imiresGP(aei) for some integers mi 0*
* withP imidi= d,
so ae- imiaei2 I. Thus, for any map f :R(G) -!Div(G) of -rings we have f(ae) =*
* imif(aei),
so if f(aei) 2 Div+di(G) for all i then f(ae) 2 Div+d(G). Using this, we see th*
*at XCh(G) is the scheme
of -ring maps R(G)=I -!Div(G) such that f(aei) 2 Div+di(G) for i = 1; : :;:t.
We next give two results that help us to understand R(G)=I without computing *
*R(G).
Proposition 9.5.Let h be the number of conjugacy classes of elements g 2 G whos*
*e order is a
power of p. Then R(G)=I ' Zh as Abelian groups.
Proof.We already know that R(G)=I is a free Abelian group, so we just need to d*
*etermine its
rank, so it is enough to show that C R(G)=I ' Ch. Let C be the set of conjuga*
*cy classes
of p-power order, and let C0 be the set of all other conjugacy classes. Let U(*
*G) be the ideal
of virtual representations V 2 R(G) whose character is zero on C0. It is well-*
*known that C
R(G) ' F (C q C0; C). This isomorphism carries U(G) to F (C; C) and I to F (C0;*
* C) so the map
C U(G) -!C R(G)=I is an isomorphism. Clearly dimCC U(G) = |C| = h, and the c*
*laim_
follows. |*
*__|
Remark 9.6. The proof shows that U(G) is a subgroup of finite index in R(G)=I. *
*This index
need neither be a power of p nor coprime to p, as one sees by taking G = 3 and *
*p = 2 or 3; the
index is 2 in both cases.
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 13
Proposition 9.7.There is a natural isomorphism Zp R(G)=I = KU0p(BG) (where KUp *
*is the
p-adic completion of the complex K-theory spectrum.)
Proof.Let KUG be the usual G-spectrum for equivariant K-theory, and let LG be i*
*ts p-completion.
It is well-known that KU0G(S0) = R(G), which is free Abelian of finite rank, an*
*d it follows that
L0G(S0) = R(G)^p= R(G) Zp. We give this ring and all its quotients the p-adic*
* topology,
or equivalently the profinite topology, which is compact. The argument of the *
*Atiyah-Segal
completion theorem shows that KU0pBG = L0EG is the completion of R(G)^pat the a*
*ugmen-
tation ideal JG. By a compactnessTargument, we deduce that the map R(G)^p-! KU*
*0pBG is
surjective; the kernel is J1G := kJkG. Now let P be a Sylow p-subgroup, so b*
*y the same ar-
guments KU0pBP = R(P )^p=J1P. It is well-known that JNP pR(P )^pfor N 0 (us*
*e the
fact that xp - p(x) 2 pR(P ) for all x 2 R(P ), for example) and it follows th*
*at J1P = 0, so
KUp0BP = R(P )^p. We now have a diagram as follows.
J1G ________J1P=w0
v |
| |
| |
|u |u
I^pv________R(G)^pw______R(Pw)^p
| |
| |'
| |
|uu |u
KU0pBG v____KU0pBP:_w
We have seen that the columns are short exact. As Zp is flat over Z and I, R(G*
*) and R(P )
are finitely generated Abelian groups, we see that the middle row is left exact*
*. The bottom
horizontal map is injective by a transfer argument. By a diagram chase we deduc*
*e that I^p= J1G,
so KU0pBG = R(G)^p=I^p= (R(G)=I) Zpas claimed. *
* |___|
10.Finiteness
It is a fundamental fact that the scheme X(G) is finite over X, or equivalent*
*ly that E0BG is
a finitely generated module over E0. This is proved in the present generality a*
*s [4, Corollary 4.4];
the argument is the same as in [5]. In this section we show that XCh(G) is also*
* finite over X.
We also study some auxiliary schemes that come up in the proof, as they turn ou*
*t to be useful in
specific computations.
Definition 10.1.For any v 0 we put
G(v) = ker(pv:G -!G):
In terms of our coordinate, we have G(v) = spf(E0[[x]]=[pv](x)). Next, recall t*
*hat G(v) G is
a divisor of degree pnv, where n is the height of G. For any m 0 we let G(v; m*
*) be m times
this divisor, considered as a subscheme of G; in other words G(v; m) = spf(E0[[*
*x]]=[pv](x)m ). For
any formal scheme Y over X such that OY is a free module of finite rank r over *
*OX , we define
Y d=d to be spfof the d'th symmetric tensorQpower of OYPover OX . If {e1; : :;:*
*er} is a basis
for OY over OX then the monomials eff:= ri=1effiiwith iffi= d form a basis f*
*or OY d=d, so
this ring is free of rank r+d-1dover OX . We will use this construction in the*
* cases Y = G(v)
andvY = G(v; m). Finally, we define Z(v; d) to be the scheme of divisors D 2 Di*
*v+d(G) such that
p D = d[0].
Theorem 10.2. The scheme Z(d; v) is finite and flat over X, of degree pndv. The*
*re are closed
inclusions
G(v)d=d -!Z(v; d) -!G(v; d)d=d -!Div+d(G):
The first two of these are infinitesimal thickenings, in other words the corres*
*ponding maps of rings
are surjective with nilpotent kernel. If OX is a field (necessarily of characte*
*ristic p) then Z(v; d)
is the fibre of the nv-fold relative Frobenius map
FDnviv+d(G)=X:Div+d(G) -!Div+d((FXnv)*G);
so
nv
OZ(v;d)= OX [[c1; : :;:cd]]=(cpi ):
14 N. P. STRICKLAND
Proof.First suppose that OX is a complete regular local ring. (In the topologic*
*al context, this
occurs when E is Landweber exact.) Consider the following diagram:
Z(v; d)v____Div+d(G)_wusu__Gd__s
| |
| pv| |pv
| |u |u
|u |u |u
X _______Div+d(G)wiusu__Gd__s
In the right hand square, all the corresponding rings are complete regular loca*
*l rings. A finite
injective map of such rings always makes the target into a free module over the*
* source [1, 2.2.7
and 2.2.11].vThe maps ss* and (pv)* are finite injective maps of degrees d! and*
* pndv. It follows v
that ( p )* is finite and injective, and thus (as deg(fg) = deg(f) deg(g) in th*
*is context) that p
is flat of degree pndv. The left hand square is a pullback by definition, and i*
*t follows that Z(v; d)
is flat of degree pndvover X. Using [4, Proposition 5.2], it is not hard to ded*
*uce that this result
remains true even if OX is not regular.
Next, let x be a coordinate on G. Then {xi | i < pnv} is a basis for OG(v)ov*
*er OX , and
{xi| i < pnmv} is a basis for OG(v;m). Using this we obtain bases for the rings
A = OG(v)d= OX [[x1; : :;:xd]]=([pv](xi))
and
A0= OG(v;m)d= OX [[x1; : :;:xd]]=([pv](xi)m )
that are permuted by d, and the orbit sums give bases for the rings
B = OG(v)d=d= Ad
and
B0= OG(v;m)=d= (A0)d :
Using these, it is easy to see that the map B0 -! B is surjective, so the map G*
*(v)d=d -!
G(v; m)d=d is a closed inclusion. A similar argument shows that G(v; m)d=d is a*
* closed sub-
scheme of Div+d(G).
Next, put
J = ker(A0-! A) = ([pv](x1); : :;:[pv](xd)):
This is clearly a nilpotent ideal, and ker(B0-! B) = B0\ J which is a nilpotent*
* ideal in B0. Thus
our map G(v)d=d -!G(v; m)d=d is an infinitesimal thickening.
It is clear that G(v)d=d is contained in Z(v; d). Next, let W (v; d) be the p*
*reimagePof Z(v; d)
in Gd, or equivalently the scheme of d-tuples a_= (a1; : :;:ad) 2 Gd such that *
* i[pvai] = d[0]. If
a_2 W (v; d) then for each iPwe have pvai2 d[0] so ai2 d:(pv)-1[0] = d:G(v) = G*
*(v; d). This means
that a_2 G(v; d)d and thus i[ai] 2 G(v; d)d=d, so the map W (v; d) -ss!Z(d; v*
*) -! Divd(G)
factors through G(v; d)d=d. As ss is faithfully flat, it follows that Z(v; d) *
* G(v; d)d=d as
claimed.
We now have maps
G(v)d=d i-!Z(v; d) j-!G(v; d)d=d k-!Div+d(G):
We know that ji, k and kj are closed inclusions and that ji is an infinitesimal*
* thickening. It
follows easily that i, j and k are closed inclusions and i and j are infinitesi*
*mal thickenings.
Now suppose that X is a field of characteristicnp.vWe then have an iterated F*
*robenius map
FXnv:X -!X corresponding to the ring map a 7! ap and thus a formal group G0= (*
*FXnv)*G over
X. The map FGnvgives rise to a map f = FGnv=X:G -!G0. As G has height n, the ma*
*p pv:G -!G
factors as G f-!G0-g!G, wherengvis an isomorphism. This is just the geometric s*
*tatement of the
fact that [pv](x) = fl(xp ) for some invertible power series fl. By definition*
* Z(v; d) is the fibre
of the map Div+d(G) -!Div+d(G) induced by pv:G -!G, and it follows easily that *
*it is also the
fibre of the map Div+d(G) -! Div+d(G0) induced by f. It is easy to identify th*
*is with the map
FDnviv+. If we use the usual generators for the coordinate rings of Div+d(G) an*
*d Div+d(G0) then
d(G) nv *
*nv __
the corresponding ring map sends ck to cpk , so OZ(v;d)= OX [[c1; : :;:cd]]=(cp*
*k ). |__|
Corollary 10.3.The scheme XCh(G) is finite over X.
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 15
Proof.Let V1; : :;:Vh be the irreducible representations of G, and let d1; : :;*
*:dhQbe their degrees.
As in the proof of Proposition 5.3, we see that XCh(G) is a closed subscheme of*
* iDiv+di(G).
Now let pv beQthe p-part of the exponent of G. We see from Lemma 4.3 that XCh(G*
*) is actually
contained in iZ(v; di), which is finite over X by the theorem. It follows tha*
*t XCh(G)_itself is
finite, as claimed. *
* |__|
Corollary 10.4.For any divisor D 2 Div(G)(A) there exist w 0 such that k(D) =*
* l(D)
whenever k; l 2 Zpwith k = l (mod pw).
Proof.Wevcan reduce easily to the case where D 2 Div+d(G) for some d 0. Lemma *
*4.3 tells us
that p D = d[0] for some v, so D 2 Z(v; d)(A) G(v; d)d=d(A). Next note that A*
* is a discrete
OX -algebra so p is nilpotent in A, say pr = 0. It followssthat [pr](x) = f(xp)*
* for some power series
f with f(0) = 0, and thus that [prs](x) is divisible by xp . Thus, for large u *
*we have [pu](x) = 0
(mod xd), so [pu+v](x) = 0 (mod [pv](x)d), so G(v; d) G(u+v). If we put w = u+*
*v this tells us
that D 2 G(w)d=d, and the action of k on G(w)d=d clearly depends only on the c*
*ongruence_
class of k mod pw, as required. *
* |__|
11.Generalised character theory
In [5], Hopkins, Kuhn and Ravenel describe Q E0BG in terms of "generalised c*
*haracters".
In this section we will give an analogous but less precise description of Q C(*
*E; G).
To explain the HKR theory, write = (Qp=Zp)n and * = Hom (; Qp=Zp) = Znp. (El*
*sewhere
these are denoted by and *, but there are enough 's in this paper already.) L*
*et (v) be
the subgroupPof killed by pv, and let Level(v; G) be the scheme of maps OE: (v*
*) -! G such
that a2(v)[OE(a)] G(v) in Div+(G). See [9] for more information about these *
*schemes. Put
Dm = OLevel(m;G)and D = lim-!Dm and L = Q D. This is a free module of countabl*
*e rank
m
over Q OX . If G is a universal deformation (as in the case considered by HKR)*
* then it can
be described more explicitly: the Weierstrass preparation theorem implies that *
*[pv](x) is a unit
multiple of a monic polynomial gv(x) of degree pnv, and L is obtained from Q O*
*X by adjoining
full set of roots for gv(x) for all v.
Now let (G) be the set of G-conjugacy classes of homomorphisms * -!G, and let*
* F ((G); L)
be the ring of all functions u: (G) -!L (with pointwise operations). HKR constr*
*uct an isomor-
phism
o :L OX E0BG -!F ((G); L):
They work with a particular admissible cohomology theory E, but it is not hard *
*to extend their
result to all admissible theories; see`[4, Proposition 5.2 and Appendix B] for *
*some pointers.
Now consider the -semiring N[] = dd=d and the -ring Z[]. If we give * its *
*p-
adic topology then every subgroup of finite index is open and any continuous ho*
*momorphism
* -! GLn(C) factors through a finite quotient of *. Using this we can identify*
* N[] with
the semiring of continuous representations of *, and Z[] with the corresponding*
* ring of virtual
representations.
Definition 11.1.We say that a -ring homomorphism f :R(G) -!Z[] is positive if f*
*(R+(G))
Z[]+. We write Ch(G) for the set of -semiring homomorphisms R+(G) -!Z[]+, or eq*
*uiva-
lently the set of positive -ring homomorphisms R(G) -!Z[].
Remark 11.2. The arguments of Lemma 4.3, Corollary 5.5 and Proposition 9.1 show*
* that any
positive homomorphism f :R(G) -!Z[] of -rings automatically sends R+d(G) to Z[]*
*+dand I
to 0 (where I is the kernel of the restriction map to a Sylow subgroup). There *
*is also an evident
analogue of Remark 9.4 for Ch(G).
Remark 11.3. From the definitions we know that the -operations determine the Ad*
*ams op-
erations. Conversely, it is well-known and easy to check that the Adams operat*
*ions determine
the -operations rationally. As Z[] is torsion-free, it follows that a ring map *
*f :R(G) -! Z[]
preserves the -operations iff it preserves the Adams operations. The correspond*
*ing statement
for homomorphisms R(G) -!Div(G) is false, however.
16 N. P. STRICKLAND
Theorem 11.4. There are natural maps : (G) -!Ch(G) and oCh:LC(E; G) -!LE0BG
making the following diagram commute:
*
L C(E; G) ______Lw1E0BG
oCh| |o
| |
|u |u
F (Ch(G); L)____wF*((G); L):
(Here the tensor products are taken over E0.) Moreover, the map oCh is surjecti*
*ve with nilpotent
kernel.
Proof.For brevity we will write C(L; G) = L C(E; G) and L0BG = L E0BG. This i*
*s a slight
abuse because these functors do not arise from a spectrum L. We also let v be a*
*ny integer greater
than or equal to the p-adic valuation of the exponent of G.
Any homomorphism u: * -! G factors through *=pv = (v)* and thus is automatica*
*lly
continuous (for the discrete topology on G). It thus gives a positive homomorph*
*ism u*:R(G) -!
Z[], and it is well-known that this depends only on the conjugacy class of u, s*
*o this construction
gives a natural map : (G) -!Ch(G).
It is easy to see using Adams operations that any positive homomorphism f :R(*
*G) -! Z[]
actually lands in the subring Z[(v)]. Suppose we have a level structure OE: (v*
*) -! G(A).
As (v) is a finite Abelian group, this gives rise as in Theorem 7.1 to a positi*
*ve homomorphism
R(*=pv) = Z[(v)] -!Div(G)(A), which we can compose with f to get a positive hom*
*omorphism
R(G) -! Div(G)(A), or in other words a point of XCh(G)(A), which we call aeCh(f*
*; OE). This
construction produces a map aeCh: Ch(G) x Level(v; G) -! XCh(G) of formal schem*
*es over X,
corresponding to a map ae*Ch:C(E; G) -!F (Ch(G); Dv) F (Ch(G); L). After tenso*
*ring by L
we obtain the required map oCh:C(L; G) -!F (Ch(G); L).
We next recall the definition of o. Suppose that u: *=pv -! G and OE 2 Level*
*(v; G)
Hom((v); G) = X(*=pv). We then have a point ae(u; OE) := X(u)(OE) 2 X(G). This *
*construction
gives a map ae: (G) x Level(v; G) -! X(G) and thus a map ae*:E0BG -! F ((G); Dv)
F ((G); L). After tensoring by L we obtain the required map o.
One can check from the definitions that the following diagram commutes:
XCh(G)uu________________X(G)_u
| |
aeC|h |ae
| |
| |
Ch(G) x Level(m; G)u____(G)_xxLevel(m;1G):
It follows easily that the diagram in the statement of the theorem commutes.
To understand oCh more explicitly, let OE 2 Level(v; G)(Dv) be the universal *
*example of a level
structure. For any element a 2 (v) we then have a point OE(a) 2 G(Dv) andPthus *
*an element
xa := x(OE(a)) 2 Dv. These elements satisfy xa+b = xa +FQxb and G(v) = a2(v)*
*[OE(a)] as
divisors, or equivalently [pv](t) is a unit multiple of a(t - xa) in Dv[[t]].*
* It is also known that
xaQ- xb is invertible in L whenever a 6= b (because it is a unit multiple of xa*
*-b, which divides
c6=0xc = [pv]0(0) = pv). Let the representations Viand the elements cik2 C(E; *
*G) be as in
the proof of Proposition 5.3. If f 2 Ch(G) and f(Vi) = [a1]+: :+:[ad] 2 N[(v)] *
*then oCh(cik)(f)
is the k'th symmetric function in the variables xa1; : :;:xad, and this charact*
*erises oCh.
We next show that oCh is surjective. For any u 2 Ch(G) we define fflu: C(L; *
*G) -! L by
fflu(c) = oCh(c)(u), and we put Iu = ker(fflu) C C(L; G). By the Chinese Remain*
*der Theorem, it
will suffice to show that Iu + Iv = C(L; G)Pwhenever u 6= v. If uP6= v we can c*
*hoose V 2 R+(G)
such that u(V ) 6= v(V ) 2 N[]. If u(V ) = ama[a] and v(V ) = ana[a] then w*
*e must have
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 17
mb6= nb for some b, and without loss we may assume mb> nb. Define
ka = min(na; ma)
X
C = ka[a]
Xa
A = (na - ka)[a]
Xa
B = (ma - ka)[a]:
a
*
* Q
We can write A in the form [a1]+: :+:[ad] with ai6= b for all i. We also write *
*fA(t) = i(t-xai),
so fA(xb) is invertible in L. On the other hand, the representation V 2 R+d(G) *
*gives rise in a
tautological way to a divisor DV 2 Div+d(G)(C(E; G)) with equation fDV (t) 2 C(*
*E; G)[t], say.
We have fflufDV (t) = fA+C (t) = fA(t)fC(t) and fflvfDV (t) = fB(t)fC(t). The p*
*olynomial fC(t)
is monic and thus is not a zero-divisor, so fA(t) = fB(t) (mod Iu + Iv). We ev*
*idently have
fB(xb) = 0 so fA(xb) = 0 (mod Iu + Iv). As fA(xb) is invertible in L, we deduce*
* that Iu + Iv = 1
as required.
Finally, we must show that the kernel of oCh is nilpotent. This kernel is th*
*e intersection of
the ideals Iu, so by well-known arguments it suffices to show that every prime *
*ideal in C(L; G)
contains Iu for some u. To see this, put R = C(L;vG) and let p C R be a prime *
*ideal. If
D 2 Div+d(G)(C(E; G)) is a divisor satisfying p (D) = d[0], then Theorem 10.2 *
*implies that
D 2 G(v; d)d=d and thus that D d2:G(v)Qas divisors, orvequivalently fD (t) div*
*ides [pv](t)d2,
which is a unit multiple in Dv[t] of a2(v)(t - xa)p . Now let K be the field*
* of fractions of
R=p, and note that xa - xb is invertible inQL and thus in K when a 6= b. As K[t*
*] is a unique
factorisation domain, wePsee that fD (t) = a(t - xa)ma in K[t] for a unique s*
*ystem of integers
ma. We define w(D) = ama[a] 2 Z[(v)]+d. In particular, if V 2 R+d(G) we ca*
*n let DV
be the tautologically associated divisor over C(E; G) and put u(V ) = w(DV ). *
*One can check
that this gives a homomorphism u: R+(G) -!Z[] of -semirings, or in other words *
*an element_
u 2 Ch(G). From the construction it is automatic that Iu p. *
* |__|
Example 11.5.If G is Abelian, it is easy to see that Ch(G) = Hom (G*; ) ' Hom (*
**; G) =
(G) and that is an isomorphism.
Example 11.6.Consider the symmetric group G = k. This acts in an obvious way on*
* Ck, and
we call this representation ss. It is known that ss generates R(G) as a -ring. *
*Thus, an element
f 2 Ch(G) is determined by the value f(ss) 2 Z[]+k.
As discussed in [12], the set (G) can be identified with the set of isomorphi*
*sm classes of sets of
order k with an action of *. For any finite subgroup A < we have a homomorphis*
*m * -!A*
and thus an action of * on A*. Note that 0* is just a single point with trivial*
* action. We write
m:A* for the disjoint union of m copies`of A*. If T is a finite *-set, then T c*
*an be written in an
essentially unique wayPin the form imi:A*i.
`If we write [A] =P a2A[a] 2 Z[] then by working through the definitions we f*
*ind that
( imi:A*i)(ss) = imi[Ai], which effectively determines .
It is now easy to exhibit cases in which is not injective. For example, supp*
*ose that p = 2 and
n > 1 and k = 6. We can then find two distinct, nonzero elements a; b 2 (1) and*
* put c = a + b.
Let A, B and C be the groups generated by a, b and c respectively, and put V = *
*A+B = {0; a; b; c}.
Then
(V *q 2:0*)(ss) = (A*q B* q C*)(ss) = 3[0] + [a] + [b] + [c];
so is not injective. In Section 15 we will give examples where is not surject*
*ive.
12.Calculating Ch(G)
In this section we define sets 0(G) and 00(G) which in some cases may be easi*
*er to compute
than (G) or Ch(G), and we define natural maps
(G) _____ww0(G) v_____wCh(G) v_____w00(G):
Definition 12.1.Let C be the set of conjugacy classes of elements of p-power or*
*der in G. We
let the multiplicative monoid Z act on * in the obvious way, and on C by k:[g] *
*= [gk]. We say
that two homomorphisms u; v :* -!G are pointwise conjugate if u(a) is conjugate*
* to v(a) for
18 N. P. STRICKLAND
all a 2 *. We recall the definitions of (G) and Ch(G) and define new sets 0(G) *
*and 00(G)
as follows:
(G) = Hom (*; G)=conjugacy
0(G) = Hom (*; G)=pointwise conjugacy
00(G)= {Z-equivariant continuous maps* -!C}
Ch(G) = { positive -ring homomorphismsR(G) -!Z[]}:
Proposition 12.2.There are natural maps as follows:
(G) _____ww0(G) v_____wCh(G) v_____w00(G):
Proof.There are evident natural maps
(G) _____ww0(G) v_____w00(G):
We have also already constructed a map : (G) -! Ch(G). If u; v :* -! G are poi*
*ntwise-
conjugate then the induced maps from class functions on G to class functions on*
* * are evidently
the same, so the induced maps R(G) -! Z[] are the same, so (u) = (v). This show*
*s that
factors through the projection (G) -!0(G).
We next define a map :Ch(G) -! 00(G). Suppose that u 2 Ch(G) and a 2 * =
Hom(; S1). Then a extends in a natural way to give a C-algebra map ^a:C[] -! C*
* and
thus a ring map (1C u)^a:C R(G) -! C. Using the fact that C R(G) is the set*
* of C-
valued class functions on G, we see that Hom C-Alg(C R(G); C) can be identifie*
*d with the
set of conjugacy classes in G. Thus there exists h 2 G (unique up to conjugati*
*on) such that
(1 u)(^a(V )) = OVm(h) for all V 2 R(G). We can choose m so that u(V ) 2mZ[(m)*
*] for all V , and
then we have OV (hp ) = O pmV(h) = Odim(V()h) = dim(V ) for all V , so hp = 1.*
* This means
that the conjugacy class [h] lies in C, so we can define (u)(a) = [h]. We leave*
* it to the reader
to check that this gives a map :Ch(G) -! 00(G) as claimed. The maps ^a:Z[] -! *
*C (as a
runs over *) are jointly injective, an it follows that is injective. One can a*
*lso check that the
composite 0(G) -!Ch(G) -!00(G) is just the obvious inclusion, which implies tha*
*t_the map
0(G) -!Ch(G) is injective. |*
*__|
13.Special divisors
In this section we study "special" divisors, which are related to the special*
* unitary group in the
same way that arbitrary divisors are related to the full unitary group.
Definition 13.1.A divisor D 2 Div+d(G) is special if dD = [0]. We write SDiv+d(*
*G) for the
scheme of special divisors.
Proposition 13.2.We have OSDiv+d(G)= OX [[c2; : :;:cd]]. In the topological sit*
*uation this can be
identified with E0BSU(d).
Proof.Put A = OGd = OX [[x1; : :;:xd]] and A0 = ODiv+d(G)= Ad = OX [[c1; : :;:*
*cd]]. Here ci
P *
* P F
is the i'th elementary symmetric function, and in particular c1 = ixi. Put c*
*01= i xi 2
A0. If we regard d as a map Div+d(G) -! Div+1(G) = G then c01= x O d, so we se*
*e that
OH3 = A=c01and OSDiv+d(G)= A0=c01. Next, observe that the inclusion A0 -! A in*
*duces an
inclusion A0=(c21; c2; : :;:cd) -! A=(x1; : :;:xd)2. We have c01= c1 (mod (x1;*
* : :;:xd)2) so c01=
c1 (mod c21; c2; : :;:cd). It follows easily that A0 = OX [[c01; c2; : :;:cd]]*
* and_thus_that A0=c01=
OX [[c2; : :;:cd]]. *
* |__|
We next put Hd = ker(Gd +-!G). If we let q :Gd -!Div+d(G) be the usual projec*
*tion (which
is finite and faithfully flat, with degree d!) then Hd = q-1 SDiv+d(G). It fol*
*lows that the map
q :Hd -! SDiv+d(G) is also finite and faithfully flat, with the same degree. I*
*t clearly factors
through Hd=d := spf(OdHd), and one would like the induced map q :Hd=d -!SDiv+d(*
*G) to be
an isomorphism. However, quotient constructions in algebraic geometry are never*
* as simple as
one would like, and we do not know whether this is true in general; certainly i*
*t becomes false
if we remove our assumption that G has finite height. For example, consider th*
*e case where
G is the additive group over F2 and d = 2; then 2a = 0 for all a 2 G so 2 acts *
*trivially on
H2 = {(a; -a) | a 2 G} so the map q :Hd=d -!SDiv+d(G) has degree two. However, *
*we do have
the following partial result.
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 19
Proposition 13.3.If d is invertible in OX then SDiv+d(G) = Hd=d.
Proof.As d is invertible in OX , multiplication by d is an automorphism of G. *
*Define maps
*
* P
G x Hd f-!Gd g-!G by f(a; b1; : :;:bd) = (a + b1; : :;:a + bd) and g(b1; : :;:b*
*n) = ibi=d, and
then define h: Gd -!G x Hd by h(b_) = (g(b_); b1- g(b_); : :;:bd- g(b_)). Clear*
*ly h is inverse to f, so
f is an isomorphism, giving an isomorphism OGd = OG bOHd = OHd[[x]] of rings. I*
*f we let d act
trivially on G then everything is equivariant, so we have OdGd= OdHd[[x]], so_D*
*iv+d(G)_= Gd=d =
G x (Hd=d). |__|
14.The group 4
We now consider the case where G = 4 and E is the 2-periodic Morava E-theory *
*spectrum
of height 2 at the prime 2. We shall show that the map C(E; G) -! E0BG is an is*
*omorphism.
To be more explicit, we need to name some representations. Note that 4 acts on*
* C4 with a
one-dimensional fixed subspace; we let ae be the representation of G on the quo*
*tient space. We
also write ffl for the sign representation. We let K = E=I2 denote the 2-period*
*ic Morava K-theory
spectrum.
Theorem 14.1. Let c2; c3 2 E0B4 be the Chern classes of the representation ffla*
*e, and let w be
the Euler class of ffl. Then C(E; 4) = E0B4, and this is a free module of rank *
*17 over E0, with
the following monomials as a basis:
1 c2 c22 c32 c3
w wc2 wc22 wc3
w2 w2c2 w2c22 w2c3
w3 w3c2 w3c22 w3c3:
Moreover, we have
C(K; 4) = K0B4 = C(E; 4)=I2 = K0[w; c2; c3]=J;
where J is generated by the following elements:
w4 ; c23; c2c3 ;
c42+ w2c32+ wc22+ w2c3;
wc32+ w2c2+ wc3:
We will prove this in a number of stages. In Section 14.1 we assemble the fac*
*ts that we need
about the representations of 4, and in Section 14.2 we deduce that the map : (4*
*) -!Ch(4)
is a bijection. We then recall some formulae for the relevant formal group law,*
* and in Section 14.4
we use them to analyse the structure of an auxiliary scheme denoted SDiv+3(G0)C*
*. This allows
us to complete our determination of C(K; 4) in Section 14.5, with the help of s*
*ome theory of
Gr"obner bases. We find in particular that C(K; 4) is a Gorenstein ring, which *
*enables us to use
the inner products defined in [10] to show that the map :C(K; 4) -!K0B4 is inj*
*ective; this
is explained in Section 14.6. We know from [5] that K(n)*B4 is concentrated in *
*even degrees,
and it follows that E0B4 is a free module over E0 of rank |(4)| = 17; see [11] *
*for more details.
In Section 14.7 we combine these various facts to prove the theorem.
14.1. Representation theory. Our first task is to understand the structure of R*
*(4). We have
already defined the characters ffl and ae. It is a standard calculation that th*
*ere is another irreducible
character oe of dimension 2 such that the character table is as follows:
____________________________________________________________________
| class | size | 1 ffl oe ae fflae |
|___________|________|______________________________________________ |
| 14 | 1 | 1 1 2 3 3 |
| | | |
| 122 | 6 | 1 -1 0 1 -1 |
| | | |
| 22 | 3 | 1 1 2 -1 -1 |
| | | |
| 13 | 8 | 1 1 -1 0 0 |
| | | |
| 4 | 6 | 1 -1 0 -1 1 |
|___________|_________|_____________________________________________|
20 N. P. STRICKLAND
The ring structure, Adams operations and -operations are described in the fol*
*lowing table.
ffl2 = 1 k(ffl) = fflk 2(oe) = ffl
ffloe = oe 2(oe) = 1 - ffl + oe 2(ae) = fflae
oe2 = 1 + ffl + oe 3(oe) = 1 + ffl 3(ae) = ffl
oeae = ae + fflae 2(ae) = 1 + oe + ae - fflae
ae2 = 1 + oe + ae + fflae 3(ae) = 1 + ffl - oe + ae:
(The first two columns are easily checked by looking at the characters, and the*
* last column follows
using the standard formulae relating Adams operations to -operations.)
Let P be a Sylow 2-subgroup (a dihedral group of order 8) and I be the kernel*
* of the restriction
map R(4) -! R(P ); one checks that I = (oe - 1 - ffl). Put o = fflae 2 R+3(4); *
*one checks that
k(o) = fflkk(ae) and so 2(o) = o and 3o = 1. We have
R(4)=I = Z{1; ffl; o; fflo} = Z[ffl; o]=(ffl2- 1; o2 - 1 - (1 + ffl*
*)(1 + o)):
The operation k acts as the identity on this ring when k is odd, and we have
2(ffl)= 1
2(o)= 2 + ffl + fflo - o:
Proposition 14.2.The set Ch(4) can be identified with the set of pairs (d; u) 2*
* (1) x Z[]+3
such that
2d = 0
3(u) = [0]
-1(u)= u
2(u) + u= 2[0] + [d] + [d]u:
Similarly, XCh(4) can be identified with the scheme of pairs (d; D) 2 G(1) x Di*
*v+3(G) such that
2d= 0
3(D) = [0]
-1(D) = D
2(D) + D= 2[0] + [d] + [d]D:
Proof.Given a positive homomorphism f :R(4) -! Z[], let d 2 be the element suc*
*h that
f(ffl) = [d] and put u = f(o). We know from Remark 11.2 that f(I) = 0 and it fo*
*llows easily from
our description of R(4)=I that d and u have the properties listed. Conversely, *
*given d and u as
described, we can define a homomorphism f :R(4)=I -!Z[] of additive groups by
f(1)= [0]
f(ffl)= [d]
f(o)= D
f(fflo)= [d]D:
It is straightforward to check that this gives a homomorphism of -rings, and th*
*at these construc-_
tions give the required bijection. The argument for XCh(4) is essentially the s*
*ame. |__|
14.2. Generalised character theory. We next work out the generalised character *
*theory (as re-
called in Section 11) of 4. The set (4) can be described in terms of *-sets as *
*in Example 11.6.
We can thus write (4) as the disjoint union 0q : :q:4, where
o 0 consists of the set 4:0* := 0*q 0*q 0*q 0*.
o 1 consists of the sets 2:0*q A*, where A ' Z=2 (so |1| = 2n - 1).
o 2 consists of the sets A* q B*, where A ' B ' Z=2, and A may be equal to B.*
* We have
|2| = 1_2|1|(|1| + 1) = 2n-1(2n - 1).
o 3 consists of the sets A* where A ' (Z=2)2. We have |3| = (2n - 1)(2n-1 - *
*1)=3 (by
counting the number of linearly independent pairs in (Z=2)n and dividing by*
* |GL2(Z=2)| =
6).
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 21
o 4 consists of the sets A* where A ' Z=4. There are 22n- 2n points in of or*
*der exactly
4, and each subgroup in 4 contains precisely two of these, so |4| = (22n - *
*2n)=2 =
2n-1(2n - 1).
Proposition 14.3.The map : (4) -!Ch(4) is a bijection.
Proof.Define
i= (i) Ch(4):
` P
Recall from Example 11.6 that ( imi:A*i)(ss) = imi[Ai]. An easy case-by-case*
* check shows
that the sets iare disjoint and that the maps : i-! iare bijections. It will th*
*us be enough
to show that the union of the sets iis the whole of Ch(4).
Suppose we have an element f 2 Ch(4), with f(ffl) = [d] and f(o) = u = [a] + *
*[b] + [c] say.
Let A, B and C be the cyclic subgroups generated by a, b and c respectively. Pu*
*t v = f(ss) =
[0] + [d]u = [0] + [a + d] + [b + d] + [c + d], and recall that this determines*
* f, because ss generates
R(4) as a -ring. As 4o = 3[0] we have 4a = 4b = 4c = 0. By Proposition 14.2, w*
*e have
2d= 0
a + b +=c0
[a] + [b] +=[c][-a] + [-b] + [-c]
[2a] + [2b] + [2c] + [a] +=[b]2+[[c]0] + [d] + [a + d] + [b + d] + [c*
* + d]:
Suppose that 2(u) 6= 3[0]. Without loss of generality we may assume that 2a 6=*
* 0 so -a 6= a.
The third equation implies that -a 2 {b; c}, so we may assume that -a = b. As a*
* + b + c = 0 we
must have c = 0. Recall also that 4a = 0 so 2a = -2a. Putting all this in the l*
*ast equation and
cancelling 2[0] gives
2[2a] + [a] + [-a] = 2[d] + [d + a] + [d - a]:
Note that 2a and d have order 2, but a, -a, d + a and d - a do not. It follows *
*that we must have
2a = d and thus v = [0] + [a] + [2a] + [3a]. We conclude that f = (A*) 2 4.
We may thus assume that 2(u) = 3[0], so 2a = 2b = 2c = 0. Suppose that d = *
*0. As
a + b + c = 0 we see that D := {0; a; b; c} is a subgroup of , of order 2e say *
*(so e 2 {0; 1; 2}).
This implies that v = 22-e[A] = (22-e:D*)(ss), so f = (22-e:D*) 2 0q 2q 3.
We may thus assume that 2a = 2b = 2c = 2d = 0 and d 6= 0. The equation 2(u)*
* + u =
2[0] + [d] + [d]u then reduces to
[0] + [a] + [b] + [c] = [d] + [a + d] + [b + d] + [c + d]:
It follows that d 2 {a; b; c} and without loss we may assume that d = a. Note t*
*hat c = a+b = d+b
(because a + b + c = 0). If b = 0 this gives c = a = d so v = 3[0] + [a], so f *
*= (2:0*q A*). The
same argument works if c = 0, so we reduce to the case where a = d 6= 0 and b a*
*nd c are also
nonzero. We then have
v = [0] + [a + d] + [b + d] + [c + d] = 2[0] + [c] + [b] = [B] + [*
*C];
so f = (B* q C*) 2 2. |_*
*__|
14.3. The formal group law. Let G be the formal group associated to E, and let *
*G0 be its
restriction to the special fibre X0 X, or equivalently the formal group associ*
*ated to K. This
has a standard coordinate giving rise to a formal group law F over OX0 = K0 = F*
*4, which is in
fact defined over F2. We will need the following formulae:
[2](x)= x4
[-1](x)= x + x4+ x10+ x16+ x22 (mod x32)
x +F y= x + y + x2y2 (mod x4y4):
The first of these is well-known and the second can be proved by straightforwar*
*d computation; for
the third, one can adapt the method of [9, Section 15] to the case p = 2.
22 N. P. STRICKLAND
14.4. The scheme SDiv+3(G0)C. Let C be the group (of order 2) generated by -1,*
* so
SDiv+3(G0)C = {D 2 Div+3(G0) | 3D = [0] and -1D = D}:
We have seen that OSDiv+3(G0)= F4[[c2; c3]]; our next task is to determine th*
*e quotient ring
OSDiv+3(G0)C.
Proposition 14.4.We have
OSDiv+3(G0)C= F4[[c2; c3]]=(c2c3; c23) = F4[[c2]] F4:c3:
Proof.Put A = F4[[x; y; z]] and
d = x +F y +F z
c1= x + y + z
c2= xy + yz + zx
c3= xyz:
Put A0= A3 = F4[[d; c2; c3]], so A is free of rank 6 over A0. Put B0= A0=dA0and
B = A=dA = B0A0A = F4[[c2; c3]] = OSDiv+3(G0):
For any element u 2 A we write __u= ( -1)*(u), so u 7! __uis a ring map and __u*
*= [-1](u) for
u 2 {x; y; z; d}. Put C = B=(_c2-c2; _c3-c3)B and C0= B0=(_c2-c2; _c3-c3)B0= OS*
*Div+3(G0)C. The
claim is that the ideal in C0generated by c3 is free of rank one over F4, and t*
*hat C0=c3C0= F4[[c2]],
so that C0= F4[[c2]] F4:c3.
We will think of Div+2(G0) as being embedded in Div+3(G0) by the map D 7! D +*
* [0], so
ODiv+2(G0)= ODiv+3(G0)=c3 = F4[[d; c2]]:
There is a faithfully flat map G0 -!SDiv+2(G0) sending a to [a]+[-a], and clear*
*ly -1([a]+[-a]) =
[a] + [-a] so SDiv+2(G0) SDiv+3(G0)C. It follows that Div2(G0) \ SDiv+3(G0)C =*
* SDiv+2(G0),
and thus that C0=c3C0= F4[[c2]] as claimed.
This implies that we must have _c2- c2 = c3r2 and _c3- c3 = c3r3 for some r2;*
* r3 2 B0.
Now work in B=(x; y; z)7. We have
z= __x+F _y= x + y + x4+ x2y2 + y4
__x= x + x4
_y= y + y4
_z= x + y + x2y2
c2= x2+ xy + y2 + x5+ x4y + x3y2 + x2y3 + xy4 + y5
c3= x2y + xy2 + x5y + x3y3 + xy5
_c 4 4
2- c2= c2c3 = x y + xy
_c 2 4 2 2 4
3- c3= c3 = x y + x y :
We also find that the ideal c3:(c2; c3)2 maps to zero in this ring. Using this,*
* we find that r2 = c2
(mod (c2; c3)2) and r3 = c3 (mod (c2; c3)2), so B0 = F4[[r2; r3]] and B0=(r2; r*
*3) = F4. It follows_
that OSDiv+3(G0)C= B0=(r2c3; r3c3) = B0=(c2c3; c23)3 as claimed. *
* |__|
Now put Y = {D 2 SDiv+3(G0)C | 4D = 3[0]}, and let U G0 be the divisor 32[0*
*]. We know
that ck( 4D) = ck(D)16so
OY = OSDiv+3(G0)C=(c161; c162; c163) = F4[c2; c3]=(c162; c23; c2*
*c3):
We can also study Y using the maps
ff: U -!Y ff(a) = [a] + [-a] + [0]
fi :G0(1)2 -!Y fi(a; b) = [a] + [b] + [a + b]:
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 23
The map ff gives a ring map ff*:OY -! F[x]=x32, with
ff*(c1)= x + __x= x4+ x10+ x16+ x22
ff*(c2)= x__x= x2+ x5+ x11+ x17+ x23
ff*(c3)= 0:
The map fi gives a ring map fi*:OY -! F[x; y]=(x4; y4). If we put z = x +F y = *
*x + y + x2y2 then
fi*(c1)= x + y + z = x2y2
fi*(c2)= xy + yz + zx = x2+ xy + y2 + x2y2(x + y)
fi*(c3)= xyz = xy(x + y) + x3y3:
Proposition 14.5.The maps ff* and fi* are jointly injective (in other words, ke*
*r(ff*)\ker(fi*) =
0). Moreover, we have c1 = c22+ c82in OY .
Proof.Recall that OY = F4[c2; c3]=(c162; c23; c2c3), so {ci2| 0 i < 16} q {c3}*
* is a basis for OY
over F4. As ff*(c2) = x2 (mod x3), it is easy to see that ff*(ci2) = x2i(mod x2*
*i+1) and that these
elements are linearly independent in OU = F4[x]=x32. Moreover, we have
fi*(1)= 1
fi*(c2)= x2+ xy + y2 + x2y2(x + y)
fi*(c22)= x2y2
fi*(c32)= x3y3
fi*(ci2)= 0 fori > 3:
It is easy to check that fi*(c3) does not lie in the span of these elements, an*
*d to deduce that ff*
and fi* are jointly injective as claimed. Thus, to show that c1 = c22+ c82we ne*
*ed only check that __
ff*(c1) = ff*(c22+ c82) and fi*(c1) = fi*(c22+ c82), which is a straightforward*
* computation. |__|
14.5. The ring C(K; 4). Consider a pair (d; D) 2 G0(1) x Y . This gives us a di*
*visor [d]D 2
Div+3(G0) defined over the ring
OG0(1) OY = F4[w; c2; c3]=(w4; c162; c23; c2c3):
Here of course c2 and c3 are the usual invariants of the divisor D, but the div*
*isor [d]D also
has invariants ck([d]D) lying in the above ring. In order to apply the descript*
*ion of XCh(4) in
Proposition 14.2, we will need to understand these invariants.
Proposition 14.6.We have
c1([d]D)= c22+ c82+ w + c42w2
c2([d]D)= c2+ (1 + c32+ c92+ c3)w2
c3([d]D)= c3+ c2w + (c22+ c82)w2 + (1 + c3+ c32+ c92)w3:
Proof.First recall that u +F v = u + v + u2v2 (mod u4v4) and w4 = 0 so w +F v =*
* w + v + w2v2
for any v.
Next note that [d]ff(a) = [d]([a] + [-a] + [0]) = [d + a] + [d - a] + [d], so*
* ff*(ck([d]D)) =
ck([d + a] + [d - a] + [d]) is the k'th elementary symmetric function of {w +F *
*x; w +F __x; w}, for
example
ff*c1([d]D)= w + (w + x + w2x2) + (w + __x+ w2__x2)
= x4+ x10+ x16+ x22+ w + x8w2 + x20w2
= ff*(c22+ c82) + w + ff*(c42)w2:
By similar computations, our other two equations also become true when we apply*
* ff*.
In the same way, we have [d]fi(a; b) = [d + a] + [d + b] + [d + a + b], so fi*
**ck([d]D) is the k'th
elementary symmetric function of the list {w +F x; w +F y; w +F x +F y}, or equ*
*ivalently the list
{w + x + w2x2; w + y + w2y2; w + x + y + w2x2+ w2y2 + x2y2}:
24 N. P. STRICKLAND
We thus have
fi*c3([d]D)= (w + x + w2x2)(w + y + w2y2)(w + x + y + w2x2+ w2y2 + x2y2)
= (x2y + xy2 + x3y3) + (x2+ xy + y2 + x3y2 + x2y3)w +
x2y2w2 + (1 + x2y + xy2)w3
= fi*(c3) + fi*(c2)w + fi*(c22+ c82)w2 + fi*(1 + c3+ c32+ c92)w*
*3:
By similar computations, our other two equations also become true when we apply*
* fi*. As ff* and __
fi* are jointly injective, it follows that our equations hold in OG0(1)xYas cla*
*imed. |__|
Proposition 14.7.Let J be the ideal in F4[w; c2; c3] generated by the elements
w4 ; c23; c2c3 ;
c42+ w2c32+ wc22+ w2c3;
wc32+ w2c2+ wc3:
Then C(K; 4) = F4[w; c2; c3]=J. Moreover, the following monomials form a basis *
*for this ring
over F4, so it has dimension 17.
1 c2 c22 c32 c3
w wc2 wc22 wc3
w2 w2c2 w2c22 w2c3
w3 w3c2 w3c22 w3c3
Proof.Proposition 14.2 is equivalent to the statement that
X(4) = {(d; D) 2 G0(1) x Y | D + 2(D) = 2[0] + [d] + [d]D}:
This means that C(K; 4) is the largest quotient of OG0(1)xYover which we have g*
*(t) = 0, where
g(t) = fD (t)f 2D(t) - t2(t + w)f[d]D(t):
Here we write fD (t) = t3+ c1(D)t2+ c2(D)t + c3(D) and similarly for our other *
*divisors. As usual
we write ck for ck(D), and we recall from Proposition 14.5 that c1 = c22+ c82. *
*We also recall that
ck( 2D) = ck(D)4, so that
f 2D(t) = t3+ c41t2+ c42t + c43= t3+ c82t2+ c42t:
P 3
The polynomial f[d]D(t) = k=0ck([d]D)t3-k can be read offPfrom Proposition 14*
*.6. Putting all
this together and expanding it out, we find that g(t) = 4k=1rkt6-k, where
r1 = c82+ c42w2
r2 = c42w3 + (c92+ c32+ c3)w2 + (c82+ c22)w + (c102+ c42)
r3 = (c82+ c22)w2 + (c122+ c92+ c62)
r4 = (c82+ c22)w3 + c2w2 + c3w + c52:
We thus have
C(K; 4) = F4[w; c2; c3]=(w4; c162; c23; c2c3; r1; r2; r3; r4):
As 1 + c62+ c122+ w3 is invertible, we can replace r2 by
r02:= (1 + c62+ c122+ w3)r2 = c42+ w2c32+ wc22+ w2c3;
which is one of the relations in the statement of the theorem. As w4 = 0 we hav*
*e (r02)2 = r1 and
(r02)4 = c162, so the relations r1 and c162are redundant. Similarly, we can rep*
*lace r4 by the relation
r04:= r4+ (c2+ w2 + c24w3)r02= wc32+ w2c2+ wc3;
which is another of the relations in the statement of the theorem. One can chec*
*k that
r3 = (1 + c32+ c62)(c22(1 + (1 + c62)(w3 + c22w2))r02+ c2r04);
so r3 is redundant. We deduce that
C(K; 4) = F4[w; c2; c3]=(w4; c23; c2c3; r04; r02)
as claimed.
We next show that the 17 monomials listed form a basis for this0quotient ring*
*. We order the
set of monomials in w, c2 and c3 by saying that ci2cj3wk < ci02cj3wk0iff i < i0*
*or (i = i0and j < j0)
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 25
or (i = i0and j = j0 and k < k0). We claim that our relations form a Gr"obner b*
*asis for J with
respect to this ordering. We first recall briefly what this means. The list of *
*leading terms of our
relations is (w4; c23; c2c3; c32w; c42). A polynomial is said to be top-reducib*
*le if any of its monomials
is divisible by one of these leading terms; if so, we can subtract off a multip*
*le of the corresponding
relation to cancel the monomial, a process called top-reduction. Clearly, if a *
*polynomial can be
reduced to zero by iterated top-reduction then it must lie in J, but the conver*
*se need not hold
for an arbitrary list of generators of an arbitrary ideal. Let a and b be any t*
*wo of our relations,
let a0and b0be their leading terms, and let c0be the greatest common divisor of*
* a0and b0. The
corresponding syzygy is the element c := (a0=c0)b - (b0=c0)a 2 J. To say that o*
*ur relations form
a Gr"obner basis means precisely that all these syzygies can be reduced to zero*
* by iterated top-
reduction. This can be checked by direct computation. For example, the syzygy o*
*f r04and r02is
the element c2r04-wr02= c32w3+c2c3w +c3w3. The first monomial is divisible by t*
*he leading term
of r04, so we can top-reduce by subtracting w2r04to get c2c3w + c2w4. We can th*
*en do two more
top-reductions by subtracting w times the relation c2c3 and c2 times the relati*
*on w4 to get 0, as
required. Now observe that the 17 monomials listed in the statement of the theo*
*rem are precisely
those that are not top-reducible. It follows from the theory of Gr"obner bases *
*that_they form a
basis for C(E; 4), as claimed. *
* |__|
Corollary 14.8.C(K; 4) is a Gorenstein ring, and the element w3c3 generates the*
* socle.
Proof.One sees easily from the relations listed that w, c2 and c3 annihilate w3*
*c3, so w3c3 lies in the
socle. Now let a be an arbitrary element of the socle. It will be convenient to*
* put e = c32+wc2+c3
(so that we = c3e = 0) and to use the basis given in the Proposition but with c*
*32replaced by e.
Using the equation wa = 0 we see immediately that a lies in the span of {w3; w3*
*c2; w3c22; e; w3c3}.
Using the equation c3a = 0 and the fact that c3c2 = c3e = c23= 0 we find that t*
*he coefficient of w3
is zero, so a = ffw3c2+ fiw3c22+ fle + ffiw3c3 say. One can check that w3c32= w*
*3c3 and c2e = w3c2,
so
0 = c2a = ffw3c22+ fiw3c3+ flw3c2;
so ff = fi = fl = 0, so a = ffiw3c3. This shows that the socle is one-dimension*
*al,_so the ring is
Gorenstein as claimed. *
* |__|
14.6. A transfer argument.
Proposition 14.9.The map :C(K; 4) -!K0B4 is injective.
Proof.Note that every nontrivial ideal in C(K; 4) contains the socle, so it wil*
*l suffice to show
that the socle is not contained in ker(), or equivalently that w3c3 6= 0 in K0B*
*4. Let P be the
Sylow subgroup in 4; it will be enough to show that w3c3 has nontrivial image i*
*n K0BP . Put
V = P \ A4; one can check that this consists of the identity and the three tran*
*sposition pairs,
so it is isomorphic to C22. Recall that the series <2>(x) is defined to be [2]*
*(x)=x, which in our
case is just x3. As w is the Euler class of ffl and V = ker(ffl: P -! C2), stan*
*dard arguments show
that trPV(1) = <2>(w) = w3. This means that w3c3 = trPV(c3). To see that this i*
*s nonzero, we
use the canonical bilinear form on K0BP defined in [10]. This satisfies Frobeni*
*us reciprocity, so
(trPV(c3); 1)P = (c3; 1)V . If we let x and y be the Euler classes of two of th*
*e nontrivial characters
of V , then K0BV = F4[x; y]=(x4; y4) and the Euler class of the third characte*
*r is x +F y =
x + y + x2y2. One checks that the restriction of ae to V is the regular represe*
*ntation minus the
trivial representation, which is the sum of the three nontrivial characters. Th*
*is implies that the
restriction of c3 to V is xy(x +F y) = x2y + xy2 + x3y3. Using [10, Corollary 9*
*.3] we see that
(xiyj; 1)V is 1 if i = j = 3 and 0 otherwise, so (c3; 1)V = 1. As (w3c3; 1)P = *
*(c3;_1)V_= 1 we see
that w3c3 6= 0, as claimed. *
* |__|
14.7. The proof of Theorem 14.1. This is now easy. We know from [11] that E0B4 *
*is a free
module of finite rank over E0. It follows by well-known arguments that K0B4 = (*
*E0B4)=I2,
which is free of the same rank over K0 = E0=I2 = F4. The rank is also the same *
*as the rank of
L E0B4 over L, and generalised character theory tells us that this is equal to*
* |(4)| = 17.
Thus, the source and target of the map :C(K; 4) -! K0B4 both have rank 17 over*
* F4 and
Proposition 14.9 tells us that the map is injective, so it must be an isomorphi*
*sm. Now consider
the map :C(E; 4) -!E0B4. This is an isomorphism modulo I2, so by Nakayama's le*
*mma it
is surjective. As E0B4 is free it is a split surjection, so C(E; 4) = E0B4N say*
*. This implies
that C(K; 4) = K0B4N=I2N, so by counting ranks we see that N=I2N = 0, so by Nak*
*ayama
again we see that N = 0. Thus C(E; 4) = E0B4 as claimed. We know from Propositi*
*on 14.7
26 N. P. STRICKLAND
that our list of 17 monomials is a basis for K0B4 over F4, and it now follows t*
*hat it is also a
basis for E0B4 over E0.
15.Extraspecial p-groups
In this section we define a class of "extraspecial" p-groups (where p is an o*
*dd prime), and
show that for these groups the map : (G) -!Ch(G) is injective but not surjectiv*
*e. It follows
using Theorem 11.4 that the map :C(E; G) -!E0BG cannot be an isomorphism. We h*
*ave not
investigated the situation more deeply than this.
Let V be an elementary Abelian p-group of rank 2d equipped with a nondegenera*
*te alternating
form b: V x V -! Fp. We will say that a subspace W V is isotropic if b(u; v) *
*= 0 for all
u; v 2 W .
Let G be the set Fpx V with the group operation (x; u):(y; v) = (x + y + b(u;*
* v); u + v). This
has order p2d+1and exponent p, and it fits in a central extension
Z = Fp j-!G q-!V:
In fact Z is the centre of G, and the non-central conjugacy classes are the fib*
*res of q over V \ {0},
so they all have order p. This gives p + p2d- 1 conjugacy classes altogether.
We can evidently view R(V ) = Z[V *] as a sub -ring of R(G).
Definition 15.1.For any nontrivial character i :Z -! S1, let OE(i) be the class*
* function on G
defined by
( d
OE(i)(g) = p i(g) ifg 2 Z
0 otherwise.
We also write aeV for the regular representation of V , and aeG for the regular*
* representation of G.
The following result is standard, but we give a proof for completeness.
Proposition 15.2.For each i 2 Z* \ {1}, the class function OE(i) is an irreduci*
*ble character.
Moreover, we have
R(G) = Z[V *] Z{OE(i) | i 2 Z* \ {1}}:
Proof.Choose a maximal isotropic subspace W V , so W ' Fdp. Put H = q-1W G, w*
*hich is
isomorphic to Z x W as a group because W is isotropic. Let r :H -!Z be the proj*
*ection and put
oe = indGHss*i. We claim that oe = OE(i). To see this, first note that H is nor*
*mal in G, so oe(g) = 0
for g 62 H. Next, suppose that g 2 H \ Z, say g = (x; w) with w 2 W \ {0}. Le*
*t U be such
that V = W U, so U ' FdpandP(0; u)-1(x; w)(0; u) = (x - 2b(u; w); w). From the*
* definitions
we see that oe(x; w) = ui(x - 2b(u; w); 0). The map u 7! (x - 2b(u; w); 0) is*
* a surjection from
U to Z, each of whose fibres has the same order, and i :Z -! S1 is a nontrivial*
* homomorphism;
it follows easily that oe(x; w) = 0, as required.PFinally, suppose that g 2 Z, *
*say g = (x; 0). Then
(0; u)-1(x; 0)(0; u) = (x; 0) so oe(x; 0) = ui(x; 0) = pdi(x; 0). This shows*
* thatPoe = OE(i) as
claimed, so OE(i) is a character. One checks easily that = |G|-*
*1 z2Z p2d = 1, so
OE(i) is irreducible. As i runs over Z* \ {1} this gives p - 1 distinct irredu*
*cibles of degree pd,
and V *gives a further p2ddistinct irreducibles of degree 1. We have seen that *
*G has p2d+ p - 1
conjugacy classes and thus p2d+ p - 1 irreducible characters, so our list is co*
*mplete. It_follows
that R(G) = Z[V *] Z{OE(i) | i 2 Z* \ {1}} as claimed. *
* |__|
Lemma 15.3. Let C be cyclic of order p. Then
8i d-1j ii j i d-1jj
< p + 1_ pd - p aeC ifp|k
k(pd-1aeC) = : 1k=p_ipdpj k k=p
p k aeC otherwise
P p-1
Proof.Let O be a generator of C*, so R(C) = Z[O]=(Op - 1) and aeC = j=0Oj. W*
*e have
OaeC = aeC and so k(aeC) = k(OaeC) = Okk(aeC). If 0 < k < p then Ok is also a g*
*enerator, and
it follows that k(aeC) is an integer multiple of aeC. On the other hand, it is *
*easy to check that
0(aeC) = p(aeC) = 1. If we put A = Z{1; aeC} then A is a subring of R(C) (with *
*ae2C= paeC) and
t(aeC) 2 A[t] so t(pd-1aeC) = t(aeC)pd-1also lies in A[t], say k(aeC) = nk+ mka*
*eC. Moreover, if
we work mod aeC we have t(aeC) ~=1 + tp so t(pd-1aeC) ~=(1 + tp)pd-1. Thus, if *
*p divides k then
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 27
i d-1j
nk = pk=p , and if p does not divide k then nk = 0. Moreover, by counting dime*
*nsions we see
i dj *
* __
that nk + pmk = pk for all k. The lemma now follows easily. *
* |__|
Proposition 15.4.
OOE(i)= OE(i)
(
OE(i)OE()= aeV ifi = 1
pdOE(i) otherwise
kO= Ok
( d
kOE(i)= p ifp|k
OE(ik) otherwise
8i d-1j ii j i d-1jj
< p + _1_2dpd - p aeV ifp|k
k(OE(i))= : 1k=p_ipdpjk k k=p
pd k OE(i ) otherwise
Proof.Everything except for k(OE(i)) can be done by easy manipulation of charac*
*ters. For the
remaining case, it suffices to check that the claimed equations hold when restr*
*icted to any cyclic
subgroup C G. First consider the case C = Z, so aeV restrictsionjC to the triv*
*ial representation
of degree p2d. Then OE(i) becomes pdi, so kOE(i) becomes pdkik. Using this, it*
* is easy to check
that the equations hold when restricted to Z.
Now suppose instead that C G is a cyclic group not contained in Z (which imp*
*lies that
|C| = p). Then aeV |C = p2d-1aeC and OE()|C = pd-1aeC for all 2 Z* \ {1}. Usin*
*g Lemma 15.3 we __
deduce that our equations for k(OE(i)) are correct when restricted to C, as req*
*uired. |__|
Definition 15.5.For any homomorphism ff: V *-!, put
X
cff= [ff(O)] 2 Z[(1)]+p2d
O2V *
and
Uff= {u 2 Z[(1)]+pd| u p-1(u) = cff}:
We also put U = {(ff; u) | u 2 Uff}.
Theorem 15.6. There is a natural bijection Ch(G) = U. The map : (G) -!U is inje*
*ctive,
and the image is the set of pairs (ff; u) 2 U such that the image of the dual m*
*ap ff*:* -!V is
isotropic.
The proof will follow after a lemma.
Lemma 15.7. Let ff: V *-! be a homomorphism with image A of order pe.P If e > *
*d then
Uff= ;. If e d then Uffis the set of elements of the form u = pd-e c2C[c], wh*
*ere C runs over
the cosets of A in (1).
P +
Proof.Put c0ff= a2A[a] 2 Z[(1)]pe so that cff= p2d-ec0ff. Suppose u 2 Uffand *
*that b 2 u.
Put v = [-b]u, so v 2PUffand 0 2 v. Thus 0 2 p-1(v) also, so v v p-1(v) = cff*
*= p2d-ec0ff,
so we can write v = a2Ana[a] for suitable natural numbersPna. By looking at t*
*he multiplicity
of [0] in the equationPv p-1(v) = p2d-ec0ffwe see that an2a= p2d-e. On the ot*
*her hand, as
v 2 Z[(1)]+pdwe have ana = pd. It follows that
X X X X
(na - pd-e)2 = n2a- 2pd-e na + p2d-2e 1 = p2d-e- 2p2d-e+ p2d-e= 0;
a a a a
*
* P
so na = pd-e for all a. If we now let C be the coset b + A we find that u = pd*
*-e c2C[c]. __
Conversely, it is trivial to check that any element of this form lies in Uff. *
* |__|
Proof of Theorem 15.6.Let i be the usual character x 7! e2ssix=pof Z = Z=p. Giv*
*en f :R(G) -!
Z[] it is clear that the restriction of f to R(V ) = Z[V *] gives a homomorphis*
*m ff: V *-!(1)
28 N. P. STRICKLAND
, and we put u = f(OE(i)) 2 Z[]+pd. As f is a -ring homomorphism we have
X
u p-1(u) = f(OE(i)OE(ip-1)) = f(aeV ) = [ff(O)];
O
so (ff; u) 2 U.
Conversely, suppose we start with (ff; u) 2 U. Let e and C be as in Lemma 15.*
*7. We define a
homomorphism f :R(G) -!Z[] of additive groups by f(O) = [ff(O)] for O 2 V *and
X
f(OE(ik)) = k(u) = pd-e [c]
c2kC
for k 2 Z \ pZ. It is easy to check that this is a ring homomorphism that sends*
* R+k(G) to Z[]+k
and commutes with the Adams operations. As Z[] is torsion free it follows that *
*f commutes
with -operations as well, so f 2 Ch(G). Clearly these constructions give the re*
*quired bijection
Ch(G) = U.
Now suppose we have a homomorphism : * -! G. Then (u) = (!(u); oe(u)) for so*
*me
functions ! :* -!Fp and oe :* -!V . As and the projection q :G -!V are homomor*
*phisms
we see tht oe is a homomorphism. Let W V be the image of oe, and put e = dimF*
*pW . As
the image of must be commutative, it is not hard to see that W is isotropic, s*
*o e d. As
q-1W ' Fp x W as groups, we see that ! is also a homomorphism. If we conjugate *
*(!; oe) by
(x; u) 2 G we get the homomorphism (! + o; oe) where o(t) = 2b(u; oe(t)). As b *
*is a perfect pairing,
o can be any map * -!Fp that factors through oe, so (!; oe) is conjugate to (!0*
*; oe) if and only if
!|ker(oe)= !0|ker(oe). Now let oe*:V *-! be the dual of oe and put A = oe*(V **
*), so |A| = pe. We
also have a map !*:F*p-! and thus a point t = !*(i) 2 (1). In R(Fpx W ) = Z[F*p*
*] Z[W *]
we have
X
OE(i)|FpxW = pd-ei aeW = pd-e i ;
2W*
P
and it follows that *OE(i) = pd-eP a2A[t + a] 2 Z[]. Thus, if we write [] for t*
*he conjugacy
class of then [] = (oe*; pd-e a2A[t + a]) 2 U. It follows that [] determines *
*oe, and it also
determines t modulo A, so it determines ! modulo oe*(Hom (V; Fp)), so it determ*
*ines the conjugacy
class []. This proves that is injective as claimed. We leave it to the reader *
*to check_that the
image is as described. *
* |__|
16.An apparently more precise approach
There are some senses in which the -operations do not capture all possible in*
*formation about
the representation theory of G, and it is reasonable to wonder whether a more a*
*ccurate approx-
imation to X(G) could be defined by taking more information into account. In th*
*is section we
show that this is not the case: we construct an approximation Y (G) using all p*
*ossible operations,
and show that it is the same as XCh(G).
Definition_16.1.Let G be the category of Lie groups and continuous homomorphism*
*s, and let
Gbe the quotient category in which conjugate homomorphisms are identified. LetQ*
*N be the set
of finite sequences n_= (n1; : :;:nr) with ni2 N. For n_2 N we put GL(n_) = iG*
*L (ni; C) and
Y __
R(n_; G) = R+ni(G) = G(G; GL(n_)):
i
*
* __
We make N into a category by putting N (n_; m_) = G(GL_(n_); GL(m_)),_and we le*
*t N be the
category with the same objects and with morphisms_N (n_; m_) = G(GL (n_); GL(m_*
*)); clearly this
gives a covariant functor n_7! R(n_; G) from N to sets. *
* Q
Next,Plet T (n_) be the evident maximal compact torus in GL (n_),Qso T (n_) '*
* Nj=1S1 where
N = ri=1ni. Let W (n_) be the Weyl group of T (n_), so W (n_) ' ini. We can*
* thus form the
scheme
Y
D(n_) = Hom (T (n_)*; G)=W (n_) = Div+ni(G):
i
By elementary arguments in representation theory we see that any homomorphism f*
* :GL (n_) -!
GL(m_) is conjugate to one that sends T (n_) into T (m_), and that the resultin*
*g map T (n_) -!T (m_)
is unique up_to the action of W (m_). Using this, it is easy to make the assign*
*ment n_7! D(n_) into
a functor N -! bXX.
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 29
Finally, we define a functor Y (G) from discrete OX -algebras to sets by putt*
*ing
__
Y (G)(A) = [N ; Sets](R(-; G); D(-)(A)):
Theorem 16.2. There is a natural isomorphism Y (G) ' XCh(G).
Before proving this, we relate Y (G) to an auxiliary model involving unitary *
*groups rather than
general linear groups.
Definition 16.3.Let eGbe the quotient of G in which homomorphisms u; v :U -!V a*
*re identified
if u|K is conjugate to v|K for every compact subgroup K U. (For example, the h*
*omomorphism
u: GL(1) -! GL(1) given by u(z) = |z| becomes trivial in eG.) Let eNbe the cate*
*gory with the
same objects as N and morphisms eN(n_; m_) = eG(GL (n_); GL(m_)). As G and T (n*
*_) are compact, it
is clear that the functors R(-; G) and D(-) factor through eN, and thus that
Y (G)(A) = [Ne; Sets](R(-; G); D(-)(A)):
Lemma 16.4. Let K be a compact Lie group, and let v; w: K -!U(d) be continuous *
*homomor-
phisms. If v and w are conjugate in GL(d), then they are conjugate in U(d).
Proof.The statement can easily be translated as follows: Let V and W be finite-*
*dimensional
vector spaces over C equipped with actions of G and invariant Hermitian inner p*
*roducts. Then if
there exists an equivariant isomorphism f :V -! W , then f can be chosen to pre*
*serve the inner
products.
To see this, we first_recall some facts about invariant Hermitian products. F*
*or_any*complex
vector_space V we let V be the same set with the conjugate action of C, and let*
* V be the dual_*
of V. The set of Hermitian products fi on V bijects with the set of isomorphism*
*s fi0:V -! V
satisfying certain symmetry and positivity conditions. For any representation_V*
**one can always
choose an invariant Hermitian product so V is equivariantly isomorphic to V . F*
*or each irreducible_*
representation S we fix a Hermitian product fiS on S; Schur's lemma implies tha*
*t Hom K(S; S ) =
Cfi0Sand that any other invariant Hermitian product is a positive scalar multip*
*le of fiS.
Now let fi be_a_Hermitian product on V and suppose that V = V0 V1 and Hom K(V*
*1; V0) = 0.
Then Hom K(V1; V*0) = 0 and
__* __* * *
Hom K(V0; V1) = Hom K(V 1; V0) = Hom K(V1; V0) = 0
__* *
* __*
so the equivariant isomorphism fi0:V -! V must have the form fi00 fi01for some*
* fi0i:Vi-! Vi.
This implies that V0 and V1 are orthogonal with respect to fi.
Now let S1; : :;:Stbe the distinct irreducible representations that occur in *
*V . Then there is a
unique decomposition V = V1: :V:t, where Vi' CdiSifor some diand thus HomK (Vi;*
* Vj) = 0
when i 6= j._*By the previous paragraph, the subspaces Vi are orthogonal to eac*
*h other. As
HomK (Si; Si) = CfiSi, we find that the restriction of fi to Vi has the form fl*
*i fiSi for some
Hermitian product fli on Cdi. By Gram-Schmidt, the space (Cdi; fli) is isomorph*
*ic to Cdiwith
its usual Hermitian product, so (Vi; fi|Vi) is equivariantly and isometrically *
*isomorphic to the
orthogonal direct sum of di copies of (Si; fiSi). This means that the numbers d*
*i determine_the
isometric isomorphism type of V , and the lemma follows immediately. *
* |__|
Lemma 16.5. There are natural bijections
eN(n_; m_) = __G(U(n_); GL(m_)) = __G(U(n_); U(m_));
Q
where U(n_) = iU(ni) GL(n_).
Proof.It is easy to reduce to the case where the list m_ has length 1, say m_ =*
*_(d). As any
representation_of U(n_) admits a Hermitian inner product, we see that the map G*
*(U(n_); U(d)) -!
G(U(n_); GL(d)) is surjective. It is also injective by Lemma 16.4. Similarly, b*
*y considering invariant
Hermitian products we see that if K is compact and u: K -! GL(n_) then u is con*
*jugate to a
homomorphism K -! U(n_). By applying this to the inclusion map, we see that an*
*y compact
subgroup of GL(n_) is conjugate to a subgroup of U(n_). It follows that any two*
* homomorphisms
v; w: GL(n_) -!GL (d) are identified in eG(GL (n_); GL(d)) iff their_restrictio*
*ns_to U(n_) are conjugate,
so we have a well-defined and injective restriction map eN(n_; d) -!G (U(n_); G*
*L(d)). It is an easy
consequence of the theory of roots and so on that any representation of U(n_) e*
*xtends uniquely to__
a complex-analytic representation of GL(n_), so our restriction map is also sur*
*jective. |__|
30 N. P. STRICKLAND
Proof of Theorem 16.2.Consider a point g 2 Y (G)(A), in other words a natural t*
*ransformation
gn_:R(n_; G) -!D(n_)(A) for n_2 N . By putting together the maps
gd:R+d(G) = R(d; G) -!D(d)(A) = Div+d(G)(A);
we get a function f :R+(G) -!Div+(G)(A). Next, for any d; e 0 we have projecti*
*ons GL(d) -
GL(d; e) -!GL (e) and we can use the resulting maps to identify R((d; e); G) wi*
*th R+d(G)xR+e(G)
and D(d; e)(A) with Div+d(G)(A) x Div+e(G)(A) and g(d;e)with gd x ge. There are*
* evident maps
: GL(d; e) -!GL (d + e) and : GL(d; e) -!GL (de), and using the naturality of g*
* with respect
to these maps_we find that f is a semiring homomorphism. Similarly, we have map*
*s k: GL(d) -!
GL( dk) in G and the naturality of g with respect to these maps implies that f *
*commutes with
-operations. It is clear that f(R+d(G)) Div+d(G)(A), so f 2 XCh(G)(A). We defi*
*ne a map
ae: Y (G) -!XCh(G) by ae(g) = f. Because gn_= gn1x : :x:gnr we see that ae is i*
*njective.
Now suppose we start with a point f 2 XCh(G)(A). Let gd:R(d; G) -! D(d)(A) b*
*e the
restriction of f :R(G) -!Div(G)(A), and put
gn_= gn1x : :g:nr:R(n_; G) -!D(n_)(A):
Q
We need to checkQthat this gives a natural transformation. As R(m_; G) = iR(*
*mi; G) and
D(m_)(A) = iD(mi)(A), it suffices to check naturality for maps u: n_-!d in eN,*
* or equivalently
(by Lemma 16.5) for homomorphisms u: U(n_) -! GL(d). We need to show that the l*
*eft hand
square in the following diagram commutes:
R(n_; G)______wR+d(G)u*v_____R(G)_w
| | |
gn|_ gd| |f
| | |
|u |u |u
D(n_)(A)_____wDiv+d(G)u*v__Div(G)(A):_w
The right hand square commutes and the two right hand horizontal maps are injec*
*tive so it
suffices to show that the two composite maps R(n_; G) -!Div(G)(A) are the same.*
* We call these
two maps ff(u) and fi(u). Let F be the set of all functions from R(n_; G) to Di*
*v(G)(A), thought
of as a ring with pointwise operations. It is formal to check that ff(u + v) = *
*ff(u) + ff(v) and
ff(uv) = ff(u)ff(v), so ff is a homomorphism of semirings from R+(U(n_)) to Div*
*(G)(A). It can thus
be extended to aNring map R(U(n_)) -! Div(G)(A), and the same applies to fi. It*
* is well-known
that R(U(n_)) = iR(U(ni)) so it suffices to check that ff = fi on R(U(ni)) fo*
*r all i. This reduces
us to the case where n_= (e) say. It is also well-known that R(U(e)) = Z[1; : :*
*;:e][(e)-1], so it
suffices to check that ff(j) = fi(j), which is true because f is a homomorphism*
* of -rings.
This shows that g 2 Y (G)(A), and clearly ae(g) = f. Thus ae is surjective a*
*nd_hence an
isomorphism. |_*
*_|
17.A result on restrictions of characters
Theorem 17.1. Let G be a finite group with a normal subgroup N such that |N| is*
* coprime to
|G=N|. Then the restriction map R+(G) -!R+(N)G is surjective.
The proof will follow after some preliminary results.
Lemma 17.2. Let H be a group, and let W; X; Y be H-sets, with equivariant maps *
*W -f!X -q Y .
Then there is an equivariant map "f:W -! Y with qf"= f iff for each w 2 W there*
* exists y 2 Y
with q(y) = f(w) and stabH(y) stabH(w).
Proof.Write W as a disjoint union of orbits. *
* |___|
Lemma 17.3. Let G and N be as above, and let ae: N -!GL(V ) be an irreducible r*
*epresentation
of N whose character is stable under G. Then there is a homomorphism oe :G -!GL*
*(V ) extending
ae.
Proof.Suppose g 2 G, and define aeg:N -!GL(V ) by aeg(x) = ae(gxg-1). By hypoth*
*esis, this has
the same character as ae, so there exists an intertwining operator :V -! V suc*
*h that aeg(x) =
-1ae(x) for all x 2 N. As V is an irreducible representation of N we see that A*
*utN(V ) = C
CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY 31
and thus is unique up to multiplication by a scalar matrix. We can thus define*
* a map OE: G -!
P GL(V ) by OE(g) = []; this is a homomorphism making the following diagram com*
*mute.
N v_________G_w
| |
a|e |OE
| |
|u |u
GL(V ) _____PwGL(Vw):ss
Put n = |N| and d = dimC(V ). As V is irreducible we know that d divides n. Put*
* Y = {ff 2
GL(V ) | det(ff)n = 1}, and note that ss :Y -! P GL(V ) is surjective and ae(N)*
* Y . Let N2 act
on G by (x; y):g = xgy-1 and on GL(V ) by (x; y):ff = ae(x)ffae(y)-1.
We claim that there is an N2-equivariant map i :G -!Y such that ssi = OE and *
*i = ae on N.
Clearly G = N q (G \ N) as N2-sets and ae: N -!Y is N2-equivariant, so it suffi*
*ces to define i on
G\N. Fix g 2 G\N, and choose as before. After multiplying by a suitable scalar*
*, we may assume
that det() = 1 so 2 Y . By Lemma 17.2, it will suffice to show that stabN2(g) *
* stabN2().
Suppose that (x; y) stabilises g, so xgy-1 = g, so y = g-1xg. By the definitio*
*n of we have
ae(y) = -1ae(x), or in other words ae(x)ae(y)-1 = , so (x; y) stabilises , as r*
*equired.
Now define :G2 -!Y by (g; h) = i(h)i(gh)-1i(g). Clearly ss(g; h) = 1, and th*
*e kernel of
ss :Y -! GL(V ) is the group Cnd of nd'th roots of unity, so we can regard as *
*a map G2 -!Cnd.
As i_is equivariant, it is easy to check that (xg; hy) = (g; h) for x; y 2 N, s*
*o we get an induced
map :(G=N)2 -!Cnd. One also sees directly that for g; h; k 2 G=N we have
_ _ -1_ _ -1
(h; k) (gh; k) (g; hk) (g; h) = 1;
_
so is a 2-cocycle. On the other hand nd divides n2 and thus is coprime to |G*
*=N|, so we
have H2(G=N; Cnd) = 0. We can thus choose a function ! :G=N -! Cnd such that (*
*g; h) =
!(h)!(gh)-1!(g) for all g; h 2 G. By putting g = h = 1 we see that !(1) = 1 and*
* thus !(x) = 1
for x 2 N. We define oe(g) = !(g)-1i(g); this clearly gives a homomorphism G -!*
* GL(V_) with
oe|N = ae, as required. *
* |__|
Proof of Theorem 17.1.For each irreducible representation ae of N, let ae0denot*
*e the sum of the
inequivalent G-conjugates of ae. Any G-invariant character is a direct sum of c*
*opies of the characters
of the representations ae0, so it suffices to show that ae0extends to a represe*
*ntation of G. Let H be
the stabiliser of Oae, so N CH G. Lemma 17.3 implies that ae can be extended t*
*o a representation
oe of H, and one sees from the Mackey formula that resGNindGH(oe) ' ae0, so ind*
*GH(oe)_is the required
extension of ae0. *
* |__|
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