16.3.19*
*93
LOCALIZATIONS OF UNSTABLE A - MODULES
AND EQUIVARIANT MOD p COHOMOLOGY
by
Hans-Werner Henn, Jean Lannes and Lionel Schwartz
Introduction
Let p be a fixed prime and G be either a compact Lie - group (not necessarily c*
*onnected)
or a discrete group of finite virtual cohomological dimension (f.v.c.d. for sh*
*ort), e.g. the
general linear group over the ring of S-integers in a number field or the mappi*
*ng class group
of an orientable surface. In [Q1] Quillen considered the category A(G) whose ob*
*jects are the
elementary abelian p - subgroups of G and whose morphisms are given as composit*
*ions of
inclusions and conjugations. He showed that the restriction homomorphisms from *
*the mod
p cohomology H*BG of the classifying space BG to H*BE for E 2 A(G) induce a map
qG : H*BG -! limA(G)opH*BE such that the kernel of qG consists of nilpotent ele*
*ments
and such that for each element in the limit a sufficiently large pn - th power *
*is in the image
of qG . (As usual, A(G)op denotes the opposite category of A(G), so that E 7! H*
**BE is a
covariant functor on A(G)op.)
In this paper we will consider H*BG as an unstable module over the mod p Steenr*
*od algebra
A (unstable module for short) and show how this structure can be used to refine*
* Quillen's
result and describe a finite sequence of approximations to H*BG starting with Q*
*uillen's map
qG and ending with a genuine isomorphism. To do this we consider the full subc*
*ategory
Niln of the category U of all unstable modules; Niln is the smallest subcategor*
*y of U which
contains all n - fold suspensions and is closed with respect to forming extensi*
*ons and taking
filtered colimits. Here is an explanation for our terminology. The unstable mod*
*ule underlying
an unstable algebra (e.g. the mod p - cohomology of a space) is an object in Ni*
*l1 if and only if
all its elements are nilpotent in the usual sense. The subcategory Niln is loca*
*lizing, i.e. there
exist a localization functor Ln and a natural transformation n : idU -! Ln, the*
* localization
away from Niln. Quillen's map qG is actually localization away from Nil1. The s*
*equence of
approximations referred to above will be the sequence of localizations away fro*
*m Niln and
the following result shows that this sequence of localizations is in many cases*
* actually finite.
THEOREM 0.1 [He]. Let K be an unstable algebra which is finitely generated as a*
*n algebra.
Then the localization away from Niln is an isomorphism for all sufficiently lar*
*ge n.
The localization LnH*BG and the localization map n can be roughly described as *
*follows.
As noted above the map 1 arose from the product of restriction homomorphisms H**
*BG -!
1
Q * * Q *
E2A(G) H BE, and L1H BG was given by the subalgebra of E2A(G)H BE consisting
of those families of elements {xE }E2A(G) which satisfy the compatibility condi*
*tions imposed
by the morphisms in Quillen's category A(G). The description of n is similar. L*
*et CG (E)
denote the centralizer of E in G and (H*BCG (E)) |x|; if p is odd and e 2 {0; 1} then fieP ix = 0 for all 2i + e > |*
*x|. Here |x| denotes
the degree of the element x.
The category of unstable modules and A - linear maps will be denoted by U.
1.2._If M is an A-module, then its suspension M is the A-module whose underlyin*
*g graded
vector space is defined by (M)n = Mn-1 and whose A-module structure is given by*
* x =
(-1)|x|x, 2 A. (As usual x 2 Mn-1 is denoted x if considered as element in (M)*
*n.)
If M is unstable, then M is also unstable, but, of course, the converse fails, *
*and this failure
is the point of departure for this paper.
DEFINITION 1.3. An unstable A-module N is called n-nilpotent iff every finitely*
* generated
submodule admits a finite filtration such that each filtration quotient is an n*
*-fold suspension.
1.4._We note that 1 - nilpotent is nilpotent in the sense of [HLS1,HLS2]. Furt*
*hermore n -
nilpotent in the sense of this definition is (n - 1) - nilpotent in the sense o*
*f [S1]. There is also
a characterization of n - nilpotent modules in terms of Steenrod operations as *
*follows [S1,S2]:
If p = 2 the operations
Sqk : Nm ! N2m-k ; x 7! Sqm-k x
have to be locally nilpotent (i.e. act nilpotently on each element x) for all 0*
* k < n; if p is
odd we write k = 2l + e with e 2 {0; 1} and replace Sqk by
Pl: N2m+e ! N2mp+e-2l(p-1); x 7! P m-lx :
4
If n = 1 and N is the mod-p-cohomology of a space this means that the p - th po*
*wer map is
locally nilpotent in the classical sense. This fact justifies the terminology.
1.5._The full subcategory of U whose objects are the n - nilpotent modules will*
* be denoted by
Niln. Clearly this is the smallest "Serre class" in U which contains all n - fo*
*ld suspensions
and is closed with respect to taking colimits. For our purposes it is very impo*
*rtant that these
subcategories can be described in terms of certain injective objects in U as fo*
*llows.
If V is an elementary abelian p-group, we will often write H*V for the mod p -c*
*ohomology
of its classifying space BV . Furthermore we have the Brown-Gitler modules J(k)*
* which can
be characterized as representing objects for the functors M 7! (Mk)* where (Mk)*
** denotes
the dual of Mk, the subspace of M of elements of degree k.
THEOREM 1.6 [LZ1]. The unstable modules H*V J(k) are injective in U for all ele*
*mentary
abelian p-groups V and all natural numbers k.
Note that for k = 0 this is just the injectivity of H*V which was essentially p*
*roved in [C],
[Mil].
THEOREM 1.7 [LS,S1,S2]. An unstable module N is n-nilpotent iff Hom U(N; H*V J(*
*k)) =
0 for all V and all k < n.
1.8._In this section we start investigating the quotient category U=Niln which *
*is defined as
follows:
The objects of U=Niln are the same as those of U.
For L and M in U=Niln we have Hom U=Niln(L; M) = colimHom U(L0; M=M0), where the
colimit is taken over all pairs (L0; M0) where L0 is a submodule of L such that*
* L=L0 is n -
nilpotent and M0 is an n - nilpotent submodule of M.
1.9._Let I k. Clearly H*V J(k) is not Nilk - reduced.
b) If 0 ! M1 ! M2 ! M3 is an exact sequence with M2 being Niln - closed and M3 *
*being
Niln - reduced, then M1 is Niln - closed.
c) By (a) and (b) H*V B is Niln - closed for each B 2 U which is trivial in de*
*grees n.
d) If M is Nil - closed, then n-1M is Niln - closed.
e) If F is in F d0M.
3.8._It is clear that Ker n-1;M= Kern;M injects into Ker on-1;M and that we can*
* identify
f 1 in this way to become convinced of this). It is often b*
*etter to
try to describe LnM inductively. This observation gives additional impetus to t*
*urn attention
towards analyzing the maps on;M. Although our analysis will not be complete it*
* will be
helpful in particular cases (cf. II.5).
We consider the map f 0 is fixed and ff is in obA(G)] = morA(G) then
(ff) : H*GX -! H*dff (H*CG(rff)Xrff) d0(H*GX) this map is a monomorphism and for n > d1(H*GX)*
* this
map is an isomorphism and hence H*GX is determined by H*CG (E) in degrees less *
*or equal
to d1(H*GX). More precisely H*GX is determined by the " funct* *or" (in sense 1.14) a(g) u d0H*BG. Then the maps (idE ) for E 2 A(G) i*
*nduce
a monomorphism Y
H*BG -! H*BE (H*BCG (E)) d0(H*BG).
Proof:_If x is in N(G) then x restricts trivially to H*BE for all E 2 A(G) and *
*hence
(idE )(x) 2 H*BE (He*BCG (E)) m,
then H*V B is in Cm;d if d = dimV .
THEOREM 1.4. Let V be an elementary abelian p-group of rank d and M a smooth *
*V -
manifold of dimension m (possibly with boundary). Assume that the subspace of M*
* consisting
of all points with isotropy group of rank i has finitely many components for ea*
*ch 0 i d.
Then H*VM is in Cm;d.
Clearly a smooth compact V - manifold satisfies the assumptions of Theorem 1.4 *
*and hence
Theorem 1.1 follows. In fact, the proof will show that in this case the Ik in D*
*efinition 1.2.
can be required to be finite direct sums.
Proof:_This depends heavily on a result of Duflot ([D1]) which we will briefly *
*recall.
Duflot considers the filtration
; = M<0 M<1 M<2::: M 0.
If d = 0 we have H*MW;d = eH*TW;d. Because W acts trivially and V=W acts freely*
* on MW;d
we get
H*VMW;d ~=H*V H*V=W H*V=W(MW;d) ~=H*V H*V=W H*((V=W )\MW;d)
and we are done by 1.3.3 above. _*
*__
|*
*__|
Duflot treats only the case p odd in her paper. However, her proof carries over*
* verbatim to
the case p = 2 if Chern classes are replaced by Stiefel Whitney classes.
2. The invariants d0 and d1 in the case of a general compact Lie group
2.1._Throughout this section we will consider the following data.
U is a compact Lie group, S an elementary abelian p - subgroup of G and T a clo*
*sed subgroup
of U containing S such that H*BT is (via restriction) a finitely generated free*
* H*BU - module
and H*BS is (via restriction) a finitely generated free H*BT - module.
Furthermore we assume that one of the following equivalent conditions holds.
a) The homomorphisms ss0S ! ss0T and ss0T ! ss0U are onto (and hence ss0T and s*
*s0U are
p-groups).
b) The natural action of ss0T on H*(T=S) and of ss0U on H*(U=T ) is nilpotent.
c) The natural action of ss0T on H*(T=S) and of ss0U on H*(U=T ) is trivial.
32
The implications (a) ) (b) and (c) ) (a) are easy. (For the second one note tha*
*t triviality
of the action in degree 0 imply that T=S and U=T are connected.) For (b) ) (c)*
* one can
use the Eilenberg - Moore spectral sequence of the fibrations BS ! BT with fibr*
*e T=S and
BT ! BU with fibre U=T to see that for each the cohomology of the total space s*
*urjects
onto the cohomology of the fibre which implies that the action is trivial.
2.2_Examples._a) For any prime p we can take U the unitary group U(n) or the sp*
*ecial
unitary group SU(n) or the symplectic group Sp(n): in each of these cases T can*
* be taken
as a maximal torus of U and S as p - torus, i.e. as the subgroup of T consistin*
*g of elements
of order p.
b) For any prime p we can take U to be a torus, T = U and S the p - torus of T .
c) For p = 2 we can take U the orthogonal group or the special orthogonal group*
* and T = S
a 2 - torus.
d) If (U1; T1; S1) and (U2; T2; S2) satisfy (2.1) then (U1 x U2; T1 x T2; S1 x *
*S2) does, too.
THEOREM 2.3. Let G be a compact Lie group and M a compact smooth G - manifold
(possibly with boundary). Assume that the triple (U; T; S) satisfies 2.1.(a) - *
*2.1.(c) and that
G embedds into U. Then
a) d0H*GM dimM + dimU=G
b) d1H*GM dimM + dimU=G + max{dim U=T; dimT }.
Theorem 2.3 and the examples above show how estimates for d0 and d1 can be obta*
*ined from
knowledge of the representation theory of G.
Theorem 2.3 is not immediately applicable to d0H*BG and d1H*BG if G is a discre*
*te_group __
of f.v.c.d.. However, for such groups there is often a finite quotient group G *
*and a finite G
- CW - complex X such that H*BG is isomorphic to H*_GX. Important examples are *
*given
by general linear groups over rings of S - integers in a number field [Se] and *
*mapping class
groups of orientable_surfaces [Ha]. Now such an X has the equivariant homotopy *
*type of a
compact smooth G - manifold M with boundary and Theorem 2.3 applies to H*_GM ~=*
*H*BG.
Theorem 2.3 will be a consequence of the following result which is essentially *
*taken from
[Q1,sect.6].
PROPOSITION 2.4. Assume U is a compact Lie group and T is a closed subgroup suc*
*h that
H*BT is (via restriction) a finitely generated free H*BU - module and such that*
* the action
of ss0U on H*(U=T ) is trivial. Let G be a closed subgroup of U and let X be an*
*y G - space.
Then the following (coequalizer) diagram of G - spaces
pr2 pr
X x (U=T ) x (U=T ) -!-!X x (U=T ) -! X
pr1
33
(in which pr1 resp. pr2 resp. pr denote the appropriate projection maps and G a*
*cts diagonally
on the products) induces an equalizer diagram of unstable algebras
* pr*2
H*GX pr-!H*G(X x (U=T )) -!-!H*G(X x (U=T ) x (U=T )) :
pr*1
_*
*__
|*
*__|
Proof_of_2.3:_Consider the action of the group G x S on M x U given by ((g; s);*
* (m; u)) 7!
(gm; gus-1). Both G and S act freely and hence we get isomorphisms
H*G(M x (U=S)) ~=H*GxS(M x U) ~=H*S(G\(M x U))
and similarly
H*G(M x (U=S) x (U=S)) ~=H*GxSxS(M x U x U) ~=H*SxS(G\(M x U x U))
(with G still acting diagonally.)
Furthermore the S x S - action on G\(M x U x U) is still free when restricted t*
*o either of
the two factors and hence 2.4 gives us an exact sequence
(2:5) 0 -! H*GM -! H*S(G\(M x U)) -! H*S((S x G)\(M x U x U)) :
From this sequence and II.1.1 we get the claimed estimate for d0 , i.e.
(2:6) d0H*GM dimM + dimU=G
and also
(2:7) d1H*GM dimM + 2 dimU - dimG :
In order to get the claimed estimate for d1 we consider now the action of G x T*
* on M x U
given as above by ((g; t); (m; u)) 7! (gm; gut-1). The same reasoning as above *
*gives us an
exact sequence
0 -! H*GM -! H*T(G\(M x U)) -! H*T((T x G)\(M x U x U))
and hence by I.3.6.b)
d1H*GM max{d1H*T(G\(M x U)); d0H*T((T x G)\(M x U x U))} :
Now we apply (2.6) and (2.7) with G and U both replaced by T , and M replaced b*
*y G\(MxU)
resp. (T x G)\(M x U x U). We obtain
d1H*T(G\(M x U)) dimM + dimU=G + dimT
d0H*T((T x G)\(M x U x U)) dimM + 2 dimU - dimT - dimG
and we are done. _*
*__
|*
*__|
34
The proof shows that there are extensions of 2.3 to noncompact manifolds simila*
*r to Theorem
II.1.4.
3. The invariants d0 and d1 in the case of the symmetric construction
3.1._Throughout this section the prime p will be 2. We consider the following p*
*roperties of a
class D of unstable modules contained in Nil1:
(3.1.1) If M 2 D and M0 M then M0 2 D.
(3.1.2) If M1 and M2 are in D then M1 M2 is in D.
(3.1.3) If M 2 D and R is reduced then M R 2 D.
DEFINITION 3.2. For a class D contained in Nil1 we say that an unstable module *
*M belongs
to the class C(D) iff
(1) M is reduced and
(2) the cokernel of the localization map 1;M belongs to D.
Because the cokernel of 1;M is always in Nil1, the assumption that D is contain*
*ed in Nil1
is no restriction (in connection with Definition 3.2).
The proof of the following result is straightforward.
PROPOSITION 3.3.a) Suppose D satisfies (3.1.1). Let 0 ! M ! M0 ! M1 be an exact
sequence of unstable modules such that M0 is Nil - closed and M1 is in D. Then *
*M belongs
to C(D).
b) Suppose D satisfies (3.1.1). Let 0 ! M ! M0 ! M1 be an exact sequence of uns*
*table
modules such that M0 belongs to C(D) and M1 is reduced. Then M belongs to C(D).
c) Suppose D satisfies (3.1.2). If M1 and M2 are in C(D) then M1 M2 is in C(D).
d) Suppose D satisfies (3.1.2) and (3.1.3). If M1 and M2 are in C(D) then M1 M*
*2 is in
C(D).
_*
*__
|*
*__|
3.4._Each of the following examples of classes D has all three properties (3.1.*
*1) - (3.1.3).
a) D = Nil1. In this case C(D) consists of all reduced modules. (i.e. d0 = 0)
b) D = Niln. In this case C(D) consists of all modules with d0M = 0 and L1M ~=L*
*2M ~=
::: ~=LnM.
c) D equal to the class of modules which are both 1 - nilpotent and Niln+1 - re*
*duced. In this
case C(D) consists of all modules with d0M = 0, d1M n.
35
d) D equal to the class of modules which are both n - nilpotent and Niln+1 - re*
*duced (or
equivalently are n - fold suspensions of Nil1 - reduced modules). In this case *
*C(D) consists
of all modules with d0M = 0, d1M n and L1M ~=L2M ~=::: ~=LnM.
e) D = 0 the class consisting of the trivial module. Then C(D) is just the clas*
*s of nilclosed
modules. (i.e. d0 = d1 = 0)
Here is the main result of this section. We denote the symmetric group on n -le*
*tters by Sn
and the symmetric construction ESn xSn Xn on a space X by SnX.
PROPOSITION 3.5. Suppose D is a class satisfying conditions (3.1.1) - (3.1.3). *
*Let X be a
space such that H*X belongs to C(D). Then H*SnX belongs to C(D).
The case that H*X is Nil - closed, which is example 3.4.e) above, was already p*
*roved in
[GLZ]. The proof in the general case uses the same strategy as in [GLZ].
Proof_of_3.5:_We will give the proof for n = 2. The general case follows then a*
*s in [GLZ].
Consider the homomorphisms H*S2X ! H*X H*X and H*S2X ! H*BS2 H*X
induced by the projection q : ES2 x X x X ! S2X resp. the Steenrod diagonal :
BS2 x X ! S2X. The image of q* are the invariants (H*X H*X)S2 with respect to *
*the
action of S2 on H*X H*X given by permuting the factors.
The study of the image of * leads to the functor R1 : U ! U of W.M.Singer [Si].*
* If we
identify H*BS2 with F2[u] then R1MPcan be described as the F2[u] - submodule of*
* F2[u]M
generated by the elements St1m := iui Sq|m|-im for all m 2 M (cf. [LZ2]). I*
*n case
M = H*X this submodule agrees with the image of *. The functor R1 comes with a *
*natural
surjection ae : R1M ! M where denotes the "doubling functor"; ae sends uR1M to*
* 0 and
St1m to m, the "double of m". As M is isomorphic to bH0(S2; M M) (with S2 again
acting by permuting the factors and Hb0 denoting 0 - th Tate cohomology, i.e. *
*"invariants
divided by norms") we have a natural map (M M)S2 ! M and we get the following
diagram
*
H*S2X -! R1H*X
?? ?
yq* ?yae
(H*X H*X)S2 -! H*X
which can be checked to be a pull-back diagram (cf. [Mi], [Z]). This diagram l*
*eads to the
definition of a functor S2 : U ! U which associates to an unstable A - module M*
* the fibre
product of (M M)S2 and R1M over M. Now 3.5 will follow from the next result.
_*
*__
|*
*__|
PROPOSITION 3.6. Suppose D is a class satisfying conditions (3.1.1) - (3.1.3). *
*Let M be
an unstable A - module which belongs to C(D). Then S2M belongs to C(D).
36
Proof:_Because of 3.3 it suffices to show that R1M belongs to C(D) whenever M i*
*s in C(D).
This in turn is a consequence of the fact that R1M is Nil - closed whenver M is*
* Nil - closed
[GLZ, Lemme 2.1.2.].
In fact, assume M 2 C(D). Then we have an exact sequence 0 ! M ! L1M ! D ! 0
with D 2 D. Now R1 is exact and hence we get an exact sequence 0 ! R1M ! R1L1M !
R1D ! 0, hence by definition of R1 an exact sequence 0 ! R1M ! R1L1M ! H D and
we are done. _*
*__
|*
*__|
3.7._We close with some remarks. Proposition 3.5 applied to the class D of exam*
*ple 3.4.c)
implies the following: if d0M = 0, d1M n then d0S2M = 0, d1S2M n. More genera*
*lly
one can show d0S2M 2d0M, d1S2M d0M + d1M for each unstable module M.
The situation for p odd is more subtle. Although H*SnX is reduced whenever H*X*
* is
reduced ([Q2]) Proposition 3.5 does not hold for general D. For example H*SnX n*
*eed not
be Nil - closed if H*X is Nil - closed [GLZ].
4. Examples I
4.1._Let G and H be two compact Lie groups and consider mod p cohomology for an*
*y prime
p. Then I.3.6 gives
d0H*B(G x H) = d0H*BG + d0H*BH
d1H*B(G x H) = max{d0H*BG + d1H*BH; d1H*BG + d0H*BH} :
4.2._Let G be a finite abelian group and p be any prime. Because of 4.1 we onl*
*y have to
consider the case that G is of order a power of p.
If G = Z=p then d0 = d1 = 0 by I.1.6 and (the easy part) of I.1.7.
If G = Z=pn for some n > 1, then d0 = 1, d1 = 2. To see this note that in this*
* case we
have a splitting of A - modules H*BG ~=HevenBG HoddBG into the even and odd pa*
*rt.
Furthermore we have HoddBG ~=HevenBG and hence it suffices to show d0HevenBG = *
*0,
d1HevenBG = 1 which follows easily from the obvious exact sequence 0 ! HevenBG !
H*BZ=p ! HoddBG ! 0. Observe that these values for d0H*BG and d1H*BG agree with
the upper bounds obtained from II.2.3 (using an embedding of G into U(1).)
4.3._If p is any prime then H*BS1 can be identified with the even part of H*BZ=*
*pn (n > 1)
and hence d0(H*BS1) = 0, d1(H*BS1) = 1.
If p = 2 then there is an exact sequence 0 ! H*BS3 ! H*BS1 ! 2H*BS3 ! 0 from
which we deduce d0(H*BS3) = 0, d1(H*BS3) = 2. Note that in these examples the u*
*pper
bounds obtainable from II.2.3 are sharp.
37
If p is odd then H*BS3 is a direct summand of H*BS1 and we find d0(H*BS3) = 0,
d1(H*BS3) = 1.
4.4._If p = 2 and G = D8, the dihedral group of order 8, then d0 = d1 = 0 by II*
*.3.5. For
the dihedral groups D2n of order 2n we get the same invariants as their mod 2 c*
*ohomology
happens to be isomorphic (in K) to the one of D8 (cf. [MiP]).
4.5._If p = 2 and G = Sn, the symmetric group on n letters, then again d0 = d1 *
*= 0 by II.3.5.
4.6._Let p = 2 and G be a quaternion group Q2n of order 2n. Then G embedds into*
* S3 and
hence II.2.3 implies d0 3, d1 5. If we look at the known computation of H*BG*
* (cf.
[MiP]), which can be interpreted as giving an isomorphism of unstable algebras *
*H*BG ~=
H*BS3 H*(S3=G), and if we apply the tensor product formula I.3.6 (together wit*
*h 4.3
above) we see that these estimates are sharp.
These invariants can also easily be read off from the 2 - local stable splitting
BQ2n ' BSL2Fq _ -1(BS3=BN) _ -1(BS3=BN)
described in [MiP]. (Here Fq is a finite field of odd order q such that Q2n is *
*a 2 - Sylow
subgroup of SL2Fq, i.e. q2 - 1 = 2nq0 with q0 odd, and N is the normalizer of a*
* maximal
torus in S3.) If u4 is the periodicity operator in H4BG then H*BSL2Fq ~=F2[u4] *
* 3F2[u4]
and H*-1(BS3=BN) ~=F2[u4] J(2) (each time this is an isomorphism in U) and hen*
*ce
we get
d0(H*BSL2Fq) = 3; d1(H*BSL2Fq) = 5
d0(H*-1(BS3=BN)) = 2; d1(H*-1(BS3=BN)) = 4 :
4.7._Letnp-=12 and G = SD2n the semidihedralngroup-of2order 2n with n > 3, i.e.*
* G =
. Now G embedds into U(2) (via the induce*
*d repre-
sentation of a faithful representation of the cyclic subgroup generated by x) a*
*nd hence II.2.3
gives the upper bounds d0 4, d1 6.
On the other hand if Fq is a finite field with q 3 mod 4 and q2 - 1 = 2n-1q0 w*
*ith q0 odd
then G is a 2 - Sylow subgroup of GL2Fq and Martino [Ma] (see also [MaP]) showe*
*d that
there is a 2 - local stable splitting
BSD2n ' BGL2Fq _ -1(BS3=BN) :
We will see in II.5.4 below that
d0(H*BGL2Fq) = 0; d1(H*BGL2Fq) = 2 ;
and hence 4.6 implies
d0(H*BSD2n) = 2; d1(H*BSD2n) = 4 :
38
5. Examples II; the localizations L1, L2 and L3 for some general linear groups
In this section we will illustrate our theory in the case of the mod p cohomolo*
*gy of the groups
GL(n; ) for various rings which will be taken from the following list consisti*
*ng of: the
real numbers R, the complex numbers C, the finite fields Fq or the ring of S - *
*integers in an
algebraic number field. In all these cases Theorem I.5.4 holds.
5.1._Fix a prime p. If we denote the Quillen category of GL(n; )) by A(n; ) and*
* Rector's
category of H*BGL(n; ) by R(n; ), then we get isomorphisms
L1H*BGL(n; ) ~= lim H*BE ~= lim H*BV :
E2A(n;) (V;')2R(n;)
5.2._From now on we will always assume that satisfies in addition the followin*
*g assumptions:
a) p is invertible in .
b) contains all p - th roots of unity.
c) Projective modules over are free.
In this case the categories R(n; ) are very simple. By (a) - (c) every represe*
*ntation of
an elementary abelian p -group V on n is isomorphic to a direct sum of one dime*
*nsional
representations.
After choosing an embedding of Z=p into the group of units x and using I.5.3 w*
*e can
identify the objects in R(n; ), i.e. the faithful representations of elementar*
*y abelian p -
groups V , with formal sums OnOO with O running through the characters of V , t*
*he nO
being nonnegative integers such that OnO = n and such that the set of O with nO*
* > 0 spans
V *, the group of characters of V . Furthermore, if ' = OnOO (resp. '0= On0OO) *
*are faithful
representations of V (resp. W ) then Hom R(n;)('; '0) = {ff 2 Hom (V; W )|' = O*
*n0OOff}.
In particular we see that the categories R(n; ) are independent of . We will de*
*note them
by Rn.
Now choose n - linearly independent characters O1, O2, ..., On of an n - dimens*
*ional Fp -
vector space Vn and consider the representation '0 := O1 + O2 + ::: + On as ele*
*ment in Rn.
Then it is easy to see that for each ' 2 Rn the set Hom Rn('; '0) is non-empty *
*and the action
of the group AutRn ('0), i.e. of the symmetric group Sn on n letters, on this s*
*et is transitive.
This implies that L1H*BGL(n; ) can be identified with the invariants (H*BVn)Sn *
*or with
(H*BEn)Sn where En is the subgroup of all diagonal matrices in GL(n; ) whose di*
*agonal
entries are p -th roots of unity.
5.3._Next we consider L2H*BGL(n; ). For simplicity we specialize to the case p *
*= 2 and
take to be either the real numbers R , the complex numbers C , a finite field *
*Fq of odd
order or the ring Z[1_2]. In all these cases conditions (a) - (c) of 5.2 above *
*are satisfied.
39
We abbreviate H*BGL(n; ) by M(n; ) or by M(n) if is irrelevant or clear from t*
*he
context. Our analysis will use the sequence
0 -! mk2M(n; ) -! L2M(n; ) -o1!L1M(n; ) -! mc1M(n; )
of I.3.8 and for this we will determine the exact sequence of functors in F<2
<2o1
0 -! k2M(n; ) -! f<2 L2M(n; ) f-! f<2 L1M(n; ) -! c2M(n; ) -! 0 :
In fact, by I.4 it is clear that we can work with the corresponding functors on*
* the category
Rn (instead of E) for which we will still use the same notation.
5.3.1._We will list the values of these functors in the table below.
For this we will introduce the following notation: Fn will denote the functor f*
*<2 L1M(n; )
(considered as functor on Rn); by 5.2 above this functor is independent of . Th*
*e constant
functor on Rn with value F2 will be denoted by F2_; this is also the subfunctor*
* of Fn con-
sisting of the homogeneous elements of degree 0. The quotient of Fn by this sub*
*functor is a
suspension of a functor which we will call eFn.
______________________________________________________________________________*
*__<2<2
|___________________|__k2M(n)_____|__f___L2M(n)___|__f___L1M(n)___|_c2M(n)____*
*__|
|_________R_________|______0______|______Fn_______|______Fn_______|_____0_____*
*__|
|_________C_________|______0______|______F2_______|______Fn_______|____Fen____*
*__|
|__Fq,_q__3_mod_4___|______0______|______Fn_______|______Fn_______|_____0_____*
*__|
|__Fq,_q__1_mod_4___|____Fen______|___F2_Fen______|______Fn_______|____Fen____*
*__|1
|_______Z[_2]_______|____Fen______|___Fn__Fen_____|______Fn_______|_____0_____*
*__|
We pause to explain this table:
a) The entries for = R are obtained from the a priori knowledge that H*BGL(n; *
*R) is Nil
- closed and hence equal to all its localizations.
b) The functors f<2 L2M(n; ) are determined byQI.5.2: if ' = OnOO is in Rn then*
* the
centralizer G' is isomorphic to the product OGL(nO; ) and hence we find
O
f<2 M(n; )(OnOO) ~=( H*BGL(nO; ))<2 :
O
c) Now (b) together with Fn = f<2 L2H*BGL(n; R) implies that
O
Fn(OnOO) ~=( (H*EnO)SnO )<2
O
where EnO is the appropriate diagonal elemenentary abelian 2 - subgroup of GL(n*
*O; ) as in
5.1 above.
40
d) The entries for = C are obtained from the vanishing of H1BGL(n; C) for all *
*n.
e) In all cases, the determinant induces an isomorphism between the mod 2 homol*
*ogy
H1(BGL(m; ); F2) and x F2. Hence we get H1BGL(m; Fq) ~=F2 and H1BGL(m; Z[1_2])
~=F2 F2.
f) Then one showsNthat the map f<2 o1(OnOO)Nabove identifies with the restricti*
*on homo-
morphism from ( O H*BGL(nO; ))<2 to ( O (H*EnO)SnO )<2 and hence is an isomor*
*phism
if = Fq with q 3 mod 4, trivial if = Fq with q 1 mod 4 and an epimorphism*
* if
= Z[1_2]. It follows that the remaining values of k2M(n) and c2M(n) are as cl*
*aimed.
We leave it now to the reader to verify that the remaining values of f<2 L2M(n)*
* are also as
stated. Note that the direct sum decompositions not only respect the A - module*
* structure
but also the "comodule structure" on these functors.
5.3.2._We already know that m<2(Fn) ~= (H*BVn)Sn . So in order to *
*describe
L2H*BGL(n; ) we need to understand the unstable A - modules
m<2(Fen) ~=m(Fen) ~= limReFnr H*d ~= Hom Ropn((Fen)*; H*)
] n E
where (Fen)* is the contravariant functor whose value on ' 2 Rn is (Fen('))*. O*
*f course, H*
is the functor whose value on ' : K -! H*V is H*V .
The group AutRn ('0) ~=Sn acts on (Fen)*('o) ~=(F2)n by permuting the summands,*
* i.e.
(Fen)*('0) ~=F2[Sn-1\Sn] as right F2[Sn] - module. Furthermore it is easy to se*
*e that the
natural map O
FpHom Rn('; '0) (Fen)*('0) -! (Fen)*(')
FpAutRn('0)
is an isomorphism for each ' 2 Rn.
Consequently we get
m(Fen) ~=Hom F2[Sn](F2[Sn-1\Sn]; H*Vn) ~=(H*Vn)Sn-1 :
For = C left exactness of m<2 yields an exact sequence
0 -! L2H*BGL(n; C) -! (H*BVn)Sn -! (H*BVn)Sn-1 :
It is not hard to check that this identifies L2H*BGL(n; C) with the squares in *
*(H*BVn)Sn ,
and hence one can conclude that H*BGL(n; C) is Nil2 - closed. (Of course, this *
*can also be
seen more directly!)
The computation of L2H*BGL(n; ) for the remaining values of is now immediate.
5.4._In the remainder of this section we will outline how our methods can be us*
*ed to recover
Quillen's computation of the mod 2 - cohomology of GL(n; Fq) if q 3 mod 4. We*
* denote
the elementary abelian 2 - subgroup of GL(n; Fq) consisting of all diagonal mat*
*rices with
41
diagonal entries 1 as above by En. Then H*BEn is a polynomial algebra F2[x1; ::*
*:; xn]. Let
w1; :::; wn be the elementary symmetric polynomials in the xi. In [Q3] Quillen *
*proved
THEOREM 5.4.1. The restriction homomorphism res : H*BGL(n; Fq) -! H*BEn maps
H*BGL(n; Fq) isomorphically onto the subalgebra of H*BEn generated by the eleme*
*nts w2i
and Sqi-1wi.
5.4.2._As in [Q3] the input in our approach is the knowledge of H*BGL(n; Fq) fo*
*r n 2. For
n = 1 we have trivially H*BGL(1; Fq) ~=H*BE1.
For n = 2 the situation is more subtle. For the convenience of the reader we g*
*ive a brief
outline of a proof which is somewhat different from Quillen's.
The mod 2-cohomology of SL(2; Fq) is well known to be isomorphic to F2[u4] E(v*
*3) where
u4 is a polynomial generator of degree 4 and E(v3) is an exterior algebra on a *
*generator of
degree 3.
Consider the spectral sequence of the extension SL(2; Fq) ! GL(2; Fq) ! (Fq)x .*
* We claim
that this spectral sequence collapses at its E2 -term. Indeed, because the exte*
*nsion is split
exact, there are no non-trivial differentials which end on the base; in particu*
*lar, v3 is a
permanent cycle. The element u4 is also a permanent cycle; it is the restrictio*
*n of the second
Chern class c2 of the complex representation of GL(2; Fq) which is induced from*
* a faithful
2-dimensional representation ae of a 2-Sylow subgroup S of GL(2; Fq). (Note th*
*at S is a
semidihedral group and ae can be taken as in II.4.7.)
Now we consider the restriction homomorphism res : H*BGL(2; Fq) -! H*BE2. It is
straightforward to check that c2 restricts to w22and the generator coming from *
*H1 of the
base (let us call it e1) to w1. The element Sq1w2 is a bit more subtle; by a co*
*unting argument it
is hit if and only if res : H*BGL(2; Fq) -! H*BE2 is injective. The injectivity*
* of res follows
from I.5.7; the spectral sequence computation shows that c2e1 is not a zero div*
*isor, clearly
c2e1 restricts trivially to all nonmaximal elementary abelian p subgroups and f*
*urthermore we
have CG (E2) = E2.
5.4.3._We reinterpret the result for n = 2 as follows.
The restriction homomorphism res maps H*BGL(2; Fq) to the invariants (H*BE2)S2.*
* In
fact, this map is the Nili- localization for i = 1; 2 by 5.2 and 5.3 above. We *
*want to describe
the image of res differently.
For this let C be the central Z=2 in GL(2; Fq) consisting of the matrices Id. T*
*he image of
the restriction from (H*BE2)S2 to H := H*BC consists of the squares, i.e. of H *
*and by
5.4.2 the image of res consists exactly of the elements in (H*BE2)S2 which furt*
*her restrict
to 4-th powers in H, i.e. of 2H. Now the quotient of H by 2H can be identified *
*with
2H which embedds into 2H and we conclude that there is an exact sequence
0 -! H*BGL(2; Fq) -! (H*BE2)S2 -ffi!2H
42
in which the second arrow is induced by res and ffi is the composition of the t*
*wo maps
(H*BE2)S2 -! H -! 2H. In particular we see that the invariants d0 and d1 take t*
*he
values 0 and 2 in the case of H*BGL(2; Fq). In fact, H*BGL(2; Fq) belongs to t*
*he class
C(D) of II.3.4.d.
5.4.4._Now we use Quillen's argument that a 2-Sylow subgroup S of GL(n; Fq) is *
*contained in
the wreath product GL(2; Fq)oSm if n = 2m resp. in (GL(2; Fq)oSm )x(Fq)x if n =*
* 2m+1.
Then Proposition II.3.5 and 5.4.3 imply d0H*BGL(n; Fq) = 0 and d1H*BGL(n; Fq) *
*2. In
particular the localization away from Nil3 is an isomorphism.
5.4.5._Finally we will describe H*BGL(n; Fq) ~=L3H*BGL(n; Fq) for n > 2. As in *
*5.3 we
abbreviate H*BGL(n; Fq) by M(n) and we use the sequence
0 -! 2mk3M(n) -! L3M(n) -! L2M(n) -! 2mc3M(n)
of I.3.8 and the exact sequence of functors
<3o2
0 -! 2k3M(n) -! f<3 L3M(n) f-! f<3 L2M(n) -! 2c3M(n) -! 0 :
Again we will consider this as an exact sequence of functors defined on Rn.
AsN in 5.3.1 the map Nf<3 o2(OnOO) is given by the restriction homomor*
*phim from
( O H*BGL(nO; ))<3 to ( O (H*EnO)SnO )<3.
In order to evaluate this we need to know H2BGL(m; Fq). We already know that d*
*0 = 0
and hence H*BGL(m; Fq) embedds via restriction into (H*BEm )Sm . In particular *
*we get
k3M(n) = 0. The class w1 is in the image of res and hence w21is as well. So i*
*t remains
to consider w2. However, the computation in case GL(2; Fq) showed that w2 is n*
*ot in the
image of res for m = 2 and hence the same must be true for all m 2. Consequent*
*ly we get
c3M(n)(OnOO) ~=Fr2 if r is the number of O with nO > 1.
Now consider m(c3M(n)) ~=Hom ERopn((c3M(n))*; H*) where (c3M(n))* is the functo*
*r defined
via (c3M(n))*(') = (c3M(n)('))*. This functor can be described as follows. Le*
*t Vn-1 =
(Z=2)n-1 and choose a dual basis O1, O2, ... On-1 of Vn*-1. Consider the obje*
*ct '1 :=
2O1 + n-1i=2Oi of Rn. Then (c3M(n))*('1) ~=F2 and the natural map
O
F2Hom Rn('; '1) (c3M(n))*('1) -! (c3M(n))*(')
F2AutRn('1)
is an isomorphism for each ' 2 Rn.
Consequently we get
m(c3M(n)) ~=Hom F2AutRn('1)(F2; H*) ~=(H*Vn-1)AutRn('1):
43
The group AutRn ('1) identifies with n-2 acting on Vn-1 by permuting the last n*
* - 2
summands. Using left exactness of m<3 we arrive at an exact sequence
0 -! H*BGL(n; Fq) -! (H*BVn)Sn -! 2H (H*Vn-2)Sn-2 :
Then one checks that the last map in this exact sequence is the composition of *
*the inclusion
from (H*BVn)Sn to (H*BV2)S2 (H*BVn-2)Sn-2 with ffi id(H*BVn-2)Sn-2 (where ffi*
* is
as in 5.4.3 above). Finally it is an algebraic exercise to identify the kernel*
* of the map
(H*BVn)Sn -! 2H (H*Vn-2)Sn-2 with the subalgebra described in Theorem 5.4.1.
_*
*__
|*
*__|
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78-07, 1978
Hans-Werner Henn Jean Lannes Lionel Schwartz
Mathematisches Institut Centre de Mathematiques Universite de Paris-Nord
der Universit"at de l Ecole Polytechnique Institut Galilee
Im Neuenheimer Feld 288 Plateau de Palaiseau Mathematiques
D-6900 Heidelberg F-91128 Palaiseau Cedex UA 742 du CNRS
Fed. Rep. of Germany France F-93430 Villetaneuse
France
46