cohomology of groups - International Mathematical Union

Actes, Congrès intern. Math., 1970. Tome 2, p. 47 à 51.
COHOMOLOGY OF GROUPS
by Daniel Q U I L L E N *
This is a report of research done at the Institute for Advanced Study the
past year. It includes some general results on the structure of the ring H* (BG , Z/pZ)
when G is a compact Lie group, a theorem computing this ring for a large number
of interesting finite groups, and applications to algebraic Af-theory consisting of
a definition of AT-groups KtA for / > 0 agreeing with those of Bass and Milnor
and their computation when A is a finite field.
1. The spectrum of H*(BG,
Z/pZ).
Let G be a compact Lie group (e.g. a finite group) and let H*(BG) be the
cohomology ring of its classifying space with coefficients in Z/pZ where p is
a fixed prime number. According to Venkov (and Evens for finite G) the ring
# * ( 5 G ) is finitely-generated, hence its Poincaré series 2 (dim z/pZ Hn(BG)) f1
is a rational function of t and one may define the dimension dim H*(BG) to
be the order of the pole of this function at t = 1. For example if A = (Z/pZ)r
is an elementary abelian p-group ([p]-group for short) of rank r9 then
dim H*(BA) = r.
The following for finite G has been conjectured independently by Atiyah and
Swan.
PROPOSITION 1. - dim H*(BG) = the maximum rank of a [p]-subgroup
of
G.
To prove this one follows the method used by Atiyah-Segal to prove the
completion theorem in equivariant ^-theory and first generalizes it to G-spaces,
which for the sake of simplicity I suppose to be smooth compact G-manifolds
with boundary. Let XG be the associated fibre space over BG with fibre X and
set H*(X) = H*(XG).
PROPOSITION 1' - dim HG(X) = the maximum rank of a [p]-subgroup
G fixing some point of X.
of
To prove this one can replace the pair (G, X) by (U, Y) where U is a
unitary group containing G and Y = U x GX ; then one can reduce to the case
(A , Y) where A is the subgroup of elements of order p in a maximal torus of
U , because H%(Y) is a finitely-generated free //y(y>module. Hence one can
(*) Supported by The Institute for Advanced Study and the Alfred P. Sloan Foundation.
48
D. QUILLEN
C 1
suppose that G is a [p]-group, in which case the result can be checked by using
the spectral sequence
E% = HS(X/G,
Gx -> HG(Gx)) => HG+t(X) .
The same technique can be used to prove the following result.
THEOREM - Consider the [p]-subgroups A of G as the objects of a category
in which a morphism from A to A1 is a component of the set of g such that
gAg~lCA'9
and let
u :
H*(BG)-*]imH*(BA)
be the homorphism induced by restriction. Then every element of Ker (u) is
nilpotent and if z is an element of the inverse limit then zpn G Im (u) for large n.
In other words 'up to extraction of p-th roots' a cohomology class of BG is
the same as a family of cohomology classes for each [p]-subgroup compatible
with conjugation and restriction. One should compare this result with Brauer's
theorem asserting that the analogous map with character rings and the category
of elementary subgroups is an isomorphism when G is finite.
This theorem and some commutative algebra permit one to deduce the
following description of the space Spec H*(BG) of prime ideals in H*(BG) (i.e.
inverse images of prime ideals in the commutative ring H*(BG)nd = H*(BG)/ideal
of nilpotent elements). If A is a [p]-subgroup of G, let
h = Ker {H*(BG) -+
H*(BA)ted}.
Then ^4 ->- ^^ gives an order-reversing bijection between conjugacy classes of
[p]-subgroups and those homogeneous prime ideals of H*(BG) which are closed
under the Steenrod operations. In particular the irreducible components of
Spec H*(BG) are in one-one correspondence with maximal [p]-subgroups up to
conjugacy. If TA is the v )set of prime ideals containing $A but not fc4, for
A1 < A, then there is a s tification
Spec H*(BG)
=L1TA
into irreducible locally closed subspaces indexed by the conjugacy classes of
[p]-subgroups. Moreover
TA = (Spec
S(Av)[eAl])/N(A)
where N(A) is the finite group of components of the normalizer of A in G,
where S(Ay) = H*(BA)Kd is the symmetric algebra of the dual of A over Z/pZ,
and eA is the product of the non-zero elements of A.
2. Computations using etale cohomology and the Lang isomorphism.
One knows (Chevalley, Steinberg) that a large number of interesting finite
groups occur as the group G° of fixpoints of an endomorphism a of a connected algebraic group G defined over an algebraically closed field k. For example
if G is defined over a finite subfied k0 of k then the group of rational points
G(k0) is the group of fixpoints of the Frobenius endomorphism associated to
this finite field of definition. Since Ga is finite there is an inseparable isogeny
COHOMOLOGY OF GROUPS
49
G/Ga -> G
gG'-tgiog)-1
(the Lang isomorphism when a is a Frobenius endomorphism), hence G/Ga and G
are homeomorphic for the etale topology. This suggests that H*(BGa) (coefficients
in Z//Z where / is a prime number different from the characteristic of k) might
be computed by using the analogue in etale cohomology of the Leray spectral
sequence of the "fibration" (G/Ga, BGa, BG)9 because the rings H*(BG) and
H*(G) are usually known, e.g. by lifting G to characteristic zero.
Before going on I should explain what is meant by BG in this context. Let
% be the topos of sheaves for the etale topology on the category of all algebraic
fc-schemes. Identifying a fc-scheme with the sheaf it represents, G becomes a
group object of % and so it has a "classifying topos" %G consisting of objects of
^ endowed with G-action (Grothendieck, réédition of SGAA). If X is a fc-scheme
endowed with a G-action, let XG be the object of *SG it gives rise to, and denote
by HG(X) the cohomology of XG with coefficients in the constant sheaf Z//Z ;
write BG instead of e G where e = Spec k. The Leray spectral sequence for the
map XG-+BG9 or as I shall say of the fibration (X, XG, BG) takes the form
(1)
E2
=H*(BG)®H*(X)=*H*(X)
provided the map X -> e is cohomologically proper, which is the case for X = G
because the map factors into a sequence of principal Ga and G m bundles and
the proper map G/B -* e.
Taking X to be G acting on itself by left translations gives a spectral sequence
(2)
E2 = H* (BG) ® H* (G) => H* (e) .
Assume that this spectral sequence has the nice form studied by Borei in his
thesis, namely H*(G) has a simple system of transgressive generators, whence
H*(BG) = S(V) is a polynomial ring and the transgression sets up an isomorphism
of the primitive subspace P of H*(G) and V[— 1] (the [— 1] means degrees are
shifted down by one). When X = Gt 9 the G-scheme obtained by letting G act
on itself by the rule g(gx) = ggx(og)~l9 (1) takes the form
(3)
E2 = H*(BG) 9 H*(G) ** H*(G/Ga) =
H*(BGa).
on account of the Lang isomorphism. To determine the differentials in (3), let
Gs be the (G x G)-scheme obtained by letting G x G act on G by the rule
( # i . £2) = SiStfi"1 a n d consider the map of spectral sequences associated to
the map (G, ( G % , 5 G ) - > (G , (Gs)GxG,B(G
x G )). In the latter spectral sequence
a primitive element z of H* (G) transgresses to v ® 1 — 1 ® v if z transgresses
to v in (2), consequently in (3) z transgresses to v — o*(v). Thus the spectral
sequence (3) can be determined completely and it yields the following.
THEOREM - Let G be a connected algebraic group defined over an algebraically closed field k, and let a be an endomorphism of G such that Ga is finite.
Assume that the etale cohomology 77* (G) (coefficients in Z/£Z, J2 prime ¥= char (k))
has a simple system of transgressive generators for the spectral sequence (2)
(e.g. if H*(G) is an exterior algebra with odd degree generators), and that the
50
D. QUILLEN
C1
subspace V of generators for the polynomial ring H* (BG) can be chosen so as
to be stable under a*. Let Va and Va be the kernel and cokernel of the endomorphism id — a* of V. Then there is an isomorphism of graded Z/fiZ — vector
spaces
H*(BGa)=S(Va)*h
(Va[- 1])
which is an algebra isomorphism if I is odd.
This theorem may be used to determine the mod £ cohomology rings of
the classical groups over a finite field kQ (at least additively when £ = 2) provided
fi is different from the characteristic. For example if k0 has q elements and r
is the order of q mod fi, then H*(BGLn(kQ), Z/fiZ) is the tensor product of a polynomial ring with one generaror of degree 2/V and an exterior algebra with one
generator of degree 2/> — 1 for / < / < [n/r], except that this is only true additively when fi = 2.
3. Applications to algebraic X-theory
Let A be a ring, GL (A) its infinite general linear group, and E (A) the subgroup generated by elementary matrices. As E (A) is perfect, by attaching 2- and
3-cells to BE (A) to kill its kundamental group without changing homology one
can construct a map / : BGL(A) -> BGL(A)+ such that irt(f) kills E(A) and
such that / as a map in the homotopy category of pointed spaces is universal
with this property. Set KtA = 7r,BGL(A)+ for / > 1 ; it is not hard to show
that this definition agrees with those of Bass and Milnor.
The representable functor on the homotopy category
K(X ;A) = [X, K0A x BGL(A)+]
deserves to be called ^-theory with coefficients in A, because it enjoys many
of the properties of topological AT-theory. For example it is the degree zero
part of a connected generalized cohomology theory. Indeed Graeme Segal has
recently associated such a cohomology theory to any category with a coherent
commutative associative composition law, and the cohomology theory in question come from the additive category of finitely-generated projective ^4-modules .
Also K(X \A) is naturally a X-ring when A is commutative.
If one wants to compute the groups KtA by standard techniques of homotopy
theory (e.g. unstable Adams spectral sequence for the //-space BGL(A)+) it is
necessary to know the homology of BGL(A)+, which is the same as that of
BGL(A). For example KtA ® Q is isomorphic to the primitive subspace of
Ht(BGL(A)9Q). For a finite field enough is known about the homology to
do the computation :
Let kQ be a finite field with q elements, let k be an algebraic closure of
kQ, and let 0 : k* -* C* be an embedding. By modular character theory one
knows how to associate to a representation of a group G over k0 a virtual complex representation fixed under the Adams operation tyq by using 0 to lift eigenvalues. Lifting the standard representation of GLn(k0) on k%9 one obtains a map
BGLn(k0) -> E^!q9 where the latter space is the fibre of the endomorphism
^q — id of BU. This map kills elementary matrices, hence gives rise to a map
in the pointed homotopy category
COHOMOLOGY OF GROUPS
51
BGL(k0)+ -*EVq
which depends only on the choice of 0.
THEOREM
1. — The map (*) is a homotopy equivalence.
This is proved by showing that the map induces an isomorphism on homology
and using the Whitehead theorem. The homotopy groups of E^q may be computed
by using Bott periodicity, so one obtains the formulas
K2i(k0) = 0
/>1
The functorial behavior of these groups as the finite field varies may be determined
in similar fashion, and it leads to the following :
THEOREM
2.— If k is an algebraic closure of F p , then
K2i(k) = 0
V,W=
i>\
© Qfi/Zg
i> 1
fifp
and the Frobenius automorphism of k over Fp acts on K2i_x (k) by multiplying
by p*. If kt is any subfield of k, then the extension-of-scalars homomorphism
induces an isomorphism
Gal (*/*,)
K2i_x(k,) ~ K2i_,(k)
l/l>
in terms of which the restriction-of-scalars homomorphism
u* : K2i_x(k2\-+
Ku^kJ
associated to a finite extension u : kx -* k2 is given by the norm from Ga\(k/k2)invariants to GdX (kIk^-invariants.
Massachusetts Institute of Technology
Dept. of Mathematics
Cambridge, Massachusetts 02139
U.S.A.