HIGHER LIMITS OF FUNCTORS ON CATEGORIES
OF ELEMENTARY ABELIAN p-GROUPS
by Bob Oliver
Recently, certain categories based on elementaryab elian p-groups (i.e., fin*
*ite p-
groups of the form E= (Cp)k) have played an important role as indexing categor*
*ies
for approximating spaces. The construction of approximations to classifying spa*
*ces
in [JM], and the realization of a certain Dickson algebra as the cohomology alg*
*ebra
of a space in [DW2],both depended on the computation of higher derived functors
of inverse limits over such categories. The purpose of this paper is to give a *
*general
procedure for doing this involving the Steinberg representation of GL n(Fp). O*
*ne
consequence is an upper bound for the degrees in which higher limits over such
categories can be nonvanishing.
As one example, consider the category Ap(G), defined for any compact Lie gro*
*up
G as follows. An object in Ap(G) is a nontrivial elementary abelian p-subgroup
1 6= E G. For any pair E1; E2of such subgroups, Mor Ap(G)(E1;E2) is the set of
monomorphisms from E1 to E2 which are composites of inclusions and conjugations
in G. This is the category which (for finite G) was used by Quillen [Q1],for ap*
*prox-
imating H (BG;Fp) up to nilpotence. More recently, it was used by Jackowski &
McClure [JM] as an indexing category for approximating the classifying space BG
itself as a homotopy direct limit of(frequently) simpler spaces. The higher der*
*ived
functors of inverse limits of certain covariant functors from Ap(G) to Ab playe*
*dan
important role in [JM]. One consequenceof the results here is that for any p-lo*
*cal
covariant functor F on Ap(G), limi(F) = 0 for all i p-rk(G) (see Theorem 1).
Higher limits over certain orbit categories were handled in [JMO]by first fi*
*ltering
the functors in such a waythat each of the quotient functors vanishes except on
one single isomorphism class of objects, then analyzing the higher limits of th*
*ose
quotient functors,and finally using long exact sequences to recover the higher *
*limits
of the original functor. That process seemsquite complicated, but it turned out*
* to
be very effective for making computations in the orbit categories used, not onl*
*y in
[JMO], but also in later papers by the same authors. The main idea of this pape*
*r is
to use a similar filtering techniqueto get information about the higher limits *
*over
categories of elementary abelian groupssuch as Ap(G).
Typeset by AMS-TEX
If A is the category of nontrivial subgroups of some fixed elementary abelia*
*n p-
group A,and if F : A! Ab is the functor which sendsthe full group A to Z and *
*the
proper subgroups to 0, then limi(F) is zero when i 6= rk(A) 1, and is isomorph*
*ic
A
in a natural way to the dual ofthe Steinberg representation when i = rk(A) 1.
This follows easily from Lemma 2 below, together with the classical description*
* of
Steinberg representations as homology groups of Tits buildings. What is surpris*
*ing
is that the higher limits can also be described interms of Steinberg representa*
*tions
in more complicated cases. The following theorem is stated here only for Ap(G);
but (as will be seen below) works equally well for other categories of elementa*
*ry
abelian p-groups.
For any elementary abelian p-group E, we let StE denote the Steinberg repre-
sentation of GL (E).
Theorem 1. Fix a prime p and a compact Lie group G. Write A = Ap(G) for
short.
(i) AssumeF : A! Z(p)-mod vanishes except on one (conjugacy class of) obje*
*ct
E. Set k =rk(E), and = Aut A(E) GL (E). Then
ae 0 if i 6= k 1
limi(F ) =
A Hom (StE ;F (E)) if i = k 1.
(ii) For each k 1, set
ffi
Ek = Ek(G) = E2 Ob (A) : rk(E) = k (isomorphisms ):
Then for any functor F : A! Z(p)-mod, lim (F ) is isomorphicto the homology of
A
a cochain complex (C (F); ffi), where
Y
Ci(F ) = Hom AutA(E)(StE; F (E)):
E2Ei+1
In particular, limi(F ) = 0 for i rk p(G).
The two parts of Theorem 1 will be proven _ also for certain other categories
of elementary abelian p-groups _ as Propositions 4 and 5 below.
Theorem 1deals only with covariant functors on Ap(G). In other words, the
limits are always taken inthe direction of the smallest subgroups. This is the *
*type
of limit which arises in [JM]; but is the opposite of the limits used by Quille*
*n [Q1]
to approximate H (BG;Fp).
One of themain results in [JM] was a theorem which says that any Mackey
functor F defined on Ap(G) is acyclic: i.e.,limi(F) = 0 for all i 0. Theorem 1*
* is