UPPER BOUNDS ON HIGHER LIMITS
(Bob Oliver)
Theorem 1. Fix V = (Z=2)r, r 2. Let X be a finite V -complexof dimension n. *
*Let FX :
A2(V)! Ab be the functor FX (E) = HV(XE ).Then lim (FjX) vanishesif i r,or*
* if i = r 1
A2(V)
or r 2 and j >(2r 1)n r, orif 1 i r 3 and j> (2r 1)n r+ 1.
The proof of Theorem 1 will use Lemma 2 and Proposition 4 below.
Lemma 2. Let V be an elementary abelian 2-group, and let X Y be a pairof V -co*
*mplexes
such that XV = Y V. Let x1; : :;:xs2V r 0 be any setof elements such that eve*
*ry isotropy
subgroup of XrY is contained in Ker(xi) for some i. Set n = dim(X). Then for ev*
*ery ff2HV(Y),
xn1 xnsff2Im[HV (X)! HV (Y )].
Proof. For each i = 0; : :;:n, set Xi= X(i)[Y. Then HV (X0) surjects ontoHV (Y *
*) (projection onto
a direct summand). And forall i 0, x1 xrHV (Xi+1;Xi) = 0.
The proof of Theorem 1 will be reduced to putting upper bounds on the degrees*
* of quotients of
the followingQform. The following notation will be useful: for any V and any su*
*bset D V r0, set
ssD = x2Dx 2 F2[V ].
Proposition 3. Fix V = (Z=2)k(k 2). Assume, for some fixed N 1, ideals Ix F2[*
*(V=x) ]
are given, for all x2V r0, such that
(1) (ss(V=x) r0)N 2 Ix
(2) For all x 6= y,(ss(V =x)r(V=y))N Iy Ix F2[V ].
Then the quotient group
i " j. D E
(IxF2[V ]) (ssV r(V=x))N Ix F2[V ]jx2 Vr 0
x2Vr0
vanishes in degrees greater than(2k 1)N k.
In order to simplify the notation,Proposition 3 will be proven in the followi*
*ng dualized version.
For convenience in notation, when V = (Z=2)kis given, then for any i k, Siwill *
*denote the set of
subspaces of V of rank i.
Proposition 3'. Fix V = (Z=2)k(k 2). Assume,for some fixed N 1,that ideals IE*
* F2[E]
are given, for all E2 Sk1 , such that
(1) (ssEr0)N 2 IEfor each E, and
(2) For any pair E1;E22 Sk1 , (ssE1rE2)N IE2 IE1 F2[V ].
Typeset by AMS-*
*TEX
Then the quotient group
i " j. D E
(IE F2[V ]) (ssVrE)N IE F2[V ]jE2 Sk1
E 2Sk1
vanishes in degrees greater than(2k 1)N k.
Proof. Assume first that k = 2, and set V = hx;yi. Then
Ihxi= xa F2[x];Ihyi= yb F2[x] and Ihx+yi= (x + y)c F2[x]
for some a; b; c 0. The quotient group
i j.D E
xayb(x + y)cF2[x;y] xNyN(x + y)c; xN yb(x + y)N ; xay N(x + y)N
!
has Poincare series!
!! 1 ta+b+c t2N+a t2N+b t2N+c+ 2t3N ;
(1 t)2
which is a polynomial of degree 3N-2.So the proposition holds in this case.
Now assume k > 2,and assume the proposition holds for rank k 1. ForQany m : V*
* r0!
f0; : :;:Ng, we say that the system fIE g is subordinate to m if IE x2Er0xm(x*
*) F2[E] for all
E 2Sk1 . Clearly,any fIEg is subordinate to the zero function. And if fIEg is s*
*ubordinate to the
constant function N, then IE =(ssEr0)N F2[E] for all E, and the quotient is tri*
*vial.
It thus suffices to prove the following. Assume m is maximal among functions *
*with the property
that fIEg is subordinate to m; and assume thatm is not the constant function N *
*(otherwise, as
noted above, weare done).! Fix ysuch that m(y)!< N, and define m0: V r 0! f0; *
*: :;:Ng by
setting m0(y) = m(y) +!1, and m0(x) = m(x) for!x6=y.Define ideals JE by setting
!! i Y 0!!j
! JE= IE" xm!(x)F2[E] (all E2 Sk1.)
!! x2Er0 !!
We claim that the proposition!holds!for fIEg!if!it holds for fJEg.
! !
To see this, it suffices!to show that the!cokernel of the map
!T !T
!! JE F2[V] !! E2Sk1IE F2[V ]
': h(sE2Sk1s ! h(ss
VrE)N JE F2[V]iVrE)N IE F2[V ]i
vanishes in dimensions greater than (2k 1)N k. Note first that IE =JE if y 62*
* E, and that
yIE JE IE if y2E. Define a system fIE=yg of ideals over F2[V=y] by setting
h i
IE=y= Im ym(y)IE! F2[E =y]
for all E 2 Sk1 containing y. Then the IE=ysatisfy the hypotheses of Propositio*
*n 3' (with N
replaced by 2N), and
Coker(') =
i " j .D E
m(y) IE=y F2[V=y] (ssV=yrE=y)2N IE=y F2[V =y] :
E=y2Sk2 (V=y)
So by the induction hypothesis, Coker(') vanishes in dimensions greater than
m(y) + (2k1 1) 2N (k 1) (2k 1)N k; (m(y) N 1)
and this finishes the proof.
Let A2(V) denote the category of all subgroups of V (including the trivial su*
*bgroup).