UPPER BOUNDS ON HIGHER LIMITS
(Bob Oliver)
Theorem 1. Fix V ~= (Z=2)r, r 2. Let X be a finite V -complex of dimension n.*
* Let FX*:
A2(V ) -!Ab be the functor FX*(E) = H*V(XE ). Then lim- (FXj) vanishes if i *
*r, or if i = r - 1
A2(V )
or r - 2 and j > (2r- 1)n - r, or if 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 pair of V *
*-complexes
such that XV = Y V. Let x1; : :;:xs2V *r 0 be any set of elements such that e*
*very isotropy
subgroup of Xr Y is contained in Ker(xi) for some i. Set n = dim(X). Then for e*
*very ff2H*V(Y ),
xn1. .x.nsff2 Im[H*V(X) -!H*V(Y )].
Proof. For each i = 0; : :;:n, set Xi= X(i)[Y . Then H*V(X0) surjects onto H*V(*
*Y ) (projection onto
a direct summand). And for all i 0, x1. ..xr.H*V(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 *r 0, set
ssD = x2Dx 2 F2[V *].
Proposition 3. Fix V ~=(Z=2)k (k 2). Assume, for some fixed N 1, ideals Ix F*
*2[(V=x)*]
are given, for all x2V r 0, 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
(Ix.F2[V *]) (ssV *r(V=x)*)N .Ix . F2[V *]|x2V r 0
x2V r0
vanishes in degrees greater than (2k - 1)N - k.
In order to simplify the notation, Proposition 3 will be proven in the follow*
*ing dualized version.
For convenience in notation, when V ~=(Z=2)k is given, then for any i k, Si wi*
*ll 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 E 2 Sk-1, such that
(1) (ssEr0)N 2 IE for each E, and
(2) For any pair E1; E2 2 Sk-1, (ssE1rE2)N . IE2 IE1 . F2[V *].
Typeset by AM *
*S-TEX
1
Then the quotient group
i " j. D E
(IE.F2[V *]) (ssV rE)N .IE.F2[V *]|E 2 Sk-1
E2Sk-1
vanishes in degrees greater than (2k - 1)N - k.
Proof. Assume first that k = 2, and set V = . Then
I= xa . F2[x]; I= yb. F2[x] and I= (x + y)c. F2[x]
for some a; b; c 0. The quotient group
i j. D *
* E
xayb(x + y)cF2[x; y] xN yN (x + y)c; xN yb(x + y)N ; xayN (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.Q For any m*
* : V r 0 -!
{0; : :;:N}, we say that the system {IE} is subordinate to m if IE x2Er0xm(x*
*). F2[E] for all
E 2 Sk-1. Clearly, any {IE} is subordinate to the zero function. And if {IE} is*
* subordinate to the
constant function N, then IE = (ssEr0)N . F2[E] for all E, and the quotient is *
*trivial.
It thus suffices to prove the following. Assume m is maximal among functions *
*with the property
that {IE} is subordinate to m; and assume that m is not the constant function N*
* (otherwise, as
noted above, we are done). Fix y such that m(y) < N, and define m0 : V r 0 -! {*
*0; : :;:N} 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 E 2 Sk*
*-1.)
x2Er0
We claim that the proposition holds for {IE} if it holds for {JE}.
To see this, it suffices to show that the cokernel of the map
T T
JE . F2[V ] E2Sk-1IE . F2[V ]
' : ___E2Sk-1__________<(ss-! __N________________ N
V rE)<(.sJEs.VF2[Vr]>E) . IE . F2*
*[V ]>
vanishes in dimensions greater than (2k - 1)N - k. Note first that IE = JE if *
*y 62 E, and that
yIE JE IE if y2 E. Define a system {IE=y} of ideals over F2[V=y] by setting
h i
IE=y= Im y-m(y)IE -! F2[E=y]
for all E 2 Sk-1 containing y. Then the IE=ysatisfy the hypotheses of Proposit*
*ion 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=y2Sk-2(V=y)
So by the induction hypothesis, Coker(') vanishes in dimensions greater than
m(y) + (2k-1- 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 s*
*ubgroup).
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Proposition 4. Fix V = (Z=2)k and N 1. Let
F : A2(V ) -!(ideals inF2[V *])
be a functor, which satisfies the following three conditions:
(1) F (V ) = F2[V *].
(2) For all E V , F (E) = IE . F2[V *] for some ideal IE F2[(V=E)*]. (Regar*
*d F2[(V=E)*] as a
subring of F2[V *].)
(3) For all E$ E0 V ,
N 0 0
ss(V=E)*r(V=E0)*. F (E ) F (E) ( F (E ))
Then lim-i(F ) for i> 0, and Coker[F (1) -!lim-0(F )], all vanish in degrees gr*
*eater than (2k- 1)N - k.
Proof. We say that F satisfies the rank t intersection condition if for any E *
*V with rk(E) t,
"
F (E) = F (E0):
E0%E
By Proposition 3, i " j.
CokerF (1) -! lim-(F ) ~= F (E) F (1)
A2(V ) E2S1
T
vanishes in degrees greater than (2k- 1)N - k. So we can replace F (1) by E2S1*
* F (E), and assume
that the rank 0 intersection property holds.
Assume now that F satisfies the rank t - 1 intersection property, but not the*
* rank t property, for
some 1 t k. We may assume inductively that the proposition holds for functors*
* which satisfy the
rank t intersection property; and for functors on groups of lower rank. If t = *
*k, then F is constant,
and lim-i(F ) = 0 for all i > 0. So we may assume t k- 1.
T
Fix W V of rank t such that F (W ) $ E%W F (E). Choose ideals
"
IW = J0 J1 . . .Js = IE . F2[(V=W )*];
E%W
such that for each i, Ji=Ji-1~=miF2 for some mi 0. By Proposition 3 again,
mi (2k-t- 1) . N - (k - t)
for each i. Define functors F0 = F F1 : : :Fs by inductively setting
8
> Ji. F2[V *] if E = W
: T 0
E0%EFi(E ) if E $ W .
Then for each 1 i s, (Fi=Fi-1)(W ) ~=miF2[W *]; and as a functor on A2(W ) Fi*
*=Fi-1 satis-
fies the hypotheses of Proposition 3 (but with N replaced by 2k-tN). So for j *
* 1, the limit
lim- (Fi=Fi-1) vanishes in degrees above
A2(W)
mi+ (2t- 1) . 2k-tN - t (2k-t- 1)N - k + t + (2t- 1) . 2k-tN - t = (2k - *
*1)N - k:
3
In particular, the higher limits of Fs=F vanish above degree (2k - 1)N - k.
Continuing this procedure with the other rank t subgroups, we embed F F so t*
*hat F satisfies
the rank t + 1 intersection property, and so that the higher limits of F=F vani*
*sh in degrees above
(2k- 1)N - k. Since the same holds for F by the induction hypothesis, this fini*
*shes the proof of the
proposition.
Proof of Theorem 1. Recall first that for i r = rk(V ), lim-i(F ) = 0 for any *
*functor on A2(V ).
The theorem clearly holds if the action of V fixes X, or if X is the union of*
* XV and a finite V -set.
So we may consider the case where X = Y [' (V=W xDm ), with rk(W ) = k < r and *
*0 < m n,
and where the theorem holds for Y .
Consider the following short exact sequences of functors on A2(V ):
0 -!K -!FX -! FIm -!0; 0 -!FIm -!FY -! C -!0; (*
*1)
and
0 -!C -!FX;Y -!K -!0: (*
*2)
Here, K, FIm, and C are the kernel, image, and cokernel, respectively, of the r*
*estriction map from
FX -! FY . The relative functor FX;Y has the form
ae0 if E 6 W
FX;Y(E) = H*V(XE ; Y E) ~=
H*V(XW ; Y W) ~=m F2[W *]if E W ,
and thus can be regarded as a constant functor on A2(W ). Note also that
aeIm[H* (Y E) -!H*+1(XE ; Y E)]if E W
C(E) ~= V V
0 if E 6 W ;
and that C and K can also be regarded as functors on A2(W ). In addition,
* W *+1 W W
C(W )~=Im HV (Y ) -!HV (X ; Y )
~=Im H*V=W(Y W) -!H*+1V=W(XW ; Y W) ~=m F2 F2[W *];
so that either C(W ) ~=m F2[W *], or C = 0 (as a constant functor). In the seco*
*nd case the result
is clear, since lim-i(FX;Y) = 0 for i > 0 (FX;Y being a constant functor).
Now, for any E W , set
h i
IE = Im H*V=E(Y E) -!H*+1V=E(XW ; Y W) ~=m F2[(V=E)*] ;
regarded as an ideal in F2[(V=E)*]. Then C(E) = IE . F2[W *]. Also, for all E$ *
*E0 W ,
n.2r-1 0
ss(W=E)*r(W=E0)* C(E ) C(E)
by Lemma 2. Hence, by Proposition 4,
* 0 * 0 *
lim-i(C*) (1 i < r) and CokerHV (Y ) -!lim-(FY ) -!lim-(C )
vanish in degrees greater than
(m - 1) + (2k - 1)(n . 2r-k) - k = (2r- 2r-k+ 1)n - (k + 1) - (n - m) (2r-*
* 1)n - r:
It follows that Ker[lim-i(FIm) -! lim-i(FY )], for 1 i< r, also vanishes in d*
*egrees greater than
(2r- 1)n - r.
Finally, since lim-i(FX;Y) = 0 for i > 0, we see that lim-k-1(K) = 0; and tha*
*t lim-i(K) for
1 i k - 2 ( r - 3) vanishes in degrees greater than (2r- 1)n - r + 1.
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