HIGHER LIMITS OF FUNCTORS ON CATEGORIES
OF ELEMENTARY ABELIAN p-GROUPS
by Bob Oliver
Recently, certain categories based on elementary abelian 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; E2 of 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 a*
*pprox-
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 de*
*rived
functors of inverse limits of certain covariant functors from Ap(G) to Ab playe*
*d an
important role in [JM]. One consequence of the results here is that for any p-l*
*ocal
covariant functor F on Ap(G), lim-i(F ) = 0 for all i p-rk(G) (see Theorem 1).
Higher limits over certain orbit categories were handled in [JMO] by first f*
*iltering
the functors in such a way that 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 seems quite complicated, but it turned ou*
*t 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 technique to get information about the higher limits*
* over
categories of elementary abelian groups such as Ap(G).
Typeset by AM S-T*
*EX
1
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 sends the full group A to Z an*
*d the
proper subgroups to 0, then lim-i(F ) is zero when i 6= rk(A) - 1, and is isomo*
*rphic
A
in a natural way to the dual of the 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 in terms of Steinberg represent*
*ations
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) Assume F : A!- Z(p)-mod vanishes except on one (conjugacy class of) obj*
*ect
E. Set k = rk(E), and = Aut A(E) GL (E). Then
ae 0 if i 6= k - 1
lim-i(F ) ~=
A Hom (StE ; F (E)) if i = k - 1.
(ii) For each k 1, set
ffi
Ek = Ek(G) = E 2 Ob (A) : rk(E) = k (isomorphisms ):
Then for any functor F : A!- Z(p)-mod, lim-*(F ) is isomorphic to the homology*
* of
A
a cochain complex (C*(F ); ffi), where
Y
Ci(F ) ~= Hom AutA (E)(StE ; F (E)):
E2Ei+1
In particular, lim-i(F ) = 0 for i rkp(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 1 deals only with covariant functors on Ap(G). In other words, the
limits are always taken in the 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 the main results in [JM] was a theorem which says that any Mackey
functor F defined on Ap(G) is acyclic: i.e., lim-i(F ) = 0 for all i 0. Theore*
*m 1 is
2
intended in part to supplement that result, in that it provides a means to comp*
*ute
higher limits for functors which are not Mackey functors.
The abstract definition of lim-*(F ) in terms of an injective resolution of *
*F is not
C
very useful when making specific calculations. The following lemma describes th*
*ese
higher limits as the homology of an explicit cochain complex. Alternatively, it*
* can
be thought of as saying that they are the cohomology of a certain sheaf over the
nerve of C.
Lemma 2. Let C be any small category, and let F : C!- Ab be any covariant
functor. Then lim-*(F ) ~=H*(C*(C; F ); ffi), where
C
Y
Cn (C; F ) = F (xn) (1)
x0!...!xn
for all n 0; and where for U 2 Cn (C; F ),
' Xn
ffi(U)(x0 ! : :-:!xn!- xn+1 ) = (-1)iU(x0 ! : :b:xi.!.x.n+1)
i=0
+ (-1)n+1 F (')(U(x0 ! . .!.xn)):
(2)
Proof. Let C-mod denote the category of covariant functors from C to Ab. For a*
*ny
F in C-mod ,
lim-(F ) ~=Mor C-mod(Z_; F );
C
where Z_denotes the constant functor with values Z. So if (P*; @) is any projec*
*tive
resolution of Z_in C-mod , then lim-*(F ) is the cohomology of the cochain comp*
*lex
Mor C-mod(P*; F ) ; Mor (@; F ) :
For each n -1, define the functor Pn : C!- Ab as follows. For each object
x in C, let Pn(x) be the free abelian group with basis the set of all sequences
x0!- : : :-!xn!- x of morphisms in C ending in x. For any morphism f in C,
Pn(f) is defined by composition in the obvious way. Note that P-1 ~= Z_. Defi*
*ne
boundary maps @ : Pn ! Pn-1 by setting
Xn
@([x0 ! . .!.xn!- x]) = (-1)i[x0 ! : :b:xi.!.x.n-! x]:
i=0
3
For each x, the chain complex
@ @ @ @
: : :-!P2(x)!- P1(x)!- P0(x)!- P-1 (x)!- 0
Id
is split by the maps ([. .!.xn!- x] 7! [. .!.xn!- x -! x]); and hence is exa*
*ct.
Thus, (P*; @) is a resolution of Z_. Also, for any F ,
Y
Mor C-mod(Pn; F ) ~= F (xn):
x0!...!xn
This shows that Pn is projective, and that Mor C-mod(P*; F ); Mor (@; F ) is *
*isomor-
phic to the complex (C*(C; F ); ffi) defined in (1) and (2) above.
The description of higher limits given in Lemma 2 is well known, but we have
been unable to find it in the literature _ aside from a rather obscurely formul*
*ated
version in [BK, XI.6.2].
As was noted above, Theorem 1 holds for a wide range of categories based on
elementary abelian p-groups; and similar results seem likely to hold for other,*
* related
categories. Hence, for the sake of other possible applications, we want to pro*
*ve
Theorem 1 _ or at least its essence _ in as much generality as possible.
The natural setting for filtering functors and reducing to "single object fu*
*nctors"
seems to be that of what we here call ordered categories. We define an ordered
category to be a category where all endomorphisms are automorphisms. This is the
condition formulated by L"uck in [L"u] (where he called them "EI-categories"). *
*If C
is such a category, then the set of isomorphism classes in C is partially order*
*ed by
the relation [x] [y] if Mor (x; y) 6= ;. And if C has only finitely many isomo*
*rphism
classes of objects, then it is easy to see that for any F : C!- Ab, F can be f*
*iltered
by as sequence 0 = F0 F1 . . . Fk = F , such that each Fi=Fi-1 vanishes
except on one isomorphism class of objects.
In order to formulate the results presented here, some more structure on the
category is needed. We define a category with subobjects to be a pair C I of
categories such that Ob (I) = Ob (C), and such that the following two conditions
are satisfied:
(a) | Mor I(x; y)| 1 for any pair of objects x; y; and
(b) each morphism f 2 Mor C(x; y) can be written in a unique way as a compos*
*ite
f = i O a, where a 2 IsoC(x; x0) for some x0, and i 2 Mor I(x0; y).
The idea for categories with subobjects comes, of course, from the categories o*
*f sets,
groups, etc. with monomorphisms, where the subcategories consist of inclusions *
*of
subobjects. With this in mind, for any category with subobjects (C; I), and any
a i
morphism f : x -!~ x0 ,! y as above, we write f(x) or Im (f) for the object x0.
=
4
Similarly, we write x y if Mor I(x; y) 6= ;. Note that an inclusion x ,i!y is*
* an
isomorphism in C only if x = y and i = Idx.
The categories Ap(G) defined above, and other categories based on elementary
abelian p-groups, are all ordered categories, and can all be made into categori*
*es with
subobjects in an obvious way. In contrast, the orbit categories dealt with in [*
*JMO]
are also ordered, but cannot be given the structure of categories with subobjec*
*ts.
For any category with subobjects (C; I), and any object x in C, we let C*x C*
*x C
denote the full subcategories
Ob (Cx) = {y 2 Ob (C) : y x} and Ob (C*x) = {y 2 Ob (C) : y $ x}:
These are equivalent to the full subcategories of objects y such that [y] [x],*
* or
[y] [x] and [y] 6~= [x], respectively. It is these last categories which appea*
*r when
one studies functors on C which vanish except on objects isomorphic to x. The
main idea is to compare them with the categories Ix = Cx \ I and I*x= C*x\ I of
subobjects of x. Note that the automorphism group Aut C(x) acts in a natural way
on Ix and I*x_ this follows from property (b) in the definition _ but not on Cx
or C*x.
Proposition 3. Let (C; I) be any (small) ordered category with subobjects. Fix
an object x in C, and set = Aut C(x). Let F : C!- Ab be any (covariant) funct*
*or
such that F (y) = 0 for y 6~= x, and regard F (x) as a Z[]-module. Then the fol*
*lowing
hold.
(1) H*(BCx; BC*x)~=H*(E x BIx; E x BI*x)
(2) There is an isomorphism
lim-*(F ) ~=H*(BIx; BI*x; F (x));
C
where H*(-; -) denotes Borel cohomology with twisted coefficients. This isomor-
phism is induced by the chain homomorphism *, where
X
n : Hom Z[] Cq(E) Cp(BIx; BI*x); F (x)
p+q=n
Y
----! Cn (C; F ) ~= F (x)
x0!...!xn-1-! x
satisfies the formula (for yi $ x)
i f1 fp-1 fp fl1 flq j
n(U) y0 -! : :-:--! yp-1 -! x -! x!- : :-:! x
i *
* j
= flq . .f.l1.U 1; fl1; fl2fl1; : :;:flq . .f.l1 Im(fp . .f.1) ,! : :,:!*
* Im (fp) ,! x :
5
(3) There is a spectral sequence
Epq2~=Hp(; Hq(BIx; BI*x; F (x))) =) lim-p+q(F ):
C
(4) Assume, for some ring R Q, that F (x) is an R-module, and *
* that
H*(BIx; BI*x; R) is R[]-projective. Then for each i 0,
*
lim-i(F ) ~=Hom Hi(BIx; BIx); F (x) :
C
Proof. Let Cx C be the full subcategory whose objects are those y such that
[y] [x], i.e., such that Mor C(y; x) 6= ;. From the formula in Lemma 2, it is
clear that lim-*(F ) ~=lim* (F ). Also, lim* (F ) ~=lim* (F ) since the categor*
*ies are
C - Cx - Cx - Cx
equivalent (every object in Cx is isomorphic to an object of Cx). So we are re*
*ally
working entirely within the category Cx.
Consider the subcategory C1x Cx defined by setting Ob (C1x) = Ob *
*(Cx),
Mor C1x(y; y0) = Mor C(y; y0) if y $ x, and Mor C1x(x; x) = {Idx}. Let act on*
* C1x
via the identity on objects, via the identity on Mor (C*x), and via composition*
* on
Mor C(y; x) for y$ x.
Step 1 Let
C(i) : C*(BIx; BI*x) ----! C*(BC1x; BC*x)
be the inclusion. Define a retraction
r : C*(BC1x; BC*x)!- C*(BIx; BI*x)
by setting
f0 f1 fk
r y0 -! y1 -! . .-.!yk -! x
= fk . .f.0(y0) ,! fk . .f.1(y1) ,! . .,.! fk(yk) ,! x :
Both of these are homomorphisms of chain complexes, rOC(i) is the identity on
C*(BIx; BI*x), and r is Z[]-linear. We first show that these homomorphisms indu*
*ce
a Z[]-linear isomorphism
~=
H*(r) : H*(BC1x; BC*x) -! H*(BIx; BI*x): (5)
6
Set X* = Coker (C(i)) = C*(BC1x; BC*x[BI*x). We must show that X* is exact.
To do this, define D : Xn!- Xn+1 by setting
f0 f1 fk
D y0 -! y1 -! . .-.!yk -! x
Xk fk...fi
= (-1)i y0!- . .-.!yi ----! fk . .f.i(yi) ,! . .,.! fk(yk) ,! x :
i=0
f0 fk
Fix an element [y0 -! . .-.!yk -! x] as above, and set y0i= fk . .f.i(yi). Th*
*en
D@ y0 !- . .-.!yk!- x
k !
X
= D (-1)i y0!- . .b.yi.-.!.yk!- x
X i=0
0 0
= (-1)i+j y0!- . .-.!yj!- yj ,! . .b.y0i.,.!.yk ,! x
j*i
and
@D y0 !- . .-.!yk!- x
0 1
Xk
= @ @ (-1)i y0!- . .-.!yj!- y0j,! . .,.! y0k,! x A
j=0
X
= (-1)i+j y0!- . .b.yi.-.!.yj!- y0j,! . .,.! y0k,! x
ij
Xk i
+ y0!- . .-.!yj-1!- y0j,! . .,.! y0k,! x
i=0
0 0 j
- y0!- . .-.!yj!- yj+1 ,! . .,.! yk ,! x :
And since [y00,! . .,.! y0k,! x] vanishes in C*(BC1x; BC*x[BI*x), this gives
(D@ + @D) y0!- . .-.!yk!- x = - y0!- . .-.!yk!- x :
Thus, D@ + @D = -Id, and so X* is exact.
7
Step 2 Let C*(E) denote the usual chain complex for E: Cn(E) is the free
abelian group with basis consisting of (n + 1)-tuples (fl0; : :;:fln), and act*
*s by
right multiplication. Define
~=
: C*(E) C*(BC1x; BC*x) -----! C*(BCx; BC*x)
by setting
i f j
fl0; fl1; :;:f:lm y0!- : :-:!yk-1!- x =
(fl0Of) fl1fl-10 flm fl-1m-1
y0!- : :-:!yk-1 ----! x ----! x!- : :-:----! x :
Here, yi2 Ob (C*x) for each i. Note that we have dropped the degenerate simplic*
*es;
at least those in BC1xwhich involve Idx. Clearly, factors through an isomorphi*
*sm
on C*(E) Z C*(BC1x; BC*x) (it sends a basis to a basis), and commutes with
boundary maps. It thus induces an isomorphism
~=
* : H*(E x BC1x; E x BC*x) -! H*(BCx; BC*x): (6)
Together with Step 1, this proves point (1).
Consider the cochain complex (C*(C; F ); ffi) of Lemma 2, whose cohomology is
lim-*(F ). Define isomorphisms
X i j
bn : Hom Z[] Cq(E) Cp(BC1x; BC*x) ; F (x)
p+q=n
~= Y
----! Cn (C; F ) ~= F (x)
x0!...!xn-1-! x
by setting (for yi $ x)
i f fl1 flq j
nb (U) y0!- : :-:!yp-1!- x -! x!- : :-:! x
i *
* f j
= (flq . .f.l1).U 1; fl1; fl2fl1; : :;:flq . .f.l1 y0!- : :-:!*
*yp-1!- x ;
or equivalently
i f j
-b1n(V ) fl0; : :;:flq y0!- : :-:!yp-1!- x
i fl0f fl1fl-10 flqfl-1q-1j
= fl-1q.V y0!- : :-:!yp-1 --! x ----! x!- : :-:---! x :
8
These commute with the coboundary homomorphisms, and hence induce an iso-
morphism
lim-*(F ) ~=H* Ex BC1x; Ex BCx; F (x)
C
where the homology groups are taken with twisted coefficients. And point (2) now
follows upon composing this with the isomorphism of Step 1.
The last two points follow from the usual spectral sequence for the cohomolo*
*gy
of the Borel construction.
Before proving Theorem 1, we first look at two other types of categories bas*
*ed
on elementary abelian p-groups.
For any space X such that H*(X; Fp) is Noetherian, define the category Ap(X)
as follows. An object in Ap(X) is a pair (E; ), where E is an elementary abeli*
*an
p-group and : BE!- X is a homotopy class of maps such that H*( ; Fp) is a
finite morphism (i.e., it makes H*(BE; Fp) into a finitely generated module over
H*(X; Fp)). A morphism in Ap(X) from (E1; 1) to (E2; 2) is a monomorphism
' : E1ae E2 such that 2OB' ' 1. By a theorem of Dwyer & Zabrodsky [DZ],
for any compact Lie group G, Ap(BG) is equivalent to the category Ap(G) defined
earlier.
For any unstable noetherian algebra K over the Steenrod algebra Ap, let Ap(K)
denote the category whose objects are pairs (E; ), where E 6= 1 is an elemen-
tary abelian p-group and : K!- H*(BE; Fp) is a finite Ap-algebra homomor-
phism. A morphism from (E1; 1) to (E2; 2) is a monomorphism ' : E1ae E2
such that H*(B'; Fp) O 2 = 1. These categories were first defined and used by
Rector [Re]. By a theorem of Lannes ([La1, Theoreme 0.4] or [La2, Theoreme 0.4]*
*),
Ap(X) ~= Ap(H*(X; Fp)) if X is simply connected and H*(X; Fp) is noetherian.
Dwyer & Wilkerson, in [DW1] and [DW2], have shown the usefulness of Ap(K),
and the importance of higher limits of functors over Ap(K), when trying to dete*
*r-
mine whether K can be realized as the cohomology algebra of a space.
For convenience, an object of any of the categories Ap(G), Ap(X), or Ap(K)
will be denoted (E; ), where is an inclusion E ,! G, a map BE!- X, or a
homomorphism K!- H*(BE; Fp), respectively. Recall that for any elementary
abelian p-group E ~= (Z=p)k, we write StE to denote the Steinberg representation
of GL (E).
Proposition 4. Assume A is one of the categories Ap(G) for a compact Lie group
G, Ap(X) for a space X such that H*(X; Fp) is noetherian, or Ap(K) for an unsta*
*ble
noetherian algebra K over the Steenrod algebra Ap. Let
F : A ----! Z(p)-mod
9
be a (covariant) functor which vanishes except on the isomorphism class of the
object (E; ). Set k = rk(E), and let = Aut A(E; ) GL (E). Then
ae0 if i 6= k - 1
lim-i(F ) =
A Hom (StE ; F (E; ))if i = k - 1.
Proof. Let IE I*Edenote the poset categories of nontrivial subgroups, and prop*
*er
subgroups, of E, with the induced actions of GL (E). Then BIE is contractible: *
*it
is the nerve of a category with final object. So by definition, for any elemen*
*tary
abelian p-group E ~=(Z=p)k with k 1,
aeH (point) if k = 1oe ( StE if i = k - 1
Hi(BIE ; BI*E) ~= ie ~=
Hi-1 (BI*E) if k > 1 0 if i 6= k - 1
as modules over Z[GL (E)] ~= Z[GLn(Fp)] (cf. [Lu, x1.13], where BI*E is denoted
SII(E)). Also, IE and I*Ecan be identified with the subcategories I*(E; ) I(E; )
Ap(X) used in Proposition 3. So the corollary will follow immediately from Prop*
*o-
sition 3 once we check that (StE )(p)is projective as a Z(p)[GL (E)]-module.
The projectivity of the Steinberg module is well known, but we have been una*
*ble
to find a reference which also directly links the Steinberg module to the homol*
*ogy
of the Tits building BI*E. So instead, we note the following proof, in the case
rk(E) 2. For any p-subgroup 1 6= P GL (E), (BI*E)P is the nerve of the poset
SP consisting of all P -invariant proper subgroups of E. Then EP 2 SP (i.e.,*
* it is
a proper subgroup), and A \ EP = AP 2 SP for any P -invariant proper subgroup
A. This shows that SP is "conically contractible" in the sense of Quillen [Q2, *
*x1.5],
and in particular that (BI*E)P is contractible.
Now assume P is a Sylow p-subgroup of GL (E), and let Y BI*Ebe the union of
the fixed point sets of all nontrivial subgroups 1 6= P 0 P . Then Y is contrac*
*tible,
since it a union of contractible spaces all of whose intersections are contract*
*ible.
So StE is the unique homology group of the chain complex C*(BI*E; Y ), and this
is a complex of free Z[P ]-modules. Thus StE is Z[P ]-stably free, and (StE )*
*(p) is
Z(p)[GL (E)]-projective.
For any k 1 and any elementary abelian p-group E of rank k + 1, let bI*E I*E
be the subcategory of objects of codimension at least 2 in E; i.e., the categor*
*y of
proper subgroups of E of rank at most k - 1. Define
M
RE : StE ----! StA
[E:A]=p
10
to be the composite
@ * * M
StE ~= Hk(BIE ; BI*E) ----! Hk-1 (BIE ; BbIE) ~= StA :
[E:A]=p
Alternatively, RE can be thought of as (up to sign) the homomorphism induced by
truncating chains of subgroups.
Proposition 4 now implies the following:
Proposition 5. Fix a prime p, and let A be one of the rings Ap(G), Ap(X), or
Ap(K) as in Proposition 4. For each k 1, set
Ek = Ek(X) = {(E; ) 2 Ob (A) : rk(E) = k}=(isomorphisms )
Then for any functor
F : A!- Z(p)-mod;
lim-*(F ) is isomorphic to the homology of a cochain complex (C*(F ); ffi), whe*
*re
A
Y
Ci(F ) ~= Hom AutA (E)(StE ; F (E; )):
(E; )2Ei+1
In particular, if r denotes the p-rank of G, or the Krull dimension of H*(X; Fp*
*) or
K, then lim-i(F ) = 0 for i r.
A
The coboundary maps ffi are defined as follows. Fix an element c 2 Ci-1(F ),
and choose some (E; ) 2 Ei+1. Then the projection of ffi(c) onto the factor
Hom AutA(E)(StE ; F (E)) is the composite
RE M c(A) M F(incl)
StE --! StA ----! F (A; |A) -----! F (E; ): (1)
[E:A]=p [E:A]=p
Proof. By assumption (or by definition of Krull dimension), rk(E) r for any
(E; ) in A. Define subfunctors F = F0 F1 F2 . . .Fr = 0 by setting
aeF (E; ) if rk(E) > i
Fi(E; ) =
0 if rk(E) i.
By Proposition 4, for each i,
ae Ci(F ) if j = i
lim-j(Fi=Fi+1) ~=
0 if j 6= i.
11
The long exact sequences for extensions of functors now show that lim-*(F ) is *
*the
A
cohomology of some cochain complex
ffi 1 ffi ffi r-1
0!- C0(F ) --! C (F ) --! : :-:-! C !- 0:
It remains to check the formula for the boundary homomorphisms. Fix an object
(E; ) in A of rank i+ 1, and write I = IE for short. The inclusion : I ,! A
induces homomorphisms * : lim-*(Fj)!- lim* (Fj|I) for each j, and similarly for
A - I
the quotient functors. In particular, this yields the following commutative dia*
*gram
ffi
Ci-1(F ) ----! Ci(F )
? ?
* ?y * ?y
ffi2 i
lim-i-1(Fi-1=Fi) ----! lim (Fi=Fi+1)
~Q I - I
= [E:A]=pHom(StA;F(A)) ~=Hom (StE;F(E))
And ffi2 is in turn induced by the coboundary homomorphism
@ * * M
RE : StE ~=Hi(BIE ; BI*E)!- Hi-1(BIE ; BbIE) ~= St(A);
[E:A]=p
which shows that ffi has the form given in (1).
For functors on Ap(X) or Ap(G) which are not p-local, the chain complex must
be replaced by a spectral sequence. The same arguments as used above show:
Proposition 6. Fix a prime p, let A = Ap(X) or Ap(G) as before, and set
Ek = Ek(X) = {(E; ) 2 Ob (Ap(X)) : rk(E) = k}=(isomorphisms )
for each k 1. Then for any covariant functor
F : A!- Ab;
there is a spectral sequence
Y
Eij1~= Hj Aut A (E); Hom (StE ; F (E) =) lim-i+j(F );
(E; )2Ei+1 A
where d1 has the form described in Proposition 5 above.
I would like to thank Hans-Werner Henn for first suggesting that I look more
closely at higher limits over these categories, and for correcting some mistake*
*s in
earlier versions of this paper.
12
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*