DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY
THEORY
LAWRENCE BREEN AND ROMAN MIKHAILOV
1. Introduction
A great number of methods for computing the homotopy or the homology of a top*
*ological space
begin with a mod p reduction, and this has proved to be very eOEcient even thou*
*gh one then has to
deal with an extension problem when reverting to integer coeOEcients. However, *
*such methods are not
wellsuited when one considers spaces which are of an algebraic nature, such as*
* EilenbergMac Lane
spaces. That a purely functorial approach is possible in such a case was alread*
*y apparent in the classical
paper of S. Eilenberg and S. Mac Lane [22], in which they calculate directly th*
*e integral homology of
EilenbergMac Lane spaces in low degrees. Their results are expressed in terms *
*of what they called
the inew and quite bizarre functorsj ( ) and R( ). These functors became more *
*intelligible with
the advent of the DoldPuppe theory of derived functors of nonadditive functor*
*s [19], as they could
then be interpreted as leftderived functors of the second exterior power funct*
*or and the second divided
power functor respectively. Higher analogues of these new functors subsequently*
* appeared in related
contexts in a number a places, particularly in the Ph.D. theses of Mac Lane's s*
*tudents R. Hamsher
[28] and G. Decker [16]. However, this line of research was not vigorously purs*
*ued, though one should
mention in this context the work of H. Baues [2] and of the orst author [9], as*
* well as the unpublished
preprint of A. K. Bousoeld [8].
In the present text, we compute in this functorial spirit certain unstable ho*
*motopy groups of Moore
spaces M(A, n) and in particular of the corresponding spheres Sn = M(Z, n) . Th*
*is approach to the
computation of the homotopy groups of spheres is of particular interest, since *
*much more structure is
revealed when these homotopy groups are described as special values at the grou*
*p A = Z of a certain
functor. Our method is in some sense quite classical, since it relies on D. Kan*
*'s construction of the
loop group GK of a connected simplicial set K and on E. Curtis' spectral sequen*
*ce determined by
the lower central series oltration of GK. The initial terms in this spectral se*
*quence were described by
Curtis in terms of the derived functors of the Lie functors Ln [15]. In additio*
*n, Curtis showed that
these Lie functors are endowed with a natural oltration whose associated graded*
* components are built
up from more familiar functors.
It follows from this description that a key ingredient in such an approach mu*
*st be a good understand
ing of the derived functors of the functor Ln. We are able to achieve this in l*
*ow degrees, where this is
made possible by the fact that this Curtis decomposition of the Lie functors re*
*duces this problem to
the computation of derived functors of iterates of certain elementary functors *
*(particularly the degree
r symmetric functor SP r, and the related rth exterior algebra and rth divided *
*power functors r and
r). In order to deal with such iterates, we require a composite functor spectr*
*al sequence along the
lines of the standard Grothendieck spectral sequence [45], but for a pair of co*
*mposable nonadditive
functors such as those mentioned above. Such a nonadditive composite functor s*
*pectral sequence was
deoned by D. Blanc and C. Stover [6], but here we give a formulation of its ini*
*tial terms which is better
suited to our computational objectives. In fact this spectral sequence degenera*
*tes in our context at E2,
and may rather be thought of, for the functors which we consider, as a symmetri*
*zation of the Kunneth
1
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 2
formula and its higher analogues a la Mac Lane [35], [36]. In this way we are a*
*ble to go beyond the
computation of the iterates of 2 already considered in this context by the sec*
*ond author [39].
In our quest for the explicit values of the DoldPuppe derived functors of th*
*e Lie algebra functors
Lr for certain values of r, we deal with a number of questions of independent i*
*nterest. First of all,
starting from the description by F. Jean, a student of the orst author, of the *
*derived functors of SP r
and r [31], we give a complete description of these derived functors (as well *
*as of the divided power
functor r) for r = 2, by a method dioeerent from that of H. Baues and T. Piras*
*hvili in [5]. We then
go on to give a similarly complete and functorial description of the correspond*
*ing derived functors of
SP 3, 3 and 3, and we deduce from this a functorial description of the derive*
*d functors LiL3(A, n)
for all i and n. We also compute certain derived functors of the quartic compos*
*ite functors 2 2 and
2 2 and deduce from them certain values of the derived functors LiL4(A, n).
In order to achieve a suOEciently precise understanding of some of these deri*
*ved functors of L4, we
were led to introduce an analogue for Lie functors of the decalage morphisms. T*
*he latter are determined
by the existence of Koszul complexes, which relate to each other the derived fu*
*nctors of SP r, r and
r. Just as r may be viewed as a superanalogue1 of SP r, we need to introduce*
* a superanalogue Lrs
of the Lie functor Lr. While there no longer exists a decalage isomorphism betw*
*een these two functors,
there is a canonical pension map between them, which we call the semidecalage *
*morphism, and which
allows us to give a reoned description of their derived functors in certain cas*
*es.
We also rely at some point on the knowledge of the homology of the complex Cn*
*(A) dual to the de
Rham complex orst introduced in the present context by V. Franjou, J. Lannes an*
*d L. Schwartz in
[25]. We give in particular in an appendix an explicit calculation of the value*
* of the homology groups
H0Cn(A) for all n announced by Jean in [31], as well as a description of all th*
*e homology groups
HiCn(A) when n < 8. The occurence of 8torsion when n = 8, as well as that of a*
* Lie functor when
n = 6, suggests that no simple description of these groups can be expected for *
*a general n.
In #11, we use these tools in order to achieve our goal of computing algebrai*
*cally certain homotopy
groups of nspheres and Moore spaces M(A, n). The task at hand is twofold. The *
*orst part consists,
as we have said, in computing the initial terms of the Curtis spectral sequence*
*, and for this we rely on
our knowledge of the derived functors of certain Lie functors and their supera*
*nalogues. The second
part consists in understanding certain dioeerentials in the spectral sequence. *
*We rely here upon various
methods, some based on the functoriality of our construction and on the fact th*
*at the dioeerentials
are now natural transformations rather then simply group homomorphisms, and oth*
*ers more classical
in spirit (the suspension of a Moore space, the comparison of a Moore space wit*
*h the corresponding
EilenbergMac Lane space, ...). We have at times in this onal section made use*
* of known results
concerning the homotopy of Moore spaces for specioc groups A, whenever this all*
*owed us to progress
with our own investigations. It is quite striking to observe how far one can go*
* in the description of
these homotopy groups, with only the knowledge of derived functors of quadratic*
* and cubical functors
as the basic input.
In this onal section we proceed in logical order, beginning with the homotopy*
* groups of S2 and
M(A, 2) and then moving on to M(A, n) for increasing values of n. As an illustr*
*ation of our methods,
we begin by explaining how one may obtain in this way the known values of the 3*
*torsion of ssi(S2),
for values of i up to 14. To have gone further, so as to retrieve the classical*
* results of H.Toda [44]
up to i = 22, would have obliged us to delve further into the analysis of the s*
*pectral sequence. We
also recover, and reinterpret, some results of Baues and his collaborators [2],*
* [21], [3]. In particular,
we obtain by our methods the results of Baues and Buth [3] concerning ssi(M(A, *
*2)) for i = 4, and
____________
1We prefer to call this the superanalogue, rather than the graded analogue a*
*s is more customary, since all our functors
are graded.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 3
give an improved description of the unpublished results of Dreckmann for i = 5 *
*[21] (see also Baues
and Goerss [4]). Specializing to the case A = Z=p, we obtain further informatio*
*n about the groups
ssi(M(Z=p, 2)) whenever the prime p is odd. By suspending these calculations, t*
*his gives us in particular
a fully functorial description of the graded components associated to a natural*
* oltration of the group
ss6(M(A, 3)). As a consequence of these computations, we can recover the value*
* of ss5(M(Z=3, 2)),
a signiocant case of the extension by Neisendorfer [40] to the prime p = 3 of C*
*ohen, Moore and
Neisendorfer's study [12] of the homotopy of Moore spaces.
As a onal example, we examine the low degree homotopy groups of M(Z=3, 5) sin*
*ce this allows
us to exhibit in a simple context some of the techniques on which we relied thr*
*oughout this section.
In fact, the reader may wish to begin with this case, before going on to the mo*
*re delicate unstable
computations which precede it.
As will be apparent from this description of our paper, a number of our resul*
*ts in homotopy theory
have already appeared in one form or another in the litterature, where they are*
* proved by very diverse
methods. Our aim here is to show that these can all be obtained by a uniform me*
*thod, based solely
on functorial techniques from homological and homotopical algebra with integer *
*coeOEcients. We ex
pect that such an approach to these questions will not only allow one to comput*
*e specioc additional
homotopy groups, but more importantly will shed some new light upon their globa*
*l structure.
2.Derived functors
2.1. Graded functors. Let Ab be the category of abelian groups and A an object *
*of Ab. For any
chain complex Ci, we will henceforth denote by C[n] the chain complex deoned by
C[n]i:= Cinfor all i.
In particular, the chainLcomplex A[n] is concentrated in degreeLn. In addition *
*to the graded tensor
power functorL := n 0 n, the symmetric power functor SP := n 0SP n, and th*
*e exterior power
functor := n 0 n (the quotient of A by the ideal generated by elements x *
* x for all x 2 A),
we will consider over Z the following somewhat less wellknown functors:
L
1. The divided power functor: (see [42]) * = n 0 n : Ab ! Ab. The graded *
*abelian group
*(A) is generated by symbols fli(x) of degree i 0 satisfying the following r*
*elations for all x, y 2 A:
1) fl0(x) = 1
2) fl1(x) = x
` '
s + t
3) fls(x)flt(x) = fls+t(x)
s
X
4) fln(x + y) = fls(x)flt(y), n 1
s+t=n
5) fln(x) = (1)nfln(x), n 1.
In particular, the canonical map A ' 1(A) is an isomorphism. The degree 2 comp*
*onent 2(A) of
*(A) is the Whitehead functor (A). It is universal for homogenous quadratic *
*maps from A into
abelian groups. The following additional relations in *(A) are consequences of*
* the previous ones:
flr(nx) = nrflr(x), n 2 Z;
rflr(x) = xflr1(x);
xr = r!flr(x);
flr(x)yr = xrflr(y).
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 4
In addition, a direct computations implies that
r(Z=n) ' Z=n(r, n1,)
where the extended g.c.d (r, n1 ) is deoned by (r, n1 ) := limm!1 (r, nm ).
2. The Lie functor L : Ab ! Ab (see [14]). The tensor algebra A is endowed wi*
*th a ZLie algebra
structure, for which the bracket operation is deoned by
[a, b] = a b  b a, a, b 2 (A).
One deones nfold brackets inductively by setting
[a1, . .,.an] := [[a1. .,.an1], an] (2*
*.1)
L
We will denote A, viewed as a ZLie algebra, by (A)Lie. Let L(A) = n 1Ln(A)*
* be the subLie
ring of (A)Liegenerated by A. Its degree 2 and 3 components are generated by t*
*he expressions
a b  b a and a b c  b a c + c a b  c b(*
*2.a2)
where a, b, c 2 A. L(A) is called the free Lie ring generated by the abelian gr*
*oup A. It is universal for
homomorphisms from A to ZLie algebras. The grading of A determines a grading *
*on L(A), so that
we obtain a family of endofunctors on the category of abelian groups:
Li: Ab! Ab, i 1.
For any free group F and i 1, one has the natural MagnusWitt isomorphism ([3*
*7], [46])
fli(F )=fli+1(F ) ' Li(Fab). (2*
*.3)
where fli(F ) is the ith term in the lower central series of F .
3. The Schur functors. We will also consider the Schur functors
Jn, Y n, En : Ab! Ab, n 2
deoned by
Jn(A) = ker{A SP n1(A) ! SP n(A)}, n 2, (2*
*.4)
Y n(A) = ker{A n1(A) ! n(A)}, n 2,
En(A) = ker{A n1(A) ! n(A)}, n 2 .
In particular,
J2(A) = E2(A) ' 2A and Y 2(A) ' 2(A) (2*
*.5)
whenever A is free. The functors Y n(A) are the Zforms of the Schur functors S*
*~(V ) associated to the
partition ~ = (2, 1 . .,.1) of the set (n) (see [24] exercise 6.11, [23] ch. 8 *
*(19)). The functors Jn(A)
and En(A) are two distinct Z forms of the Schur functors S~ associated to the *
*partition ~ = (n  1, 1)
of (n), which is the conjugate partition of ~.
The functors Jn and their derived functors were considered by E. Curtis [15] *
*and J. Schlesinger [43].
Just as the quotient of a Lie ring L by the ideal generated by all brackets is *
*an abelian Lie ring, one
can consider the metabelian Lie rings. These are the quotients of L by the idea*
*l generated by brackets
of the form [[ ], [ ]]. The following proposition asserts that the functors Jn,*
* restricted to free abelian
groups A, are metabelian Lie functors
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 5
Proposition 2.1. [43] Let A be a free abelian group and n 2. The 4term seque*
*nce
0 ! Ln(A) \ L2L2(A) ! Ln(A) pn!A SP n1(A) rn!SP n(A) ! 0, (2*
*.6)
is exact, where rn is the multiplication, and the map pn is deoned by
pn : [m1, . .,.mn] 7! m1 m2. .m.n m2 m1m3, . .m.n, where mj2 A 8j.
The projection
Ln(A) ___////_Jn(A) (2*
*.7)
of Ln(A) onto its image in A SP n1A will also be denoted pn, so that if we s*
*et
J"n(A) := Ln(A) \ L2L2(A),
the sequence (2.6)splits into a pair of short exact sequences
pn n
0____//_"Jn(A)_//_Ln(A)__//_J (A)__//_0 . (2*
*.8)
0 ____//_Jn(A)__//_A SP n1_//_SP n(A)__//_0 (2*
*.9)
In particular,
L2(A) ' 2(A) and L3(A) ' J3(A) (2*
*.10)
since "Jn(A) is trivial for n = 2, 3.
Let us now recall some the natural transfomations between these functors:
fn :SP n(A) ! n A (2*
*.11)
X
a1. .a.n 7! ai1 . . .ain
oe2 n
gn : n(A) ! n A (2*
*.12)
X
a1^ . .^.an 7! sign(oe) ai1 . . .ain
oe2 n
hn : n(A) ! n A (2*
*.13)
X
flr1(a1) . .f.lrk(ak)7! ai1 . . .ain
(i1,...,in)
In these deonitions of fn and gn, we have set ij:= oe(j), whereas the (i1, . *
*.,.in) in the deonition of
hn range over the set of ntuples of integers for which j occurs rj times (1 *
*j k). When A is free
abelian, the induced morphism
hn : n(A) ! ( nA) n,
from n(A) to the group of tensors invariant under the action of the symmetric *
*group is an isomorphism
for all n 1. By the universal property of the algebra *(A), hn may also be c*
*haracterized as the
map determined by the divided power algebra structure on A := n( nA), where t*
*he product in
this algebra is deoned by the shuOe product, and the divided powers are charact*
*erized by the rule
fln(a) := a . . .a 2 nA for all a 2 A.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 6
2.2. Derived functors. Let A be an abelian group, and F an endofunctor on the c*
*ategory of abelian
groups. Recall that for every n 0 the derived functor of F in the sense of Do*
*ldPuppe [19] are deoned
by
LiF (A, n) = ssi(F KP*[n]), i 0
where P* ! A is a projective resolution of A, and K is the DoldKan transform, *
*inverse to the Moore
normalization functor
N : Simpl(Ab) ! C(Ab)
from simplicial abelian groups to chain complexes [45] Def. 8.3.6. We denote by*
* LF (A, n) the object
F K(P*[n]) in the homotopy category of simplicial abelian groups determined by *
*F K(P*[n]), so that
LiF (A, n) = ssi(LF (A, n)) .
We set LF (A) := LF (A, 0) and LiF (A) := LiF (A, 0) for any i 0. When the fu*
*nctor F is additive,
the LiF (A) are isomorphic by iterated suspension to Li+nF (A, n) for all n, an*
*d coincide with the
usual derived functors of F . As examples of these constructions, observe that *
*the simplicial models
LF (L ! M) of LF A and F K((L ! M)[1]) of LF (A, 1) associated to the twoter*
*m AEat resolution
0 ! L f!M ! A ! 0 (2*
*.14)
of an abelian group A are respectively of the following form in low degrees:
@0,@1,@2 @0,@1
F (s0(L) s1(L) s1s0(M))!!!F (L s0(M))!!F (M) (2*
*.15)
 
where the component F (M) is in degree zero, and
@0,...,@3 @0,@1,@2
F (s0(L) s1(L) s2(L)!!! !!
s1s0(M) s2s0(M) s2s1(M))!F (L s1(M) s0(M)) ! F (M) (2*
*.16)
where the component F (M) is in degree 1. It follows from the deonition of homo*
*logy that LiZ(A, n) '
Hi(K(A, n); Z) for all n, where K(A, n) is an EilenbergMacLane space associate*
*d to the abelian group
A .
1. Derived functors of n [36]. For n 1, and abelian groups A1, . .,.An, we d*
*eone2
` L L '
Tori(A1, . .,.An) := Hi A1 . . .An , i 0.
L
where A B is the derived tensor product of the abelian groups A and B in the *
*derived category of
abelian groups, as in [45] #10.6. In particular,
Tor0(A1, . .,.An) ' A1 . . .An and Tori(A1, . .,.An ) = 0 i *
* n.
One sets
Tor(A1, A2) := Tor1(A1, A2) and Tor[n](A) := Torn1(A, . .,.A) (n cop*
*ies of.A)
While computations of such iterated Tor functors for specioc abelian groups A a*
*re elementary, an
explicit functorial description of the multifunctors Toriis more delicate. The*
* functorial short exact
sequence
0 ! Tor(A1, A2) A3 ! Tor1(A1, A2, A3) ! Tor(A1 A2, A3) ! 0 ,
splits unnaturally [35], [36]. The EilenbergZilber theorem determines natural *
*isomorphisms
Li n A ' Tori(A, . .,.A), i 0.
____________
2In [35], Mac Lane uses the notation Trip(A1, A2, A3) for the group Tor1(A1, *
*A2, A3) and Tor(A1, . .,.An) for
Torn1(A1, . .,.An).
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 7
The group Tor[n](A). It is generated by the nlinear expressions oh(a1, . .a.n)*
* (where all ailive in the
subgroup hA of elements a of A for which ha = 0 (h > 0), subject to the socall*
*ed slide relations
ohk(a1, . .,.ai, . .a.n) = oh(ka1, . .,.kai1, ai, ki+1, . .,.*
*kan)(2.17)
for all i whenever hkaj = 0 for all j 6= i and hai= 0. The associativity of the*
* derived tensor product
functor implies that there are canonical isomorphisms
Tor[n](A) ' Tor(Tor[n1](A), A), n 2.
The description of derived functors Li n A for a general i follows from that of*
* Tor[n](A). For every
abelian group A, n 1, 1 i n  1, the group Li n (A) is by [36] the quotie*
*nt :
Li n (A) ' Tor[i+1](A) ( ni1(A))=Jac ,
where Jac is the subgroup of generalized Jacobitype relations, generated by t*
*he elements
i+2X
(1)koh(x1, . .,.cxk, . .,.xi+2) xk xi+3 . . .xn
k=1
for all x1, . .,.xn 2 A.
2. Derived functors of SP n. The map n ! SP ninduces a natural epimorphism
Tor[n](A) ! Ln1SP n(A) (2*
*.18)
which sends the generators oh(a1, . .,.an) of Tor[n](A) to generators fih(a1, .*
* .,.an) of
Sn(A) := Ln1SP n(A) .
The kernel of this map is generated by the elements oh(a1, . .,.an) with ai= aj*
* for some i 6= j. It is
shown by Jean in [31] that
LiSP n(A) ' (LiSP i+1(A) SP n(i+1)(A))=JacSP, (2*
*.19)
where JacSP is the subgroup generated by elements of the form
Xi+2
(1)kfih(x1, . .,.^xk, . .,.xi+2) xky1. .y.ni2.
k=1
with xi 2 hA and yj 2 A for all i, j. The oltration of Z(A, n) by powers of the*
* augmentation ideal
determine a oltration on the homology groups Hr(K(A, n)), whose associated grad*
*ed pieces are the
LrSP s(A, n) [9].
3. Derived functors of n. For any abelian group A and n 1, we set
n(A) := Ln1 n(A).
Consider the action of the symmetric group n on Tor[n](A), deoned by
oeoh(a1, . .,.an) = sign(oe) oh(aoe(1), . .,.aoe(n))
where ha1 = . .=.han = 0, ai2 A, oe 2 n. We denote this action by ffln. The n*
*atural transformations
gn induces functorial isomorphisms between n(A) and the fflninvariants in To*
*r[n](A) [9], [31] th. 2.3.3:
ffl
n(A) ' (Tor[n](A)) n.
In particular, for all n > 0,
n(Z=r) ' Z=r . (2*
*.20)
In addition, the morphisms oh which describe the Tor functors now symmetrize to*
* homomorphisms
~nh: n( hA) ! n(A) (2*
*.21)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 8
for h 1 and the group n(A) is generated by the elements
!hi1(x1) * . .*.!hij(xj) := ~h(fli1(x1) . .f.lij(xj)) (2*
*.22)
P
with ik 1 for all k, and kik = n. These satisfy relations which may be thou*
*ght of, as in [9], as
symmetrized versions of the slide relations (2.17). The following description o*
*f the derived functors
Li n is given in [31] Theorem 2.3.5:
Li n(A) ' ( i+1(A) ni1(A))=Jac . (2*
*.23)
Here Jac is the subgroup generated by the expressions
Xj
!hi1(x1) * . .*.!hik1(xk) * . .*.!hij(xj) xk ^ y1^ . .y.n*
*i2
k=1
P j
for all h, with k=1= i + 2. In particular, this implies that for any onite cy*
*clic group A,
Li n(A) = 0 i 6= n  1. (2*
*.24)
4. Derived functors of n. Not all is known about derived functors of the div*
*ided power functors.
For an abelian group A, the double decalage isomorphism (described in (2.40)bel*
*ow) determines a
composite isomorphism
L1 2(A) ' L5SP 2(A, 2) ' H5(K(A, 2), Z)
so that L1 2(A) is isomorphic to the functor R(A) of EilenbergMac Lane [22] [2*
*2] #22, deoned as
(Tor(A, A) 2(2A))=S,
where S is the subgroup generated by elements
oh(x, x), x 2 hA, h 2 N,
fl2(x + y)  fl2(x)  fl2(y)  o2(x, y), x, y 2 2A.
More generally, we set
Rn(A) := Ln1 n(A) ,
so that R2(A) = R(A), even though this is inconsistent with the notation in [16*
*]. The sequence
0 ! SP 2(A) ! 2(A) ! A Z=2 , (2*
*.25)
is exact for any abelian group A, and derives to the short exact sequence
0 ! L1SP 2(A) ! L1 2(A) ! Tor(A, Z=2) ! 0.
Analogous short exact sequences were obtained in [31] #3.1 for the functor 3 :
0 ! L1SP 3(A) ! L1 3(A) ! (Tor(A, Z=2) A Z=2) Tor(A, Z=3) ! 0(*
*2.26)
0 ! L2SP 3(A) ! L2 3(A) ! Tor(A, Z=2) Tor(A, Z=2) ! 0
2.3. Koszul complexes. ([41], [29] I (4.3.1.3)). Let f : P ! Q be a homomorphis*
*m of abelian groups.
For n 1 and any k = 0, . .,.n  1 consider the maps
~k+1: k+1(P ) SP nk1(Q) ! k(P ) SP nk(Q)
deoned, for pi2 P and qj2 Q, by:
k+1X
~k+1: p1^ . .^.pk+1 qk+2. .q.n7! (1)k+1ip1^ . .^.^pi^ . .^.pk+1 f(pi) *
*qk+2. .q.n.
i=1
The associated Koszul complex is deoned by
~n1 n1 ~1 n
Kosn(f) : 0 ! n(P ) ~n!^n1(P ) Q ! . .!.P SP (Q) ! SP (Q) ! 0(.*
*2.27)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 9
Dually, one deones maps
~k+1: k+1(P ) nk1(Q) ! k(P ) nk(Q), k = 0, . .,.n  1
by setting
~k+1: flr1(p1) . .f.lrk(pk) q1^ . .^.qnk17!
Xk
flr1(p1) . .f.lrj1(pj) . .f.lrk(pk) f(pj) ^ q1^*
* . .^.qnk1 (2.28)
j=1
These maps determine a dual Koszul complex:
n ~n1 n1 ~1 n
Kosn(f) : 0 ! n(P ) ~! n1(P ) Q ! . .!.P (Q) ! (Q) ! 0 (2*
*.29)
The complexes Kosn(f) and Kosn(f) are the total degree n components of the Kosz*
*ul complexes
(P ) SP (Q) and (P ) (Q) associated to a given homomorphism f : P ! Q. F*
*or a twoterm AEat
resolution (2.14)of an abelian group A, the complexes Kosn(f) and Kosn(f) repre*
*sent the derived
category objects LSP n(A) and L n(A) respectively (see for example [32]). In pa*
*rticular, when P is
free abelian and f the identity arrow, both complexes are acyclic.
L
For n 2, the derived category object LSP n1(A) A may be represented, for*
* some 2term AEat
resolution f : L ! M of A, by the tensor product of Kosn(f) and L ! M, in other*
* words as a total
complex associated to the bicomplex
n1(L) L____//_._._.//_L SP n2(M) __L_//_SP n1(M) L
fflffl fflffl fflffl
n1(L) M____//_._._.//_L SP n2(M) __M_//_SP n1(M) M
L
The diagram LJn(A) ! LSP n1(A) A ! LSP n(A) in the derived category may ther*
*efore be
represented by the following diagram of (horizontal) complexes:
" n
Y n(L)"O`___//_._._._______//L SP n1(M)" _M_____________//_`J"(M)` (2*
*.30)
fflffl " fflffl fflffl
n1(L) LO___//_._._.//_L SP n2(M) M SP n1(M)___L//_SP n1(M) M
fflfflfflffl fflfflfflffl fflfflfflff*
*l
" n
n(L)O______//._._.__________//L SP n1(M)_______________//SP (M)
The upper line
Y n(L)____//_. ._._//_L SP n1(M) M___////_Jn(M) (2*
*.31)
in this diagram is a Koszul complex for the functors Jn and Y n. A similar diag*
*ram, whose lower line
is the dual Koszul complex (2.29)for n = 3 is described in appendix B below. Fo*
*r Koszul complexes
associated to more general Schur functors, see [33] lemma 1.9.1, [1].
2.4. Pensions and decalage. Consider the homomorphisms ([7] 7.4)
jn : n(A) n(B) ! SP n(A B),
n : n(A) SP n(B) ! SP n(A B).
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 10
These are characterized as the unique homomorphisms for which the corresponding*
* diagrams
jn n n n n
n(A) n(B) _____//_SP (A B) n(A) SP (B) ____//_SP (A B)
gn gn fn hn fn fn
fflffl fflffl fflffl fflffl
~n n n n ~n n
( n(A)) ( n(B))_____// (A B) ( (A)) ( (B))____// (A B)
(2*
*.32)
commutes, with ~n deoned by
~n : (a1 . . .an) (b1 . . .bn) 7! (a1 b1) . . .(an bn), ai2 A, *
*bi2 B.
The map jn is explicitly given by the formula
X
(a1^ . .^.an) (b1^ . .^.bn) 7! sign(oe)(a1 boe(1)) . .(.an (2*
*boe(n))..33)
oe2 n
It is shown in [7] 7.6 that for any free simplicial abelian group X, the maps*
* jn and n induce the
isomorphisms of homotopy groups
ssi( n(X) nK(Z, 1)) ! ssi(SP n(X K(Z, 1))), i 0, (2*
*.34)
ssi( n(X) SP nK(Z, 2)) ! ssi(SP n(X K(Z, 2))), i 0. (2*
*.35)
so that there are natural isomorphisms
L n n
L n(A, m) (Z, 1) ' LSP (A, m + 1),
L n n
L n(A, m) SP (Z, 2) ' LSP (A, m + 2).
in the derived category. These derived pairings induce for n 1, by adjunctio*
*n with the volume
element ncycle in n(Z, 1)n = n(Zn) and the corresponding element in SP n(Z, *
*2) respectively, a
pair of functorial pension morphisms
L j(A, n)[j] ! LSP j(A, n + 1) (2*
*.36)
L j(A, n)[j] ! L j(A, n + 1) (2*
*.37)
in the derived category. The maps induced on homotopy groups are are of the form
Li j(A, n) ' Li+jSP j(A, n + 1) (2*
*.38)
Li j(A, n) ' Li+j j(A, n + 1) (2*
*.39)
The inverses of these maps are iterated boundary maps arising from the exactnes*
*s of the Koszul
complexes and are known as decalage isomorphisms [29] I 4.3.2. Composing the la*
*st two determines a
double decalage isomorphism:
Li j(A, n) ' Li+2jSP j(A, n + 2) . (2*
*.40)
Similarly, it follows from the existence of Koszul sequences of type (2.31)th*
*at there exist decalage
isomorphisms
LiY j(A, n) ' Li+jJj(A, n + 1) (2*
*.41)
between the derived functor of Jj and Y jfor all j, n 0.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 11
3.The de Rham complex and its dual
Let A be an abelian group. For n 1, let Dn*(A) and Cn*(A) be the complexes *
*of abelian groups
deoned by
Dni(A) = SP i(A) ni(A), 0 i n,
Cni(A) = i(A) ni(A), 0 i n,
where the dioeerentials di: Dni(A) ! Dni1(A) and di: Cni(A) ! Cni1(A) are:
Xi
di((b1. .b.i) bi+1^ . .^.bn) = (b1. .^.bk.b.i.) bk ^ bi+1^ . *
*.^.bn
k=1
Xi
di(b1^ . .^.bi X) = (1)kb1^ . .^.^bk^ . .^.bi bkX
k=1
for any X 2 ni(A). The complex Dn(A) is the degree n component of the classic*
*al de Rham complex,
orst introduced in the present context of polynomial functors in [25] and denot*
*ed n in [26]. The dual
complexes Cn(A) were considered in [31]. We will call them the dual de Rham com*
*plexes.
We will now give a functorial description of certain homology groups of these*
* complexes Cn(A).
Proposition 3.1. Let A be a free abelian. Then
(1)[26] For any prime number p, H0Cp(A) = A Z=p, and HiCp(A) = 0, for all *
*i > 0;
(2)[31] There is a natural isomorphism
M
H0Cn(A) ' n=p(A Z=p).
pn, p prime
A proof of proposition 3.1 (2) is given in Appendix A.
The higher homology groups HiCn(A) are more complicated. The following table,*
* which is a con
sequence theorem A.1 of Appendix A, gives a complete description of HiCn(A) for*
* n 7 and A free
abelian:
n__________H0Cn(A)__________________H1Cn(A)___________H2Cn(A)___H3Cn(A)___
8  4(A Z=2) * * *
7  A Z=7 0 0 0
6  2(A Z=3) 3(A Z=2) 2(A Z=3) L3(A Z=2) 3(A Z=2) 0
5  A Z=5 0 0 0
4  2(A Z=2) 2(A Z=2) 0 0
3  A Z=3 0 0 0
2  A Z=2 0 0 0
Table 1
For example, the isomorphism
f : 2(A Z=2) ! H1C4(A) (3*
*.1)
is deoned, for representatives a, b 2 A of ~a,~b2 A Z=2, by
f : ~a ~b7! a afl2(b)  b bfl2(a).
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 12
3.1. Comparing the de Rham and Koszul complexes. For any free abelian group A, *
*consider
the following natural monomorphism of complexes:
" n
Kosn(A) : n(A) O___//_ n1(A) A___//_. ._._//_A SP n1(A)__////_SP *
*(A)
" `
    
    
fflffl   fflffl ffl*
*ffl
"
Cn(A) : n(A) O___//_ n1(A) A___//_. ._.__//A n1(A)_____//_ n(A)
(3*
*.2)
Let us denote the cokernel of this map by Dn(A). We have
Dn(A) : n2(A) W2(A) ! n3(A) W3(A) ! . .!. A Wn1(A) ! Wn(*
*A)
where
Wn(A) = coker{SP n(A) ! n(A)}
Since Koszul complex is acyclic, it follows that
HiCn(A) ' HiDn(A), n 0.
Proposition 3.1 implies that the sequence:
0 ! A A Z=2 ! W3(A) ! A Z=3 ! 0 (3*
*.3)
is exact (and splits naturally). For every n 2, m 0, we obtain the natural *
*exact sequence:
0 ! LSP n(A, m) ! L n(A, m) ! LWn(A, m) ! 0 (3*
*.4)
Passing to homotopy groups and applying the decalage isomorphisms (2.38), (2.39*
*), this yields the
long exact sequence
. .!.LiSP n(A, m) ! Li+2nSP n(A, m + 2) ! LiWn(A, m) !
Li1SP n(A, m) ! Li+2nSP n(A, m + 2) ! Li1Wn(A, m) ! .*
* . .(3.5)
Let X be a free abelian simplicial group and k 1, n 2 be integers. If ssi(X*
*) = 0, i < k, then by
[19], Satz 12.1 (
fori < n, when k = 1,
ssi(SP n(X)) = 0, (3*
*.6)
fori < k + 2n  2, providedk > 1.
We will make use of exact sequence (3.5)and of the assertion (3.6)in order to c*
*ompute derived functors
of polynomial functors of low degrees.
4.Derived functors of quadratic functors
For every abelian group A, the exactness of the sequence (2.25)implies that W*
*2(A) ' A Z=2.
Since this functor is additive, it follows immediately that
8
>:
0, i 6= m, m + 1
for all m. Let us deone a new functor ~2(A) by:
~2(A) := 2(A) Tor(A, Z=2). (4*
*.1)
The long exact sequence (3.5), the connectivity result (3.6), and the decalage *
*formulas (2.38)and (2.39)
produce the following complete description of the derived functors of the symme*
*tric power functor SP 2.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 13
Proposition 4.1.
8 2
>>>SP (A), i = 0, n = 0
>>>S (A), i = 1, n = 0
>>> 2
>>> 2(A), i = 2, n = 1
>>> n1_
>>>> 2(A), i = 2n, n 6= 0 even
>>> 2
>>>~ (A), i = 2n, n 6= 1 odd
>>>R2(A), i = 2n + 1, n 6= 0 even
>>>
>>: 2(A), i = 2n + 1, n odd
0 for all other i.
We will only sketch the proof of this computation in the present quadratic si*
*tuation, and will discuss
the more elaborate case of cubical functors in the following section. These qua*
*dratic results were also
obtained in [5] #4 by a dioeerent method (see also [39] #A.15).
Proof: The orst two equations follow from the deonitions. By double decalage (*
*2.40), there is an
iterated isomorphism
2(A) = L0 2(A, 0) ' L4SP 2(A, 2)
which determines the sixth equation above for n = 2. The general case of the s*
*ixth equation then
follows by induction when we consider the isomorphism L2nSP 2(A, n) ' L2n+2SP 2*
*(A, n + 2) from
(3.5)for n even. Decalage also implies that
2A ' L2SP 2(A, 1)
and the sequence (3.4)then determines a short exact sequence
0 ! L2SP 2(A, 1) ! L2 2(A, 1) ! Tor(A, Z2) ! 0
Consider the following diagram, in which the vertical arrows are the suspension*
* maps:
0______//L1SP 2A______//L1 2(A)___//Tor(A, Z2)__//_0
0  
fflffl fflffl 
0____//_L2SP 2(A,_1)_//_L2 2(A,_1)//_Tor(A, Z2)_//_0
The lefthand vertical arrow is trivial by [19] corollary 6.6, since all elemen*
*ts of L1SP 2A are in the
image of the arrow (2.18)(n = 2). The lower sequence is therefore split since i*
*t is a pushout by the
trivial map, and a diagram chase makes it clear that this splitting is functori*
*al. This proves the seventh
equation in proposition (4.1)for n = 3 by the double decalage isomorphism
L2 2(A, 1) ' L6SP 2(A, 3)
The general case of the sixth equation now follows by induction since (3.4)and *
*decalage imply that
LiSP 2(A, n) ' Li 2(A, n) ' Li+4SP 2(A, n + 2)
for all n. A similar discussion, in the next degree, shows that the seventh and*
* eight equations are also
satisoed. The remaining fourth and ofth equations are proved by considering onc*
*e more the sequence
(3.5), and observing that the functors LiSP 2(A, n) vanish by [19] whenever i i*
*s suOEciently large.
As a corollary, one onds that this computation (and even the inductive reason*
*ing that led to it) can
be carried over by the decalage isomorphisms (2.38), (2.40)to the derived funct*
*ors of 2 and 2. We
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 14
simply state the result:
8
>>> 2(A), i = 0, n = 0,
>>> n1_
>>>A Z=2, i = n + 1, n + 3 . .,.n + 2[ 2 ] + 1, i 6= 2n
>>>Tor(A, Z=2), i = n + 2, n + 4, . .,.n + 2[n1_]
>< 2
2(A), i = 2n, n odd
Li 2(A, n) = (4.2)
>>>~2(A), i = 2n, n 6= 0 even
>>>
>>>R2(A), i = 2n + 1, n odd
>>> 2(A), i = 2n + 1, n even
>:
0 for all other i.
8 n1
>>>A Z=2, i = n, n + 2, . .,.n + 2[___2], n > 0
>>>Tor(A, Z=2), i = n + 1, n + 3, . .,.n + 2[n1_] + 1, n > *
*0, i 6= 2n
>>> 2
>< 2(A), i = 2n, n even
Li 2(A, n) = ~2(A), i = 2n, n odd (4*
*.3)
>>>
>>>R2(A), i = 2n + 1, n even
>>> (A), i = 2n + 1, n odd
>: 2
0 for all other i.
5. The derived functors of certain cubical functors
It follows from (3.3) that
W3(A) = (A A Z=2) (A Z=3), (5*
*.1)
and this derives to an isomorphim
L L L
LW3(A) ' (A A Z=2) (A Z=3) (5*
*.2)
in the derived category, from which the values of LiW3(A) follow immediately (c*
*onsistently with the
two equaitons (2.26)). This implies that
8
>>>A Z=3, i = 1
>>>
< A A Z=2 Tor(A, Z=3), i = 2
LiW3(A, 1) = Tor1(A, A, Z=2), i = 3 (5*
*.3)
>>>
>>>Tor2(A, A, Z=2), i = 4
: 0, for all other i
and, for m > 1: 8
>>>A Z=3, i = n
>>>
>>>>Tor1(A, A, Z=2), i = 2n + 1,
>>>
>>:Tor2(A, A, Z=2), i = 2n + 2,
0, for all other i.
We will now use this computation in order to determine the derived functors o*
*f SP 3in all degrees.
Let X be a free abelian simplicial group. The natural map of simplicial groups
E : SP 3(X) K(Z, 2) ! SP 3(X K(Z, 2))
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 15
induces the pairing map [8]:
ffl3 : ssiSP 3(X) H6K(Z, 2) ! ssi+6SP 3(X K(Z, 2))
Observe that the following diagram is commutative:
ffl3 3
ssiSP 3(X)______//ssi+6SP (X K(Z, 2))
OO
 
 
fflffl 
ssi 3(X)__'__//ssi+6( 3(X) SP 3K(Z, 2))
where the right hand vertical arrow is the homomorphism (2.35). It follows from*
* (3.5), (5.3)and (5.4)
that the maps
ffl3 : LiSP 3(A, n) ! Li+6SP 3(A, n + 2)
are isomorphisms for , i 6= n  1, n, n + 1, n + 2, 2n  1, 2n, 2n + 1, 2n + 2.*
* In addition the sequence
0 ! L2n+2SP 3(A, n) ! L2n+8SP 3(A, n + 2) ! Tor2(A, A, Z=2) !
L2n+1SP 3(A, n) ! L2n+7SP 3(A, n + 2) ! Tor1(A, A, Z=2) !
L2nSP 3(A, n) ! L2n+6SP 3(A, n + 2) ! A A Z=2 !
L2n1SP 3(A, n) ! L2n+5SP 3(A, n + *
*2) (5.5)
is exact by (3.5). Furthermore, for n > 1,
Ln+7SP 3(A, n + 2) ' Tor(A, Z=3)
Ln+6SP 3(A, n + 2) ' A Z=3
by (3.6). Finally, according to [8] corollary 4.3 , the pension maps
"3 : LiSP 3(A, n) ! Li+6SP 3(A, n + 2)
are split injections, for all i 0 and all n > 1. The long exact sequence (5.5*
*) therefore decomposes
into short exact sequences:
0 !L2n+2SP 3(A, n) ! L2n+8SP 3K(A, n + 2) ! Tor2(A, A, Z=2) ! 0
0 !L2n+1SP 3(A, n) ! L2n+7SP 3K(A, n + 2) ! Tor1(A, A, Z=2) ! 0
0 !L2nSP 3(A, n) ! L2n+6SP 3(A, n + 2) ! A A Z=2 ! 0
and an isomorphism
L2n+5SP 3(A, n + 2) ' L2n1SP 3(A, n).
Example 5.1. Since the values taken by the derived functors of W3 in (5.3)and (*
*5.4)are distinct, we
must consider the implications of(5.3)separately. Observe that exact sequence (*
*3.5) and the equations
(5.3)imply that
L11SP 3(A, 3) = 3(A),
L8SP 3(A, 3) = A A Z=2 Tor(A, Z=3),
L7SP 3(A, 3) = A Z=3,
and that the groups LiSP 3(A, 3) for i = 9, 10 live in the long exact sequence
0 ! L1 3(A) ! L10SP 3(A, 3) ! Tor2(A, A, Z=2) @!
3(A) "3!L9SP 3(A, 3) ! Tor1(A, A, Z=2) !*
* 0 . (5.6)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 16
The diagram
" g3
3(A)O______________________// 3(A)
o o
fflffl h3 fflffl
L3SP 3(A, 1)___//_L3 3(A,_1)_//_L3 3(A, 1)
commutes, where the lefthand vertical arrow is the map (2.38), and the righth*
*and one the corre
sponding obvious decalage map for tensor powers. It follows that the composite *
*map
ffl3 : 3A ! L3 3(A, 1) ' L9SP 3(A, 3)
is injective so that the boundary map @ in (5.6) is trivial. The complete descr*
*iption of the LiSP 3(A, 3)
is therefore given by
8
>>> 3(A), i = 11
>>A Z=3, i = 7
:0, i 6= 7, 8, 9, 10, 11
and the exactness of the sequences
0 ! L1 3(A) ! L10SP 3(A, 3) ! Tor2(A, A, Z=2) ! 0
0 ! 3(A) ! L9SP 3(A, 3) ! Tor1(A, A, Z=2) ! 0 .
The discussion in example 5.1 does not only apply to the derived functors LiS*
*P 3(A, n) with n = 3.
The corresponding assertion for a general n is the following one:
Theorem 5.1. Case I: n 3 is odd.
8
>>>A Z=3, n + 4 i < 2n + 2, i  n 0 mod 4
>>>
>>>Tor(A, Z=3), n + 4 i < 2n + 2, i  n 1 mod 4
>>>A A Z=2, i = 2n + 2, n 1 mod 4,
><
Tor(A, Z=3) A A Z=2, i = 2n + 2, n 3 mod 4
LiSP 3(A, n) =
>>>Tor1(A, A, Z=2), 2n + 3 i 3n  2, i  n 2 mod 4,
>>>
>>>A Z=3 Tor1(A, A, Z=2), 2n + 3 i 3n  2, i  n *
*0 mod 4,
>>> 3(A), i = 3n + 2,
>:
0, for all other i.
In addition, the folllowing sequences are exact:
0 ! Tor(A, Z=3) A A Z=2 ! LiSP 3(A, 3) ! Tor2(A, A, Z=2) ! 0,
2n + 4 i 3n  1, i  n 1 mod 4
0 ! A A Z=2 ! LiSP 3(A, n) ! Tor2(A, A, Z=2) ! 0,
2n + 3 i 3n  1, i  n 3 mod 4,
0 ! L1 3(A) ! L3n+1SP 3(A, n) ! Tor2(A, A, Z=2) ! 0
0 ! 3(A) ! L3nSP 3(A, n) ! Tor1(A, A, Z=2) ! 0.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 17
Case II: n > 3 is even.
8
>>>A Z=3, n + 4 i < 2n + 2, i  n 0 mod 4
>>>
>>>Tor(A, Z=3), n + 4 i < 2n + 2, i  n 1 mod 4
>>>A A Z=2, i = 2n + 2, n 0 mod 4,
>>>
>>
>>>Tor1(A, A, Z=2), 2n + 3 i 3n  1, i  n 3 mod 4,
>>>L (A), i = 3n + 1,
>>> 1 3
>>>R3(A), i = 3n + 2,
:0, for all other i.
In addition, the following sequences are exact:
0 ! A Z=3 A A Z=2 ! LiSP 3(A, n) ! Tor2(A, A, Z=2) ! 0,
2n + 4 i 3n  2, i  n 0 mod 4
0 ! A A Z=2 ! LiSP 3(A, n) ! Tor2(A, A, Z=2) ! 0,
2n + 4 i 3n  2, i  n 2 mod 4,
0 ! 3(A) ! L3nSP 3(A, n) ! Tor2(A, A, Z=2) ! 0.
The corresponding description of the derived functors of 3 and 3 now follow*
* by the decalage
isomorphisms (2.38), (2.39).
6.Some derived functors of SP 4.
We will now make use of the computation of the homology of the dual de Rham c*
*omplex C4(A)
in proposition 3.1 in order to investigate some of the derived functors of SP 4*
*. For A a free abelian
group, we now consider the following diagram with exact rows and columns, which*
* extends diagram
(3.2)when n = 4:
"
4(A)O___//_ 3(A) _A__//_ 2(A)" `SP 2(A)_//_A "SP`3(A)////_SP"4(A)`
 "  fflffl fflffl fflffl
4(A)O___//_ 3(A) _A___// 2(A) 2(A)____//A 3(A)____// 4(A)
fflfflfflffl fflfflfflfflfflfflfflf*
*fl
2(A) A Z=2__~_//_A W3(A)___//W4(A)
By proposition 3.1, this determines a functorial diagram of exact sequences
SP 2(A) " A` Z=2 (6*
*.1)
" fflffl
H1D4(A)O____//_(A W3(A))=im(~)_//_W4(A)__////_ 2(A Z=2)
fflfflfflffl
A A Z=3
The map (3.1) deones canonical isomorphisms
H1D4(A) ' H1C4(A) ' 2(A Z=2) ' L1SP 2(A Z=2).
Let us deone a map
ffi : 2(A Z=2) ! SP 2(A) A Z=2 ( (A W3(A))=im(~))
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 18
as follows, where a, b are representatives in A of the classes ~aand ~b:
ffi : ~a^ ~b7! aa ~b bb ~a, a, b 2 A, ~a,~b2 A Z=2.
It follows from this discussion that diagram (6.1)induces a short exact sequence
SP 2(A) A Z=2
0 ! ________________ A A Z=3 ! W4(A) ! 2(A Z=2) ! 0.
2(A Z=2)
The oltration on 4(A) provided by this description of W4(A) is consistent with*
* that in [31] proposition
3.1.2. Together with long exact sequence (3.5), it allows one to compute derive*
*d functors of the functor
SP 4by comparing them to those of 4 and taking into account the double decalag*
*e. For example, one
onds:
L9SP 4(A, 3) ' A Z=2,
L10SP 4(A, 3) ' A A Z=3 2(A Z=2) Tor(A, Z=2),
L10SP 4(A, 4) ' A Z=2,
L11SP 4(A, 4) ' Tor(A, Z=2).
7.Lie and superLie functors
We will now consider the structure theory of Lie and superLie functors.
7.1. The third Lie functor. For any free abelian group A, consider the Koszul r*
*esolution:
0 ! 3(A) ! 2(A) A f!A SP 2(A) ! SP 3(A) ! 0, (7*
*.1)
in which the map f is deoned by
f : a ^ b c 7! a bc  b ac, a, b, c 2 A
It decomposes as
2(A) A__u_//_A A A__v_//_A SP 2(A) (7*
*.2)
where
u : a ^ b c 7! a b c  b a c  c a b + c b (a*
*,7.3)
v : a b c 7! a bc, a, b, c 2 A.
Since the expressions u(a ^ b c generate L3(A), the long exact sequence (7.1)*
*decomposes as a pair
of short exact sequences
0 ! 3(A) ! 2(A) A ! L3(A) ! 0 (7*
*.4)
0 ! L3(A) p3!A SP 2(A) ! SP 3(A) ! 0 , (7*
*.5)
In particular the map
p3
L3(A)____////_J3(A)
[a, b,Oc]//_(a, b, c)
induced by p3 (2.7)is an isomorphism.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 19
Remark 7.1. i) The sequences (7.4)and (7.5)both remain exact for an arbitrary g*
*roup A. Indeed,
(7.5)derives for any A to long exact sequences, and the arrow
` L '
ss1 A LSP 2A  ! L1SP 3A
is surjective, as follows from the presentation (2.19)of L1SP 3A.
ii) There is a natural isomorphism
L3(A) ' E3(A) := ker{ 2(A) A ! 3(A)} , (7*
*.6)
as follows from the following prolongation of part of diagram (3.2)for n = 3:
` L L ' ` '
" L L
ss1 A A Z=2O___//_ss1 A A Z=2 Tor(A, Z=3) (7*
*.7)
" fflffl fflffl
L3(A)O______//SP 2(A) _A_______________////_SP 3(A)
fflffl" fflffl fflffl
E3(A)O_______// 2(A) _A_________________// 3(A)______________////_A Z=3
fflfflfflffl fflfflfflffl 
A A Z=2_________//(A A Z=2) A Z=3_____//A Z=3
7.2. The Curtis decomposition. We will now consider higher Lie functors. Curtis*
* gave in [14] (see
also [39]) a decomposition of the functors Ln(A) into functors SP n, Jn and the*
*ir iterates. For example,
when A is a free abelian group we have the following decompositions in low degr*
*ees:
L1(A) =A (7*
*.8)
L2(A) =J2(A)
L3(A) =J3(A)
L4(A) =J2(J2(A)) J4(A)
L5(A) =(J3(A) J2(A)) J5(A)
L6(A) =J3(J2(A)) J2(J3(A)) (J4(A) J2(A)) J6(A)
L7(A) =(J3(A) SP 2(J2(A))) (J5(A) J2(A)) (J2(J2(A)) J3(A))
(J4(A) J3(A)) J7(A)
L8(A) =J2(J2(J2(A))) J2J4 (J3(A) J2(A) J3(A))
J5(A) J3(A) J4(J2(A)) (J4(A) SP 2(J2(A)))
(J6(A) J2(A)) J8(A)
. . .
We will refer to these descriptions of the Lie functors as their Curtis decom*
*positions. It should
be understood that the splittings into direct sums displayed here are not funct*
*orial, and that all that
exists functorially are oltrations of the Ln(A), whose associated graded compon*
*ents are the expressions
displayed. As a matter of convenience, we will nevertheless refer to these expr*
*essions as summands of
the Lie functors. We have already come across the cases n = 2, 3 of these decom*
*positions (prop. 2.1).
The next two cases are the short exact sequences
0 ! 2 2(A) ! L4(A) p4!J4(A) ! 0 (7*
*.9)
0 ! 2(A) J3(A) ! L5(A) p5!J5(A) ! 0, (7*
*.10)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 20
where the lefthand arrows are respectively deoned by
(a ^ b) ^ (c ^7d)! [[a, b], [c, d]]
(a ^ b) (c, d,7e)! [[a, b], [c, d, e]] .
It is a general fact that the onal term in the decompostion of Ln(A) is always *
*Jn(A), the projection
of Ln(A) onto Jn(A) being the map pn (2.7).
7.3. SuperLie functors. We will now deone superLie functors
Lns: Ab ! Ab, n 1 . (7*
*.11)
We consider the tensor algebra A as a graded ZLie algebra under the bracket o*
*peration
{a, b} = a b  (1)abb a, a, b 2 (A), (7*
*.12)
for a, b 2 A where   : (A) ! N is the obvious grading function. We denote t*
*his graded Lie algebra
ZLie algebra by (A)Lies. The bracket satisoes the skewsymmetry relation
{x, y} = (1)xy{y, x}, (7*
*.13)
and the super Jacobi identity given by
(1)zx{x, {y, z}} + (1)xy{y, {z, x}} + (1)yz{z, {x,*
*(y}}7=.0,14)
where x, y, z are homogeneous elements of (A)Lies. Deone Ls(A) be the subgrad*
*edLie ring of (A)Lies
generated by A. We will call the Lie ring Ls(A) the free graded superLie ring *
*generated by the abelian
group A. The grading of (A) determines a grading on Ls(A), so that we obtain i*
*n this way a series of
endofunctors (7.11). In particular, L2s(A), is the subgroup of A A, generated b*
*y elements a b+b a
for all a, b 2 A, so that there is a natural isomorphism
SP 2(A)' L2s(A)
ab 7! {a, b}
analogous to (2.10).
7.4. The third superLie functor. We will now adapt the discussion of section 7*
*.1 to the context
of the superLie functors. Here L3s(A) is the subgroup of A A A generated b*
*y the elements
a b c + b a c  c a b  c b a, a, b, c 2 A,(7*
*.15)
which are the super analogues of the generators (2.2)of L3(A). Let us now show *
*that there is a natural
isomorphism
L3s(A)' Y 3(A) (7*
*.16)
{{a, b}, c}7! a b ^ c + b a ^ c, a, b, c 2 A.
Consider, for any free abelian group A, the Koszul resolution Kos3(A) (2.29)
f~ 2 3
0 ! 3(A) i! 2(A) A ! A (A) ! (A) ! 0, (7*
*.17)
where the maps ~fand i are deoned by
~f: fl2(a) b 7! a a ^ b, a, b 2 A
(
fl3(a) 7! fl2(a) a, a 2 A
i :
fl2(a)b 7! fl2(a) b + ab a, a, b 2 A.
The map ~ffactors as
2(A) A __~u//_A A A_~v//_A 2(A) (7*
*.18)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 21
where
~u: fl2(a) b 7! a a b  b a a,
~v: a b c 7! a b ^ c, a, b, c 2 A.
It follows that
~u(ab c) = {{a, b}, c}, a, b, c 2 A
so that (7.17)decomposes as a pair of short exact sequences
0 ! 3(A) ! 2(A) A ! L3s(A) ! 0
0 ! L3s(A) ! A 2(A) ! 3(A) ! 0 (7*
*.19)
In view of the presentation (2.23)of L1 3A, these two sequences remain exact wh*
*en A is an arbitrary
abelian group.
7.5. Higher superLie functors. There exists a decomposition of superLie funct*
*ors, analogous to
the Curtis decomposition of Lie functors, which we will now describe in low deg*
*rees. We begin by
deoning inductively an extension of the Lie superbracket (7.12)to leftnormali*
*zed nfold brackets, by
setting
{a1, . .,.an} = {{a1, . .,.an1}, an}.
For all n 2 and any abelian group A, a natural epimophism ~pn: Lns(A) ! Y n(A*
*), is deoned by
p~n: {a1, a2, . .,.an} 7! a1 a2^ . .^.an + a2 a1^ a3^ . .^.an, a1, . .,.*
*an(27A..20)
Proposition 7.1. For any free abelian group A, the sequence of abelian groups
0 ! 2 2(A) j!L4s(A) ~p4!A 3(A) ! 4(A) ! 0
is exact, with j and ~p4respectively deoned by
j : fl2(a1) ^ fl2(a2) 7! {a1, a2, a1, a2}, a1, a2 2 A (7*
*.21)
~p4: {a1, a2, a3, a4} 7! a1 a2^ a3^ a4+ a2 a1^ a3^ a4, a1, a2, a3, *
*a4 2 A
The relations (7.13) and (7.14) imply that
{a1, a2, a1, a2} = {a2, a1, a2, a1}, a1, a2 2 A.
In addition
j : (fl2(a + b)  fl2(a)  fl2(b)) ^ fl2(c) 7! {a, c, b, c}  {b, c,*
* a, c}, a, b, c 2 A
so that the map j is welldeoned. Let us deone a functor eY nby the short exact*
* sequence
j n ~pn n
0 ____//_eY(nA)_//_Ls(A)__//_Y (A)__//_0 . (7*
*.22)
Proposition 7.1 asserts in particular that
eY(4A) = 2 2(A), (7*
*.23)
so that we have the following superanalogue for n = 4 of the short exact seque*
*nce (2.8):
j 4 p~4 4
0____//_ 2 2(A)__//_Ls(A)__//_Y (A)__//_0 . (7*
*.24)
Similarly, the short exact sequence (7.22)for n = 5, which is the superanalo*
*gue of the decomposition
(7.10)of L5(A), is described more precisely by the short exact sequence
~p5 5
0 ____//_Y 3(A) 2(A)h//_L5s(A)//_Y (A)__//_0 , (7*
*.25)
where the arrow h is deoned by
h : {a, b, c} 7fl2(d)! {a, b, c, d, d}
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 22
7.6. Relations between Lie and superLie functors. Let A be a free abelian grou*
*p, and consider,
for n 2, the natural monomorphisms
cn : Ln(A) ! nA,
zn : Lns(A) ! nA
which respectively deone the Lie and superLie functors. For n = 3, we have by*
* (7.3), (7.15), for
a, b, c 2 A:
c3 : [a, b, c] 7! a b c  b a c  c a b + c b a
z3 : {a, b, c} 7! a b c + b a c  c a b  c b *
*a.
For any pair of free abelian groups A and B, and n 2, we deone a pair of mo*
*rphisms
On : Lns(A) n(B) ! Ln(A B) (7*
*.26)
~On: Ln(A) n(B) ! Lns(A B) (7*
*.27)
for a1, . .,.an 2 A and b1, . .,.bn 2 B, by
X
On : {a1, . .,.an} b1^ .7.^.bn! sign(oe)[a1 boe1, . .,.an *
*boen]
oe2 n
X
~On: [a1, . .,.an] b1^7.!.^.bnsign(oe){a1 boe1, . .,.an boen}*
* .
oe2 n
Theorem 7.1. The following diagrams, with arrows deoned by (2.10) and (2.32)are*
* commutative:
On n n n O~n n
Lns(A) n(B)____//_L (A B) L (A) (B) ____//_Ls(A B) (7*
*.28)
zngn cn cngn zn
fflffl fflffl fflffl fflffl
~n n n n ~n n
( nA) ( nB) ____//_ (A B) ( A) ( B) ____//_ (A B)
Proof.Let us begin by considering the orst diagram (7.28) for n = 2. The commut*
*ativity of diagram
O2
SP 2(A) 2(B)________//_ 2(A B)
z2g2 c2
fflffl fflffl
~2
(A A) (B B)____//_(A B) (A B)
can be checked directly: for any a1, a2 2 A, b1, b2 2 B:
~2O (z2 g2)(a1a2 b1^ b2) = c2O O2(a1a2 b1^ b2)
= (a1 b1) (a2 b2)  (a1 b2) (a2 b1) + (a2 b1) (a1 b2)  (a2 b2*
*) (a1 b1) .
By induction on n, we ond that
X
zn gn({a1, . .,.an} b1^ . .^.bn) = sign(oe) zn{a1, . .,.an} boe1 .*
* . .boen=
oe2 n
X
sign(oe) zn1{a1, . .,.an1} an boe1 . . .boen
oe2 n
X
+ (1)n sign(oe) an zn1{a1, . .,.an1} b*
*oe1 . . .boen
oe2 n
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 23
Hence
~n O (zn gn)({a1, . .,.an} b1^ . .^.bn) =
0 1
Xn X
~n1@ sign(j, j) zn1{a1, . .,.an1} boe1 . .(.boen*
*1Aan bj)
j=1 j={oe1,...,oen1}2{1,...,^j,...,n}
0 *
* 1
nX X
+(1)n (an bj) ~n1@ sign(j, j) zn1{a1, . .,.an1} boe1*
* . . .boen1A
j=1 j={oe1,...,oen1}2{1,...,^j,...,n}
Xn X
= ( sign(j, j) (cn1[a1 boe1, . .,.an1 boen1] (*
*an bj) 
j=1j={oe1,...,oen1}2{1,...,^j,...,n}
(an bj) cn1[a1 boe1, . .,.an1 boen1])) =
X
sign(oe) cn[a1 boe1, . .,.an boen] = cn O On({a1, . .*
*,.an} b1^ . .^.bn)
oe2 n
and the commutativity of the orst diagram (7.28) is proved. The commutativity o*
*f the second diagram
is proved in a similar manner: for n = 2, it coincides with diagram (2.32) and*
* one then simply
repeats the previous computation for a general n, with appropriate changes in t*
*he signs of the various
expressions.
For any pair of abelian groups A, B, we deone as follows a natural arrow:
fin : Y n(A) n(B) ! Jn(A B) (7*
*.29)
X
fin : ~pn{a1, . .,.an} b1^ . .^.bn 7! sign(oe) pn[a1 boe1, . .,.an b*
*oen],
oe2 n
a1, . .,.an 2 A, b1, . .*
*,.bn 2 B.
Proposition 7.2. There is a natural commutative diagram with exact columns
fin n
Y n(A) " `n(B)____________//_J (A" `B)
 
 
fflffl j0n fflffl
A n1(A) n(B)___//_(A B) SP n1(A B)
 
fflfflfflffl fflfflfflffl
jn n
n(A) n(B)____________//SP (A B) ,
where the map jn is deoned by (2.33)and j0nis by
X
j0n: a1 a2^ . .^.an b1^ . .^.bn 7! sign(oe) (a1 boe1) (a2 boe2)*
* . .(.an boen) .
oe2 n
The proof of this proposition follows directly from the deonition of the vari*
*ous maps.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 24
8. Derived functors of Lie functors
It is asserted in [43] that if p is an odd prime then the groups Ln+kLp(Z, n)*
* are ptorsion for all k,
and in particular
(
Z=p, k = 2i(p  1)  1, i = 1, 2, . .,.[n=2]
Ln+kLp(Z, n) = (8*
*.1)
0, otherwise
In the next three subsections, we will give a direct proof of this fact for p =*
* 3, in other words show
that (
Z=3, k = 4i  1, i = 1, 2, . .,.[n=2]
Ln+kL3(Z, n) = (8*
*.2)
0, otherwise
and we will more generally compute in theorem 8.1 the derived functors Ln+kL3(A*
*, n) for a general
abelian group A. Note that the derived functors of the Lie functors Lq are com*
*plicated when q a
composite number, and we refer to [39], page 280 for a description of these in *
*low degrees. One onds
for example that (
Z=2, i = 4, 5, 7
LiL8(Z, 1) =
0, i 6= 4, 5, 7
The gap in the homotopy groups which occurs here for i = 6 is the illustration *
*of a general phenomenon
which, as we will see for example in (8.16), also occurs in more elaborate cont*
*exts.
Returning to the p = 3 case, let us orst observe that for any abelian group A*
* the exact sequence
(7.5) derives to a long exact sequence
` L '
. .!.Li+1L3(A, n) ! ssi+1 A[n] LSP 2(A, n)! Li+1SP 3(A, n) !
` L '
LiL3(A, n) ! ssi A[n] LSP 2(A, n)! LiSP 3(A, n) ! .*
* . .(8.3)
In addition, the isomorphism (2.41)implies that
LiY 3(A, n) ' Li+3L3(A, n + 1), for all i . 3 (8*
*.4)
8.1. The derived functors LiL3(A). The Koszul sequence (2.31)associated for n =*
* 3 to a AEat 2term
resolution L ! M of an abelian group A may be written as follows:
0 ! L3s(L) ! L M L ffi!L M M ! L3(M) ! L3(A) ! 0
where
ffi(l m l0) = l m f(l0) + l0 f(l) m  l f(l0) m, l, l0*
*2 L, m 2 M.
The three middle terms constitute a complex which represents the object LL3(A) *
*of the derived
category. In particular, if one considers the resolution Z m!Z of Z=m, one onds*
* that
(
Z=m, i = 1,
LiL3(Z=m) =
0, i 6= 1
By (2.19), the natural transformation Tor(S2(A), A) ! S3(A) is an epimorphism. *
*More generally, the
exact sequence (8.3) provides the following description of the derived functors*
* of L3:
L2L3(A) = ker{Tor(S2(A), A) ! S3(A)}
0 ! ker{S2(A) A ! L1SP 3(A)} ! L1L3(A) ! Tor(SP 2(A), A) ! 0
LiL3(A) = 0, i > 2.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 25
8.2. The derived functors LiL3(A, 1). Similarly, the short exact sequence (7.19*
*)derives to a long
exact sequence
` L '
. .!.Li+1L3s(A, n) ! ssi+1 A[n] L 2(A, n)! Li+1 3(A, n) !
` L '
LiL3s(A, n) ! ssi A[n] L 2(A, n)! Li 3(A, n) ! .*
* . .(8.5)
analogous to (8.3). For n = 0 this reduces to the exact sequence
0 ! L2Y 3(A) ! Tor( 2(A), A) ! 3(A) !
` L '
L1Y 3(A) ! ss1 L 2(A) A ! L1 3(A) ! 0 *
* (8.6)
This exact sequence is consistent with the results of [9] prop. 6.15, and with *
*the presentation (2.23)
of the groups Li n(A). The latter implies that the composite arrow
` L '
L1 2(A) A ! ss1 L 2(A) A ! L1 3(A) (8*
*.7)
is an epimorphism. The Kunneth formula, together with the exact sequence (8.6),*
* determines a 3step
oltration of L1Y 3(A). Taking into account the isomorphism (8.4) for n = 0, we *
*obtain the following
description of the derived functors LiL3(A, 1):
L3L3(A, 1) ' Y 3(A) (8*
*.8)
gr1L4L3(A, 1) ' gr1L1Y 3(A) ' coker{Tor( 2(A), A) ! 3(A)}
gr2L4L3(A, 1) ' gr2L1Y 3(A) ' ker{ 2(A) A ! L1 3(A)}
gr3L4L3(A, 1) ' gr3L1Y 3(A) ' Tor( 2(A), A)
L5L3(A, 1) ' L2Y 3(A) ' ker{Tor( 2(A), A) ! 3(A)}
Remark 8.1. The natural map
` L '
Tor( 2(A), A) ' ss2 L 2A A ! 3(A) (8*
*.9)
in the exact sequence (8.6)is in general neither injective nor surjective. This*
* can be seen by considering
the generators !hi(x) (2.22)of the groups n(A). We know by [9] (5.14) that the*
* diagram
2(hA) A_____// 3(hA) (8*
*.10)
2   3
~h 1 ~h
fflfflfflfflfflffl
Tor( 2(A), A)____//_ 3(A)
is commutative. It follows from the relation fl2(x)x = 3 fl3(x) in 3(A) that, *
*with the notation intro
duced in (2.22), the corresponding relation
!h2(x) * x = 3 !h3(x)
is satisoed in 3(A) for all x 2 hA. In particular, this implies that the arrow*
* (8.9)is trivial for h = 3
and A = Z3. Moreover, it is asserted in [3] that
L1Y 3(Z=3) = Z=9. (8*
*.11)
We refer to proposition B.1 for a proof by our methods of this assertion.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 26
8.3. The derived functors LiL3(A, 2). The decalage`isomorphisms'(2.38) and Kunn*
*eth formula yield
L
the following description of the groups ssr LSP j(A, 2) A[2]:
` L '
ss2j+2 LSP j(A, 2) A[2]' j(A) A
` L '
0 ! Li2 j(A) A ! ss2j+i LSP j(A, 2) A[2]! Tor(Li3 j(A), A) ! 0,
i = 3, . .,.j + 1
` L '
ss3j+2 LSP j(A, 2) A[2])' Tor(Lj1 j(A), A)
From (2.25)and (5.2)we have the following commutative diagram, a prolongation*
* of (7.7):
` '
" 2 L
S2(A) AO__//_ss1 LSP (A) _A_________________________________////_Tor(SP"*
*2(A), A)
XX *
* `
 XXXXXXXXXX 
  XXX,,,,X 
 q L1SP 3(A) 
  " ` 
 ` fflffl '  
fflffl" L  fflff*
*l
R2(A) AO___//ss1 L 2(A) A__________________________________//Tor( 2(A)*
*, A)
XXX 
 XXXXXXXX  
  t XXXXXX,,X  
  1 L1 3(A) 
fflfflfflffl fflfflfflffl  fflff*
*l
Tor(A, Z=2) AO_"__//coker(q)ZZZZ_______________ ________________//Tor(A Z=*
*2, A)
ZZZZZZZZZ,,ZZZ fflfflfflffl
(Tor(A, Z=2) A Z=2) Tor(A, Z=3)
(8*
*.12)
By a diagram chase, there is a canonical isomorphism coker(t1) ' Tor(A, Z=3). W*
*e obtain the following
commutative diagram, in which the middle vertical sequence is the long exact se*
*quence (8.3)for n = 2:
` '
" 2 L
R2(A) AO___//_ss7 LSP (A, 2) A[2]////_Tor( 2(A), A) (8*
*.13)
 t2
fflffl fflffl
L1 3(A)___________L7SP 3(A, 2)
fflffl
L6L3(A, 2)
` fflffl '
L
2(A) A______ss6 LSP 2(A, 2) A[2]

fflffl fflffl
3(A)____________L6SP 3(A, 2)
fflffl fflffl
A Z=3____________L5L3(A, 2)
Here t2 := t1 O q[2], up to a double decalage map. There is a functorial direct*
* sum decomposition of
the term L6L3(A, 2) = L3Y 3(A, 1), as can be seen from the following diagram, i*
*n which the vertical
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 27
arrows are suspension maps:
"
Tor(A, Z=3)O__//_L6L3(A,_2)////_L3(A)
offlffl fflffl
Tor(A, Z=3)~__//_L7L3(A, 3)
We refer to theorem 8.1 below for this description of L7L3(A, 3). We may extend*
* the middle column
of diagram (8.13)by the following exact sequence:
0 ! L8L3(A, 2) ! Tor(R2(A), A) ! R3(A) ! L7L3(A, 2) ! ker(t2) ! 0.
The diagram
"
Tor(S2(A), A)O__//_Tor(R2(A),_A)_//_Tor2(A, A,_Z=2)////_ker{S2(A) A ! R2(A*
*) A}
fflfflfflffl fflffl 
"
S3(A)O_________//_R3(A)_____////_Tor2(A, A, Z=2)
then implies that
ker{Tor(S2(A), A) ! S3(A)} = ker{Tor(R2(A), A) ! R3(A)}
and
coker{Tor(R2(A), A) ! R3(A)} = ker{S2(A) A ! R2(A) A}.
Taking once more into account the decalage isomorphisms (8.4), this provides a *
*complete description
of the functors LiL3(A, 2):
L5L3(A, 2) = L2Y 3(A, 1) = A Z=3 (8*
*.14)
L6L3(A, 2) = L3Y 3(A, 1) = L3(A) Tor(A, Z=3)
gr1L7L3(A, 2) = gr1L4Y 3(A, 1) = ker{S2(A) A ! R2(A) A}
gr2L7L3(A, 2) = gr2L4Y 3(A, 1) = ker{R2(A) A ! L1 3(A)}
gr3L7L3(A, 2) = gr3L4Y 3(A, 1) = Tor( 2(A), A)
L8L3(A, 2) = L5Y 3(A, 1) = L2L3(A).
For any other values of i, LiL3(A, 2) = 0.
As an illustration of these results, we will now give an explicit description*
* of the isomorphism
A Z=3 ! L2Y 3(A, 1) = L2L3s(A, 1) (8*
*.15)
occuring in the orst equation of (8.14), even though this will not be used in t*
*he sequel. Consider
the simplicial model (2.16)of LL3s(A, 1) associated to a free resolution (2.14)*
*of A. The isomorphism
(8.15) is induced by the map
A Z=3 ! L3s(L s1(M) s0(L))=@0(\3i=1@i)
deoned, for a liong a to M of ~a2 A Z=3, by
~a7! {s1(a), s0(a), s1(a)}.
In order for this map to be welldeoned, we must verify that
3{s1(a), s0(a), s1(a)} 2 @0(ker(@1) \ ker(@2) \ ker(@3)).
This is true since the element
j = 3{s2s0(a), s1s0(a), s2s0(a)}  {s2s1(a), s1s0(a), s2s0(a)} + {s2s1(a), s2*
*s0(a), s1s0(a)} 2
L3s(s0(A1) s1(A1) s2(A1) s1s0(A0) s2s0(A0) *
* s2s1(A0))
satisoes the equations @i(j) = 0, i = 1, 2, 3 and @0(j) = 3{s1(a), s0(a), s1(a)*
*}.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 28
8.4. The derived functors LiL3(A, n) forn 3. In each of the three following c*
*ommutative dia
grams, the exactness of the upper short exact sequence follows from proposition*
* 4.1 and the exactness
of the lower one from theorem 5.1. For n 3 odd:
` L ' ` '
" 2 L
ss1 2(A) AO___//_ss3m+1 LSP (A, n) A[n]////_Tor2(A, A, Z=2)

fflffl" fflffl 
L1 3(A)O____________//_L3n+1SP 3(A,_n)_____////_Tor2(A, A, Z=2)
` '
" 2 L
2(A) AO____//_ss3n LSP (A, n) A[n]////_Tor1(A, A, Z=2)
 
fflffl" fflffl 
3(A)O___________//L3nSP 3(A,_n)_____////_Tor1(A, A, Z=2)
and for n > 3 even:
` '
" 2 L
2(A) AO__//_ss3n LSP (A, n) A[n]////_Tor2(A, A, Z=2)
 
fflffl" fflffl 
3(A)O__________//_L3mSP 3(A,_n)_____////_Tor2(A, A, Z=2)
The sequence (8.3) therefore determines the following exact sequences:
Case I: n odd 3
0 ! L3n+2L3(A, n) ! Tor( 2(A), A) ! 3(A) !
` L '
L3n+1L3(A, n) ! ss1 2(A) A ! L1 3(A) !
L3nL3(A, n) ! 2(A) A ! 3(A) ! L3n1L3(A*
*, n)
Case II: n even
0 ! L3n+2L3(A, n) ! Tor(R2(A), A) ! R3(A) !
` L '
L3n+1L3(A, n) ! ss1 L 2(A) A ! L1 3(A) !
L3nL3(A, n) ! 2(A) A ! 3(A) ! L3n1L3(*
*A, n)
We may now summarize this discussion as follows:
Theorem 8.1. Case I: n odd
8
>>>ker{Tor( 2(A), A) ! 3(A)}, i = 3n + 2,
>>> ` L '
>>>gr1L3n+1L3(A, n) = ker{ss1 L 2(A) A ! L1 3(A)}
><
LiL3(A, n) = gr2L3n+1L3(A, n) = coker{Tor( 2(A), A) ! 3(A)}
>>> 3
>>>Y (A), i = 3n
>>>A Z=3, n + 3 i < 3n  1, i n + 3 mod 4
>:
Tor(A, Z=3), n + 4 i 3n  1, i n mod 4
and LiL3(A, n) = 0 for all other values of i.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 29
Case II: n even 8
>>>ker{Tor(R2(A), A) ! R3(A)}, i = 3n + 2
>>> ` L '
>>>gr1L3n+1L3(A, n) = ker{ss1 L 2(A) A ! L1 3(A)}
><
LiL3(A, n) = gr2L3n+1L3(A, n) = coker{Tor(R2(A), A) ! R3(A)}.
>>> 3
>>>Tor(A, Z=3) L (A), i = 3n
>>>A Z=3, n + 3 i 3n  1, i n + 3 mod 4
>:
Tor(A, Z=3), n + 4 i < 3n  1, i n mod 4
and LiL3(A, n) = 0 for all other values of i.
Note that in the computation of L3nL3(A, n) for n odd, we relied on the surject*
*ivity of the map (8.7).
Example. The previous discussion shows in particular that
8
>:
0, otherwise
A functorial description of some of these groups will be given in lemma 11.1.
For a given a free abelian group A and an integer n 2 the composite map
Ln(A) ! n(A) ! Ln(A)
is simply multiplication by n. It follows that for an odd prime p, the derived *
*functors LiLp(Z, n) are
pgroups (see (8.1)). The torsion part of the derived functors can be usually b*
*e determined by general
arguments. Recall that by Bousoeld [7], corollary 9.5, if T : Ab ! Ab is a fun*
*ctor of onite degree
which preserves direct limits, then LqT (A, n) is a torsion group for every abe*
*lian group A, unless q is
divisible by n.
The pcomponents of the derived functors of Lp and Jp are connected by the fo*
*llowing relation ([43]
proposition 4.7): for every prime p, there are natural isomorphisms
` L ' ` L '
ssi Z=p LLp(Z, n) ' ssi Z=p LJp(Z, n), i 0, n 2.
However the formulas for the full derived functors LiJp(Z, n) are more complica*
*ted than those for the
functors LiLp(Z, n) (8.1). For example, we know by theorem 8.1 that
LL5(Z, 3) ' K(Z=5, 10)
so that by (4.2)
L 3
LJ2(Z, 3) LJ (Z, 3) ' K(Z=3, 12).
On the other hand LJ3(Z, 3) ' K(Z=3, 6) by (8.1), and the values of the derived*
* functors LiJ5(Z, 3)
now follow from those of L5(Z, 3) and the Curtis decomposition of L5. One onds:
8
>:
0, i 6= 10, 13
One can compute the groups LiJ5(Z, n) for a general n by similar methods . One *
*onds that LiJ5(Z, n)
is isomorphic to LiL5(Z, n) for i 1 and even n, and that LiJ5(Z, n) contains*
* only 3torsion and
5torsion elements whenever n is odd. A similar computation detects a nontrivi*
*al 11torsion element
in L29J13(Z, 3), whereas the corresponding groups LiL13(Z, 3) are 13torsion fo*
*r all i.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 30
9.Derived functors of composite functors
Consider a pair of composable functors
_G__//_F__//_
A B C
between abelian categories, and in which the categories A and B have enough pro*
*jectives. In addition,
we assume that G(A) is of onite projective dimension for each object A 2 A. Whe*
*n these functors
are additive, the composite functor spectral sequence [45] #5.8 describes the d*
*erived functors of the
composite functor G O F in terms of those of F and G, under the condition that *
*the objects G(P )
are F acyclic for any projective object P of A (we will refer to this as the F*
* acyclicity hypothesis).
We will now carry out a similar discussion when G and F are no longer additive,*
* in which case chain
complexes must be replaced by simplicial abelian groups.
Let P* be projective resolution of an object A 2 A. Following the notations o*
*f [45] #5.7, we construct
a CartanEilenberg resolution P*.*of the simplicial object G(P*), with PBp,*(re*
*sp. PZp,*, resp.PHp,*) the
chosen projective resolution of BpG(P*) (resp. ZpG(P*), resp.LpG(A)). By the *
*DoldKan corre
spondence this yields in particular a projective bisimplicial resolution of LG(*
*A)) which we will also
denote by P*.*, as well as a corresponding projective simplicial model PHp,*for*
* the EilenbergMac Lane
spaces K(LpG(A), p). The two oltrations on the complex associated to the bisimp*
*licial object F P*.*
determine a pair of spectral sequences with common abutment ssn(LF O LG(A)). Th*
*e initial terms of
the orst of these are given by
M
E2p,q= Lp+qF ( K(LqiG(A), qi)) (9*
*.1)
P
iqi=q
as in [6]. When the functor F is of onite degree, we may decompose this initia*
*l term according to
crosseoeects of F [19] #4.18, so that the spectral sequence can be expressed a*
*s:
M M
E2p,q= Lp+qF[r](Lq1G(A), q1 . ..LqrG(A), qr) =) ssp+q((LF O LG*
*)(A))(.9.2)
r 1q1+...+qr=q
The F acyclicity hypothesis, which here asserts that LiF (G(P )) = 0 for any p*
*rojective abelian group
P and all i > 0, implies that the morphism (LF O G)(P ) ! F G(P ) is a quasii*
*somorphism for any
projective object P in A, and so is the induced map
(LF O LG)(A) ! L(F G)(A) . (9*
*.3)
The spectral sequence (9.2)can now be written as:
M M
E2p,q= Lp+qF[r](Lq1G(A), q1 . ..LqrG(A), qr) =) Lp+q(F G)*
*(A)(.9.4)
r 1q1+...+qr=q
Replacing the object A by the shifted derived category object A[n], in other *
*words by the Eilenberg
Mac Lane object K(A, n), we may now compute under the same hypotheses the deriv*
*ed functors
Lr(F G)(A, n) for all n. The F acyclicity hypothesis implies inductively that *
*the quasiisomorphism
(9.3)determines a quasiisomorphism
(LF O LG)(A, n) ! L(F G)(A, n) (9*
*.5)
for all n 0, since we can choose as a simplicial model for K(A, n) the bisimp*
*licial model
. ..! K(A3, n  1) ! K(A2, n  1) ! K(A, n  1) ! {e}
and work componentwise. Since no change is necessary in the discussion of the s*
*pectral sequence (9.2)
when passing from the case n = 0 to the general situation, we onally obtain for*
* any positive n, when
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 31
F is of onite degree and the F acyclicity hypothesis is satisoed, a functorial*
* spectral sequence:
M M
E2p,q= Lp+qF[r](Lq1G(A, n), q1 . ..LqrG(A, n), qr) =) Lp+q(F(G*
*)(A,9n)..6)
r 1q1+...+qr=q
We now restrict ourselves to a special case, that in which F and G are endof*
*unctors on the category
of abelian groups (or more generally the category of Rmodules, with R a princi*
*pal ideal domain
or even a hereditary ring). By construction, the total complex of P*,*and the c*
*omplex qPHq,*are
both projective and have as homology qLqG(A)[q], viewed as a complex with triv*
*ial dioeerentials. It
follows as in [20] #II.4 that this identiocation of their homology may be reali*
*zed by a chain homotopy
equivalence between the complexes, which in turn induces a simplicial homotopy *
*equivalence between
the corresponding simplicial groups P and qPHq,*. The induced homotopy equival*
*ence between F (P)
and F (PHq,*) makes it clear that in this case the E2 term of the spectral sequ*
*ence (9.6)is (non
canonically) isomorphic to its abutment. It follows that the spectral sequence *
*degenerates at the E2
level, so that this proves the following proposition:
Proposition 9.1. Let F and G be a pair of endofunctors on the category of abeli*
*an groups, with F of
onite degree. Suppose that for any projective abelian group P , LqF (G(P )) = 0*
* whenever q > 0. Then
the spectral sequence (9.6)degenerates at E2, and the graded components associa*
*ted to the oltration on
the abutment Lm (F G)(A) of the spectral sequence are described by the formula:
M M
grpLp+q(F G)(A, n) ' Lp+qF[r](Lq1G(A, n), q1 . ..LqrG(A,(*
*n),9qr).7)
r 1q1+...+qr=q
When F is one of the functor SP s, s or s, such an assertion may also be de*
*duced, under the
F acyclicity hypothesis, from the formula [29] V (4.2.7) of Illusie.
9.1. The derived functors of 2 2. As an illustration of proposition 9.1, we wi*
*ll now compute the
derived functors of the functor Li( 2 )(A, n) for all for n = 0, 1, 2. Such res*
*ults are of interest to us,
since 2 2 is the orst composite functor arising in the decomposition (7.8)of t*
*he Lie functors LnA.
We know by (4.2)that
8 8
>< 2(A), i = 1 >: >:
0, i 6= 0, 1 0, i 6= 2, 3
8
8 >>R2(A), i = 7
>>> 2(A), i = 5 >>>
<~2(A), i = 4 >< 2(A), i = 6
Li 2(A, 2) = Li 2(A, 3) = Tor(A, Z=2), i = 5
>>>A Z=2, i = 3 >>
:0, i 6= 3, 4, 5 >>>>A Z=2, i = 4
:0 i 6= 4, 5, 6*
*, 7.
8 8
>>> 2(A), i = 9 >>R2(A), i = 11
>>>2 >>>
<~ (A), i = 8 >< 2(A), i = 10
Li 2(A, 4) = Tor(A, Z=2), i = 6 Li 2(A, 5) = Tor(A, Z=2), i = 7, 9
>>> >>
>>>A Z=2, i = 5, 7 >>>A Z=2, i = 6, 8
:0 otherwise >:0 otherwise.
with ~2(A) deoned in (4.1). Proposion 9.1 yields the following table for the fu*
*nctors Li 2 2(A):
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 32
2 OO 0 R2 2(A)

1  2 2(A) 2 2(A) Tor( 2(A), 2(A))

p = 0  2 2(A) 2A 2(A)
______________________________________________//
 q = 0 1
Table 2. grp(Lp+q( 2 2)(A))
The functors Ln( 2 2)(A) can be read ooe from the p + q = n line of this tabl*
*e. In particular, there
is a (nonnaturally split) short exact sequence
0 ! 2(A) 2(A) ! L1( 2 2)(A) ! 2 2(A) ! 0
*
* L
which may be viewed as a symmetrized version of a Kunneth formula for ss1( 2(A)*
* 2(A).
We now pass to the derived functors Ln( 2 2(A, 1)). The corresponding table o*
*f values of these
derived functors may be read ooe from the values (9.8)and (9.1)of the derived f*
*unctors of 2:
OO
4  0 R2R2(A) 0 0

3  2 2(A) 2R2(A) 0 0

2  ~2 2(A) Tor(R2(A), Z=2) 0 0


1  2(A) Z=2 R2(A) Z=2 0 Tor( 2(A), R2(A))

p = 0  0 0 0 2(A) R2(A)
___________________________________________________________//_
 q = 2 3 4 5
Table 3. grp(Lp+q( 2 2)(A, 1))
When one also takes into account the values of Li (A, n) for n = 4, 5, one on*
*ds the following values
for the derived functors of 2 2(A, 2):
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 33
6 OO 0 0 R2 2(A) 0 0 0 0

5  0 2~2(A) 2 2(A) 0 0 0 0
4 R2(A Z=2) ~2~2(A) Tor( 2(A),Z=2)0 0 0 0

3  2(A Z=2) ~2(A) Z=2 2(A) Z=2 0 0 0 0
2  A Z=2 Tor(~2(A),Z=2)Tor( 2(A),Z=2)0 0 0 0

1  A Z=2 ~2A Z=2 2(A) Z=2 0 Tor(A Z=2,~2(A))Tor(A Z=2, 2(A))To*
*r(~2(A), 2(A))
p=0  0 0 0 0 (A Z=2) ~2(A) A Z=2 2(A) ~2(A)*
* 2(A)
_____________________________________________________________________________/*
*/
 q=3 4 5 6 7 8 9
Table 4. grp(Lp+q( 2 2)(A, 2))
9.2. The derived functors of 2 2. We will now carry out a similar discussion f*
*or the derived
functors of 2 2. By (4.3),
8 8> 2(A), i = 3
>><
~2(A), i = 2,
Li 2(A) = 2(A), i = 0 Li 2(A, 1) = (9*
*.9)
>: >>A Z=2, i = 1
0, i 6= 0, 1 >:
0, i 6= 1, 2, 3
Since 2(P ) is torsionfree for any torsionfree group P , the derived functor*
*s of 2 2 may be computed
by formula (9.7). The following tables may now be deduced from (9.8)and (9.9):
OO

2  R2R2(A)

1  2 2(A) 2R2(A) Tor( 2(A), R2(A))

p = 0  2 2(A) 2(A) R2(A)
________________________________________________//_
 q = 0 1
Table 5. grp(Lp+q( 2 2)(A))
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 34
OO
4  0 0 R2 2(A) 0 0

3  0 2~2(A) 2 2(A) 0 0
2 R2(A Z=2) ~2~2(A) Tor( 2(A),Z=2) 0 0

1  2(A Z=2)~2(A) Z=2 2(A) Z=2 Tor(A Z=2,~2(A))Tor(A Z=2,T2(A))or(~2(A),*
* 2(A))
p=0  0 0 A Z=2 ~2(A) A Z=2 2(A) ~2(A) 2(A)
__________________________________________________________________________//_
 q=1 2 3 4 5
Table 6. grp(Lp+q( 2 2)(A, 1))
The coincidence, up to changes in degree, between certain terms in table 5 an*
*d those in table 3, and
(more strikingly) between certain terms in table 6 and those in table 4 is expl*
*ained by the decalage
isomorphisms (2.39)between the derived functors of 2 and those of 2.
10.Derived functors of superLie functors
In view of (7.16), the decalage isomorphisms(2.41) and formulas (8.2) imply t*
*hat for (n 1):
(
Z=3, k = 4i + 1, i = 0, 1, . .,.[n1_]
Ln+kL3s(Z, n) = 2 (1*
*0.1)
0, otherwise
We will now examine the relations between the derived functors of Ln and Lns. F*
*or any free simplicial
abelian group A*, the maps On and ~On(7.26)induce arrows :
` L '
O*n: ssm LLns(A*) L n(Z, 1)! ssm (LLn(A) Z[1]), m 0
` L '
~O*n: ssm LLn(A*) L n(Z, 1) ! ssm (LLns(A) Z[1]), m 0
For A* = K(A, m), these determine by adjunction pension maps
O*n: Lm Lns(A, k) ! Lm+n Ln(A, k + 1)
~O*n: Lm Ln(A, k) ! Lm+n Lns(A, k + 1).
which may be viewed as generalized decalage transformations, even though the ma*
*ps O*nare no longer
isomorphisms. We will for this reason refer to such maps as semidecalage morph*
*isms. Similarly, the
pairing fin (7.29)determines a family of pension isomophisms (2.41)which we now*
* denote in:
in n
Lm Y n(A, k)__//_Lm+n J (A, k + 1) . (1*
*0.2)
Proposition 7.2 now implies the following assertion:
Theorem 10.1. The following diagram is commutative:
Lm Lns(A, k)_______//_Lm Y n(A, k)
O*nfflffl inofflffl
Lm+n Ln(A, k + 1)__//_Lm+n Jn(A, k + 1)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 35
We will now consider the boundary maps:
`m : Lm Jn(A, k) ! Lm1J"n(A, k), (1*
*0.3)
~`m: Lm Y n(A, k) ! Lm1Yen(A, k). (1*
*0.4)
induced by the short exact sequences (2.8)and (7.22). The following propositio*
*n is a corollary of
theorem 10.1:
Proposition 10.1. For m, n 1, and an abelian group A, the following diagram, *
*in which the vertical
arrows are decalage and semidecalage morphisms, commutes:
~`m+1 ~pn
Lm+1Y n(A, k)__________//Lm eY(nA,_k)_______//Lm Lns(A,_k)_______//Lm Y n(A,*
* k)
infoflffl f_nflffl O*nfflffl infoflff*
*l
`m+n+1 n pn n
Lm+n+1Jn(A, k + 1)_____//_Lm+n "Jn(A, k_+_1)//_Lm+n L (A, k +_1)_//_Lm+n J (A, *
*k + 1)
(1*
*0.5)
Let A be an abelian group and n 2. The following diagram, in which the ver*
*tical arrows are
decalage maps, is commutative:
` L '
ss2n LSP n1(A, 2) A[2]_//_L2nSP n(A,_2)////_L2n1Jn(A, 2) (1*
*0.6)
offlffl o o
` L ' fflffl fflffl
ssn L n1(A, 1) A[1]____//_Ln n(A,_1)__////_Ln1Y n(A, 1)
offlffl offlffl offlffl
n1(A) A_____________//_ n(A)_______////_H0Cn(A)
It follows in particular, by proposition 3.1, that there exists a natural isomo*
*rphism
M
L2n1Jn(A, 2) ' n=p(A Z=p), (1*
*0.7)
pn
which describes explicitly the righthand terms in diagram (10.6).
10.1. The fourth Lie and superLie functors. We will now discuss certain derive*
*d functors of the
functors L4 and L4s. Recall that by (7.9)and (7.23),
J"4(A) ' 2 2(A), eY(4A) ' 2 2(A).
for any free abelian A. By (10.7), the righthand vertical arrows in diagram (1*
*0.6)for n = 4 are:
L7J4(A, 2)~__//_L3Y 4(A,~1)u//_ 2(A Z=2). (1*
*0.8)
Proposition 10.2. For every abelian group A, the arrow
~`3
L3Y 4(A, 1)___//_L2 2 2(A, 1)
is a natural isomorphism between a pair of functors, both naturally isomorphic *
*to 2(A Z=2).
Proof.Let us orst verify that the map ~`3is surjective. Consider the simplicia*
*l model (2.16)of
L 2 2(A, 1) determined by a AEat resolution (2.14)of A. We deone a map
2(A Z=2) v!L2 2 2(A, 1)
explicitly as follows:
2(A Z=2) ! 2 2(A1 s0(A0) s1(A0))=@0(ker(@1) \ ker(@2) \ ker(@3*
*))
fl2(~a) 7! fl2(s0(a)) ^ fl2(s1(a))
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 36
for some lift a to M of ~a2 A Z=2. Under the natural transformation 2 2 ! L*
*4s(7.21), the image
of fl2(~a) goes to the element
{so(a), s1(a), s0(a), s1(a)}
in the term L4s(L s0(M) s1(M)) of the corresponding simplicial model for LL*
*4s(A, 1). The element
o := {s1s0(a), s2s0(a), s1s0(a), s2s0(a)}  {s1s0(a), s2s0(a), s1s0(a), s2s1(*
*a)} 2
L4s(s0(A1) s1(A1) s2(A1) s1s0(A0) s2s0(A0) *
* s2s1(A0))
satioes the equations @i(o) = 0, i = 1, 2, 3 and
@0(o) = {s0(a), s1(a), s0(a), s1(a)} 2 L4s(A1 s0(A0) s1(A0)).
It follows that the map L2 2 2(A, 1) ! L2L4s(A, 1) is trivial so that, by exact*
*ness of the upper line of
diagram (10.5), the arrow ~`3is surjective.
We will now give a more explicit description of the target of ~`3. We have a *
*natural isomorphism
v : L2 2 2(A,_1)~//_ 2(A Z=2)
since the only nontrivial total degree 2 term in table 6 is the expression 2(*
*A Z=2) in bidegree (1,1).
We now have a pair of arrows u (10.8)and v, which provide natural isomorphisms *
*between both the
source and target of ~`3and the group 2(A Z=2). We may now assume that A is *
*onitely generated.
In that case both source and target of the surjective map ~`3are onite groups o*
*f the same order, so
that ~`3is an isomorphism.
We know by the description of homotopy groups of L 2 2(A, 2) in table 4 that *
*there is a natural
projection L6 2 2(A, 2) ! 2(A Z=2). The following proposition is a consequen*
*ce of propositions
10.1 and 10.2:
Proposition 10.3. The boundary arrow `7 lives in a diagram of short exact seque*
*nces
2(A) Z=2 L7J4(A, 2)
" ` " ` OO
  OOO'O
 `7 OOOO
fflffl fflffl O''
" w
ker(w)O_______//_L6 2 2(A, 2)__////_ 2(A Z=2)

fflfflfflffl
Tor(~2(A), Z=2)
where w is the mapping to 2(A Z=2) arising from the edgehomomorphism in the*
* spectral sequence
(9.6)described in table 4.
Proof.The associated graded terms 2(A) Z=2 and Tor(~2(A), Z=2) in the line p*
* + q = 6 of table
4 give us the required description of ker(w). The previous discussion provides *
*us with a commutative
diagram
2(A Z=2)_'u_//_L3Y 4(A,_1)'~//_L2( 2 2)(A, 1)
`3 " `
' _4
fflffl`7 fflffl w
L7J4(A, 2)___//_L6( 2 2)(A,_2)//_ 2(A Z=2)
where the injectivity of the map _4 (10.5)is obtained by examining the behavior*
* of the decompositions
(9.7)of its source and target under decalage. It remains to show that the compo*
*site map
w O `7 : L7J4(A, 2) ! 2(A Z=2)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 37
is an isomorphism. When A is free abelian of onite rank, L6 2 2(A, 2) ' 2(A *
*Z=2), so that the
injective map
_4 : 2(A Z=2) ,! L6 2 2(A, 2)
is a monomorphism between two onite groups of the same order. It follows that t*
*he map w O `7 is an
isomorphism whenever A is free abelian, and therefore an epimorphism for an arb*
*itrary abelian group
A. Returning to the case of an abelian group A of onite rank, we conclude that *
*the epimorphism w O`7
is an isomorphism, since source and target are onite groups of the same order. *
*This implies that the
corresponding assertion is true for an arbitrary abelian group.
In the sequel, we will also need the following result, which follows since th*
*e only nontrivial terms
contributed by the Curtis decomposition to LiL4(A, 2) for i < 7 are those provi*
*ded by the derived
functors of 2 2:
Corollary 10.1. There is a canonical decomposion of L6L4(A, 2) as the direct su*
*m of the two following
terms:
gr1L6L4(A, 2) = Tor(~2(A), Z=2) = Tor( 2(A), Z=2) Tor2(A, Z=2, Z=2)
gr2L6L4(A, 2) = 2(A) Z=2.
11.Homotopical applications
11.1. Moore spaces and the Curtis spectral sequence. In this section we will re*
*view the Curtis
spectral sequence, which will be our main tool for homotopical applications of *
*our theory. Recall that
for any abelian group A and n 2, a Moore space in degree n is deoned to be a *
*simply connected space
X with Hi(X) ' A for i = n, and eHi(X) = 0, i 6= n. Such a Moore space will be *
*denoted M(A, n)
when in addition an isomorphism Hn(X) ' A is oxed. The homotopy type of M(A, n)*
* is determined
by the pair (A, n), since homology equivalence implies homotopy equivalence for*
* simplyconnected
spaces. When A is free abelian with a chosen basis, a Moore space M(A, n) can b*
*e constructed as
a wedge of nspheres, labelled by basis elements of A. For an arbitrary abelian*
* group A and n 2,
an ndimensional Moore space is constructed as follows: choose a 2step free re*
*solution (2.14)of A
with chosen bases. M(A, n) can then be deoned as the cone [27] VI 2 of the indu*
*ced map between the
wedges of spheres M(L, n) ! M(M, n). For any homomorphism of abelian groups f :*
* A ! B, it is
possible to construct a map OE : M(A, n) ! M(B, n) such that Hn(OE) = f. Howeve*
*r, the construction
of the map OE is not canonical and the construction of Moore spaces is nonfunc*
*torial. The canonical
class in Hn(M(A, n), A) induces a map
M(A, n) ! K(A, n) (1*
*1.1)
which is welldeoned up to homotopy.
We will now recall the construction of the Curtis spectral sequence. Let G be*
* a simplicial group.
The lower central series oltration on G gives rise to the long exact sequence
. .!.ssi+1(G=flr(G)) ! ssi(flr(G)=flr+1(G)) ! ssi(G=flr+1(G)) ! ssi(G=fl*
*r(G)) ! . . .
This exact sequence deones a graded exact couple, which gives rise to a natural*
* spectral sequence E(G)
with the initial terms
E1r,q(G) = ssq(flr(G)=flr+1(G))
and dioeerentials
dir,q: Eir,q(G) ! Eir+i,q1(G). (1*
*1.2)
According to [15], for K a connected simplicial set and G = GK the associated*
* Kan construction
[38] #26, this spectral sequence Ei(G) converges to E1 (G) and rE1r,qis the gr*
*aded group associated
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 38
to the oltration on ssq(GK) induced by the lower central series oltration on K.*
* Since GK is a loop
group of K, this spectral sequence may be written as
E1r,q(K) := ssq(flr(GK)=flr+1(GK)) =) ssq+1(K). (1*
*1.3)
The groups E1(G) are homology invariants of K. By the MagnusWitt isomorphism (*
*2.3), the spectral
sequence can be rewritten as
E1r,q(K) = ssq(Lr(eZK, 1)) =) ssq+1(K). (1*
*1.4)
since the abelianization GKab:= GK=fl2GK of GK corresponds to the reduced chain*
*s eZK on K, with
degree shifted by 1. When K = M(A, n), eZK corresponds to an EilenbergMac Lane*
* space K(A, n)
so that the spectral sequence is simply of the form
E1r,q= LqLr(A, n  1) =) ssq+1(M(A, n)) . (1*
*1.5)
In particular, (
A, q = n  1
E11,q= ssq(K(A, n  1)) =
0, q 6= n  1
For a additional information regarding this spectral sequence, see [15], [39] c*
*h. 5.
11.2. The 3torsion of ssn(S2). As a orst illustration of our techniques, we wi*
*ll now discuss the
3torsion components of the homotopy groups of the sphere S2. For this, conside*
*r the 3torsion parts
of the various terms in the spectral sequence (11.5), with A = Z and n = 2:
E1r,q= LqLr(Z, 1) ) ssq+1(S2). (1*
*1.6)
From now on, we will denote by pA the ptorsion subgroup of an abelian group A *
*and by (p)A the
quotient of A by the qtorsion elements, for all primes q 6= p . We will refer *
*to this quotient group as
the (p)torsion group of A.
It is shown in [15] (see also [39] props. 5.33 and 5.35) that
(
Z, i = 2, n = 2
LiJn(Z, 1) = (1*
*1.7)
0, otherwise
This, together with the Curtis decomposition (7.8)of the Lie functors and the c*
*omputation of the
groups Li 2 2(Z, 1) in table 3, implies that there is no 3torsion in any of th*
*e expressions LqLp(Z, 1)
for p < 6. Let us show that the orst nontrivial 3torsion term in the spectral*
* sequence (11.6)occurs
in the group L5L6(Z, 1). It follows from (11.7)and the Kunneth formula that no *
*3torsion is produced
by either of the factors J6(Z, 1) and J4(Z, 1) J2(Z, 1) of L6(Z, 1), nor is a*
*ny contribution made by
J2J3(Z, 1) since J3(Z, 1) is contractible. It thus follows from (11.7)and (8.2)*
*(or (8.14)) that
(
Z=3, i = 5
LiL6(Z, 1) ' LiJ3J2(Z, 1) ' LiL3(Z, 2) ' (1*
*1.8)
0, i 6= 5.
We restate this result as:
LJ3J2(Z, 1) ' K(Z=3, 5). (1*
*1.9)
More generally, the Curtis decomposition (7.8), together with (11.7) and (10.*
*7), implies that 3
torsion in the groups LqLr(Z, 1) can only arise from components of the decompos*
*ition of the form
F J3kJ2 and their tensor products (for functors F = SP k, F = Jk), so that ther*
*e is no 3torsion in
the initial terms of (11.6)unless 6r. The analysis of the r = 18 case is simi*
*lar to that of r = 6.
The only contribution to the 3torsion in LqLr(Z, 1), for q 14, comes from th*
*e derived functors of
J3J3J2(Z, 1), and by (11.9):
LiJ3J3J2(Z, 1) ' LiL3(Z=3, 5) .
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 39
These groups were computed in (8.16), so it now follows from the connectivity r*
*esult (3.6) that
3LqLr(Z, 1) = Z=3, r = 18, q = 8, 9 (11*
*.10)
and
3LqLr(Z, 1) = 0, 5 < q < 10, r 6= 18.
We refer to [39] ch. 5 for a similar analysis of the 2torsion components in *
*the spectral sequence
(11.6).
For r 6= 12, the 3torsion components of LqLr(Z, 1) may all be computed by th*
*e previous method
so long as q 14, and indeed all of these components are trivial except for th*
*ose provided by (11.8)
and (11.10). We will now consider in detail the case of the 12th Lie functor. W*
*e will need to introduce
additional techniques in order to achieve a complete understanding of the deriv*
*ed functors of L12and
of the dioeerentials in the spectral sequence (11.6)within the range q 14.
First observe that only the functors J6J2, J3J2J2, J2J3J2, J4J2 J2J2 may give*
* any contribution to
the 3torsion in LqL12(A, 1) in degrees q 14. By (4.2)and (8.14), the derived*
* functors of J3J2J2 and
J4J2 J2J2 are all 2torsion groups for A = Z. It follows that that 3torsion in*
* LqL12(Z, 1) within our
range can only occur in degrees q = 10, 11. In fact we will now show that while*
* J6J2(Z, 1) = K(Z6, 11)
by (10.7), and so could in principle contribute to the 3torsion of L11L12(Z, 1*
*), this is not in fact the
case:
Proposition 11.1. The groups (3)LqL12(Z, 1) = 0 are trivial for all q 2.
Proof: By (11.7), we may think of the Curtis decomposition of L12(Z, 1) as re*
*ducing to a short
exact sequence
0____//_J2J3J2(Z,_1)//_L12(Z,_1)_//_J6J2(Z,_1)//_0
when only 3torsion is considered. This induces following commutative diagram o*
*f onite groups with
exact horizontal lines, and boundary maps 3j11:
O_"__//_ 6 2 _3j11//_ 2 3 2 ___////_ 12
3L11L12(Z, 1) 3L11J J (Z, 1) 3L10J J J (Z, 1) 3L10L (Z, 1) (11*
*.11)
   
 O "  3j11  
3L11L6(Z, 2)____//3L11J6(Z,_2)___//_3L10J2J3(Z,_2)_////_3L10L6(Z, 2)
 
 
2(Z3)___________//_ 2(Z3)
In this diagram, the value of 3L11J6(Z, 2) was determined by (10.7)and that of *
*3L10J2J3J2(Z, 1)
follows from (11.9)and (4.2).
Let us now consider the Curtis spectral sequence (11.4)for the space K := K(A*
*, n) for some abelian
group A:
E1r,q= Lq+1Lr(eZK(A, n), 1) =) ssq(K(A, n  1)) (11*
*.12)
We will now look at this in more detailed , for A = Z:
E1r,q= Lq+1Lr(eZK(Z, n), 1) =) ssq(K(Z, n  1)) (11*
*.13)
By Dold's theorem [18] th. 5.1, we may replace the expression ZK(Z, n) in the i*
*nitial term of (11.13)
by iK(Hei+1(Z, n), i) so that the spectral sequence becomes
E1r,q= ssqLr( iK(Hei+1(Z, n), i)) =) Z[n  1] (11*
*.14)
In particular,
E11,q= eHq+1(K(Z, n)) . (11*
*.15)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 40
We now consider the case n = 3. The lowdegree (3)torsion integral homology gr*
*oups of K(Z, 3) are
wellknown [11], [16], in fact the only nontrivial generators for such groups a*
*re the fundamental class
i3 in degree 3, the degree 7 suspension of element fl3(i2) 2 H6(K(Z, 2)), and t*
*heir product in degree
10 (under the multiplication induced by the Hspace structure of K(Z, 3)):
_____n_______3__4__5_6___7___8_9__10___11_12__

(3)HnK(Z, 3) Z 0 0 0 Z=3 0 0 Z=3 0 0
When n = 3, Dold's theorem also allows us to (nonfunctorially) compute the oth*
*er initial terms in
(11.14), since within the range of values of q 11 we may replace the expressi*
*on eZK(Z, 3) by the
product of EilenbergMac Lane spaces K(Z, 2) K(Z=3, 6) K(Z=3, 9). For r = 2*
* we must therefore
compute the homotopy of the induced LJ2(K(Z, 2) K(Z=3, 6) K(Z=3, 9)). No 3*
*torsion in the
homotopy is provided by the functor J2 applied to any of the three summands, so*
* the only nontrivial
terms are those coming from the crosseoeect terms Z[2] Z=3[6] and Z[2] Z=3*
*[9], in other words
copies of Z=3 in degrees 8 and 11 respectively.
Similarly, in looking for the 3torsion of the r = 3 initial terms of (11.13)*
*within our range of values
q 11, we need only consider the homotopy of LJ3(K(Z, 2) K(Z=3, 6)). Let us*
* record here the
functorial form of (8.16), for all n and a more restricted range of values of k:
Lemma 11.1. For any abelian group A, and integer n > 4
8
>Tor(A, Z=3) k = 4, 8
:0, k = 2, 5, 6
It follows that the summand LJ3(Z, 2) contributes a term Z=3 in degree 5 to t*
*he 3torsion of E13,q,
while the summand LJ3(Z=3, 6) contributes a pair of terms Z=3 in degrees 9 and *
*10. In addition, since
the second third crosseoeect of the functor J3 is the functor
J3[2](AB) ' (A B A) (A B B),
it contributes an additional term Z[2] Z=3[6] Z[2] to the homotopy of LJ3(K*
*(Z, 2) K(Z=3, 6)) 9
in degree 10, in other words a second factor Z=3 to the initial term E13,10of (*
*11.14).
There is no contribution to the 3torsion component of the initial terms of t*
*he spectral sequence
(11.13)for r = 4, 5, 7, 8 since none of these numbers is a multiple of 3. If we*
* leave aside the case p = 6
for the time being, the only initial terms which we still need to consider are *
*those for which r = 9.
In our range q 11, the only the summand of L9 which comes into play is J3J3 a*
*nd by (8.16)the
homotopy groups of LJ3K(Z=3, 5) contribute a pair of groups Z=3 to the 3torsio*
*n of LL9(Z, 2) in
degrees 8 and 9.
We now collect in the following table the outcome of this discussion of the (*
*3)torsion components
of the initial terms of the spectral sequence (11.13) for n = 3 :
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 41
___r______1____2_____3____4_5______6_______7_8___9__
(3)E1r,110 Z=3 0 0 0 * 0 0 *
(3)E1r,100 0 (Z=3)2 0 0 3L10L6(Z, 2)0 0 0
(3)E1r,9Z=3 0 Z=3 0 0 0 0 0 Z=3
(3)E1r,8 0 Z=3 0 0 0 0 0 0 Z=3
(3)E1r,7 0 0 0 0 0 0 0 0 0
(3)E1r,6Z=3 0 0 0 0 0 0 0 0
(3)E1r,5 0 0 Z=3 0 0 0 0 0 0
(3)E1r,4 0 0 0 0 0 0 0 0 0
(3)E1r,3 0 0 0 0 0 0 0 0 0
(3)E1r,2 Z 0 0 0 0 0 0 0 0
Table 7. The 3torsion in the initial terms for the spectral sequence (11.*
*13)when n = 3
Since all the terms in the abutment of this spectral sequence vanish (except *
*for a copy of Z in
degree 2), it follows by examining the possible dioeerentials in the spectral s*
*equence that the term
(3)L10L6(Z, 2) survives all the way to E16,10and must therefore be trivial. Dia*
*gram (11.11)now makes
it clear that (3)L11L12(Z, 1) = (3)L11L6(Z, 2) also vanishes. These were the on*
*ly possibly nonvanishing
terms within our range of degrees, so that onally:
(3)LrL12(Z, 1) = 0, r 14. (11*
*.16)
Remark 11.1. A direct computation shows that the triviality of L10L6(Z, 2) is e*
*quivalent to the
assertion the class in L6s(Z, 1)4, of the element
i ={{s2s1s0(a), s2s1s0(a), s3s1s0(a)}, {s3s2s0(a), s3s2s0(a), s3s2s*
*1(a)}}
{{s2s1s0(a), s2s1s0(a), s3s2s0(a)}, {s3s1s0(a), s3s1s0(a), s3s2s1*
*(a)}}+
{{s3s1s0(a), s3s1s0(a), s3s2s0(a)}, {s2s1s0(a), s2s1s0(a), s3s2s1*
*(a)}}
is trivial, where a is a generator of Z = ss1K(Z, 1). It would be of some inte*
*rest to ond a specioc
element in L6s(Z, 1)5 with boundary i.
We now return to the spectral sequence (11.6), where we now know that E112,q=*
* 0 for all q 14.
We will now display the entire table of initial terms in the range q 14:
_____r________6___12__18__24__30_36__42_48_____54_____162_
3L14Lr(Z, 1) 0 0 0 0 0 0 0 0 0 Z=3
3L13Lr(Z, 1) 0 0 Z=3 0 0 0 0 0 Z=3 0
3L12Lr(Z, 1) 0 0 Z=3 0 0 0 0 0 Z=3 Z=3 0
3L11Lr(Z, 1) 0 0 0 0 0 0 0 0 Z=3 0
3L10Lr(Z, 1) 0 0 0 0 0 0 0 0 0 0
3L9Lr(Z, 1)  0 0 Z=3 0 0 0 0 0 0 0
3L8Lr(Z, 1)  0 0 Z=3 0 0 0 0 0 0 0
3L7Lr(Z, 1)  0 0 0 0 0 0 0 0 0 0
3L6Lr(Z, 1)  0 0 0 0 0 0 0 0 0 0
3L5Lr(Z, 1)  Z=3 0 0 0 0 0 0 0 0 0
Table 8. The 3torsion in the initial terms of the spectral sequence*
* (11.6)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 42
The values of the various terms in this table are justioed as follows. Observ*
*e orst of all that the
vanishing of all terms E112,qterms implies that there are no nonzero terms E1r*
*,qwhenever r is a multiple
of 12. Nontrivial terms with r = 18 arise by applying the functor J3 according*
* to the rule of lemma
11.1 to the cyclic group 3E16,5= Z=3, so that they are contributed by derived f*
*unctors of the summands
J3J3J2 of L18. Applying one more functor J3 to each of the two cyclic groups E1*
*18,8and E118,9provides
us, according to the same rule, with two additional copies of Z=3 in the column*
*s r = 54. Finally, a last
composition with a J3 yields the only nontrivial term in column r = 162 within*
* our range r 14.
Our discussion makes it clear that this cyclic group has been contributed by th*
*e appropriate derived
functor of the summands J3J3J3J3J2 of L162.
It now follows from this discussion, by taking into account the possible dioe*
*erentials in the spectral
sequence, that we have obtained the following description of the 3torsion in s*
*si(S2) in the range i 11:
(
Z=3 i = 6, 9, 10
3ssi(S2) = 0 otherwise. (11*
*.17)
In addition,
ssi(S2) Z=3, i = 13, 14.
We recover in this way by purely algebraic methods certain of Toda's results [4*
*4]. In fact, it can be
shown by comparing once more once more the dioeerentials in a spectral sequence*
* for the Moore space
(11.18)with those in the corresponding spectral sequence for an EilenbergMac L*
*ane space, and by
suspension arguments, that the additional dioeerentials d3618,12: Z=3 ! Z=3 and*
* d3618,13: Z=3 ! Z=3 Z=3
in (11.6)are both monomorphisms. In this way, we recover algebraically the enti*
*re description of the
3torsion in ssn(S2) up to degree 14.
11.3. Some homotopy groups of M(A, 2). We now consider the spectral sequence (1*
*1.5)for n = 2:
E1r,q= LqLr(A, 1) ) ssq+1M(A, 2). (11*
*.18)
For r = 3, some intial terms in this spectral sequence were computed in #8.2. W*
*e will now study the
terms E14,q= LqL4(A, 1). The short exact sequences (2.8)and (7.9)derive to the *
*horizontal lines of
the two following diagrams, while the vertical ones arise from semidecalage an*
*d the computations of
the groups Li 2 2(A, 1) in table 3 :
~`1
L1Y 4(A) _________// 2 2(A)__________//_L4s(A)______////Y 4(A) (11*
*.19)
" `
'   '
fflffl fflffl fflffl fflffl
`4 2 2 4 4
L5J4(A, 1)_______//L4 (A, 1)_____//_L4L (A, 1)__////_L4J (A, 1)

fflfflfflffl
` L '
ss1 2(A) Z=2
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 43
~`2
L2Y 4(A)_______//L1 2 2(A) _______//_L1L4s(A)____//L1Y 4(A)
" `
'   '
fflffl fflffl fflffl fflffl
`6 2 2 4 4
L6J4(A, 1)_____//_L5 (A, 1)____//_L5L (A, 1)__//_L5J (A, 1)

fflfflfflffl
Tor(R2(A), Z=2)
The computation of the Li 2 2(A) also implies that there are genuine decalage i*
*somorphisms
LiLns(A) ' Li+nLn(A, 1) (11*
*.20)
for n = 4 whenever i > 2. The same is true for n = 5 and all i by comparison of*
* the derived long exact
sequences associated to the sequences (7.10)and (7.25).
This discussion provides the justiocation for the description of the columns *
*q = 1, 2, 3, 5 of the
following table of initial terms of the spectral sequence (11.18):
_q__E11,qE12,q___E13,q_________E14,q__________E15,q_
7  0  0  0  L3L4s(A)  L2L5s(A)
6  0  0  0  L2L4s(A)  L1L5s(A)
5  0  0  L2Y 3(A)Tor(R2(A),Z=2)`' L1L4s(A)L5s(A)
4  0  0  L1Y 3(A)ss1 L 2(A) LZ=2 L4s(A)0
    
3  0 R2(A)  Y 3(A) 2(A) Z=2  0
2  0  2(A)  0  0  0
1  A  0  0  0  0
_q________________E16,q________________________E17,q_________`'
7  L1L6s(A) Tor(L2Y 3(A),Z=2)ss1 Y 3(A) L 2(A) LZ=2 L7s(A)
 ` L ' ` L ' 
6  ss1 2A Z=3 ss2 Y 3(A) Z=2 L6s(A) Y 3(A) 2(A) Z=2
 ` L  '
5  2(A) Z=3 ss1 Y 3(A) Z=2 0
 
4  Y 3(A) Z=2  0
3  0  0
2  0  0
1  0  0
_q________________E18,q________________E19,q_____E110,q_`'`'
6  ss2 2(A) LZ=2 LZ=2 ss1 L4s(A) LZ=2Y 3(A) LZ=35s(A) Z=2
 ` L L '  
5  ss1 2(A) Z=2 Z=2 L4s(A) Z=2 0  0
  
4  2(A) Z=2  0  0
3  0  0  0
2  0  0  0
1  0  0  0
Table 9. The E1terms of the spectral sequence (11.18)
The description of the fourth column of table 9 follows from a diagram chase in*
* diagram (11.19)and its
derived versions. We will now show how to ond the terms of interest to us in co*
*lumns 6 and 8, by the
methods of #9. Those in the sixth column in degrees q = 4, 5 only depend on the*
* orst two summands
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 44
J3J2(A) and J2J3(A) of L6(A). The term J2J3(A) in L6(A) contributes an expressi*
*on
L4(J2J3(A, 1)) ' L4 2(Y 3(A), 3) ' Y 3A Z=2 ,
to E16,4, since L2 2(A, 1) ' 2A and L5J3(A, 2) ' A Z=3 (8.14). The same comp*
*utation provides
the corresponding factor in E16,5. Similarly, the term J3J2(A) provides the exp*
*ression 2(A) Z=3 in
E16,5, since L2 2(A, 1) ' 2A and L5J3(A, 2) ' A Z=3 (8.14). Finally, the ter*
*m E18,4comes from the
term L4J2J2J2(A, 1) in L4L8(A, 1) by the same sort of reasoning: we already kno*
*w that
L3 2 2(A, 1) ' 2(A) Z=2 .
This implies that
L4 2( 2 2(A, 1)) ' L4 2( 2A Z=2, 3)
and the result follows, since
L4 2( 2A Z=2, 3) ' L6SP 2( 2A Z=2, 4) ' H6(K( 2A Z=2, 4)) ' 2A *
*Z=2 ,
with the last isomorphism following by a direct calculation, or by reference to*
* the wellknow Eilenberg
MacLane functorial stable isomorphism
H6(K(B, 4)) ' B Z=2 .
We refer to [39] #5.5 for a more complete discussion by one of us of the derive*
*d functors of iterates of
2 when A = Z.
It is immediate from the line q = 2 of table 9 that
ss3(M(A, 2)) = 2(A) ,
a result which goes back to J.H.C. Whitehead, and that in particular a generato*
*r of 2(Z) corresponds
to the class of the Hopf map j : S3 ! S2. By comparing the spectral sequence *
*(11.18)with the
corresponding spectral sequence (11.12)for n = 3, one verioes that the dioeeren*
*tial d13,4: E13,4! E14,3
in (11.18)is trivial. The line q = 3 of table 9 then implies that there is a sh*
*ort exact sequence
0 ! L3s(A) ( 2(A) Z=2) ! ss4M(A, 2) ! R2(A) ! 0 , (11*
*.21)
a result already proved in [2], [3], where the expression L3s(A) ( 2(A) Z=2*
*) is denoted 22(A).
Similarly, the last two terms in the line q = 4 of our table, together with t*
*he factor L4s(A) from
E14,4, regroup to the expression denoted 32(A) in [3], while the direct sum of*
* the two remaining
terms on the line q = 4 correspond to the derived functor L1 22(A) of the funct*
*or 22(A) mentioned
above. By considering the restriction of the dioeerential d44,5: E44,5! E48,4in*
* our table to the factor
Tor(R2(A), Z=2) of E44,5we therefore recover the description of ss5M(A, 2) in [*
*3] as a middle term in
an exact sequence:
L2 22(A) d2! 32(A) ! ss5M(A, 2) ! L1 22(A) ! 0. (11*
*.22)
where d2 is a dioeerential in the spectral sequence from [21] (for a generalize*
*d version of this sequence,
see [4] theorem 5.1). We will verify later on in this section (see diagram (11.*
*30)) that this restriction of
d44,5is not zero. This implies that the corresponding dioeerential d2 in (11.22*
*)is also nontrivial. This
discussion is consistent with the lowdimensional homotopy groups of the Moore *
*space M(Z=2, 2) =
RP 2as known from [47]:
_____i________2___3____4______5_______6______________7___________

ssiM(Z=2, 2)Z=2 Z=4 Z=4 (Z=2) 3 (Z=2) 5 (Z=2) 2 (Z=4) 2 Z=8
Table 10
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 45
Finally, returning to the case A = Z, we also observe in table 9, in position*
*s E1p,pwith p prime, the
early occurences as pss2p(S2) of Serre's orst nontrivial ptorsion in the homo*
*topy of S2 (see also for
this [39] corollary 5.40 and the discussion pp. 280~281).
Remark 11.2. In the spectral sequence from [21] there are terms E2p,q= Lp q2(A)*
*, where q2(A) is the
qth term arising from the homotopy operation algebra. In particular, there is *
*a natural homomorphism
Lq+1s(A) ! q2(A), where the occurence of the (q+1)st superLie functor is due*
* to Whitehead products,
viewed as homotopy operations. It is natural to conjecture that the semidecala*
*ge described in theorem
10.1 connects the homotopy operation spectral sequence from [21] with the Curti*
*s spectral sequence,
with for example the existence of a commutative diagram
O*1 q
LiLqs(A)________//Li+qL (A, 1)
^d2 d1
fflffl fflffl
O*1 q+1
Li2Lq+1s(A)____//_Li+q1L (A, 1)
where ^d2is a natural map induced by the second dioeerential in the homotopy op*
*eration spectral
sequence. The lowdimensional computations which we have given here conorm this*
* conjectural con
nection between the two spectral sequences.
11.4. Some homotopy groups of M(Z=p, 2), p 6= 2. The next proposition provides *
*us with some
information regarding the derived functors of L4s(A). We begin with the followi*
*ng lemma:
Lemma 11.2. Let A = Z=p for some prime p 6= 2. The natural map Tor( 3(A), A) ! *
* 4(A) is an
isomorphism.
Proof.By [9] (5.14), there exists, for any abelian group A and integer h, a com*
*mutative diagram of
abelian groups
3( hA) hA____//_ 4( hA) (11*
*.23)
~3h1 ~h4
fflffl fflffl
Tor( 3(A), A)____// 4(A)
where the upper horizontal arrow is induced by the multiplication in the divide*
*d power algebra. The
arrows ~ihprovide, where h varies, and the slide relations are taken into accou*
*nt, presentations for
the groups Tor( 3(A), A) and 4(A) respectively. Let us now suppose that A is *
*cyclic of order p,
with a chosen generator a 2 A. In that case the only relevant integer is h = p*
*. We know that
3(Z=p) = 4(Z=p) = Z=p so that the lower horizontal map in (11.23) is a homomo*
*rphism Z=p ! Z=p.
Let us show that this morphism is nontrivial: the image of fl3(a) a in 4(Z=*
*p) :
fl3(a) a 7! 4fl4(a) 7! 4!p4(a)
and 4 !p4(a) 6= 0 since p 6= 2.
Proposition 11.2. For any integer i 0 and any odd prime p, one then has
(
Z=p, i = 1, 2
LiL4s(Z=p) =
0, i 6= 1, 2
Proof.It follows from deonition that L4s(A) = 0 for every cyclic group A. By (7*
*.24), the sequence
0 ! L3 2 2(A) ! L3L4s(A) ! L3Y 4(A) ! L2 2 2(A) !
L2L4s(A) ! L2Y 4(A) @!L1 2 2(A) ! L1L4s(A) ! L1Y 4(A) *
* (11.24)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 46
is exact. By (4.3), L 2(Z=p) = K(Z=p, 0) for p odd, so that
(
Z=p, i = 1,
Li 2 2(Z=p) = (11*
*.25)
0, i 6= 1 .
In particular, the righthand arrow in (11.24)is surjective. The deonition of Y*
* 4implies that there is
a long exact sequence
` L ' ` L '
L2Y 4(A) ! ss2 3(A) A ! L2 4(A) ! L1Y 4(A) ! ss1 3(A) A ! L1 4(A)
and an isomorphism
L3Y 4(A) ' ker{Tor( 3(A), A) ! 4(A)}.
Lemma 11.2 asserts that the group L3Y 4(Z=p) is trivial, and we know by (2.20)a*
*nd (2.24)that
(
Z=p, i = 3
Li 4(Z=p) =
0, i 6= 3.
The exactness of the sequence (11.24)then implies that
(
Z=p, i = 2,
LiY 4(Z=p) = (11*
*.26)
0, i 6= 2.
We will now show that the boundary map (11.24):
L2Y 4(Z=p)__@_//_L1 2 2(Z=p) (11*
*.27)
 
Z=p ___________//Z=p
is trivial. Consider orst the case p 6= 2, 3. One then has the following values*
* for the homology groups
H*K(Z=p, 2) :
______n________2__3___4___5__6___7___8__9__

HnK(Z=p, 2)  Z=p 0 Z=p 0 Z=p 0 Z=p 0
Table 11
The analogue of the spectral sequence (11.13)for K = K(Z=p, 2) is
E1r,q= Lq+1Lr(eZK(Z=p, 2), 1) =) ssq(K(Z=p, 1)) (11*
*.28)
Reasoning as in the proof of proposition (11.1), we ond that the initial terms *
*in this spectral sequence
are the following:
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 47
_q__E11,qE12,q_E13,q___E14,q__E15,qE16,qE17,qE18,qE19,qE110,q_
8  0 * * * * * * * * *
7  Z=p Z=p * * * * * * * *
6  0 Z=p2 Z=p2 * * * * * * *
5  Z=p Z=p Z=p L1L4s(Z=p) 0 0 0 0 0 0
4  0 Z=p Z=p 0 0 0 0 0 0 0
3  Z=p 0 0 0 0 0 0 0 0 0
2  0 Z=p 0 0 0 0 0 0 0 0
1  Z=p 0 0 0 0 0 0 0 0 0
Table 12. The E1term of the spectral sequence (11.28)for p 6= 2, 3
This spectral sequence converges to the graded group Z=p[1]. Suppose that L1L*
*4s(Z=p) = 0. In that
case E12,6 E13,66= 0 and this contradicts the fact that homotopy groups ssiK(Z*
*=p, 1) are trivial for
i 2. It follows by (11.25)and (11.26)that L1L4s(Z=p) = Z=p and the map (11.27*
*) is the zero map.
The description of all the derived functors of L4s(Z=p) for p 6= 2, 3 now follo*
*ws from the exact sequence
(11.24).
For p = 3, the situation is somewhat more complicated. We have the following *
*description of the
low degree homology of K(Z=3, 2):
_____n_________2__3___4___5__6____7____8____9__

HnK(Z=3, 2)  Z=3 0 Z=3 0 Z=9 Z=3 Z=3 Z=3
Table 13
The initial terms for the spectral sequence (11.28)for p = 3 are the followin*
*g:
_q__E11,qE12,q_E13,q___E14,q__E15,qE16,qE17,qE18,qE19,qE110,q_
8  Z=3 * * * * * * * * *
7  Z=3 Z=32 * * * * * * * *
6  Z=3 Z=32 Z=33 * * * * * * *
5  Z=9 Z=3 Z=32 L1L4s(Z=3) 0 Z=3 0 0 0 0
4  0 Z=3 Z=9 0 0 0 0 0 0 0
3  Z=3 0 0 0 0 0 0 0 0 0
2  0 Z=3 0 0 0 0 0 0 0 0
1  Z=3 0 0 0 0 0 0 0 0 0
Table 14. The E1term of the spectral sequence (11.28)for p = 3
We will now prove that L1L4s(Z=3) = Z=3. For any abelian group A, the dioeere*
*ntials d11,7and d11,8
in the corresponding spectral sequence (11.12)for n = 2 have the property that *
*the following natural
diagrams are commutative:
4 ~41 ` L '
4(A)"_~__//_` 3(A)" `A L1 4(A)____//_ss1 3(A) ,A
" `
  
fflffld11,fflffl7  d11,8 fflffl
E11,7______//_E12,6 E11,8_________//E12,7
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 48
where ~4 is the homomorphism in the Koszul complex Kos4(A 1A!A) (2.29)and ~41it*
*s orst derived
analog. This implies, in the case A = Z=3, that the dioeerentials d11,7and d11,*
*8are monomorphisms.
The assumption L1L4s(Z=3) = 0, implies that E13,66= 0 and this contradicts the *
*triviality of the sixth
homotopy group of K(Z=3, 1).
Remark 11.3. For p = 2 the description of LiL4s(Z=p) is also more complicated. *
*For example, the
group L2L4s(Z=2) contains nontrivial 4torsion elements. In the simplicial lan*
*guage, a generator of
the 4torsion subgroup is provided by the following element:
{s0a1, s1a1, s0a1, s1a1}  {s1a1, s1s0a0, s0a1, s0a1} +
{s0a1, s1s0a0, s1a0, s1a0} + {{s1a1, s1s0a0}, {*
*s0a1, s1s0a0}}
This 4torsion element corresponds to twice the 8torsion element of CohenWu [*
*13],[47] App. A,
which lives in the last summand of ss7 RP 2= ss7M(Z=2, 2) (see table 10). We wi*
*ll not discuss this
computation, since it involves more elaborate techniques than those described h*
*ere.
After these preliminaries regarding the derived functors of L4s, let us begin*
* our computations of the
homotopy of the spaces M(Z=p, 2) with the case p = 3. By remark 8.1, we know th*
*at
8
>:
0, i 6= 4, 5 .
The computation of the derived functors LqLr(Z=3, 1) for q < 7 follows easily f*
*rom the Curtis decom
position of Lr and the known values of the derived functors of L3(Z=3, 1). We d*
*isplay the result in the
following table:
_q__E11,qE12,qE13,qE14,qE15,qE16,qE17,qE18,qE19,qE110,q_
6  0 0 0 Z=3 Z=3 Z=3 0 0 0 0
5  0 0 Z=3 Z=3 0 Z=3 0 0 0 0
4  0 0 Z=9 0 0 0 0 0 0 0
3  0 0 0 0 0 0 0 0 0 0
2  0 Z=3 0 0 0 0 0 0 0 0
1  Z=3 0 0 0 0 0 0 0 0 0
Table 15. The E1term of the spectral sequence (11.5)for A = Z=3 and n*
* = 2
The dioeerentials d15,6: Z=3 ! Z=3 and d24,6: Z=3 ! Z=3 are trivial, as follo*
*ws from the comparison
between the Curtis spectral sequences for K = M(Z=3, 2) and K = K(Z=3, 2) and f*
*rom the structure
of table 15. The assumption that either d15,6or d24,6is an isomorphism would pr*
*oduce a nontrivial
term E13,6in the spectral sequence whose initial terms were given in table 14. *
*Looking at the horizontal
lines in table 15, we now see that
ss2M(Z=3, 2) = Z=3
ss3M(Z=3, 2) = Z=3
ss4M(Z=3, 2) = 0
ss5M(Z=3, 2) = Z=9
ss6M(Z=3, 2) = 27.
The homotopy groups of the spaces M(Z=p, 2) for a prime p 6= 2, 3, are simple*
*r to describe. In that
case, the initial terms of the spectral sequence (11.5)are:
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 49
_q__E11,qE12,qE13,qE14,qE15,qE16,qE17,qE18,qE19,qE110,q_
6  0 0 0 Z=p Z=p 0 0 0 0 0
5  0 0 0 Z=p 0 0 0 0 0 0
4  0 0 Z=p 0 0 0 0 0 0 0
3  0 0 0 0 0 0 0 0 0 0
2  0 Z=p 0 0 0 0 0 0 0 0
1  Z=p 0 0 0 0 0 0 0 0 0
Table 16. The E1term of the spectral sequence (11.5)for A = Z=p when p *
*6= 2, 3
and n = 2
In particular, the derived functors LiL3(Z=p, 1) are simpler for these values o*
*f p and the ptorsion in
L4L3(Z=p, 1) comes from the term ker{ 2(Z=p) Z=p ! L1 3(Z=p)} in (8.8). We ob*
*tain
ss6M(Z=p, 2) = Z=p
ss7M(Z=p, 2) = p2.
11.5. Some homotopy groups of M(A, 3). Consider the spectral sequence (11.5)for*
* n = 3:
E1r,q= LqLr(A, 2) ) ssq+1M(A, 3). (11*
*.29)
The initial terms of this spectral sequence are given in the following table (r*
*ecall that ~2 is the functor
(4.1)):
_q__E11,q___E12,q_________E13,q_______
8  0  0  L2L3(A) 
7  0  0  * 
6  0  0  L3(A) Tor(A, Z=3)
5  0  2(A)  A Z=3 
4  0  ~2(A)  0 
3  0  A Z=2  0 
2  A  0  0 
_q_______________E14,q_____________E15,q_E16,qE17,q___E18,q_
8  *  0  *  *  *
7  *  0  *  *  *
6  Tor(~2(A), Z=2) 2(A) Z=2 0  0  0  *
5 Tor1(A, Z=2, Z=2) ( 2(A) Z=2)0  0  0  A Z=2
4  A Z=2  0  0  0  0
3  0  0  0  0  0
2  0  0  0  0  0
Table 17. The initial terms of the spectral sequence (11.29)
As a result we have a natural isomorphism
ss4M(A, 3) ' A Z=2
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 50
and the following natural short exact sequence (which however is not split sinc*
*e it is known for example
that ss5M(Z=2, 3) = Z=4):
0 ! A Z=2 ! ss5M(A, 3) ! ~2(A) ! 0
The dioeerential d13,6: E13,6! E14,5is trivial, as can be seen by reduction t*
*o the case of A free
abelian of onite rank, and a comparison of the rank of E13,6with that of the ho*
*motopy group of the
corresponding wedge of spheres S3, as computed by the HiltonMilnor theorem [30*
*]. On the other hand,
the dioeerential d44,6: E44,6! E48,5can be nontrivial. It is an isomorphism fo*
*r A = Z=2, as follows from
the known description of the groups ssi(M(Z=2, 3)) = ssi( 2RP 2) for small valu*
*es of i:
_____i________3___4____5_______6_________7_________8_________9_____

ssiM(Z=2, 3)Z=2 Z=2 Z=4 Z=4 Z=2 Z=2 Z=2 Z=2 Z=2 Z=4 Z=2
Table 18
In addition, one can express the dioeerential d44,6in (11.29)as a suspension *
*by comparing the spectral
sequences (11.18)and (11.29). For this, consider the following commutative dia*
*gram, in which the
vertical arrows are suspension morphisms:
d44,5(A,2)
Tor(R2(A), Z=2)_________//_ 2(A) Z=2 (11*
*.30)
 
 
fflffl fflffl
Tor(~2(A), Z=2)_d4_______//A Z=2
4,6(A,3)
The upper arrow in this diagram is the restriction to the orst summand of the d*
*ioeerential d44,5from
(11.18), whereas the lower one is the dioeerential d44,6from (11.29). The suspe*
*nsion maps are isomor
phisms for A = Z=2. Since we know that d44,6is an isomorphism in that case, so *
*is the dioeerential d44,5
in (11.18).
The spectral sequence (11.29)determines in particular a oltration on the grou*
*p ss6M(A, 3), with the
following nontrivial associated graded components:
gr2ss6M(A, 3) = 2(A)
gr3ss6M(A, 3) = A Z=3
gr4ss6M(A, 3) = ( 2(A) Z=2) Tor1(A, Z=2, Z=2)
gr8ss6M(A, 3) = A Z=2=im(d44,6)
For A = Z, this determines precisely 12 elements in ss6(S3), which are the non*
*trivial elements in the
associated graded components gr2, gr4, gr5 listed above. Table 17 also implies *
*that there is a natural
epimorphism
ss7M(A, 3) ! L3(A) Tor(A, Z3).
As an example of this computation, consider the case A = Z=3. A simple analysi*
*s, with the help
of (8.14), gives the following description of the initial terms of the correspo*
*nding spectral sequence
(11.29):
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 51
_q__E11,qE12,qE13,qE14,qE15,qE16,qE17,qE18,qE19,qE110,q_
8  0 0 0 0 0 Z=3 0 0 Z=3 0
7  0 0 Z=3 0 0 0 0 0 0 0
6  0 0 Z=3 0 0 0 0 0 0 0
5  0 Z=3 Z=3 0 0 0 0 0 0 0
4  0 0 0 0 0 0 0 0 0 0
3  0 0 0 0 0 0 0 0 0 0
2  Z=3 0 0 0 0 0 0 0 0 0
Table 19. The initial terms of the spectral sequence (11.29)for A = *
*Z=3
We conclude that
ss7M(Z=3, 3) = ss8M(Z=3, 3) = Z=3.
11.6. Some homotopy groups of M(A, 4). The spectral sequence (11.5)for n = 4:
E1p,q= LqLp(A, 3) ) ssq+1M(A, 4). (11*
*.31)
has the following initial terms in low dimensions:
_q__E11,q____E12,q_______E13,q____
9  0  0  Y 3(A) 
8  0  0  0 
7  0  R2(A)  Tor(A, Z=3)
6  0  2(A)  A Z=3 
5  0  Tor(A, Z=2) 0 
4  0  A Z=2  0 
3  A  0  0 
_q______________E14,q____________E15,q_E16,q_E17,q________E18,q______
9  *  0  *  *  *
8  *  0  *  *  *
7 Tor2(A, Z=2, Z=2) 2(A) Z=20  0  0 Tor1(A, Z=2, Z=2, Z=2)
6  Tor1(A, Z=2, Z=2)  0  0  0  A Z=2
5  A Z=2  0  0  0  0
4  0  0  0  0  0
3  0  0  0  0  0
Table 20. The initial terms in the spectral sequence (11.5)for n =*
* 4
The suspension homomorphisms ss4M(A, 2) ! ss5M(A, 3) ! ss6M(A, 4) can be desc*
*ribed in terms
of the suspension homomorphisms between the corresponding derived functors of L*
*ie functors. The
result is expressed by the following commutative diagram (see also [2] VIII #3,*
* IX #2, XI #1):
"
L3s(A) 2(A) Z=2O_//_ss4M(A,_2)__////_R2(A).
fflffl" fflffl fflffl
A Z=2O_________//ss5M(A,_3)___////_~2(A)
foflffl fflffl fflfflfflffl
"
A Z=2O_________//ss6M(A,_4)_//_Tor(A, Z=2)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 52
Note that these horizontal short exact sequences are in general not split, sinc*
*e ss4(M(Z=2, 2)) =
ss5(M(Z=2, 3)) = Z=4. Finally, the class of the generalized Hopf obration : S*
*7 ! S4 appears in
table 20 as a generator of the torsionfree component 2(Z) of ss7(S2).
11.7. Some homotopy groups of M(Z=3, 5). In this simple example, we will illust*
*rate some lines
of reasoning by which we computed certain dioeerentials in Curtis spectral sequ*
*ences. Consider such
a spectral sequence (11.3)for n = 5, with abutment M(Z=3, 5) and initial terms *
*E1p,q= LqLp(Z=3, 4)
One onds in low degree:
_q___E11,qE12,qE13,qE14,qE15,qE16,qE17,qE18,qE19,q_
10  0 0 0 0 0 0 0 0 Z=3
9  0 Z=3 0 0 0 0 0 0 0
8  0 0 Z=3 0 0 0 0 0 0
7  0 0 Z=3 0 0 0 0 0 0
6  0 0 0 0 0 0 0 0 0
5  0 0 0 0 0 0 0 0 0
4 Z=3 0 0 0 0 0 0 0 0
Table 21. The initial terms in the spectral sequence (11.5)for n = 5 and*
* A = Z=3
We will now provide two separate justiocations for the triviality of the dioe*
*erential d12,9: Z=3 ! Z=3,
both of which were used in more complex situations in the previous paragraphs. *
*The orst argument
goes as follows. The dioeerential d12,9for n = 5 and A = Z=3 lives in the follo*
*wing commutative diagram,
in which the notation is the same as in diagram 11.30 (and the vertical arrows *
*are suspension maps):
d12,9(Z=3,5)
L9L2(Z=3, 4)_________//_L8L3(Z=3, 4)
 
fflffl fflffl
L10L2(Z=3, 5)d1_______//_L9L3(Z=3, 5)
2,10(Z=3,6)
This commutative square is actually of the form:
d12,9(Z=3,5)
Z=3 __________//_Z=3
 o
 
fflffl fflffl
0 __d1_______//_Z=3
2,10(Z=3,6)
so that the map d12,9(Z=3, 5) is trivial. As a consequence, ss9M(Z=3, 5) = ss10*
*M(Z=3, 5) = Z=3.
Here is the second proof of this assertion. Consider the natural map M(Z=3, 5*
*) ! K(Z=3, 5) (11.1)
and the corresponding map between the spectral sequences (11.5)and (11.14)for n*
* = 5 and A = Z=3.
The homology groups of K(Z=3, 5) are given by:
______n________5___6_7__8___9___10___11_

HnK(Z=3, 5)  Z=3 0 0 0 Z=3 Z=3 Z=3
Table 22
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 53
The initial terms of the spectral sequence (11.14)for n = 5 and A = Z=3 are t*
*he following:
_q____E11,q_E12,q_E13,q_E14,qE15,qE16,qE17,qE18,qE19,q_
10  Z=3 0 0 0 0 0 0 0 (Z=3)
9  Z=3 (Z=3) 0 0 0 0 0 0 0
8  Z=3 0 (Z=3) 0 0 0 0 0 0
7  0 0 (Z=3) 0 0 0 0 0 0
6  0 0 0 0 0 0 0 0 0
5  0 0 0 0 0 0 0 0 0
4 (Z=3) 0 0 0 0 0 0 0 0
Table 23. The initial terms in the spectral sequence(11.12)for A = Z=3 a*
*nd n = 5
We displayed within brackets those terms which are in the image of elements f*
*rom the corresponding
spectral sequence (11.5), as given in table 21. Since the spectral sequence (11*
*.12)converges here to the
graded group Z=3[4], it follows that the map d12,9is necessarily zero: otherwis*
*e, the element E11,10= Z=3
would contribute nontrivially to ss10(K(Z=3, 4)). It follows that the correspo*
*nding map d12,9in the
spectral sequence whose initial terms are displayed in table 21 is also trivial*
*. We deduce from this that
ss9M(Z=3, 5) = ss10M(Z=3, 5) = Z=3.
Appendix A. Homology of the dual de Rham complex
In this appendix, we give an explicit proof of the second part of Proposition*
* 3.1. We will make use
of the following fact from number theory (see [34] corollary 2):
Lemma A.1. Let n and k be a pair positive integers and p is a prime number, then
` ' ` '
pn n r
mod p ,
pk k
where r is the largest power of p dividing pnk(n  k).
Proof of Proposition 3.1 (2). Let n 2 and deone the map
M
qn : n(A) ! n=p(A Z=p)
pn
by setting:
X
qn : fli1(a1) . .f.lit(at) 7! fli1=p(~a1) . .f.lit=p(~at),
pik, for all1 k t
i1+ . .+.it= n, ak 2 A, ~ak2 A *
* Z=p.
If (i1, . .,.it) = 1, then we set
qn(fli1(a1) . .f.lit(at)) = 0, whereak 2 Afor allk.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 54
Let us check that the map qn is welldeoned. For that we have to show that
` '
j1+ j2
qn(flj1(x)flj2(x) . .f.ljt(xt)) = qn( flj1+j2(x) . .f.lj*
*t(xt))(A.1)
j1
X
qn(flj1(x1+ y1) . .f.ljt(xt)) = qn(flk(x1)fll(y1) . .f.ljt*
*(xt))(A.2)
k+l=j1
qn(flj1(x1) . .f.ljt(xt)) = (1)j1qn(flj1(x1) . .f.ljt(xt)) (A*
*.3)
Veriocation of (A.1). First suppose that pj1 + j2, p  j1. Since for every p*
*air of numbers n k,
one has ` '
__n__ n
 , (A*
*.4)
(n, k) k
we have ` '
j1+ j2 s s
= p m1, j1+ j2 = p m2, (m1, p) = (m2, p) = 1, s 1.
j1
j +jm j +j
Hence 1j122_m1= p(_1__2p). Observe that j1+j2_(A Z=p) is a pgroup, hence
p
` '
j1+ j2
flj1+j2_(~x) = 0, ~x2 A Z=p (A*
*.5)
j1 p
since ` '
j1+ j2 m2_
flj1+j2_(~x) = p~xflj1+j2_(~x) = 0.
j1 m1 p p 1
The equality (A.5) implies that
` '
j1+ j2
qn( flj1+j2(x) . .f.ljt(xt))  qn(flj1(x)flj2(x) . .f.ljt(xt)) =
j1
` '
j1+ j2 X X
flj1+j2_(~x) . .f.ljt=p(~xt)  flj1=p(~x)flj*
*2=p(~x) . .f.ljt=p(~xt) =
j1 pj p
1+j2, pjk, k>2 pjk, for all1 k t
_ ` ' `j1+j2'!
j1+ j2 ____p X
 flj1+j2_(~x) . .*
*f.ljt=p(~xt)
j1 j1_p pj p
k, for all1 k t
Let j1+ j2 = psm, (m, p) = 1. Lemma A.1 implies that
` ' `j1+j2_'
j1+ j2 p r
mod p
j1 j1_p
where r is the largest power of p, dividing (j1+ j2)j1j2=p2. Since pj1, pj2, *
*we have
` ' `j1+j2_'
j1+ j2 p s
mod p .
j1 j1_p
Hence _
` ' `j1+j2_'!
j1+ j2 p
 flj1+j2_(~x) = 0, ~x2 A Z=p
j1 j1_p p
and the property (A.1) follows.
Veriocation of (A.2). We have
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 55
X
qn(flj1(x1+ y1) . .f.ljt(xt))  qn(flk(x1)fll(y1) . .f.ljt(xt)) =
k+l=j1
X X X
flj1=p(~x1+ ~y1) . .f.ljt=p(~xt)  flk=p(~x1)fll=p(*
*~y1) . .f.ljt=p(~xt) =
pjt, t 1 k+l=j1pk,pl,pjt, t 2
X X X
flj1=p(~x1+ ~y1) . .f.ljt=p(~xt)  flk=p(~x1)fll*
*=p(~y1) . .f.ljt=p(~xt) = 0
pjt, t 1 pk,pl,pjt,kt_2p+_lp=j1_p
Veriocation of (A.3). We have
qn(flj1(x1) . .f.ljt(xt))  (1)j1qn(flj1(x1) . .f.ljt(xt)) =
X
(flj1=p(~x1)  (1)j1flj1=p(~x1)) . *
*.f.ljt=p(xt) = 0
pjk, k 1
since
(flj1=p(~x1)  (1)j1flj1=p(~x1)) = 0
(we separately check the cases p = 2 and p 6= 2).
We now know that the map qn is welldeoned. It induces a map
M
~qn: H0Cn(A) ! n=p(A Z=p),
pn
since qn(a) = 0 for every a 2 im{A n1(A) ! n(A)}.
Let A = Z, then
H0(Z) = coker{ n1(Z) Z ! n(Z)} ' coker{Z n!Z} ' Z=n.
Q s
Let n = piibe the prime decomposition of n. Then
M M
n=pi(Z=pi) = Z=psii= Z=n.
pin pin
It follows from deonition of the map ~qn, that
M
~qn: H0Cn(Z) ! n=pi(Z=pi)
pin
is an isomorphism.
For free abelian groups A and B, one has a natural isomorphism of complexes
M
Cn(A B) ' Ci(A) Cj(B)
i+j=n, i,j 0
This implies that the crosseoeect Cn(AB) of the functor Cn(A) is described by
M
Cn(AB) ' Ci(A) Cj(B)
i+j=n, i,j>0
and its homology (HkCn)(AB) = HkCn(AB) can be described with the help of Kunn*
*eth formulas:
M
0 ! HrCi(A) HsCj(B) ! HkCn(AB) !
i+j=n, i,j>0, r+s=k
M
Tor(HrCi(A), HsCj(B)) !*
* 0
i+j=n, i,j>0, r+s=k1
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 56
Hence we have the following simple description of the crosseoeect of H0Cn(A):
M
H0Cn(AB) ' H0Ci(A) H0Cj(B) (A*
*.6)
i+j=n, i,j>0
From the other hand, we have the following decomposition of the crosseoeect of*
* the functor " pn=p(A) :=
n=p(A Z=p): M
" pn=p(AB) = l(A Z=p) k(B Z=p)
l+k=n=p
Hence M M M
" pn=p(AB) = l(A Z=p) k(B Z=p) (A*
*.7)
pn pnl+k=n=p
We must now show that the maps ~qnpreserve the decompositions (A.6) and (A.7). *
*This is equivalent
to the commutativity of the following diagram:
~qi ~qjL L
H0Ci(A) H0Cj(B) _________//_pi i=p(A Z=p)" ` pj j=p(B Z=p)(A*
*.8)

o 
fflffl 
H0(Ci(A)" `Cj(B)) "0
 
 
fflffl L fflffl
H0Ci+j(A B)_______q~i+j____//p(i+j) i+j_p((A B) Z=p)
The map "0O (~qi ~qj) is deoned via the natural map
Y Y X Y Y M
flik(xk) fljk(yk) 7! flik=p(~xi) fljk=p(~yi) 2 i+j_((*
*A B) Z=p),
pik, pjk p(i+j)p
X X
xk 2 A, yk 2 B, ~xk2 A Z=p, ~yk2 B Z=p, jk = j, *
* ik = i
and the commutativity of the diagram (A.8) follows. This proves that the natura*
*l map
M
H0Cn(AB) ' H0Ci(A) H0Cj(B) !
i+j=n, i,j>0
M M M
" pn=p(AB) = l(A Z=p) k(B *
* Z=p)
pn pnl+k=n=p
induced by ~qnon crosseoeects is an isomorphism, and it follows from this that*
* ~qnis an isomorphism
for all free abelian groups A.
We will now construct a series of maps:
fn,pi: LiSP n=p(A Z=p) ! HiCn(A)
for a free abelian A and pn. We orst choose liftings (xi) to A of a given fami*
*ly of elements (~xi) 2 A Zp.
We set
fn,pi: fip(~x1, . .,.~xi+1) ~xi+2.~.x.n_p7! ji(x1, . .,.xn_p) :=
i+1X
(1)tx1^ . .^.^xt^ . .^.xi+1 flp1(x1) . .".flp1(xt).f.l.p1(xi+1)flp(x*
*t)flp(xi+2) . .f.lp(xn_p)
t=1
2 i(A) ni(A), ~xk2 A Z=p, x*
*k 2 A.
Proposition A.1. The maps fn,piare well deoned for all i, n, p.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 57
Proof.We have
ji(px1, . .,.xn_p) =
Xi+1
(1)tpx1^ . .^.^xt^ . .^.xi+1 flp1(x1) . .".flp1(xt).f.l.p1(xi+1)flp(x*
*t)flp(xi+2) . .f.lp(xn_p)
t=1
Xi+1
(1)tx1^ . .^.^xt^ . .^.xi+1 xtflp1(x1) . .f.lp1(xt) . .f.lp1(xi+1)fl*
*p(xi+2) . .f.lp(xn_p) =
t=1
di+1(x1^ . .^.xi+1 flp1(x1) . .f.lp1(xt) . .f.lp1(xi+1)flp(xi+2) *
*. .f.lp(xn_p))
di+1i
2 im{ i+1(A) ni1(A) ! (A) n*
*i(A)}
One verioes that for every 1 k n=p, one has
di+1i
ji(x1, . .,.pxk, . .,.xn_p) 2 im{ i+1(A) ni1(A) ! (A) ni*
*(A)}
It follows that the map fn,pi: LiSP n=p(A Z=p) ! ( i(A) ni(A))=im(di+1) i*
*s welldeoned.The
simplest examples of such elements are the following
j1(x1, x2) = x1 x1fl2(x2)  x2 x2fl2(x1) 2 A 2(A),
j2(x1, x2, x3) = x1^ x2 x1x2fl2(x3) + x1^ x3 x1x3fl2(x2)
 x2^ x3 x2x3fl2(x1) 2 2(A) 4(A)
By construction, the elements jilie in i(A) ni(A). In fact, let us verify *
*that
ji(x1, . .,.xn_p) 2 ker{ i(A) ni(A) di! i1(A) ni+1(A)}
Observe that
diji(x1, . .,.xn_p) =
i+1X
di (1)tx1^ . .^.^xt^ . .^.xi+1 flp1(x1) . .".flp1(xt).f.l.p1(xi+1)flp(*
*xt)flp(xi+2) . .f.lp(xn_p)
t=1
In this sum, for every pair of indexes 1 r < s i + 1 there occurs a pair of*
* terms
(1)s+rx1^ . .^.xr.^.x.s.^.x.i+1 xrflp1(x1) . .".flp1(xs).f.l.p1(xi+1)flp(*
*xs)flp(xi+2) . .f.lp(xn_p)
and
(1)s+r+1x1^ . .^.xr.^.x.s.^.x.i+1 xsflp1(x1) . .".flp1(xr).f.l.p1(xi+1)flp*
*(xr)flp(xi+2) . .f.lp(xn_p)
which cancel each other. It follows that the entire sum is equal to zero.
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 58
For the same reason, the map
i+2X
(1)kfip(x1, . .,.^xk, . .,.xi+2) xky1. .y.l7!
k=1
Xi+2 i+2X
(1)k (1)m x1^ . .^.xm.^.x.k.^.x.i+2
k=1 m=1, mk
i+2Y
( flp1(xl))flp(xm )flp(xk)flp(*
*y1) . .f.lp(yl)
l=1, l6=m,k
is trivial.
Given abelian group A and i > 0, consider a natural map
X n,p M n_
fni= fi : LiSP p(A Z=p) ! HiCn(A)
pn pn, p prime
Theorem A.1. The map fniis an isomorphism for i > 0, n 7.
Proof.Given an abelianPsimplicial group (Go, @i, si), let A(Go) be the associat*
*ed chain complex with
A(Go)n = Gn, dn = ni=0(1)n@i. Recall that given abelian simplicial groups G*
*o and Ho, the
EilenbergMacLane map
g : A(Go) A(Ho) ! A(Go Ho)
is given by
X
g(ap bq) = (1)sign(~,s) q.s. .1(ap) s~p. .s.~1(bq)
(p;q)shuOes(~, )
For any free abelian group A and inonite cyclic group B, we will show that ther*
*e is a natural commu
tative diagram with vertical Kunneth short exact sequences:
L i j L i j
i+j=n_p, r+s=kLrSP (A" Z=p) LsSP (B _Z=p)//_i+j=n, r+s=kHrC"(A) HsC (*
*B)
` `
 
 
n fflffl n,p fflffl
_ ________fk_____________//_ n
LkSP p((A B) Z=p) HkC (A B)
 
fflfflfflffl fflfflfflffl
L i j L i j
i+j=n_p, r+s=k1Tor(LrSP (A), LsSP_(B))//_i+j=n, r+s=k1Tor(HrC (A), HsC*
* (B))
(A*
*.9)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 59
where all maps are induced by maps f*,p*. Since B is a cyclic, it is enough to *
*consider the case s = 0
and summands of the upper square from (A.9)
fip,pr fjp,p0
LrSP i(A Z=p) SP j(B Z=p)_//_HrCip(A Z=p) H0Cjp(B Z=p) (A.*
*10)
" ` " `
sr hr
fflffl fflffl
fip+jp,pr
LrSP i+j((A B) Z=p)__________//_HrCip+jp((A B) Z=p) ,
where the maps sr, hr come from Kunneth exact sequences. Consider natural proje*
*ctions
u1 : LrSP r+1(A Z=p) SP ir1(A Z=p) ! LrSP i(A Z=p)
u2 : LrSP r+1((A B) Z=p) SP i+jr1((A B) Z=p) ! LrSP i+j((A B*
*) Z=p)
The map sr is deoned by
sr : u1(fip(a1, . .,.ar+1) ar+2. .a.i) b1. .b.j7! u2(fip(a1, . .,.ar+2) *
* ar+2. .a.ib1. .b.j),
ak 2 A Z=p, bl2 B *
*Z=p
We have
fip,pr fjp,p0(u1(fip(a1, . .,.ar+1) ar+2. .a.i) b1. .b.j) = jr(a1, . .*
*,.ai) flp(b1) . .f.lp(bj)
and we see that the diagram (A.10) is commutative.
The map fniis an isomorphism for a cyclic group A, since both source and targ*
*et groups are trivial.
For i = 1, n = 4, and cyclic B, we have a natural diagram
L1SP 2(A Z=2B Z=2)________//_H1C4(AB)
OO OO
o o
 
Tor(A Z=2, B Z=2)____//_Tor(H0C2(A), H0C2(B))
and the isomorphism f41follows. The proof is similar for other i, n. The only n*
*ontrivial case here is
i = 1, n = 6, for the 2torsion component of H1C6(A). In that case, the stateme*
*nt follows from the
natural isomorphism
Tor(SP 2(A Z=2), Z=2) ! Tor( 2(A Z=2), Z=2)
for every free abelian group A.
Remark A.1. For any free abelian group A and prime number p, there are canonica*
*l isomorphisms
Ln1SP n(A Z=p) ' n(A Z=p)
~i+2
LiSP n(A Z=p) = coker{ i+2(A Z=p) SP ni2(A Z=p) !
i+1(A Z=p) SP ni1(A Z=p)}
where ~iis the corresponding dioeerential in the nth Koszul complex.
When i = 1, n = 3 and A a free abelian group there is a natural isomorphism
L1SP 3(A Z=p) ' L3(A Z=p).
Observe however that the natural map
f81: L1SP 4(A Z=2) ! H1C8(A)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 60
is not an isomorphism. Indeed, every element of L1SP 4(A Z=2) is 2torsion, w*
*hereas H1C8(A) can
contain 4torsion elements, since its crosseoeect H1C8(AB) contains Tor( 2(A *
* Z=2), 2(B Z=2))
as a subgroup. The map f81is given by
fi2(~a,~b) ~c~d7! a afl2(b)fl2(c)fl2(d)  b bfl2(a)fl2(c)fl2(d*
*), a, b, c, d 2 A.
Appendix B. Derived Koszul complex
In this appendix, we illustrate our derived functor methods, by giving an exp*
*licit description of
certain objects and morphisms obtained by deriving the Koszul sequence (2.29).
Let
0 ! L ffi!M ! A ! 0
be a AEat resolution of the abelian group A. A convenient model for the derived*
* category object L n(A)
is provided by the dual Koszul complex of the morphism L !ffiM. Recall that for*
* n = 2 this is the
complex
2(L) ffi2!L M ffi1! 2(M) (B*
*.1)
with the dioeerentials
ffi2(fl2(l)) = l ffi(l)
ffi1(l m) = ffi(l) ^ m
and for n = 3 the complex
3(L) ffi3! 2(L) M ffi2!L 2(M) ffi1! 3(M)
with the dioeerentials
ffi3(fl3(l)) = fl2(l) ffi(l)
ffi3(fl2(l)l0) = ll0 ffi(l) + fl2(l) ffi(l0)
ffi2(fl2(l) m) = l m ^ ffi(l)
ffi1(l m ^ m0) = ffi(l) ^ m ^ m0
L
The derived category object L 2(A) A may be represented by the tensor product*
* of the complex
(B.1) with the complex L ffi!M, in other words by the total complex associated *
*to the bicomplex
2(L) L ____//_L M L_____// 2(M) L
  
  
fflffl fflffl fflffl
2(L) M ____//_L M M____//_ 2(M) M
in other words the complex
ffi03 ffi02 2 ffi012
2(L) L ! 2(L) M (L M L) ! (L M M) (M) L ! (M) M *
* (B.2)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 61
with dioeerentials
ffi03(fl2(l) l0) = (fl2(l) ffi(l0), l ffi(l) l0)
ffi02(fl2(l) m) = (l ffi(l) m, 0)
ffi02(l m l0) = (l m ffi(l0), m ^ ffi(l) l0)
ffi01(l m m0) = ffi(l) ^ m m0
ffi01(m ^ m0 l) = m ^ m0 ffi(l)
Recall that
L1 2(A) = 2(A) (B*
*.3)
` L '
ss2 2(A) A = Tor( 2(A), A). (B*
*.4)
Given elements a, a02 nA, let us choose its representatives m, m02 M and cross*
*cap elements l, l02 L
such that ffi(l) = nm, ffi(l0) = nm0. The maps
2(A) ! (L M)=im(ffi2)
Tor( 2(A), A) ! ( 2(L) M (L M L))=im(ffi03)
which deone the isomorphisms (B.3) and (B.4) are given by
w2(a) 7! l m + im(ffi2)
w2(a) * b 7! (fl2(l) m0, l m l0) + im(ffi3)
Next we consider the following diagram with exact rows and columns:
" ffi003 ffi002 ffi001
L3(L)O___________//"L` "M` _L_______________//L M" `M____________//_Y"3(M)
  `
fflffl"ffi03 fflffl ffi02 fflffl ffi01 fflff*
*l
2(L) LO___//_ 2(L) M (L M __L)//_(L M M) 2(M) _L__//_ 2(M) *
* M
OE3fflffl OE2fflfflfflffl OE1fflfflfflffl OE0fflf*
*flfflffl
" ffi3 ffi2 ffi1
3(L)O___________//_ 2(L) _M_______________//_L 2(M)___________// 3(M)
fflfflfflffl
L Z=3
(B*
*.5)
with
OE0(m ^ m0 m00) = m ^ m0^ m00
OE1(l m m0) = l m ^ m0
OE1(m ^ m0 l) = l m ^ m0
OE2(fl2(l) m) = fl2(l) m
OE2(l m l0) = ll0 m
OE3(fl2(l) l0) = fl2(l)l0.
and
________ 00 0 00 0 0*
* 00 0 00
ffi003(l l0^ l00) = l ffi(l ) l + l ffi(l) l  l ffi(l *
*) l  l ffi(l) l
ffi002(l m l0) = l ffi(l0) m + l0 ffi(l) m + l m ffi(*
*l0)
ffi001(l m m0) = {ffi(l), m0, m}
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 62
________
Here l l0^ l00denotes the image of the element l l0^l00under the natural epi*
*morphism L 2(L) !
L3(L). We will now make use of the fact that the dual de Rham complex
0 ! 3(L) ! L 2(L) ! 2(L) L ! 3(L)
has trivial homology in positive dimensions and hence
L3(L) = ker{ 2(L) L ! 3(L)} = coker{ 3(L) ! L 2(L)}
Consider the functor ~E3(A) := im{ 2(A) A ! 3(A)}. We have a natural short*
* exact sequence
0 ! ~E3(A) ! 3(A) ! A Z=3 ! 0
which induces the following commutative diagram
~ffi3 ffi2 ffi1
~E3(L)"___// 2(L) M___//_L 2(M)__//_ 3(M) (B*
*.6)
` 
fflfflffi3  ffi2  ffi1 
3(L)_____// 2(L) M___//_L 2(M)__//_ 3(M)
fflfflfflffl
L Z=3
From diagrams (B.5) and (B.6) we deduce the following diagram with exact arrows*
* and columns:
"
H2W" O____________//_`L2Y"3(A) (B*
*.7)
 `
` fflffl ' ` fflffl '
L L
ss2 L 2(A) A______ss2 L 2(A) A

fflffl fflffl
L Z=3 _________//_H2Q____________////_L2 3(A)

fflffl fflffl
H1W _____________//_L1Y 3(A)

` fflffl ' ` fflffl '
L L
ss1 L 2(A) A______ss1 L 2(A) A
fflfflfflffl fflfflfflffl
L1 3(A) _____________L1 3(A)
where W and Q are the upper rows in diagrams (B.5) and (B.6) respectively. We g*
*ive the following
simple example, which illustrates the inner life of the previous diagrams.
Proposition B.1. 8
>:
0, i 6= 1, 2
Proof.It follows from the description Y 3(A) = ker{ 2(A) A ! 3(A)} that
L2Y 3(A) = ker{Tor( 2(A), A) ! L2 3(A)}
Let A = Z=3 and L @!M is Z 3!Z. Let l, m be generators of L and M respectively *
*with ffi(l) = 3m.
The group Tor( 2(A), A) is generated by the homology class of the element (fl2(*
*l) m, l m l) in
the complex (B.2). We have
OE2(fl2(l) m, l m l) = fl2(l) m + ll m = 3fl2(l) m = fl2(l) *
* ffi(l) = ffi3(fl3(l))
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 63
Hence the map
Tor( 2(A), A) ! L2 3(A)
induced by the map OE2 is the zero map. This proves that
L2Y 3(Z=3) = Z=3.
It is easy to see that for our choice of L and M, the complex W has the form Z *
*!9Z ! 0 so that
H1W = Z=9, H2W = 0. In this case, the diagram (B.7) has the form
0 _______//_Z=3
 '
fflffl fflffl
Z=3 ________Z=3
" fflffl f0flffl
Z=3O___//_H2Q_____////_Z=3

fflffl fflffl
Z=9 ____//_L1Y 3(A)
fflffl fflffl
Z=3 ________Z=3
fflffl fflffl
0 0
and we see that the map H1W ! L1Y 3(A) is an isomorphism and hence
L1Y 3(Z=3) = Z=9.
The object L 2(A) of the derived category may be represented as the following*
* complex:
L L ! 2(L) (M L) ! 2(M)
Consider the following diagram:
2(L)"_`______//M" `L________// 2(M) (B*
*.8)
 (0,id)
fflffl fflffl 
L L_____// 2(L) (M _L)__// 2(M)
fflfflfflffl fflfflfflffl
SP 2(L)________//_ 2(L)
Denote by C = C(L !ffiM) the upper complex in (B.8). Diagram (B.8) implies the *
*following exact
sequence of homology groups:
0 ! H1C ! R2(A) ! L Z=2 ! H0C ! 2(A) ! 0
In particular, the ptorsion components of HiC and Li 2 are naturally isomorphi*
*c for p 6= 2.
L
The objects L 3(A) and A L 2(A) of the derived category may be represented *
*by the following
complexes:
L L L ! ( 2(L) L) (L 2(L)) (L L M) !
3(L) ( 2(L) M) (L 2(M)) ! 3(M)*
* (B.9)
L L L ! (M L L) (L 2(L)) (L L M) !
(L 2(M)) (M 2(L)) (M L M) ! M 2(M) *
*(B.10)
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 64
Consider the following diagram:
3(L)____//_M 2(L)___//SP 2(M)" _L__________//_`SP"3(M)` (B.*
*11)
  fflffl fflffl
3(L)____//_M 2(L)___// 2(M) L_____________// 3(M)
fflfflfflffl fflfflfflffl
M L Z=2____//_M Z=3 (M M Z=2)
The upper complex in (B.11) is a model for the element LSP 3(A) in the derived *
*category. Denote the
middle horizontal complex by D = D(L ffi!M). We have the natural isomorphism
H2D ' L2SP 3(A)
and the following exact sequence:
0 ! L1SP 3(A) ! H1D ! Tor(M A, Z=2) !
SP 3(A) ! H0D ! M Z=3 (M A Z=2) ! 0 (*
*B.12)
Now consider the following diagram which extends the diagram (B.5):
"
3(L)O___________//_"M`" `2(L)______________//_ 2(M)" _L___________//` 3(M)*
*" `
 
fflffl"~ffi3 fflffl ~ffi2 fflffl ~ffi1 fflff*
*l
L 2(L)O___//_M 2(L) (L M _L)_//_(M M L) L 2(M)_//_M 2(*
*M)
fOE03flffl fOE02flffl fOE01flffl fOE00*
*flffl
"ffi03 ffi02 ffi01
2(L) LO___//_ 2(L) M (L M __L)_//(L M M) 2(M) _L__//_ 2(M) *
* M
fOE3flffl fOE2flfflfflffl fOE1flfflfflffl fOE0f*
*lfflfflffl
" ffi3 ffi2 ffi1
3(L)O___________//_ 2(L) _M_______________//_L 2(M)___________// 3(M)
fflfflfflffl
L Z=3
(B.*
*13)
Here
~ffi1(m m0 l) = m m0ffi(l)
~ffi1(l fl2(m)) = ffi(l) fl2(m)
~ffi2(m l ^ l0) = (m ffi(l) l0 m ffi(l0) l, 0)
~ffi2(l m l0) = (ffi(l) m l0, l mffi(l))
~ffi3(l l0^ l00) = (ffi(l) l0^ l00, l ffi(l0) l00+ l *
* ffi(l00) l0)
and
OE00(m fl2(m0)) = m ^ m0 m0
OE01(m m0 l) = (l m m0, m ^ m0 l)
OE01(l fl2(m)) = (l m m, 0)
OE02(l m l0) = (ll0 m, l m l0)
OE02(m l ^ l0) = (0, l m l0 l0 m l)
OE03(l l0^ l00) = ll0 l00+ ll00 l0
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 65
We obtain the natural isomorphism of complexes:
` L '
H2D ________//_H2 A C______//_Tor( 2(A),_A)_//_ 3(A)
  
  
   
L2SP 3(A)____//_Tor(A, L1SP 2(A))//_Tor( 2(A),_A)_//_ 3(A)
The map
Tor(A, L1SP 2(A)) ! (M 2(L) (L M L))=im(~ffi3)
is given as follows: let a, a0, a002 A with na = na0= na00= 0 are represented b*
*y elements m, m0, m00is
M and ffi(l) = nm, ffi(l0) = nm0, ffi(l00) = nm00, then
a * a0^*a007! (m l0^ l00, l m0 l00 l m00 l0) + im(~ff*
*i3)
The map OE02induces the Koszultype map
Tor(A, L1SP 2(A)) ! Tor( 2(A), A)
deoned by
a * a0^*a007! (w(a + a00)  w(a)  w(a00)) * a0 (w(a + a0)  w(a)  w*
*(a0)) * a00.
Example. In the case (L ffi!M) = (Z n!Z), the diagram (B.13) has the following *
*form:
__3n__//
Z Z
(1,1) 1
(n,2n)fflffl fflffl
Z______//Z Z(2n,n)//_Z
(2,1) (1,1)
(n,n)fflffl(n,fflffln)
Z____//_Z _Z____//Z
3 (1,2)
fflfflnfflffl
Z______//Z
This diagram implies that the map
` L ' ` L '
H1 Z=n C(Z n!Z) ! ss1 L 2(Z=n) Z=n
is multiplication by 3 in the group Z=n.
When A = Z=2 and C = C(Z 2!Z), we have the following commutative diagram
A H1C"_________//`A " R2(A)
 `
` fflffl' ` fflffl ' ` '
L L L
H1 A C ____//_ss1 A L 2(A)_//_ss1 L 2(A) A
AA______________________
fflfflfflffl fflfflfflffl _________________________*
*__________________
' _____________________
Tor(A, H0C)_____//Tor(A, 2(A)) ____________________________*
*________________________________
________________________________________________________*
*_______________________________________
______________________________________________________*
*_________________________________________________________
___________________________________________________*
*_____________________________________________________________________________*
*_____________
________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*__
______'_____________________________________*
*_____________________________________________________________________________*
*___
As a corollary, we ond that:
DERIVED FUNCTORS OF NONADDITIVE FUNCTORS AND HOMOTOPY THEORY *
* 66
Proposition B.2. The derived Koszul map
` L ' ` L '
ss1 A L 2(A) ! ss1 L 2(A) A
is the zero map for A = Z=3 and an epimorphism for A = Z=p, where p is a prime *
*6= 3.
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LB: UMR CNRS 7539
Institut Galilee
Universite Paris 13
99, avenue JeanBaptiste Clement
93430 Villetaneuse
France
Email address: breen@math.univparis13.fr
RM: Steklov Mathematical Institute
Department of Algebra
Gubkina 8
Moscow 119991
Russia
Email address: romanvm@mi.ras.ru