ON THE COMPOSITION PAIRING OF ADAMS
SPECTRAL SEQUENCES
By R. M. F . MOSS
[Received 23 July 1966]
The aim of this paper is to generalize Theorem 2.2 of (1) by giving a
composition pairing of Adams spectral sequences. The existence of this
pairing is proved in §2, and its properties are given in Theorem 2.1.
Our approach to the Adams spectral sequence is based on that of
H. B. Brinkmann, who has shown how to set up Adams spectral sequences
for cohomology functors in arbitrary stable categories. We shall mainly
restrict our attention to a category of spectra, since the proof given of
Theorem 2.1 does not appear to generalize to an arbitrary stable category.
The category of spectra that we shall use is the one whose properties
are discussed by J. M. Boardman in (4), and in § 1 we give the basic
definitions that are required to set up Adams spectral sequences in
Boardman's category.
In § 3 we discuss briefly a dual type of Adams spectral sequence that
can arise. It will be seen that the proofs of §2 are applicable in this
dual situation, and that a theorem similar to Theorem 2.1 can be proved
if required. Finally, in § 4 we make some remarks on the proof of a pairing
theorem for the Adams spectral sequence defined in (3).
I am grateful to Professor J. F. Adams for advice and encouragement;
in particular it was he who first suggested that a pairing of this nature
could be defined.
Part of the work for this paper was done while I held a scholarship
from the Carnegie Trust for the Universities of Scotland, and I am
grateful to them for their support.
1. As in (4), we denote Boardman's category of spectra by £f\ with
SP is associated the homotopy category Sfn, and with the categories SP
and S?h are associated graded categories SP* and Sfh*. The category Sfh%
is a stable category in the sense of Puppe (8); the morphisms of degree p
in S^ from X to Y will be denoted by {X, Y}p, and a{X) will denote the
element of {SX,X}_1 that is the natural isomorphism given in E8 of (4).
In these and other respects we shall follow the notation and definitions
of Boardman, and the reader is referred to (4) for further details.
Now let s# be a graded additive category (8); then the dual category
s#° is defined as follows. For each object A in s# there is an object A0
Proe. London Math. Soc. (3) 18 (1968) 179-92
180
R. M. F. MOSS
in J / ° , and for each morphism a: A -> B of degree p in stf there is a
morphism a 0 : JB° -> A0 of codegree p (= degree — p) in J / ° . Composition
in J / ° is defined by putting
whenever a: ,4 -s- i? has degree p and jS: B -> C has degree g. We denote
by Djj: J / -> j / ° the natural bijection.
If «s/' is a stable category and s0 is a graded abelian category then a
covariant functor H: s/' —>• $0 is said to be a homology functor if it takes
exact triangles in $0' into exact triangles in stf. If H: srf' -*• $0 is such
that D^H is a homology functor then H is said to be a cohomology
functor. Any spectrum K defines a homology functor HK: SPh% -> s/K
and a cohomology functor HK: Sfh%. -> jtfK as follows. For any spectrum
Zput
{K,X}n
and H$(X) = {X, K}_n.
We define, for 6 in {X, Y}m and <p in [Y, X}m,
d*
and
HK{
where O^oc = doc and (p*a = (— l)mna<p for a in HnK(X). The category
is the category of graded right [K, K}%-modules, and the category $0K
is the category of graded left {K, iT}*-modules.
Of fundamental importance in all that follows is Definition 1.2, where
we define the Adams system of a spectrum. We base our definition of
an Adams system on that of a realization of a projective resolution as
given in (1) and (2). We shall require Adams systems for an arbitrary
cohomology functor H, and so we must first describe a class of spectra
that will act as Eilenberg-MacLane objects (2) for H. This we do in
Definition 1.1, which we now give.
DEFINITION 1.1. Let H: Sfh* -»-j^bea cohomology functor; a spectrum
K is said to be H-injective if it satisfies the following two conditions,
(i) H(K) is a projective object in stf.
(ii) For all spectra X the natural transformation
H: {X,K}t-*
Rom*(H(K),H(X))
is an isomorphism.
An .ff-injective object has the following property, which resembles the
defining property of an injective object in an abelian category. Let
K be H-injective, and let 6 e {X,K}*; then, if <p e {X,Y}^ and H(<p) is
surjective, there exists an element 6 of {Y, K}% such that Bcp = 6.
PAIRING OF ADAMS SPECTRAL SEQUENCES
181
DEFINITION 1.2. Let H: S?h* -> srf be a cohomology functor; an Adams
system for H is a sequence x = (Xn, xn) of spectra Z n , n ^ 0, and classes
xn e {Xn, Xn_1}0. This sequence is required to have the following
properties.
(i) For each n > 1, H{xn) = 0.
(ii) For each n ^ 1, there is an exact triangle
in which Kn_x is fZ-injective.
Let x = (Xn,xn) and x' = (X'n,x'n) be Adams systems for H; a map
from x' to x consists of a sequence 8 = (6n) of classes 6n G {X'n,
making the following square commute for all n ^ 1.
J
n-1
An Adams system x = (Xn, xn) is called an Adams system of a spectrum
X if X = Xo. Let X and X' be spectra having Adams systems x and x',
and let 6 e (X', X}^; then a map 8 = (8n) from x' to x is said to be induced
by 6 if 60 = 6.
In general there is no guarantee that a particular spectrum has an
Adams system for some cohomology functor H. The possible nonexistence of such a system is the main obstruction to setting up the
relevant Adams spectral sequence. It is known that certain spectra do
have Adams systems for the mod-^» cohomology functor; in particular,
any CW-complex with finite skeletons has such an Adams system.
We next pass on to the question of induced maps of Adams systems.
The details are given in Proposition 1.3, which follows. It is clear from
this proposition that any element 9 of {X',X}p will induce a map of
Adams systems.
1.3. Let x = {Xn,xn) and x' = (X'n,x'n) be Adams systems
for some cohomology functor H; then, if 60 e {X'Q,X0}p, there exists, for
each n^ 1, 6nin {X'n,Xn}p such that xn6n = 9n_1x'n.
PROPOSITION
182
R. M. F . MOSS
Moreover, let the following triangles be exact and suppose that
for each n ^ 1.
•\rt
A
n-1
71
<
y/
A
"n-V
n
*«-l N
"'n-l
kn-1
K
n-1
Then there exist elements [Mn in {K'n, Kn}p making the following diagram
commutative up to the sign shown.
V
L
n-1
^
JX
TI-1
'n-l
"n-X
Proof. The elements 6n can be constructed by induction. Suppose
dn_x has been constructed for some n ^ 1; then, as Kn_x is //-injective,
1.2(i) and 1.1 (ii) imply that the composition
is trivial. The exactness of
h
now implies the existence of an element 6n e {X^, JT,^ with the property
required.
Given these elements dn the existence of the sequence (/xn) now follows
without difficulty from J12 of (4).
We next examine the spectral sequence that results from an Adams
system. The details are given in the following theorem, which generalizes
2.1 of (1). This generalization is due to Brinkmann.
THEOBEM 1.4. Let H: SPM -> s4 be a cohomology functor, let X be a
spectrum having an Adams system, and let X' be any spectrum; then there
is a spectral sequence E(X',X) = (Es/(Xf,X),dr) of abelian groups such
that
dr: ES/(X',X)
and having the following properties.
P A I R I N G OF ADAMS SPECTRAL SEQUENCES
(i) For each s,t in Z there is a natural
(ii) There is a natural
183
isomorphism
monomorphism
whenever s + l < r < / < o o .
(iii) The group, {X',X}p, admits a filtration
{X',X}p
and for each s,tinZ
= F% => ... => $n
there is a natural
3
^n
isomorphism
Proof. Let x = {Xn, xn) be an Adams system of X, and take for each
n ^ 1 an exact triangle as follows:
Then the spectral sequence is t h a t which results from the exact couple
E
in which B* = {X',Xs}t_s and E* = {X',K8}t_s.
There is no difficulty in proving t h a t this spectral sequence satisfies
1.4(ii) and 1.4(111). That the spectral sequence is independent of t h e choice
of Adams system is immediate from 1.3 and the way in which t h e
isomorphism 1.4(i) is defined. We now proceed to define this isomorphism,
thus completing the proof of 1.4.
Define for each O O a projective object Cs by putting GSJL = Hl-S(K8);
also define cs: Cs-> C^ by cs = Htyjc^) and s: Go-> H(X) by e = H(i0).
We obtain the following projective resolution of H(X).
H(X)
Now define c*+1: Kom^(Gs,H(Xf))
-> TIom<JCs+1,H(X'))
and so obtain
:'))^kerc*+1</imc*.
by
184
R. M. F . MOSS
Since Ks is #-injective we have isomorphisms
E?(X',X) = {X\Ks}t_s^^
Kom<jCs,H(X')),
and as Hdx = c*H we obtain an isomorphism
8: E?(X',X) ->ExVj(H(X),H(X')).
The spectral sequence E{X',X) of 1.4 is the Adams spectral sequence
for X', X, and H. We shall require a more explicit description of the
jE7r-term of this sequence, and this can be given by means of a spectral
system (see (5), (6)). We may assume that X has an Adams system
x = (J^.rcJ in which
and xn is the class of the inclusion. For by H5 of (4) we can always
replace X by a homotopy-equivalent spectrum having this property.
We now extend the definition of Xn by putting Xm = * and -X,j = XQ
if — oo ^ n ^ 0; we define X{n,m) = X^X^, if n ^ m. If n' ^ n, m' ^ m,
and m' ^ n', there is a map X(n', m') -» X(n, m) induced by the inclusions
Xtf c: ^ and X^ c: Xm. The homotopy class of this map will be denoted
by x(n,m : n',m'), and we put x(n,n') = x(n,co : n',co). If I ^ m ^ n the
composition of the boundary map X(n,m) -> Xm (see J l of (4)) and the
natural map X^ -> Xm/Xl = X(m, Z) defines an element k(n, m, I) of
{X{n)m))X{m,l)}_1. Using the functor Hx> it is clear that we can obtain
a spectral system of groups from these spectra, and that this spectral
system defines the Adams spectral sequence E(X',X). In particular,
Esr,t{X',X)
=
™({X',X(s,s + r)}t_s
** >
{X',X(s,s+l)}t_s)
', X(s and d r : Es/ -> ^+'•.'+'-1 is induced by
2. We first recall the definition of the Yoneda pairing of Ext-groups.
Let M and M' be objects in a graded abelian category $0 having projective resolutions (Cn, cn) and {C'n, c'n) respectively. Then if M" is any object
in s& we can define the Yoneda pairing
ExtsJ(M',M")®ExtsJ(M,M')
-> Ext8+s'>l+l'{M,M")
as follows. Take a in ExtsJ(M,M') and a' in ~E,xtsJ\M',M"), and select
representatives / in Hom^(Cs,iW') a n d / ' in Hom^(C^,if"). Form the
P A I R I N G OF ADAMS SPECTRAL SEQUENCES
185
following diagram commutative u p to the sign shown.
Cs <- ... <- C n + S _ t
n+s
T h e n / ' / , e Hom'/(C S + S ,,M") a n d / ' / x s + s , + 1 = 0. The resulting element
in E x t is independent of all the choices made and will be denoted by oc'a.
This defines an associative and bilinear pairing.
Now take a cohomology functor H: Sfh* ->stf, and let X and X' be
spectra having Adams systems for H. Then if X" is any spectrum we can
and
E(X",X).
set u p the Adams spectral sequences E{X',X), E(X",X')
The following theorem will show how to pair the first two spectral
sequences to the third.
THEOREM
2.1. Let 2 ^ r ^ oo; then there exist associative
s
l
E /{X',X)®E?> '{X",X')
->
8
s l
pairings
l
E r+ '' + \X",X),
and these pairings have the following properties.
(i) Let a e ES/(X',X) and a' e E(>l'{X"}X'); then
fi(aa') =
(-l)™'£{a')8(a),
where p = t — s, p' = t' — s', and 6 is the isomorphism of 1.4(i).
(ii) Let a e ES/(X',X) and a' e E*'f(X",X');
then
d"r{aa') = {dra)ar + ( - l)*>a{d'ra'),
where p = t — s and dr, d'r, d"r are the differentials in the three spectral
sequences.
(iii) The pairings commute with the isomorphisms Er+1 ^ H(Er) and the
monomorphisms of 1.4(ii).
(iv) The composition pairing
{X',X}p®{X",X%,
-•
{X",X}p+p.
preserves filtration; on passing to quotients and using the isomorphisms of
1.4(iii) the pairing of E^-terms is obtained.
Proof. We take Adams systems x = (Xn, xn) and x' = (X'n, x'n) of
X and X' such t h a t Xn_x => Xn a n d X'n_x => X'n for all n. We p u t
E = E(X',X), 'E = E{X",X'), and "E = E{X",X), and use a similar
notation for all groups associated with these spectral sequences.
186
R. M. F . MOSS
The method of proof is to show that the pairing E2®'E2 -+ "E2, defined
so as to give 2.1(i), passes to the subquotient. We first make explicit
the description we shall require of Er as the quotient of a subgroup of E2.
Let Zp< = im{{X', X(s,s + r)l_s -> Eg) and
B? = im({X', X(s-r + l,s)}t_s+1 -> E#);
l
l
then Ep ~ Zp /Bpl, and if 2 ^ r ^ oo we have the following inclusions.
0 = Btf «= ... <= Bpl c Bp^ c ... c
p
The element k(s,s + r, s + r+1) induces an additive relation (7)
*£': E\f -^ Ep****-1 which itself induces dr: E8/ -> ^+'•.'+'•-1. It is easily
verified that
def KS/ = Zs/,
im KS/ =
ker /c^ = Zplv
ind ^ =
2.2. / / the pairing of E2-terms, defined by means of 2.1(i) and
the Yoneda pairing, satisfies conditions (i) and (ii) that follow, then this
pairing passes to the subquotient; the pairing of Er-terms that results satisfies
2.1(i), (ii), and (iii).
LEMMA
(i) / / 2 ^ r ^ co, Zpl'Z°'>1' <= »Z°+S'>1+1'.
(ii) Ifae Zs/ and a' e 'Zs^' then
Knr{aa') ^ (Kra)ar + ( -
Proof. It is clearly sufficient to show that the following inclusions
hold for 2 ^ r ^ 00.
(2.3)
?
Bpl'Zs;>1' ^ "Bsr+s''l+l\
(2.4)
These can be proved by induction on r, and this will include the case
r = 00, since £%,' = 5 ^ . Suppose then that 2.3 and 2.4 have been
proved for some r ^ 2 ; let a e Zp\,x and b' e 'i^+i; then b' e «r^a' for
some a' in '£s'-r/-r+i Now £ r + 1 = ker«:r and so Kra = Bphr>t+r-1', hence
by 2.4 we obtain
(Kra)a' c
Since ind«:r = Br, 2.2(ii) now gives
ab' e a{K'ra') c ( - l)*»/cr(aa') c
This proves 2.3 for r+1, and it is clear that 2.4 can be proved in a
similar manner.
P A I R I N G OF ADAMS SPECTRAL SEQUENCES
187
We next introduce the following notation which is less cumbersome
than that given in § 1 to define the spectral system. We put
if
r>2
and
Ks = X{s,s + l),
we also define the following elements.
xsr = x(s,s+l
k8r =
: s,s + r),
+ r,s + r+l),
h
xs>r = x(s + l,s + r+ 1 :
ks-r = k{s,s+ l,s + r+ 1),
J.S.00
We shall need an explicit description of the pairing of i^-terms that
we have introduced by means of the Yoneda pairing in Ext and the
identity 2.1(i). Let a e E^1 and a' e 'E(* be represented by A e {X£,K8}P
and A' E {X" ,K'S)V, and let X = (AJ be any sequence of elements making
the following diagram commutative up to the sign shown.
V
n+1 -> ...
n+1
71+S+l ->
...
We shall refer to such a sequence X as a lifting of A. It is clear that in Ext,
(cls#(A'))(cls#(A)) = cls(#(A')#(As,))5
and, as H{\')H{XS.) = (-1)™'H{\S,\'), we have in "E2,
aa' = cls(A6wV).
Now let a in Zs/ and a' in 'Z** be represented by 9 in {XQ, Xsr}p and
6' in {X",Xs,r}p, and put A = xsrd, X = x's,rd', S = ksrd, and S' = k'Jr6'. To
complete the proof of 2.1 we have only to prove Proposition 2.5, which
follows; parts (i) and (ii) of this proposition imply 2.2, and part (iii)
implies 2.1(iv).
2.5. Given elements 6 and d' as above there exist an element
and liftings X = (AJ of A and 8 = (SJ of 8 so that the
6" of {X",Xs+s,r}p+p.
following identities are satisfied.
if2^r^ao.
(i) xs+s,tTd" = AS,A'
(ii) ks+sj" = SS,A' + ( - l)*As-+rS' if 2 ^ r < oo.
(iii) Ifr = oo5 x"{0,s + s')6" = x{Q,s)dx'(O,s')d'.
PROPOSITION
R. M. F . MOSS
188
Proof. Let Yn =
as follows.
s>r, Ln
(
x
= Kn+S\lSKn+s+r,
and define y n , j n , and ln
\
ln = (kn+8'r,
-Xn+S'a).
We have here put a = o(Kn+s+r).
following triangle.
y
For each n ^ 1 — s we obtain the
"»
TTn
/
(2.6)
The lemma that follows implies that the sequence y = (Yn, yn) forms an
Adams system.
LEMMA
2.7. For each n ^ 1 — 5 the triangle 2.6 is exact, and H(yn) = 0.
Proof. Let A = ^ + s _ l j 7 l + s + r , B = Yn, and C = Kn+s+r_v
so t h a t
C <=• B <=• A. Using G6 of (4) we can identify 2.6 with a triangle of the
following form.
A/C<
B
(A/B)vSC
There is no difficulty in proving this triangle exact; the proof is a
slight generalization of that required to prove J9 of (4) (which follows
on putting A = B).
To show that H{yn) = 0 it is sufficient to show that H(ln) is a
monomorphism. This follows without difficulty from the following
diagram, in which the vertical sequence and both horizontal sequences
are exact.
> H(Xn+s)
0
H(Yn
H(Xn
P A I R I N G OF ADAMS SPECTRAL SEQUENCES
189
We now apply 1.3 to the element 6 and the Adams systems x' and y; by
so doing we obtain sequences 8 = (6n) and [L = (fxn) making the following
diagram commutative up to the sign shown.
V-l
•
K
jr,
n-1
K
n-1
(2.8)
Vn
r»-i—r
Let \n in {K'n, Kn+S}p and 8n in {K'n, Kn+s+^v_x be defined by putting
then X = (AJ lifts A and 8 = (8B) lifts S. This follows without difficulty
from our definitions and the commutativity of 2.8.
If r = oo we put 6" = Qs,d', and it is easy to verify 2.5(i) and 2.5(iii).
If r < oo, put Y'n = X'n>r, L'n = K'nvSK'n+r, and define y'n, j ' n , l'n in the
obvious manner. Let /Xn in {L'n, Ln}p be defined by
LEMMA 2.9. For each n, there exists 6n in {Y'n,Yn}p making the following
square commute.
-\ri
Jn
.
TI
H-n
Yn—^Ln
Jn
Proof. In view of 2.7 it is sufficient to show that lnfLnj'n = 0.
Now
0
190
R. M. F . MOSS
The following commutative diagram at once implies that
as required.
n+r
n+r
l
n+l ^
n+s+r
The commutativity of this diagram follows without difficulty from that
of 2.8.
To complete the proof of 2.5 we put 6" = ds,9' and apply 2.9 having
put n = s'. We obtain the following.
Taking components, we obtain 2.5(i) and 2.5(ii) for r < oo.
3. In this section we consider briefly an Adams spectral sequence which
is in some sense dual to the one whose properties were given in 1.4.
This sequence arises from a co-Adams system.
Let stf' be a stable category, $0 a graded abelian category, and
H.stf'^s/
a cohomology functor; it is clear from 1.1 how to define
//-injective objects in s/'. We now give the dual definitions.
DEFINITION 3.1. Let H: $0' -^ $0 be a homology functor; then K is
said to be H-injective if it is Z^/7-injective.
Let H: stf' -> s/ be a homology or cohomology functor; then K is
said to be H-projective if K° is
It is now clear how to define an Adams system for a homology or
cohomology functor. An Adams system for a homology functor H is
PAIRING OF ADAMS SPECTRAL SEQUENCES
191
an Adams system for the cohomology functor D^H. Dually, we have
the following definition.
3.2. Let H: stf' -> J / be a homology or cohomology functor;
a co-Adams system for H is a sequence x = (Xnl xn) of spectra Xn, n ^ 0,
and classes xn e {Xn, X n+1 } 0 such that x° = (X°, x%) is an Adams system
DEFINITION
for HD$.
Given a homology or cohomology functor H: ja/' -> $0 we can obtain,
as in 1.4, a spectral sequence from an Adams system. If the functor H
is contravariant then the spectral sequence is of the form
Ext**(#(X),#(X')) => Homf (X',X),
and if H is covariant then the spectral sequence has the form
Exty(H(X'),H(X)) => Homf (X',X).
The spectral sequence for a covariant functor H can be identified with
that which arises for the functor D^H, and so a theorem similar to 2.1
is obtained when J / = Sfh*.
Dually, we can obtain a spectral system from a co-Adams system. We
apply the functor Hx. to the exact triangles
which arise from a co-Adams system of a spectrum X. If jy is contravariant the resulting spectral sequence has the form
Ext*.*(#(X'),#(X)) => Hom*,(X,X'),
and if H is covariant then the spectral sequence is of the form
Ext*.*(#(X),#(X')) => Hom*,(X,X').
Once again we can identify the spectral sequence for a covariant functor
H with that which arises for the functor D^H. I t is clear that when
stf' = SP-hx the proof of 2.1 dualizes, and we shall obtain a pairing theorem
for these co-Adams spectral sequences. Provided the boundary homomorphisms are correctly defined, the signs of 2.1(i) and 2.1(ii) will be
as expected in this dual situation. In practice this means that we must
define dx: {Ks, X% -» {Ks+1, X%_i by dtf) = ( - l)*«Aw- A s i m i l a r s i g n
is necessary in the definition of the higher differentials, and these signs
are in accordance with the sign conventions we have been using.
So far these dual sequences have received little attention. One such
sequence of possible interest might arise from the homology functor
192
P A I R I N G OF ADAMS SPECTRAL SEQUENCES
s0
/7 : S?h* -> stf^ (where S° is the zero-dimensional sphere). Provided the
Eilenberg-MacLane spectrum K(Zp) has a co-Adams system we should
obtain a spectral sequence of the form
•Ext*?(Zp,Zp) => A*.
Here we have put X = X' = K{Zp), G* denotes the stable homotopy
ring, and A* denotes the mod-p Steenrod algebra. By dualizing 2.1 we
can assert that this spectral sequence will have a ring structure.
4. The spectral sequence defined in (3) just fails to fall into the situation
described in §1. However, Theorem 2.1 can be proved for these Adams
spectral sequences provided we replace the term '.ff-injective spectrum'
by the term 'generalized Eilenberg-Maclane object' as defined in §6 of
(3). If we then use 6.6 and 6.5 of (3) instead of l.l(i) and l.l(ii), the
proofs given in §2 will transfer without further alteration. We shall
thus obtain Theorem 2.1 for a cohomology functor of the type described
in (3). Of course we shall also have to assume that the spectra X, X',
and X" satisfy 2.1 of (3) since it is only in such circumstances that the
spectral sequences can be set up.
1.
2.
3.
4.
5.
6.
7.
8.
REFERENCES
J. F. ADAMS, 'On the structure and applications of the Steenrod algebra',
Comment. Math. Helvetici 32 (1958) 180-214.
Stable homotopy theory (Springer-Verlag, 1966).
'A spectral sequence defined using if-theory', Collogue de Topologie,
Brussels (1964) 149-66.
J. M. BOAHDMAN, Stable homotopy theory (mimeographed notes, University of
Warwick, 1965).
H. CABTAN and S. EILENBEBG, Homological algebra (Princeton, 1956).
A. DOTJADY, 'La suite spectrale d'Adams', Sdminaire Henri Cartan, 1958-59,
exposes 18 et 19.
S. MACLANE, Homology (Springer-Verlag, 1963).
D. PUPPE, 'On the formal structure of stable homotopy theory', Colloquium on
Algebraic Topology, Aarhus (1962) 65-71.
Department of Pure Mathematics
University of Hull