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Proc. Camb. PhU. Soc. (1966), 62, 365
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365
A periodicity theorem in homological algebra
B Y J. F. ADAMS
Department of Mathematics, University of Manchester
(Received 29 September 1965)
1. Introduction. In (1-3,6) it is shown that homological algebra can be applied to
stable homotopy-theory. In this application, we deal with A -modules, where A is the
mod^p Steenrod algebra. To obtain a concrete geometrical result by this method
usually involves work of two distinct sorts. To illustrate this, we consider the spectral
sequence of (1,2):
Ext5 ( (#*(F; Zp), H*(X; Zp)) => p7i%(X, Y).
Here each group Exts>( which occurs in the E2 term can be effectively computed; the
process is purely algebraic. However, no such effective method is given for computing
the differentials dr in the spectral sequence, or for determining the group extensions
by which PTT%(X, Y) is built up from the Ex term; these are topological problems.
A mathematical logician might be satisfied with this account: an algorithm is given
for computing E2; to find the maps dr still requires intelligence. The practical mathematician, however, is forced to admit that the intelligence of mathematicians is an
asset at least as reliable as their willingness to do large amounts of tedious mechanical
work. In fact, when a chance has arisen to show that such a differential dr is non-zero,
it has been regarded as an interesting problem, and duly solved; see (3,8,12). However,
the difficulty of actually computing groups Extf}'(£, M) has remained the greatest
obstacle to the method.
In the circumstances, what we need are theorems to tell us the value of certain
groups Ext%(. I have given some results in this direction in lectures delivered at the
University of California, Berkeley, in July 1961 ((5)). Unfortunately, those lectures contained only the barest hint of proof. It is the object of the present paper to give a proper
treatment of these results; I must apologize to my readers for this long delay.
This paper (like the lectures mentioned) deals only with the case^j = 2. When p is an
odd prime, the analogous questions have been investigated by Liulevicius (see (9),
especially the foot of p. 975).
To indicate the nature of the results, I will show how they apply to the special case
which is relevant in computing the stable homotopy groups of spheres. The groups
HS-'(A) are zero for t < s and known for t = s.
sl
THEOREM 1-1. We have H ' (A) = 0 provided 0 < s < t < U(s), where V(s) is the
following numerical function:
= 12s-l,
17(40+1) = 12s+2,
U(4s+2) = 12s + 4,
t/(4s + 3) = 120 + 6.
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J. F. ADAMS
This result is best possible, in the sense that the function U(s) cannot be increased.
It supersedes the corresponding result in my earlier note ((4)).
THEOREM 1*2. For each r ^ 2 there is a suitable neighbourhood of the line t = 3s in
which we have a 'periodicity' isomorphism
defined by
nr(x) = (hr+1, h%, x).
The precise inequalities on s and t for which the isomorphism is proved will be given
in section 5; see Corollaries 5-5, 5-8. The symbol (z,y,x) means the Massey product,
and the element h^H1' ^(A) is as in (3).
The 'periodicity' isomorphism nr increases the total degree t — s by 2r+1. So this
result seems to hint that there may be phenomena in the stable homotopy groups of
spheres which recur with periods 8, 16, 32, etc. It would be most interesting to have
geometric information on this point.
The theorems stated above will be proved by considering Ext%!(L; Z2), where A' runs
over suitable subalgebras of the Steenrod algebra, and L is a module more general than
Z%. In what follows, all our algebras and modules will be graded, and all their components will be finitely generated over Z2; then* components in sufficiently large
negative dimensions will be zero.
2. The Vanishing Theorem. With the ordering adopted in (5), thefirstsort of theorem
to be discussed is the Vanishing Theorem ((5), p. 62, Theorem 3).
We shall need some notation. Let A be the mod 2 Steenrod algebra; if r isfinite,let
Ar be the subalgebra of A generated by Sq1, Sq2,..., 8q2T; Ax will mean A. We assume
that L is a left module over Ar, that L is free qua left module over Ao> and that Lt = 0
for t < I. We define a numerical function by
T{4Jc)=\2k,
T(4k+l) = 12k+2, T(4:k+2) = 12&+4, T(4Jc + 3) = 124+7,
where k runs over the integers.
2-1. (Vanishing.) Ext^(£,Z 2 ) is zero ift < l + T(s).
In (5) the proof of this theorem proceeds hand-in-hand with the proof of the Approximation Theorem ((5), p. 63, Theorem 4). In the present paper, however, the proofs will
be separated; I hope this will be found simpler. The proof proceeds in stages. First we
remark that Ao can be regarded as an ^4r-module in a unique way.
THEOREM
LEMMA
2-2. The Vanishing Theorem is true in the special case r = oo, L = Ao, s ^ 4.
Proof. This lemma is essentially computational. At least two proofs may be given,
depending on how much one is willing to assume known. As a matter of fact, a good
deal is known about the groups Ext^'(22, Z2) (3,10); suppose that one is willing to assume
as much of this as is needed. Then one simply expresses the module Ao as an extension
with submodule and quotient module both isomorphic to Z2 (but differently graded);
this extension gives rise to an exact sequence of Ext groups, from which one easily
computes 'Extsjt(A0,Z2) in low dimensions, say for t — s ^ 8.
A periodicity
theorem in hornological algebra
367
On the other hand, some of the methods which have been used in computing
Ext*£*(Z2, Z2) are less elementary than others. Since the present lemma can be proved
by an elementary and explicit calculation, it might be held that this is the proper way
to do it. To do things this way one has to give an explicit A-free resolution of Ao
(preferably a minimal one); at least one must do this in low dimensions. I have done
this; it is not prohibitively laborious; and the reader may duplicate the calculation if
he wishes. However, it is hardly worth publishing.
LEMMA
2-3. The Vanishing Theorem is true in the special case r = oo, s < 4.
Proof. Suppose that L is an A -module which is free over Ao and such that L, = 0 for
t <l. Pick an .40-basis of L, and let L{v) be the sub-^0-module generated by the basis
elements of grading t^v. Then L(v) is actually a sub-J.-module of L. The module
L(v)jL(v+ 1) is an ^4,,-free module on basis elements all of the same grading, and the
present lemma holds for it, by Lemma 2-2 and addition. This allows us to perform an
induction. Suppose as an inductive hypothesis that the present lemma is true for the
module LjL(v). (Since L = L{1), the induction starts with v = I). Form the exact
sequence
L(v)IL{v+l)^-LIL(v+l)->LIL(v).
This yields an exact sequence
),Za) <- Ext*Al(LIL(v+l),Z2) <
Hence the middle groups are zero for t < I + T(s) (s < 4) and the present lemma holds
for L/L(v + 1). This completes the induction.
On the other hand, we have
by taking v sufficiently large compared with t; so what we have proved above is
sufficient to prove the lemma.
LEMMA 2-4. Let
be an exact sequence of A0-modules. If two of them are A0-free, then so is the third.
Proof. A module over Ao is the same thing as a chain complex. It is free over Ao if and
only if its homology is zero. Now the result follows from the exact homology sequence.
2-5. The Vanishing Theorem is true in the special case r = oo.
Proof. Let L be as in the data, so that L is an A -module, L is free over Ao and Lt = 0
for t < 1. Form the first four terms of a minimal A -free resolution of L, say
PROPOSITION
j e
Li*—
/-i <Zi /-r dt r-\ d, i-\ d, />
O 0 <— V1 *— O 2 <— O 3 •>— O 4 .
Since A is ^40-free, and hence each Ci is -40-free, Lemma 4 shows successively that
Imdj, Imd 2 , Imd 3 and Imd 4 are Jlo-free. Let us write M for Imd 4 ; then Lemma 2-3
(the special case r = oo, s = 4) shows that M, = 0 for t < 1+ 12.
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J. F. ADAMS
The result is true for k — 0, by Lemma 2-3; let us suppose, as an inductive hypothesis, that the result is true for some value of k (where k is the integer used in defining
T). Then we may apply the inductive hypothesis to M, and we find that the group
'ExtA+i>\L,Z2) ~
is zero if
s = 4k,
t > I + 12 + 12k,
t> l+12+l2k
+ 2,
, t> Z+12 + 12& + 4,
or
s = 4k + 3, t > Z+12+12&+7.
That is, the result is true for L with k replaced by k + 1. This completes the induction,
and proves the result.
Proof of Theorem 2-1, the Vanishing Theorem. For r = 0 there is nothing to be
proved; and we have already proved the special case r = oo (Proposition 2-5). So let us
assume that 0 < r < oo, that L is a (graded) left module over AT, that L is free over Ao,
and that Lt = 0 for t < I.
By a standard result on Hopf algebras and subalgebras, A is free qua right module
over Ar. By a standard result on change-of-rings, which is in Cartan-Eilenberg ((7),
p. 118) for the ungraded case, we have the following isomorphism:
Ext%(L, Za) s ExtA'(A ®Ar L, Z2).
This allows us to prove that Ext^(L, Z2) is zero by applying Proposition 2-5 (the case
r = oo of the Vanishing Theorem) to the module A (g)A L. Of course we have to verify
the assumptions of the Vanishing Theorem for this module. It is easy to see that
(A <S)Ar L)I = 0 for t < I. It remains only to check that A ®Ar L is free qua left module
over Ao. Actually we will prove something slightly more general, for use in sections 3,5.
We assume 0 < r < p; then Ar is a subalgebra of Ap. We assume L is a left module
over AT.
T.
PROPOSITION 2-6. If L is free qua left module over Ao, then so is
Ap%AfL.
The special case which we need to prove Theorem 2-1 is the case p = oo.
Proof of Proposition 2-6. As in the proof of Lemma 2-3, we can choose an ^40-base of L
and filter it by dimensions, thus filtering L by .4,,-submodules. Since Ap®Ar is an exact
functor, this filters Ap ®Ar L. It is now sufficient to prove the result for the special case
L = Ao.
This requires us to consider the module Ap®ArA0. By using the canonical antiautomorphism of A (which preserves Ap, Ar), we may change the question and consider
instead the module A0®ArAp, considering it as a right module over Ao. The point is
that it is easier to write down the Z2-dual of this module ((11)). We may identify A*
with the quotient of A* which has as a base the monomials ^I'^l'---^ such that
ip < 2P+2~P for each p, the remaining monomials being zero. Similarly for A*.
The module A0®ArAp is defined so that the following sequence is exact:
A periodicity theorem in homological algebra
369
Here the map /*: Ar® Ap -»• Ap is the usual product map, but the map fi: A0®AT -> Ao
is the map which makes Ao an .4r-module; that is, the ^-dimensional part of Ar
annihilates Ao for t ^ 2.
The Z2-dual of the exact sequence displayed above is the following exact sequence.
A*®A?®A*
r;®1-1®*' A*®A* <- (A0®ArAp)*
Here the map /i*: A* ->A*®A*
/i*: A$ -> A%®A? is given by
«e- 0.
is the usual coproduct map, while the map
/t*(l) = l ® l , /t*(g1) = g 1 ®l + l®SiNext we have to describe the kernel of /i* ® 1 — 1 ®fi*. First we have the elements
such that ip < 2P+2~P (for each p) and i p = 0 mod 2r+2~p (for each ^» such that p ^ r + 2).
Next we have the elements
where (il5 i 2 ,..., ie) is as before. It is not too hard to verify (using the explicit form of the
coproduct in A) that these elements are indeed annihilated by [i*®l — 1 ®/**; one can
also check (using the fact that Ap is a free left module over AT) that there are the correct
number of elements in each dimension. We conclude that the elements given constitute
a base of (A0®Ar Ap)*.
The base elements we have given are of two kinds, and the subset of elements
has an obvious interpretation. We have an epimorphism of ylr-modules
Ao -> Z2,
whence an epimorphism
and a monomorphism
A0®ArAp-^-Z2®ArAp
(^40®^r^4p)* +-
(Z2®ArAp)*.
1 ®E,\l£,i>...£j<*
The elements
represent a base of the submodule (Z2®ArAp)*.
The fact that our basis elements occur in pairs
and
£i®£ I &-££+ I®fi1+1&...#
now means that we have found an isomorphism
(A0®ArAp)*
£
A$®(Z2®ArAp)*.
It might be interesting to know if some reason for this isomorphism could be found
in the theory of Hopf algebras; but for our purposes this is not essential.
We must now consider the map
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J. F. ADAMS
defined by operating with S] on the right; we must show that Ao ®Ar Ap is acyclic under
8. We have the following commutative diagram (in which dx = xS*).
A0®Ap
>
A0®ArAp
A0®Ap
>
A0®ArAp
On dualizing, therefore, we get the following commutative diagram.
At®A*p<
(A0®ArAp)*
A
A
<
(A0®ArAp)*
If we examine the effect of l®d* on our elements
and
^i®^HI ! - • •&
we see that our isomorphism
(A0®ArAp)* s
expresses (A0®ArAp)* as the tensor product (in the usual sense) of two chain complexes, of which one (viz. A%) is acyclic. By the Kiinneth theorem, (AQ®ArAp)* is
acyclic. This proves Proposition 2-6, which completes the proof of Theorem 2-1.
3. The Approximation Theorem. The second sort of theorem to be discussed is the
Approximation Theorem ((5), p. 63, Theorem 4).
We use the same notation as before. We assume 0 < r < p; then we have an injection
i: Ar -*• Ap. As before, we assume that L is a (graded) left module over Ap, that L is
free qua left module over Ao, and that Lt = 0 for t < I. We also assume s > 0.
THEOREM 3-1. (Approximation.) The map
is an isomorphism if
%*: Ext^(Z, Z2) <~ BxtAp(L, Z2)
t<l + 2r+x + T(s - 1)
The theorem remains true for s = 0, provided we interpret T{ — 1) as 0.
Proof. Let K be the kernel of the obvious map Ap®ArL -> L, so that we have the
following6 exact sequence: .
„
. _ T
_
T
^
0->X-> Ap®ArL->
L-> 0.
Since every element of dimension less than 2 r+1 in Ap is in the subalgebra Ar, it is easy
to see that if, = Ofort < l + 2r+1. By Proposition 2 - 6 , ^ ® ^ . Lis free over ^4 0 ;also£is
free over Ao; therefore K is free over AQ, by Lemma 2-4. This will allow us to apply the
Vanishing Theorem to K.
We now argue as in the proof of Theorem 2-1. By a standard result on Hopf algebras
and subalgebras, Ap is free qua right module over Ar; by a standard result on changeof-rings, we have the following isomorphism:
Ext%(L,Z2)
?
A periodicity theorem in homological algebra
371
Moreover, we have the following commutative diagram:
Z,Z 2 ) <-
Ext°A'p(Ap®ArL,Z2)
By the Vanishing Theorem, the groups
Ext^(iT,.Z 2 )
and
^ ^ , ^ )
r 1
£ < I + 2 + + T(s - 1).
are zero for
Therefore the map i* is an isomorphism for these values of t. This completes the proof.
4. Construction of periodicity elements. In this section we shall construct certain
elements
wr€H*.*-*(At)
(r>2),
which are needed for the statement and proof of our periodicity theorems. We shall
also prove some of their properties; see Lemmas 4-3, 4-4 and 4-5. The motivation for
this work is to be found in section 5.
We first recall from, for example, (3) that H**(A) can be defined as the cohomology
of a ring of cochains, by using the cobar construction F{A*). For example, hr+1h% is
the cohomology class of the cocycle
*> = teH£il-|£J.
r
where the symbol £x appears 2 times. We propose to construct cochains cr in F(A*) for
r ^ 2 such that
„
....
8cr = z r .
(4-1)
For r = 2 we may assume it known by direct calculation that
Therefore it is possible to choose c2, and c2 is unique up to a coboundary.
Let us now suppose, as an inductive hypothesis, that we have chosen cr in such a way
that it is defined up to a coboundary. Let the \JX product in F{A*) be as in (3), p. 36.
Then we have
*,
.
Moreover, the cochain cr\jcr + zrKJxcr is defined up to a coboundary. If we evaluate
zr\j1zr by the explicit formula given in (3), p. 36, we find terms of three sorts.
(i) [gf +2 1^1... |£x] with 2r+x entries gx. This is the cocycle zr+1.
(ii) [£f+1 |^i| • • • |gi| £f+1+1 |£i| • • • | I J with a entries £,x in the first batch and b entries
gx in the second batch, where 0 < a < 2r — 1 and a + b = 2 r+1 — 1. Each such term occurs
twice, and these terms cancel.
(iii) Ef+1 &I... | ^ | gf" |£i| • • • I£il H ISil • • • |£J ^ ^ a e n t r i e s £ i m t h e fi™t batch, b in
the second batch and c in the third batch. Here 0 < a < 2r — 1 and a + b + c = 2 r+1 — 2,
so 6 + c £ 3. Since
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Camb. Philos. 62, 3
372
J. F. ADAMS
each such term is the boundary of a cochain y, where y is either
If both choices for y are possible, then they differ by a coboundary.
We now set
_,
,AC..
<V+i = <V u cr + zr u x cr + Ly.
(4-2)
Then c r+1 is defined up to a coboundary, and dcr+1 = zr+v This completes the induction.
Remark. What we propose to prove includes the following two facts.
(a) Hs>l(A) = 0 for s = 2 r + 1, t = 3.2r (Theorem 1-1). This shows that it is possible
to choose cr satisfying (4-1).
(b) H»>l(A) = 0 for s = 2r, t = 3. 2r (Corollary 5-6). This shows that (4-1) defines cr up
to a coboundary.
However, one of the properties of cr (viz. 4-4) seems easiest to prove from a semiexplicit construction.
We will now define rnr. Let i: Ar -> A be the injection map. Since the cobar construction is functorial, this induces i*\ F(A*) -*• F(A*). Since the dual of i annihilates
if*1, i* annihilates zr, and i*cr is a cocycle in ^(^4*). We define mr to be the cohomology
class of i*cr in H2r-3-*r(Ar).
4-3. Hi-12(A1) s Z2, generated by the image ofvr2.
Proof. Since Ax is so small, we may easily make an explicit resolution and check that
£?4'12(^L1) £ Z2. Let B be the subalgebra of A generated by Sq0-1; thus B is an exterior
algebra, and H**(B) is a polynomial algebra on one generator which lies in H}'Z(B).
In (3) this generator is called h2 0. By direct calculation again, we check that the
injection B -> A± induces an isomorphism
LEMMA
I t is now sufficient to check that the image of tn2 in H*- 12(B) is (h2 0 ) 4 . Next recall that
in (3) one obtains information about H**(A) by using a family of spectral sequences
(see especially (3), p. 45); here we shall require the first spectral sequence of this family,
namely that with n = 2. We propose to check that in this spectral sequence, the transgression T is denned on (h2< 0)4 and takes the value h3 A4; this will give exactly what is
wanted. We calculate as follows:
By ((3), p. 45, Lemma 2-5-2 (i)) we have
Hence
r(h2i 0 ) 2 =
(by the Cartan formula)
A periodicity theorem in homological algebra
373
(This formula is also given by (3), p. 45, Lemma 2-5-2 (i).) Hence
(by the Cartan formula)
(since hsh\ is a boundary under d2). This completes the proof.
Let us now consider the injection j : Ar ->• Ar+1.
2
LEMMA 4-4. j*wr+1 = (wr) for r 3s 2.
Proof. This follows immediately from (4-2), since zr and all the cochains y map to
zero in F{Af).
In order to introduce a 'periodicity map' in H**(A), we shall need to consider
Massey products. Let L be a left A -module. In order to avoid discussing the dependence
of our constructions on the choice of resolution, we use a standard resolution, namely
that given by the bar construction; this is defined in terms of symbols
ao[a1\a2\...\as\l],
with ai e A, I e L. This allows us to obtain ExtJ* (L, Z2) as the cohomology of a standard
cochain complex, namely that given by the cobar construction; this is defined in terms
of symbols
[«i|a 2 |...|a s | A]
with a^A*, AeL*. This cochain complex is a left module over F(A*). It is now clear
how to define Massey products. In particular, let
be the subgroup of elements e such that hffe = 0. For any such e, take a representative
cocycle x; then we have
rr I l£ ix = 8y
let nr e be the class of the cocycle
We have defined a homomorphism
and nr e is a representative for the Massey product (hr+1,~h%,e>.
Remark. As soon as we have proved that Hs>'(A) = 0 for s = 2r, t = 3. 2r (Corollary
5-6) we shall know that nre coincides exactly with this Massey product.
LEMMA 4-5. The following diagram is commutative:
0
)
i*\
i
Ext%(L,Z2)
, Z2)
\i*
^
24-2
374
J. F. ADAMS
Note. The arrow marked 'wr' is defined by multiplication on the left with wr.
Proof. i*([gr+1]y + crx) = 0 + (i*cr)(i*x).
5. The periodicity theorems. In this section we will state and prove the periodicity
theorems. The order of proof is as follows. We begin with the result for modules over
A± (Theorem 5-1); we deduce the result for modules over Ar (2 ^ r < oo) (Theorem 5-3);
from this we deduce the result for modules over A (Theorem 5-4 plus Corollary 5-7);
finally we deduce the result for H**(A).
Theorem 5-1. Let Lbea left A^module which is free over Ao. Then the homomorphism
w2: Ext°Al(L, Z2) -+ Ext£ 4 '* +12 (A Z2)
is an epimorphism for s = 0 and an isomorphism for s > 0.
Note. The homomorphism xn2 is defined by left multiplication with the image of
G7 2 €# 4 ' 12 (^4 2 ) in # 4 > 1 2 (^i) (see Lemma 4-3).
5-2. Theorem 5-1 is true in the special case L = Ao.
The proof is by direct calculation, which is fairly light since the algebra Ar is so small.
Note that the minimal resolution of Ao over A± is periodic with period 4.
LEMMA
Proof of Theorem 5-1. Let 0^*L->M-+N->0hea,n.
extension of ^-modules all
free over Ao; then we have the following commutative diagram.
^
l(N,Z2) -> ExtsA[(M,Z2)
W2
7572
-» Vxt%[(L,Z2) - ^ Ext^^N,
\m2
Z 2)
\VJ2
(The squares involving 8 are commutative because 8 is right multiplication by the
class of the extension 0^-L-^-M-^N->-0m
Ext^°(iV, L).) By the Five Lemma, if
Theorem 5-1 is true for L and N, it is true for M. By induction, Theorem 5-1 is
true for any finite extension of modules isomorphic to Ao. Hence (arguing as for
Lemma 2-3) it is true for all L.
For the next result, we assume that L is a left module over Ar (2 < r < oo), that L is
free over Ao, and that L, = 0 for t < I.
THEOREM
5-3. The homorphism
xur: E x t ^ ( A Z2) -> Ext£ 2r ''+ 3 - 2r (A Z2)
is an isomorphism for s ~& 0, t < 1 + 4s.
Note. The homomorphism zur is defined by left multiplication with the element
mTeExt22'3-2r(Z2,Z2) (see section 4).
Proof. For s = 0 both Ext groups are zero (using Theorem 2-1), so the result is
trivially true. We may therefore restrict attention to the case s > 0.
For s > 0,t — I < 0 both Ext groups are zero (using Theorem 2-1 again), so the result
is true in this case. We may now proceed by induction over t — I; as an inductive hypothesis, we assume the result known for smaller values oft —I.
A periodicity theorem in homological algebra
375
Let K be the kernel of the obvious map
Ar®AiL-+L,
so that we have an exact sequence
0 -+ K -+ Ar®AJj -+L^0.
The module Ar®A L is ^40-free by Proposition 2-6. Hence K is -40-free, by Lemma 2-4;
also Kt = 0 for t < 1 + 4. Now we can consider the following commutative diagram.
2r
(A Z a )
Since we are assuming s > 0, we have s — 1 ^ 0. The inductive hypothesis applies to K,
and shows that the second vertical arrow is an isomorphism for t < I + 4s, while the
fifth vertical arrow is an isomorphism for t < I + 4s + 4. As for the fourth vertical arrow,
ExtA'r(Ar®AiL,
Za) s ExtSj(L, Za),
These isomorphisms carry the homomorphism wr into (7zr2)2r~a (using Lemma 4-4); and
this is an isomorphism for s > 0, by Theorem 5-1. Similarly for the first vertical arrow,
which is an epimorphism. The required conclusion now follows by the Five Lemma.
This completes the induction, and proves the theorem.
For the next result we assume that L is a left .4-module, that L is free over Ao, and
that Lt = 0 for t < I.
THEOREM 5-4. For each r (in the range 2 < r < oo) the map TTT of section 4 gives an
isomorphism
valid for s > 0, t < I + min (4s, 2r+1 + T(s - 1)).
Proof. Since t < l + 2r+1 + T(s- 1), it follows (using Theorem 2-1) that
Lemma 4*5 now provides the following commutative diagram.
?
i*\
L, Zt)
\i
The maps i* are isomorphisms for t < l + 2r+1 + T(s— 1), by Theorem 3-1; the map wT
is iso for t < Z + 4s, by Theorem 5-3. This proves the theorem.
376
COROLLARY
J. F. ADAMS
5-5. For each r (in the range 2 ^ r < oo) there is an isomorphism
nr:
valid for 1 < s < t < min(
Proof. Let I(A) = T, At and let L = I(A)/A Sq1, so that L is free over ^40 and I = 2.
Then we have an isomorphism
valid for 0 < s < t. So Corollary 5-5 follows immediately from Theorem 5-4.
Proof of Theorem 1-1. L e t £ = I(A)jASq1; we have to prove that Ext^f ^ ' ( i , . ^ ) = 0
for 0 < s, £ < U(s). By applying Theorem 2-1 we would only obtain the result for
t < V(s), where
V(4Je) = 12&-3,
F(4jfc+1) = 12&+2,
V(ik + 2) = 12& + 4,
F(4i+3) = 124 + 6.
However, we may assume the result known for s = 4 by direct calculation; for larger
values of s, of the form s = 4Jc, it follows by periodicity (Theorem 5-4 with r = 2).
s l
COROLLARY 5-6. We have H < {A) = Ofor s = 4k,t= 12& (k > 0).
Proof. For 4 = 1 we may assume the result known by direct calculation. For larger
values of k it follows by periodicity (Corollary 5-5 with r = 2).
COROLLARY 5-7. The map nr of section 4 is defined by
7Trx =
(hr+l!hf,x).
Proof. This follows immediately from Corollary 5-6, as remarked in section 4.
g
COROLLARY 5-8. The isomorphism nr of Corollary 5 5 is defined by
TTrx =
{hr+vhf,x).
Proof. Let L = I (A)/A Sq1. Then the isomorphism
(for 0 < s < t) is defined by multiplication on the right with a fixed element of
Ext^°(Z 2 , L), namely the class of the extension
0 -> I (A) IA Sq1 -+ A/ASq1 ->Z2^0.
Such multiplication commutes with Massey products on the left.
Theorem 1-2 follows by combining Corollaries 5-5 and 5-8.
Remark. I t follows from our constructions, using Lemma 4-4, that our periodicity
. .„
isomorphisms satisfy
in the region where irr is valid. This may also be checked by the following manipulation.
<hr+1,hf,
(hr+1,hf,xyy
A periodicity theorem in homological algebra
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REFERENCES
ADAMS, J. F. On the structure and applications of the Steenrod algebra. Comment. Math.
Helv. 32 (1958), 180-214.
ADAMS, J. F. Thtkme de l'homotopie stable. Bull. Soc. Math. France, 87 (1959), 277-280.
ADAMS, J. F. On the non-existence of elements of Hopf invariant one. Ann. of Math. 72
(1960), 20-104.
ADAMS, J. F. A finiteness theorem in homological algebra. Proc. Cambridge Philos. Soc. 57
(1961), 31-36.
ADAMS, J. F. Stable homotopy theory (Springer-Verlag, 1964).
(6) CABTAK SEMINAR NOTES, 1958/59.
(7) CAETAN H. and EIUENBERG, S. Homological
algebra (Princeton, 1956).
(8) LruiiEvicnjs, A. The factorisation of cyclic reduced powers by secondary cohomology
operations. Mem. Amer. Math. Soc. 42 (1962).
(9) LinLEVicius, A. Zeroes of the cohomology of the Steenrod algebra. Proc. Amer. Math. Soc.
14 (1963), 972-976.
(10) MAY, J. P. Thesis (Princeton, 1964).
(11) MTLNOB, J. The Steenrod algebra and its dual. Ann. of Math. 67 (1958), 150-171.
(12) SHIMADA, N. and YAMANOSHTTA, T. On triviality of the modp Hopf invariant. Japanese
Journal of Math. 31 (1961), 1-25.