Proc. Camb. PhU. Soc. (1966), 62, 365 3 6 5POPS 62-50

Printed in Great Britain

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 tostable homotopy-theory. In this application, we deal with A -modules, where A is themod^p Steenrod algebra. To obtain a concrete geometrical result by this methodusually involves work of two distinct sorts. To illustrate this, we consider the spectralsequence 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; theprocess is purely algebraic. However, no such effective method is given for computingthe differentials dr in the spectral sequence, or for determining the group extensionsby 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 givenfor computing E2; to find the maps dr still requires intelligence. The practical mathe-matician, however, is forced to admit that the intelligence of mathematicians is anasset at least as reliable as their willingness to do large amounts of tedious mechanicalwork. 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 greatestobstacle to the method.

In the circumstances, what we need are theorems to tell us the value of certaingroups Ext%(. I have given some results in this direction in lectures delivered at theUniversity of California, Berkeley, in July 1961 ((5)). Unfortunately, those lectures con-tained only the barest hint of proof. I t is the object of the present paper to give a propertreatment 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 anodd 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 groupsHS-'(A) are zero for t < s and known for t = s.

THEOREM 1-1. We have Hs'l(A) = 0 provided 0 < s < t < U(s), where V(s) is thefollowing numerical function:

= 12s - l , 17(40+1) = 12s+2, U(4s+2) = 12s + 4, t/(4s + 3) = 120 + 6.

366 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 inwhich 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 givenin 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 thisresult seems to hint that there may be phenomena in the stable homotopy groups ofspheres which recur with periods 8, 16, 32, etc. It would be most interesting to havegeometric information on this point.

The theorems stated above will be proved by considering Ext%!(L; Z2), where A' runsover suitable subalgebras of the Steenrod algebra, and L is a module more general thanZ%. In what follows, all our algebras and modules will be graded, and all their com-ponents will be finitely generated over Z2; then* components in sufficiently largenegative dimensions will be zero.

2. The Vanishing Theorem. With the ordering adopted in (5), the first sort of theoremto 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 is finite, letAr be the subalgebra of A generated by Sq1, Sq2,..., 8q2T; Ax will mean A. We assumethat L is a left module over Ar, that L is free qua left module over Ao> and that Lt = 0for 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.

THEOREM 2-1. (Vanishing.) Ext^(£,Z2) is zero ift < l + T(s).In (5) the proof of this theorem proceeds hand-in-hand with the proof of the Approxi-

mation Theorem ((5), p. 63, Theorem 4). In the present paper, however, the proofs willbe separated; I hope this will be found simpler. The proof proceeds in stages. First weremark that Ao can be regarded as an ^4r-module in a unique way.

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 gooddeal is known about the groups Ext^'(22, Z2) (3,10); suppose that one is willing to assumeas much of this as is needed. Then one simply expresses the module Ao as an extensionwith 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 easilycomputes '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 computingExt*£*(Z2, Z2) are less elementary than others. Since the present lemma can be provedby an elementary and explicit calculation, it might be held that this is the proper wayto 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 donethis; it is not prohibitively laborious; and the reader may duplicate the calculation ifhe 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 fort <l. Pick an .40-basis of L, and let L{v) be the sub-^0-module generated by the basiselements of grading t^v. Then L(v) is actually a sub-J.-module of L. The moduleL(v)jL(v+ 1) is an ^4,,-free module on basis elements all of the same grading, and thepresent lemma holds for it, by Lemma 2-2 and addition. This allows us to perform aninduction. Suppose as an inductive hypothesis that the present lemma is true for themodule LjL(v). (Since L = L{1), the induction starts with v = I). Form the exactsequence

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 holdsfor 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 issufficient 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. I t is free over Ao if andonly if its homology is zero. Now the result follows from the exact homology sequence.

PROPOSITION 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 = 0for t < 1. Form the first four terms of a minimal A -free resolution of L, say

j e /-i <Zi /-r dt r-\ d, i-\ d, />Li*— 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 thatImdj, Imd2, Imd3 and Imd4 are Jlo-free. Let us write M for Imd4; then Lemma 2-3(the special case r = oo, s = 4) shows that M, = 0 for t < 1+ 12.

368 J. F. ADAMS

The result is true for k — 0, by Lemma 2-3; let us suppose, as an inductive hypo-thesis, that the result is true for some value of k (where k is the integer used in definingT). 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 beproved; and we have already proved the special case r = oo (Proposition 2-5). So let usassume 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 moduleover 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 caser = oo of the Vanishing Theorem) to the module A (g)A L. Of course we have to verifythe assumptions of the Vanishing Theorem for this module. It is easy to see that(A <S)Ar L)I = 0 for t < I. I t remains only to check that A ® Ar L is free qua left moduleover 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 moduleover 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 Land filter it by dimensions, thus filtering L by .4,,-submodules. Since Ap®Ar is an exactfunctor, this filters Ap ®Ar L. I t is now sufficient to prove the result for the special caseL = Ao.

This requires us to consider the module Ap®ArA0. By using the canonical anti-automorphism of A (which preserves Ap, Ar), we may change the question and considerinstead the module A0®ArAp, considering it as a right module over Ao. The point isthat 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 thatip < 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)* «e- 0.

Here the map /i*: A* ->A*®A* is the usual coproduct map, while the map/i*: A$ -> A%®A? is given by

/t*(l) = l ® l , /t*(g1) = g1®l + l®Si-

Next 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 ip = 0 mod 2r+2~p (for each ̂ » such that p ^ r + 2).Next we have the elements

where (il5 i2,..., ie) is as before. It is not too hard to verify (using the explicit form of thecoproduct in A) that these elements are indeed annihilated by [i*®l — 1 ®/**; one canalso check (using the fact that Ap is a free left module over AT) that there are the correctnumber of elements in each dimension. We conclude that the elements given constitutea 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 A0®ArAp-^-Z2®ArAp

and a monomorphism (^40®^r^4p)* +- (Z2®ArAp)*.

The elements 1 ®E,\l£,i>...£j<*

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 foundin the theory of Hopf algebras; but for our purposes this is not essential.

We must now consider the map

370 J . F . ADAMS

defined by operating with S] on the right; we must show that Ao ® Ar Ap is acyclic under8. 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

a n d ^i®^HI!- • •&

we see that our isomorphism

(A0®ArAp)* s

expresses (A0®ArAp)* as the tensor product (in the usual sense) of two chain com-plexes, of which one (viz. A%) is acyclic. By the Kiinneth theorem, (AQ®ArAp)* isacyclic. 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 theApproximation Theorem ((5), p. 63, Theorem 4).

We use the same notation as before. We assume 0 < r < p; then we have an injectioni: Ar -*• Ap. As before, we assume that L is a (graded) left module over Ap, that L isfree qua left module over Ao, and that Lt = 0 for t < I. We also assume s > 0.

THEOREM 3-1. (Approximation.) The map

%*: Ext^(Z, Z2) <~ BxtAp(L, Z2)

is an isomorphism if 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 thefollowing exact sequence: . „ . _ T T _6 ^ 0->X-> Ap®ArL-> L-> 0.

Since every element of dimension less than 2r+1 in Ap is in the subalgebra Ar, it is easyto see that if, = Ofort < l + 2r+1. By Proposition 2 - 6 , ^ ® ^ . Lis free over ^40;also£isfree over Ao; therefore K is free over AQ, by Lemma 2-4. This will allow us to apply theVanishing Theorem to K.

We now argue as in the proof of Theorem 2-1. By a standard result on Hopf algebrasand subalgebras, Ap is free qua right module over Ar; by a standard result on change-of-rings, we have the following isomorphism:

Ext%(L,Z2) ?

A periodicity theorem in homological algebra 371

Moreover, we have the following commutative diagram:

Z,Z2) <- Ext°A'p(Ap®ArL,Z2)

By the Vanishing Theorem, the groups

Ext^(iT,.Z2) and ^ ^ , ^ )

are zero for £ < I + 2r+1 + T(s - 1).

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 shallalso prove some of their properties; see Lemmas 4-3, 4-4 and 4-5. The motivation forthis work is to be found in section 5.

We first recall from, for example, (3) that H**(A) can be defined as the cohomologyof a ring of cochains, by using the cobar construction F{A*). For example, hr+1h% isthe cohomology class of the cocycle

*> = teH£il-|£J.where the symbol £x appears 2r times. We propose to construct cochains cr in F(A*) forr ^ 2 such that „ . . . .

8cr = zr . (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 evaluatezr\j1zr by the explicit formula given in (3), p. 36, we find terms of three sorts.

(i) [gf+21^1... |£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 entriesgx in the second batch, where 0 < a < 2r — 1 and a + b = 2r+1 — 1. Each such term occurstwice, 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 inthe second batch and c in the third batch. Here 0 < a < 2r — 1 and a + b + c = 2r+1 — 2,so 6 + c £ 3. Since

24 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 ux cr + Ly. (4-2)

Then cr+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 = 2r+ 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 upto a coboundary.

However, one of the properties of cr (viz. 4-4) seems easiest to prove from a semi-explicit construction.

We will now define rnr. Let i: Ar -> A be the injection map. Since the cobar con-struction is functorial, this induces i*\ F(A*) -*• F(A*). Since the dual of i annihilatesif*1, i* annihilates zr, and i*cr is a cocycle in ^(^4*). We define mr to be the cohomologyclass of i*cr in H2r-3-*r(Ar).

LEMMA 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 exterioralgebra, 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 theinjection B -> A± induces an isomorphism

It is now sufficient to check that the image of tn2 in H*- 12(B) is (h2 0)4. Next recall thatin (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 trans-gression T is denned on (h2< 0)4 and takes the value h3 A

4; this will give exactly what iswanted. 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.

LEMMA 4-4. j*wr+1 = (wr)2for r 3s 2.

Proof. This follows immediately from (4-2), since zr and all the cochains y map tozero in F{Af).

In order to introduce a 'periodicity map' in H**(A), we shall need to considerMassey products. Let L be a left A -module. In order to avoid discussing the dependenceof our constructions on the choice of resolution, we use a standard resolution, namelythat 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 standardcochain complex, namely that given by the cobar construction; this is defined in termsof symbols [«i|a2|... |as| A]

with a^A*, AeL*. This cochain complex is a left module over F(A*). I t is now clearhow to define Massey products. In particular, let

be the subgroup of elements e such that hffe = 0. For any such e, take a representativecocycle 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 (Corollary5-6) we shall know that nre coincides exactly with this Massey product.

LEMMA 4-5. The following diagram is commutative:

0 )

i

, Z2)i*\ \i*

Ext%(L,Z2) ^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 periodicitytheorems. The order of proof is as follows. We begin with the result for modules overA± (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 ofG72€#4'12(^42)in #4>12(^i) (see Lemma 4-3).

LEMMA 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.

Proof of Theorem 5-1. Let 0^*L->M-+N->0hea,n. extension of ^-modules allfree over Ao; then we have the following commutative diagram.

^ l(N,Z2) -> ExtsA[(M,Z2) -» Vxt%[(L,Z2) -^ Ext^^N, Z2)

W2 7572 \m2 \VJ2

(The squares involving 8 are commutative because 8 is right multiplication by theclass of the extension 0^-L-^-M-^N->-0m Ext^°(iV, L).) By the Five Lemma, ifTheorem 5-1 is true for L and N, it is true for M. By induction, Theorem 5-1 istrue for any finite extension of modules isomorphic to Ao. Hence (arguing as forLemma 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 isfree over Ao, and that L, = 0 for t < I.

THEOREM 5-3. The homorphism

xur: Ex 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 elementmTeExt22'3-2r(Z2,Z2) (see section 4).

Proof. For s = 0 both Ext groups are zero (using Theorem 2-1), so the result istrivially 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 resultis true in this case. We may now proceed by induction over t — I; as an inductive hypo-thesis, 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 Za)

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 thefifth 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, andthat Lt = 0 for t < I.

THEOREM 5-4. For each r (in the range 2 < r < oo) the map TTT of section 4 gives anisomorphism

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.

? L, Zt)

i*\ \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 J. F. ADAMS

COROLLARY 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 , . ^ ) = 0for 0 < s, £ < U(s). By applying Theorem 2-1 we would only obtain the result fort < 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 largervalues of s, of the form s = 4Jc, it follows by periodicity (Theorem 5-4 with r = 2).

COROLLARY 5-6. We have Hs<l{A) = Ofor s = 4k,t= 12& (k > 0).

Proof. For 4 = 1 we may assume the result known by direct calculation. For largervalues 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.

COROLLARY 5-8. The isomorphism nr of Corollary 5g5 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 ofExt^°(Z2, 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 periodicityisomorphisms 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 377

REFERENCES

(1) ADAMS, J. F. On the structure and applications of the Steenrod algebra. Comment. Math.Helv. 32 (1958), 180-214.

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