ON TRIVIALITY OF DICKSON INVARIANTS
IN THE HOMOLOGY OF THE STEENROD ALGEBRA
NGUY^E~N H. V. HU_.NG
Dedicated to Professor Nguy^e~n Duy Ti^e'n on the occasion of his sixtieth bir*
*thday
Abstract.Let A be the mod 2 Steenrod algebra and Dk the Dickson algebra
of k variables. We study the Lannes-Zarati homomorphisms
'k : Extk,k+iA(F2, F2) ! (F2 Dk)*i,
A
which correspond to an associated graded of the Hurewicz map H : is*(S0)*
* ~=
i*(Q0S0) ! H*(Q0S0) . An algebraic version of the long-standing conjectu*
*re
on spherical classes predicts that 'k = 0 in positive stems, for k > 2. *
*That the
conjecture is no longer valid for k = 1 and 2 is respectively an exposit*
*ion of
the existence of Hopf invariant one classes and Kervaire invariant one c*
*lasses.
This conjecture has been proved for k = 3 in [9]. It has been shown th*
*at
'k vanishes on decomposable elements for k > 2 in [14] and on the image
of Singer's algebraic transfer for k > 2 in [9] and [12]. In this paper*
*, we
establish the conjecture for k = 4. To this end, our main tools include *
*(1) an
explicit chain-level representation of 'k and (2) a squaring operation S*
*q0 on
(F2 Dk)*, which commutes with the classical Sq0 on ExtkA(F2, F2) through
theALannes-Zarati homomorphism.
1.Introduction and statement of results
Let H : ßs*(S0) ~=ß*(Q0S0) ! H*(Q0S0) be the Hurewicz homomorphism of
the basepoint component Q0S0 in the infinite loop space QS0 = limn nSn. Here
and throughout the paper, homology and cohomology are taken with coefficients
in F2, the field of two elements. The long-standing conjecture on spherical cla*
*sses
states as follows: Only the classes of Hopf invariant one and those of Kervaire
invariant one are detected by the Hurewicz homomorphism. (See Curtis [6], Snaith
and Tornehave [26] and Wellington [27] for a discussion.)
An algebraic version of this problem, which we are interested in, goes as fol*
*lows.
Let Pk = F2[x1, . .,.xk] be the polynomial algebra on k generators x1, . .,.xk,*
* each
of degree 1. Let the general linear group GLk = GL(k, F2) and the mod 2 Steenrod
algebra A both act on Pk in the usual way. The Dickson algebra of k variables,
Dk, is the algebra of invariants
Dk := F2[x1, . .,.xk]GLk .
____________
1The research was supported in part by Johns Hopkins University and the Nati*
*onal Research
Program, Grant N0140801.
22000 Mathematics Subject Classification. Primary 55P47, 55Q45, 55S10, 55T15.
3Key words and phrases. Spherical classes, Loop spaces, Adams spectral seque*
*nces, Steenrod
algebra, lambda algebra, Invariant theory, Dickson algebra.
1
2 NGUY^E~N H. V. HU_.NG
Since the action of A and that of GLk on Pk commute with each other, Dk is an
algebra over A. In [17], Lannes and Zarati construct homomorphisms
'k : Extk,k+iA(F2, F2) ! (F2 Dk)*i,
A
which correspond to an associated graded of the Hurewicz map. The proof of
this assertion is unpublished, but it is sketched by Lannes [16] and by Goerss *
*[8].
The Hopf invariant one and the Kervaire invariant one classes are respectively
represented by certain permanent cycles in Ext1,*A(F2, F2) and Ext2,*A(F2, F2),*
* on
which '1 and '2 are non-zero (see Adams [1], Browder [5], Lannes-Zarati [17]).
Therefore, we are led to the following conjecture.
Conjecture 1.1. 'k = 0 in any positive stem i for k > 2.
The conjecture has been proved for k = 3 in [9] and for k = 4 in a range of s*
*tems
in [14]. It has been shown that 'k vanishes on decomposable elements for k > 2
in [14] and on the image of Singer's algebraic transfer T rk : ((F2 Pk)GLk)* !
A
ExtkA(F2, F2) for k > 2 in [9] and [12].
The following is the main result of the present paper.
Theorem 1.2. '4 = 0 in positive stems.
An ingredient in our proof of this theorem is the squaring operation Sq0 on
(F2 Dk)*, which is defined in our paper [9]. The key step in the proof is to s*
*how
A
the following theorem.
Theorem 1.3. The squaring operation Sq0 on (F2 Dk)* commutes with the clas-
A
sical squaring operation Sq0 on ExtkA(F2, F2) through the Lannes-Zarati homomor-
phism 'k, for any k.
Applying this theorem, we get a proof of Theorem 1.2 by combining the com-
putation of Ext4A(F2, F2) by W. H. Lin [18] and that of F2 D4 by the author and
A
Peterson [13].
In order to prove Theorem 1.3, we need to exploit Singer's invariant-theoretic
description of the lambda algebra [24]. According to Dickson [7], one has
Dk ~=F2[Qk,k-1, ..., Qk,0],
where Qk,idenotes the Dickson invariant of degree 2k - 2i. Singer sets k =
Dk[Q-1k,0], the localization of Dk given by inverting Qk,0, and defines ^kto be
a certain ön t too large" submodule of k. He also equips ^ = k ^kwith a
differential @ : ^k! ^k-1and a coproduct. Then, he shows that the differential
coalgebra ^ is dual to the (opposite) lambda algebra of the six authors of [4].
Thus, Hk( ^) ~=T orAk(F2, F2). (Originally, Singer uses the notation +kto deno*
*te
^k. However, by D+k, A+ we always mean the submodules of Dk and A respectively
consisting of all elements of positive degrees, so Singer's notation +kwould m*
*ake
a confusion in this paper. Therefore, we prefer the notation ^k.)
The following result plays a key role in our proof of Theorem 1.3.
Theorem 1.4. ([11]) The inclusion Dk ^kis a chain-level representation of the
Lannes-Zarati dual homomorphism
'*k: (F2 Dk)i! T orAk,k+i(F2, F2).
A
TRIVIALITY OF DICKSON INVARIANTS 3
By this theorem, Conjecture 1.1 is equivalent to our conjecture on the trivia*
*lity
of Dickson invariants in the homology of the Steenrod algebra:
Conjecture 1.5. ([10]) Let D+kdenote the submodule of all positive degree ele-
ments in Dk. If q 2 D+k, then [q] = 0 in Hk( ^) ~=T orAk(F2, F2) for k > 2.
Therefore, Theorem 1.2 can be restated as follows.
Theorem 1.6. Every positive-degree Dickson invariant of four variables represen*
*ts
the 0 class in the homology, T orA*(F2, F2), of the Steenrod algebra.
Also, the theorem that 'k vanishes on the image of the (Singer) algebraic tra*
*nsfer
T rk : ((F2 Pk)GLk)* ! ExtkA(F2, F2) for k > 2 is restated as follows: Every
A
positive-degree Dickson invariant of k variables represents a class in the kern*
*el of
the algebraic transfer's dual T r*k: T orAk(F2, F2) ! (F2 Pk)GLk for k > 2 (se*
*e [10],
A
[12]). It should be noted that the algebraic transfer is computationally showed*
* to
be highly nontrivial by Singer [25] and by Boardman [3].
The paper contains four sections. Section 2 is a recollection on modular inva*
*riant
theory. Its goal is to make the paper self-contained by recalling Singer's inva*
*riant-
theoretic description of the lambda algebra and our chain-level representation *
*of
the Lannes-Zarati dual map. Section 3 and Section 4 are respectively devoted to
the proofs of Theorem 1.3 and Theorem 1.2.
Acknowledgment: The present paper was written during my visit to the Johns
Hopkins University, Maryland (USA), in the Spring semester 2001. I would like to
express my warmest thanks to Mike Boardman, Jean-Pierre Meyer, Jack Morava
and Steve Wilson for their hospitality and for providing me with ideal working
atmosphere and conditions.
2.Recollection on modular invariant theory
The purpose of this section is to make the paper self-contained. First, we su*
*m-
marize Singer's invariant-theoretic description of the lambda algebra.
Let Tk be the Sylow 2-subgroup of GLk consisting of all upper triangular k x *
*k-
matrices with 1 on the main diagonal. The Tk-invariant ring, Mk = PkTk, is call*
*ed
the M`ui algebra. In [22], M`ui shows that
PkTk= F2[V1, ..., Vk],
where Y
Vi= (c1x1 + . .+.ci-1xi-1+ xi).
cj2F2
Then, the Dickson invariant Qk,ican inductively be defined by
Qk,i= Q2k-1,i-1+ Vk . Qk-1,i,
where, by convention, Qk,k= 1 and Qk,i= 0 for i < 0.
Let S(k) Pk be the multiplicative subset generated by all the non-zero line*
*ar
forms in Pk. Let k be the localization: k = (Pk)S(k). Using the results of
Dickson [7] and M`ui [22], Singer notes in [24] that
k := ( k)Tk = F2[V1 1, ..., Vk 1],
k := ( k)GLk = F2[Qk,k-1, ..., Qk,1, Qk1,0].
4 NGUY^E~N H. V. HU_.NG
Further, he sets
v1 = V1, vk = Vk=V1. .V.k-1 (k 2),
so that k-2 k-3
Vk = v21 v22 . .v.k-1vk (k 2).
Then, he obtains
k = F2[v11, ..., vk1],
with degvi= 1 for every i.
Singer defines ^kto be the submodule of k = Dk[Q-1k,0] spanned by all mono-
mials fl = Qik-1k,k-1.Q.i.0k,0with ik-1, ..., i1 0, i0 2 Z, and i0 + degfl *
*0. He also
shows in [24] that the homomorphism
@k : F2[v11, ..., vk1]!F2[v11,æ..., vk1-1],
j1. .v.jk-1,if j =,-1
@k(vj11. .v.jkk):= v10, k-1 otherkwise,
maps ^kto ^k-1. Moreover, it is a differential onP ^ = k ^k. This module is
bigraded by putting bideg(vj11. .v.jkk) = (k, k + ji).
Let be the (opposite) lambda algebra, in which the product in lambda symbols
is written in the order opposite to that used in [4]. It is also bigraded by pu*
*tting (as
in [23, p. 90]) bideg(~i) = (1, 1 + i). Singer proves in [24] that ^ is a diff*
*erential
bigraded coalgebra, which is dual to the differential bigraded lambda algebra *
* via
the isomorphisms
^ ! *,
2.1. vj1 k jk k *
1 . .v.k 7! (~j1. .~.jk) .
Here the duality * is taken with respect to the basis of admissible monomials o*
*f .
As a consequence, one gets an isomorphism of bigraded coalgebras
2.2. H*( ^) ~=T orA*(F2, F2).
As stated in Theorem 1.4, we prove in [11] that the inclusion Dk ^kis a
chain-level representation of the Lannes-Zarati dual homomorphism
'*k: (F2 Dk)i! T orAk,k+i(F2, F2).
A
In the remaining part of this section, we recall definition of the classical *
*squaring
operation on Ext*A(F2, F2).
Liulevicius was perhaps the first person who noted in [20] that there are squ*
*ar-
ing operations Sqi : Extk,tA(F2, F2) ! Extk+i,2tA(F2, F2), which share most of *
*the
properties with Sqion the cohomology of spaces. In particular, Sqi(ff) = 0 if i*
* > k,
Sqk(ff) = ff2 for ff 2 Extk,tA(F2, F2), and the Cartan formula holds for the Sq*
*i's.
However, Sq0 is not the identity. In fact, Sq0 can be defined in terms of the l*
*ambda
algebra as follows:
0 : ! ,
2.3. Sq0(~ Sq k k
i1. .~.ik)= ~2i1+1. .~.2ik+1.
So, by dualizing, the following map
Sq0v: ^k! (^k,
j1-1_ jk-1_
2.4. Sq0 j1 jk v 2 . .v. 2 , j1, ..., jk odd,
v(v1 . .v.k) = 01, k otherwise
TRIVIALITY OF DICKSON INVARIANTS 5
is a chain-level representation of the dual squaring operation
Sq0*: T orAk(F2, F2) ! T orAk(F2, F2).
3. The squaring operations
Given a module M over the dual of the Steenrod algebra A*, let P (M) denote t*
*he
submodule of M spanned by all elements annihilated by any operations of positive
degrees in A*.
Let Vk be an F2-vector space of dimension k. As is well known, H*(BVk) ~=Pk.
Then, it is easily seen that P (F2 H*(BVk)) and F2 P H*(BVk) are respec-
GLk GLk
tively dual to F2 (Pk)GLk and (F2 Pk)GLk.
A A
In [9], we have defined a squaring operation
Sq0 : P (F2 H*(BVk)) ! P (F2 H*(BVk)) ,
GLk GLk
which is derived from Kameko's squaring operation Sq0 on F2 P H*(BVk) (see
GLk
[15], [3]). We also prove in [9, Proposition 4.2] that these two squaring opera*
*tions
commute with each other through the canonical homomorphism
j*k: F2 P H*(BVk) ! P (F2 H*(BVk))
GLk GLk
induced by the identity map on Vk.
The goal of this section is to show that the Sq0 on P (F2 H*(BVk)) commutes
GLk
with the classical squaring operation Sq0 on ExtkA(F2, F2) through the Lannes-
Zarati map 'k.
Now we recall the definitions of the above mentioned squaring operations.
As is well known, H*(BVk) is a divided power algebra
H*(BVk) = (a1, . .,.ak)
generated by a1, . .,.ak, each of degree 1, where ai is dual to xi 2 H1(BVk).
Here, the duality is taken with respect to the basis of H*(BVk) consisting of a*
*ll
monomials in x1, . .,.xk.
In [15] Kameko defines a GLk-homomorphism
Sq0 : H*(BVk) ! H*(BVk),
a(i1)1.a.(.ik)k7!a(2i1+1)1.a.(.2ik+1)k,
where a(i1)1.a.(.ik)kis dual to xi11. .x.ikk. He shows that Sq0 maps P H*(BVk)*
* to
itself. (See also [2].) The induced homomorphism, which is also denoted by Sq0,
Sq0 : F2 P H*(BVk) ! F2 P H*(BVk)
GLk GLk
is called Kameko's squaring operation.
In [9], we consider the homomorphism
Sq0D= 1 Sq0 : F2 H*(BVk) ! F2 H*(BVk)
GLk GLk GLk
6 NGUY^E~N H. V. HU_.NG
and show that it sends the primitive part P (F2 H*(BVk)) to itself. The resul*
*ting
GLk
homomorphism will be redenoted by Sq0 for short:
Sq0 : P (F2 H*(BVk)) ! P (F2 H*(BVk)).
GLk GLk
The following theorem, which is a re-statement of Theorem 1.3, is the main re*
*sult
of this section.
Theorem 3.1. For an arbitrary positive integer k, the squaring operation Sq0 on
P (F2 H*(BVk)) commutes with the classical Sq0 on ExtkA(F2, F2) through the
GLk
Lannes-Zarati homomorphism 'k. In other words, the following diagram commutes:
'k
ExtkA(F2, F2)____________-P (F2 H*(BVk))
GLk
| |
| 0 | 0
|Sq |Sq
| |
|? |?
'k
ExtkA(F2, F2)____________-P (F2 H*(BVk)) .
GLk
We will prove this theorem by showing its dual version. To this end, let us
consider the dual homomorphism of Kameko's one:
Sq0x= Sq0*: F2[x1, ...,!xk]F2[x1,(..., xk],
j1-1_2 jk-1_2
Sq0x(xj11. .x.jkk)= x1 . .x.k ,j1, ..., jk odd,
0, otherwise.
In order to explain the behavior of this homomorphism on modular invariants,
we present a homomorphism:
Sq0v: F2[V1, ...,!Vk]F2[V1,(..., Vk],
j1-1_2 jk-1_2
Sq0v(vj11. .v.jkk)= v1 . .v.k ,j1, ..., jk odd,
0, otherwise.
Obviously, this map coincides with the map in 2.4 on the intersection of their
domains.
The two homomorphisms Sq0xand Sq0vdepend on k and, when necessary, will
respectively be denoted by Sq0x,kand Sq0v,k.
Technically, the following proposition is the key point in our proof of Theo-
rem 3.1.
Proposition 3.2. Sq0xcoincides with Sq0von F2[V1, ..., Vk], for any k.
This proposition will be shown by means of the following two lemmata, which
directly come from the definitions of Sq0xand Sq0vgiven above.
Lemma 3.3. (i) Sq0x,k(ab2) = Sq0x,k(a)b, for any a, b 2 F2[x1, ..., xk].
(ii)Sq0v,k(AB2) = Sq0v,k(A)B, for any A, B 2 F2[V1, ..., Vk].
Lemma 3.4. (i) Sq0x,k(axk) = Sq0x,k-1(a), for any a 2 F2[x1, ..., xk-1].
(ii)Sq0v,k(Avk) = Sq0v,k-1(A), for any A 2 F2[V1, ..., Vk-1].
TRIVIALITY OF DICKSON INVARIANTS 7
We are now ready to prove Proposition 3.2.
Proof of Proposition 3.2. The proof proceeds by induction on k.
For k = 1, since x1 = v1, we get obviously Sq0x,1= Sq0v,1.
Let k > 1 and suppose inductively that Sq0x,k-1= Sq0v,k-1. We need to show
Sq0x,k= Sq0v,k. Let V = V1i1. .V.ikkbe an arbitrary monomial in Mk = F2[V1, ...*
*, Vk].
We consider the following two cases.
Case 1: ik is even. Recall that
k-1
Vk = Qk-1,0xk + Qk-1,1x2k+ . .+.Qk-1,k-1x2k
(see M`ui [22, Appendix]). Since Qk-1,0, ..., Qk-1,k-1, V1, ..., Vk-1 all do no*
*t depend
on xk, we have X
V = V1i1. .V.ikk= xj11. .x.jkk,
jk even
where jk is even in every monomial of the sum. Therefore, by definition of Sq0x*
*,k,
X j j
Sq0x,k(V ) = Sq0x,k(x11. .x.kk) = 0.
jk even
On the other hand, from the expansions of Vi's in terms of vj's, we get
V = V1i1. .V.ikk= v`11. .v.`kk,
where `k = ik is even. Hence, by definition of Sq0v,k,
Sq0v,k(V ) = Sq0v,k(v`11. .v.`kk) = 0.
Case 2: ik = 2n + 1. We have
V = V1i1. .V.ik-1k-1Vkik
k-1 2n
= V1i1. .V.ik-1k-1(Qk-1,0xk + Qk-1,1x2k+ . .+.Qk-1,k-1x2k )Vk .
Since V1i1. .V.ik-1k-1Qk-1,0xkVk2nis the only term in the above expansion of V *
*with
power of xk odd, we get
Sq0x,k(V ) = Sq0x,k(V1i1. .V.ik-1k-1Qk-1,0xkVk2n).
Note that V1, ..., Vk-1, Qk-1,0all do not depend on xk. Combining Lemma 3.3,
Lemma 3.4 and the inductive hypothesis, we obtain
Sq0x,k(V )= Sq0x,k(V1i1. .V.ik-1k-1Qk-1,0xk)Vkn (by Lemma 3.3)
= Sq0x,k-1(V1i1. .V.ik-1k-1Qk-1,0)Vkn (by Lemma 3.4)
= Sq0v,k-1(V1i1. .V.ik-1k-1Qk-1,0)Vkn (by the inductive hypothesis)
= Sq0v,k(V1i1. .V.ik-1k-1Qk-1,0vk)Vkn (by Lemma 3.4)
= Sq0v,k(V1i1. .V.ik-1k-1Qk-1,0vkVk2n) (by Lemma 3.3)
= Sq0v,k(V1i1. .V.ik-1k-1Vk2n+1).
The last equality comes from the expansions
Qk-1,0vk = V1. .V.k-1vk = Vk.
The proposition is completely proved.
Now we come back to Theorem 3.1.
8 NGUY^E~N H. V. HU_.NG
Proof of Theorem 3.1. We will show the commutativity of the dual diagram:
F2 Dk _____________-'*kT orA (F , F )
A k 2 2
| |
| 0 | 0
|Sq* |Sq*
| |
|? |?
F2 Dk _____________-'*kT orA (F , F ) .
A k 2 2
This will be obtained from a commutative diagram of appropriate chain-level rep-
resentations of the homomorphisms in questions.
Indeed, by definition of Sq0 on (F2 Dk)* = P (F2 H*(BVk)), the restriction
A GLk
of Sq0xon Dk is a chain-level representation of Sq0*: F2 Dk ! F2 Dk. On the
A A
other hand, from 2.4, the map
Sq0v: ^k! (^k,
j1-1_2 jk-1_2
Sq0v(vj11. .v.jkk)= v1 . .v.k ,j1, ..., jk odd,
0, otherwise.
is a chain-level representation of Sq0*: T orAk(F2, F2) ! T orAk(F2, F2). Now, *
*since
Dk Mk = F2[V1, ..., Vk], Proposition 3.2 implies the commutativity of the dia-
gram:
Dk _____________- ^k
| |
| 0 | 0
|Sqx |Sqv
| |
|? |?
Dk _____________- ^k.
By Theorem 1.4, the inclusion Dk ^kis a chain-level representation of the
Lannes-Zarati's dual map '*k. Therefore, the last commutative diagram shows the
commutativity of the previous one.
Theorem 3.1 is proved.
4. The triviality of '4
The goal of this section is to prove Theorem 1.2, the main result of this pap*
*er.
To this end, we need to recall the computation of Ext4A(F2, F2) by W. H. Lin
and that of F2 D4 by F. P. Peterson and the author.
A
Theorem 4.1. (W. H. Lin [18], see also [19, Theorem2.2]). The following classes
form an F2-basis for the vector space of indecomposable elements in Ext4A(F2, F*
*2):
(1) di= [(Sq0)i(~6~2~23+i~24~23++~2~4~5~34+i~1~5~1~7)]+1
2 Ext4,2A +2 , i 0,
(2) ei= [(Sq0)i(~8~33+i~4(~25~3++4~7~23)i++~2(~3~5~72+i~1~11~3))]
2 Ext4,2A +2 +2, i 0,
TRIVIALITY OF DICKSON INVARIANTS 9
(3) fi= [(Sq0)i(~4~0~27+i~3(~9~23++~3~5~7)4+i~22~27)]+2i+1
2 Ext4,2A +2 +2, i 0,
(4) gi+1= [(Sq0)i(~6~0~27+i~5(~9~23++~3~5~7)4+i~3(~5~9~3++3~11~23))]
2 Ext4,2A +2 , i 0,
(5) pi= [(Sq0)i(~14~5~27+i~10~9~27++~6~9~11~7)]5i+2i
2 Ext4,2A +2 +2, i 0,
(6) D3(i) = [(Sq0)i(~22~1~7~31+i~16~27~31++~14~9~7~31+6~12~11~7~31]i
2 Ext4,2A +2, i 0,
(7) p0i= [(Sq0)i(~0~39~215+ ~0~15~23~31)]
i+6+2i+3+2i
2 Ext4,2A , i 0.
To simplify notation, we will denote Qa4,3Qb4,2Qc4,1Qd4,0by Q(a, b, c, d) in *
*the fol-
lowing theorem.
Theorem 4.2. (Hu_.ng-Peterson [13]). The following elements form an F2-basis for
the vector space F2 D4:
A
(1) Q(2s - 1, 0, 0, 0), s 0,
(2) Q(2r - 2s - 1, 2s - 1, 1, 0), r > s > 0,
(3) Q(2t- 2r - 1, 2r - 2s - 1, 2s - 1,t2),> r > s > 1,
(4) Q(2r - 2s+1- 2s - 1, 2s - 1, 2s - 1,r2),> s + 1 > 2.
They are of degrees 2s+3-8, 2r+3+2s+2-6, 2t+3+2r+2+2s+1-4 and 2r+3+2s+1-4,
respectively.
Now we come back to prove Theorem 1.2.
Proof of Theorem 1.2.
In [14], F. Peterson and the author have proved that 'k vanishes on any de-
composable elements for k > 2 by showing that '* = k'k is a homomorphism of
algebras and, more importantly, that the product of the algebra k(F2 Dk)* is
A
trivial, except for the case
(F2 D1)* (F2 D1)* ! (F2 D2)*.
A A A
Therefore, we need only to show '4 vanishing on any indecomposable elements.
Let a0 denote one of the seven generators
d0, e0, f0, g1, p0, D3(0), p00,
each of which is the element of lowest stem in its own family. Furthermore, set
ai= (Sq0)i(a0), for i 0. From Theorem 3.1, we have
'4(ai) = '4(Sq0)i(a0) = (Sq0)i'4(a0).
So, in order to prove that '4(ai) = 0 for any i, it suffices to show '4(a0) = 0.
We will do this by checking that the stem of a0 is different from degrees of al*
*l the
generators of F2 D4 given in Theorem 4.2.
A
Now let us check it case by case.
10 NGUY^E~N H. V. HU_.NG
Case 1: For d0 of stem 24 + 21 - 4 = 14,
2s+3 = 8 + 14 = 22, no solution;
2r+3+ 2s+2 = 6 + 14 = 16 + 4, r = 1, s = 0, it does not satisfy s > 0;
2t+3+ 2r+2+ 2s+1 = 4 + 14 = 16 + 2, no solution;
2r+3+ 2s+1 = 4 + 14 = 16 + 2, r = 1, s = 0 it does not satisfy r > s + 1 > 2.
Case 2: For e0 of stem 24 + 22 + 20 - 4 = 17,
2s+3 = 8 + 17 = 25, no solution;
2r+3+ 2s+2 = 6 + 17 = 16 + 4 + 2 + 1, no solution;
2t+3+ 2r+2+ 2s+1 = 4 + 17 = 16 + 4 + 1, no solution;
2r+3+ 2s+1 = 4 + 17 = 16 + 4 + 1, no solution.
Case 3: For f0 of stem 24 + 22 + 21 - 4 = 18,
2s+3 = 8 + 18 = 26, no solution;
2r+3+ 2s+2 = 6 + 18 = 16 + 8, r = 1, s = 1 it does not satisfy r > s;
2t+3+ 2r+2 + 2s+1 = 4 + 18 = 16 + 4 + 2, t = 1, r = s = 0 it does not satisfy
r > s > 1;
2r+3+ 2s+1 = 4 + 18 = 16 + 4 + 2, no solution.
Case 4: For g1 of stem 24 + 23 - 4 = 20,
2s+3 = 8 + 20 = 28, no solution;
2r+3+ 2s+2 = 6 + 20 = 16 + 8 + 2, no solution;
2t+3+ 2r+2+ 2s+1 = 4 + 20 = 16 + 8, no solution;
2r+3+ 2s+1 = 4 + 20 = 16 + 8, r = 1, s = 2, it does not satisfy r > s + 1 > 2.
Case 5: For p0 of stem 25 + 22 + 20 - 4 = 33,
2s+3 = 8 + 33 = 41, no solution;
2r+3+ 2s+2 = 6 + 33 = 32 + 4 + 2 + 1, no solution;
2t+3+ 2r+2+ 2s+1 = 4 + 33 = 32 + 4 + 1, no solution;
2r+3+ 2s+1 = 4 + 33 = 32 + 4 + 1, no solution.
Case 6: For D3(0) of stem 26 + 20 - 4 = 61,
2s+3 = 8 + 61 = 69, no solution;
2r+3+ 2s+2 = 6 + 61 = 64 + 2 + 1, no solution;
2t+3+ 2r+2+ 2s+1 = 4 + 61 = 64 + 1, no solution;
2r+3+ 2s+1 = 4 + 61 = 64 + 1, no solution.
Case 7: For p00of stem 26 + 23 + 20 - 4 = 69,
2s+3 = 8 + 69 = 77, no solution;
2r+3+ 2s+2 = 6 + 69 = 64 + 8 + 2 + 1, no solution;
2t+3+ 2r+2+ 2s+1 = 4 + 69 = 64 + 8 + 1, no solution;
2r+3+ 2s+1 = 4 + 69 = 64 + 8 + 1, no solution.
Therefore, '4 vanishes on any indecomposable elements.
In summary, '4 = 0 in positive stems. Theorem 1.2 is completely proved.
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Department of Mathematics, Johns Hopkins University
3400 N. Charles Street, Baltimore MD 21218 - 2689
E-mail address: nhvhung@math.jhu.edu
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