奈良教育大学学術リポジトリNEAR
Action of A(p) on the Polynomial Algebra Zp[x4, y4]
著者 OCHIAI Shoji
journal or
publication title
奈良教育大学紀要. 自然科学
volume 30
number 2
page range 9‑10
year 1981‑11‑25
URL http://hdl.handle.net/10105/2347
Bull.監ra umv. E盈｡.T豊.盈N｡. 2 (Nat.). 1浩
Action of A(p) on the Polynomial Algebra′ Zplxi, yj
Shoji OCHIAI
(Department of Mathematics, Nara University of Education, Nara, Japan) (Received April 30, 1981)
By a polynomial algebra A over the Steenrod algebra A(l), we mean the one on which the reduced powers and the Bockstein coboundary act just as if A were the cohomology algebra of a space.
This note is to prove the following theorem.
Theorem. All polynomial algebras over A(3) on two generators in each degree 4 are isomorphic.
2 2
Let Plx4‑∑a,x害 ''yま, Pl3U‑∑毎呈蝣>; where Pl∈A(3) and a(, i,∈Z3.
i‑0 1‑0
Lemma 1. If à hi take a value in the i‑th column of the following table, Z3[xt,.yォ] is
the algebra over the Steenrod algebra by the operation defined above.
Proof. By the Adem relation and the unstable condition, we obtain PIPlxt‑2xl PIPlyt‑2y3t. Therefore a,, bt must satisfy the following system of congruences.
aibo +2aZ≡2 2<22&0十fllOl≡0
2a2bl+aib2+2ailai+a至≡0 2a2&2+fllf12=0
2&o#o+Sooi=O
biao+2boai+b至+2bfjbi≡O
blal+2boa2=O
bla2+2b芸≡2
We obtain the next solutions.
42
32
2
21
2
0
2
91
8
1
7日リ
6
日H
Tサ
1
4
日り
3
日目
2
日H
HH
1
0
19
8
76
5
43
2
1
叫 叫 的1 叫‑ o"
^S l
ハ ム o o o o c
^
(M C¥] rH O O CM
CO rH t‑1 O O T‑i
t‑h cj eg o o <m
.‑< i‑t (M O CD t‑I
ウ 山 i
‑ 1 O O O O J
( M C M O O O H c s i o o 0 0
,‑H i‑1 CD O O <M
,
‑
<
o o o o o a
蝣
r
‑
1
C
<
3
O
O
O
i
‑
H
^ j O O O O i
‑ H oa o cj o y‑i o
oa o o o i‑H <m
CJ O O O *‑I i‑I
,‑1 O H O C*J 0
,‑1 0 0 O C‑J C¥l
,‑t O O O (M i‑1
<NI O O CJ CO i‑H
C M O O H C M C M T‑ 1 O O
<M T
‑1 1
‑I
i‑I O O i‑I i‑I <N
O i‑H O (M O <M
O
<
M O H O r H
If i‑th column is ao, alt a2, bo, bu b2, we denote by P) the reduced power operation
Pl such that Plxi‑a^x¥+aix▲y<+a2yl Ply4‑btxl+bixtyl+b2y呈. This operation Pき make
Z3[xt, yt] the algebra over the Steenrod algebra.
Lemma 2. Let 7¥: Z3[xit yt ¥‑Z3[xi,一yt ¥ be an algebra automorphism defined by
g
10 Shoji Ochiai
Ty(x▲) ‑xi‑ (at/b*)y▲
Txiyl)‑y*
where ao. &o ore coefficients in the term ofP¥ (Ti is determined by P¥).
Then 7丁'PIT^PI TrlFkTi ‑I% TtlPsTi‑Pi, TTYtT^Pl hold.
Proof. These are checked by computation.
Lemma 3. There exist algebra automorphisms T2, T3 such that fallowings hold.
CTOT‑'Pき(TtTi)‑P王, 1‑7, 8, 17, 18, 19 (T2Ti)‑1PKT2Tl)‑P,圭. 1‑10, ll, 13, 14, 15, 16 (r,r,)‑>p…(T,Tj) ‑P圭, 1‑9, 12, 22, 23
(T3T2)‑1Pき(T3T2) ‑P圭, 1‑20, 21
Proof. Let /(α) be /(α)‑(ォ0‑&i)α2+(a1‑b2)α+a2 for the given Pき. If ait bt take value in the f‑th column of the table (t‑7, 8, 10, ll, 13, 14, 15, 16, 17, 18, 19), there is α奪0∈Z3 such that /(α)毒0. Let T2 and T3 be
T2(ズl)‑*4 T,{yl) ‑ズ4‑αy4
Ts(*<)‑yA T,(yi)‑xi.
By these automorphisms, above equations hold.
Proof of Theorem. Let T4 be the automorphism defined by
T4(x▲) ‑Xl+ (2/bo)y4
Tt(yt) ‑βx4‑β(2/&o)j4
where β is quadratic non‑residue of Z3 and bo is the coefficient of x¥ in P吉(ỳ)(t‑l, 2).
Then we can verify that 7TIPきTi‑P¥A(i‑l, 2) holds. This concludes the proof. It seems that it fills a gap in the proof of Theorem 7.2 [1] in the case of l‑3.
References
[ 1 ] John Ewing, On the type of associative H‑space, Preprint Series, 1970/71 No. 15. Aarhus Univer‑
sitet.
[ 2 ] Norman Steenrod, Polynomial algebra over the algebra of cohomology operation, H‑spaces, Neu‑
chatel (Suisse) Aoflt 1970, Lecture notes in Math., vol 196, SpringeトVerlag, 1970.
