奈良教育大学学術リポジトリNEAR
On the action of the Steenrod reduced powers in a Polynomial Hopf algebra over the Steenrod algebra
著者 OCHIAI Shoji
journal or
publication title
奈良教育大学紀要. 自然科学
volume 28
number 2
page range 1‑3
year 1979‑11‑15
URL http://hdl.handle.net/10105/2454
奈良教fデ入学紅紫 第28巻 第2sj‑ (〔】然)耶柑54T‑
Bull. Nara Univ. Educ, Vol.28, No.2 (Nat.),1979
0n the action of the Steenrod reduced powers in a Polynomial Hopf algebra over the Steenrod algebra
Shoji Ochiai
(Depa′･//′lent of Mat/ie7natics, Na′･a Universiy of Educ〟tioti, Nar‑a, Japan)
(Received May 1, 1979)
1. Introduction Let A be an algebra over the Steenrod algebra,し〆(/>).By an algebra over the Steenrod algebra, ｡1〆(/>), we mean the algebra on which the reduced powers act just as ifA were the cohomology algebra of a space. E. Thomas determines the action of the Steenrod squares in a polynomial algebra over the mod 2 Steenrod algebra (Thorem 2.1
〔5〕, Theorem 1.4 〔6〕). These theorem impose necessary conditions on the dimensions of the generators of the polynomial algebra that occur. N.E.Steenrod and E. Thomas proposed an problem (〔4〕 Question 3 〔7〕): Which mod p (/>, an arbitrary prime) polynomial algebras
haveL♂(/>) structures? To solve this problem, we need Thomas's theorem in the mod p
case. However, the obvious analogue of Thomas's theorem for an odd prime ♪ is false(〔7〕).
This fact makes the problem difficult to solve. The following theorem is a partial generalization of Thomas's theorem to the case of an odd prime ♪ and its proof is similar to 〔2〕,〔3〕.
THEOREM. Let A be a Hopf algebra oT′er the Steenrod algebra L♂(/>), (/>, odd prime) and
also a polynomial algebra on finite primitive homogeneous gener〟tors of even dimension. Let
a be a primitive element. Then there i∫ a primitive element b e P{A ) such that
u‑｡グ】b, be P(A).
where dim a‑2 (pk+r(p‑1)), 1≦r≦p‑1, 1≦k andP{A) denotes the メpace of the primitit,e
elements.
2. Proof of Theorem In order to prove this theorem, we need to introduce the following spaces.
L(k,i)‑可 ｣00‑(｡ダb)p, aeP(A), bfA, dim ̀i‑2(pk+i(p‑Y)¥l≦i≦p‑1, 1≦*}.
M(k,i)‑P(A)2(pk+Hp‑1サ/W).
N(k,i‑) ‑{P(A)/^> 】(n^)))S( ‑1))
Note that a primitive element is preserved under the reduced power operations. Let k(Q,r)
‑k be any fixed positive integer and for n≧1,
〃
A(n,r)‑pサA+(r‑!)(/>‑!)∑p*‑l, where i‑ is l≦γ≦♪‑1･
J ､ I
Proof of Theorem. If we c呈in show that there exist an epimorphism ヮ iV(｣O,r);0
‑ナM (｣(ォ,r);r) and a monomorphism an,r: Ar(^(ォ,r);r) ‑ M (｣(ra+l,r) ;r), the proof is
complete. Because all the primitive and decomposable elements are the />‑th power of some
l
2
Shoji Ochiaielement [1], we see M(｣(m,r);r)‑O for sufficiently large m. An existence of a monomor‑
phism ffォーi,,, implies N(k(m‑l,r);r) ‑0. By using an epimorphismヮ and a monomorp‑
hism aれ̲ we obtain N(k(m‑2,r);r)‑0. Continuing this way, we get finally Af(｣(0,r); r)
‑0. This completes the proof.
Therefore, it remains to prove the following two lemmas.
LEMMA 1. Let A be the algebra in Theorem. Then there exists a mo/lomorphism an,r'‑ N (*(ォ,0;r) ‑ M(｣(/z+l,r);r) for all the nonnegative integers n and r, 1≦r≦p‑1.
Proof. Let pn,r: P(A)2如(n,r)+r(j>‑1)) ‑ iV(&(?z,r);r) be the projection and sn,r: ‑P(A)
2(pfc(n,r)+r(p‑1)) ‑ M(｣(w+l,r);r) be the homomorphism defined by sn,*・(ォ)‑〔g p(た(nr)+r‑1)
(a)〕 a (P(A)2(pk(n,r)+r(p‑1))1 where 〔α〕 denotes the coset represented by α. Let a be an
element of the kernel pk(n,r), namely, a‑t^ff lb. From this assumption, the next equation
holds
(｡グl)p‑rLグJ>(Kn,r)+ド"00)‑(｡グl)p‑r iグ2>(*(n,r)+r‑1)(‑j^サ｣)‑(｡グ1b′)p.
Therefore, we see a e kernel sn,r. Conversely, let a be an element of the kernel sn,r. This implies
(｡ダ た(n,r)+r‑1)(V)‑(｡ダby.
Since A is the above mentioned polynomial algebra, we see a‑tダ1b′ From this fact, we get kernel 5n,r‑kernel pn,r. It we define
･x*(n,r)‥ N(*(ォ,r);r)‑Af(*O+l,r);r) by (Iri,r‑sn,r.p詰this homomorphism is the desired monomorphism.
LEMMA 2(r). Let A be the algebra in Theorem. Then there exists an epimorphismヮ.,n N (*(ォ,r);r) ‑ M(/fc(ォ,r);r) for all the nonnegative integeγS n 〈ind r l≦r≦p‑1.
This lemma is proved by the induction on r.
PROOFOF LEMMA 2(1). Recall that N(k(n,l);l) and M(｣(h,1);1) are the factor space of
P(A)2(p*(n,i)+p‑i). Let pn,pn′ be the projections
Pn Pn′
tf(*(ォ,D;D ‑ P(A)2(P*O>1)+J>‑1) ‑→M(k(n､D;l).
We will show kernel pn⊂kernel pn'. Let a be an element of the kernel pn. This implies a‑｡グib. Therefore, we get (^グO'‑'OX｡グ1)p(6)‑0.This shows kernel pn⊂kernel /> '.
Then, if we define恥,!‑/>サ′'pn‑1, this homomorphismり is clearly an epimorphism.
Now suppose Theorem is true for the positive integers less than r and for all the
nonnegative integers n.Proof of Lemma 2(r). Let a be a∈kernel pn,r, namely, a‑｡ダib. Operate(｡ダ1)p‑T on both sides, we get (｡グ蝣)サー(*)‑(｡ク.)p‑r(｡グ'&). The induction hypothesis implies｡グ1b
‑(.J^1)r^‑ From this, we see ( 1)p‑r(a)‑0 and so,a ∈ kernel /> ′ Therefore, rj外,r
defined byでn,r‑pn',r*pn¥ is an epimorphism. This completes the proof.
On the action of the Steenrod reduced powers
3
References
1. J. Milnoh and J. C. Moore, On the structureof Hopf algebra, Ann. of Math. 81 (1965), 211‑264.
2. Shoji Ochiai, On the dimension ofindecomposable elements of polynomial algebra oter the Steenrod algebra, Sci. Rep. Tokyo kyoiku Daigaku, Sec. A. vo1 9, No. 203. (1965) 24‑31.
3. Stavros G. Papastavridis, Polynomial algebras which a/e modules otノer the?nod p Steenrod algebra,
