The factorization of cyclic reduced powers by secondary cohomology operations
The factorization of cyclic reduced powers by secondary cohomology operations
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DOI：
10.1090/memo/0042
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发表时间：
1962
影响因子：
1.9
通讯作者：
A. Liulevicius
中科院分区：
文献类型：
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作者：
A. Liulevicius
(1)exta V(Zp，Zp)以某些类hi(i=0，1，…)为基等级2pi(p-1)，等级1的Ao；(2)ExtA2(Zp，Zp)的ZP基由某些类hihj，i<；j-1(i=0.1，...J=2，，...)2级(p-1)(pi+pj)，2pi(p-1)+1级(i$O)，2级(p-1)(pi+1+2pi)，2级vi(p-1)(2pi+i+pi)，2级xi(p-1)(pi+1)，4级p(p-1)+1，2级奥奥；(3)2pj+i(p-1)+1的ExtA3(Zp，Zp)中的Xjao元素不为零。利用Hopf代数的Adams谱序列证明了这一定理。通过引入Steenrod运算，证明变得更容易了。这些运算定义在Extjr S(Z，ZP)上，其中G是ZP上具有结合积和结合对角线的分次连通Hopf代数：
(1) ExtA V (ZP, Zp) has as basis certain classes hi (i= 0, 1,...) ofgrading2pI (p-1), ao of grading 1;(2) a Zp basis for ExtA2 (ZP, Zp) is furnished bycertain classes hihj, i< j-1 (i= O. 1,.... j= 2,,...) of grading 2 (p-1)(pi+ pJ), hiaco (i $ O) of grading 2pi (p-1)+ 1,, of grading 2 (p-1)(pi+ 1+ 2pi), vi of grading 2 (p-1)(2pi+ I+ pi), Xi of grading 2 (p-1)(pi+ 1), p of grading 4 (p-1)+ 1, and aoao of grading 2;(3) the elements Xjao in ExtA3 (ZP, ZP) of grading 2pJ+ I (p-1)+ 1 are nonzero. The theorem is proved by using the Adams spectral sequence for Hopf algebras. The proof is made easier by introducing Steenrod operations. These operations are defined on Extjr s (Z, Zp), where G is a graded, connected Hopf algebra over Zp with associative product and associative and commutative diagonal: