Finite complexes with A(n)-free cohomology
Finite complexes with A(n)-free cohomology
复制标题
具有无 A(n) 上同调的有限复形
DOI：
10.1016/0040-9383(85)90057-6
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发表时间：
1985
期刊：
影响因子：
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通讯作者：
S. A. Mitchell
中科院分区：
文献类型：
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作者：
S. A. Mitchell
设A是素数p，B是rood p Steenrod代数A的有限维子代数.那么我们可能会问B是否可以由一个有限复形实现：也就是说，是否存在一个有限复形，它的模p上同调同构为B的某个悬置的左B-模？然后，我们立即面临着这样一个实现的代数障碍：B能被实现为A-模吗？也就是说，B是否允许一个左A-模结构扩展它的左B-模结构？例如，A包含在本原生成元Q 0，···，Q，上的外代数F（n），并且众所周知，并且容易证明，F（n）可以实现为A-模当且仅当p是奇数或n= 0。另一个有趣而重要的子代数族是族A（n）：A（n）是由fl，pt.生成的子代数。PP '-~，其中pi=
FIX A prime p, and suppose that B is a finite-dimensional subalgebra of the rood p Steenrod algebra A. Then we may ask whether or not B can be realized by a finite complex: That is, does there exist a finite complex whose mod p cohomology is isomorphic as a left B-module to some suspension of B? We are then immediately faced with an algebraic obstruction to such a realization: Can B be realized as an A-module? That is, does B admit a left A-module structure extending its left B-module structure? For example, A contains exterior algebras F (n) on primitive generators known as Q0,•••, Q,, and it is well known, and easy to prove, that F (n) can be realized as an A-module if and only if p is odd or n= 0. Another interesting and important family of subalgebras is the family A (n): A (n) is the subalgebra generated by fl, pt..... PP'-~, with pi=