Levels in algebra and topology
Levels in algebra and topology
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代数和拓扑水平
DOI：
10.1007/bf02566358
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发表时间：
1980
影响因子：
0.9
通讯作者：
Tsit Yuen Lam
中科院分区：
文献类型：
--
作者：
Z. D. Dai;Z. D. Dai;Tsit Yuen Lam;Tsit Yuen Lam
根据定义，交换环 A 的级 s (A) 是最小整数 n，使得 -1 是 A 中 n 的平方和。（如果 -1 不是 A 中的平方和，我们定义 s (A) 为~。）根据 A 的著名定理 Pfister，如果 A 是一个域，并且如果 s (A)<~，则 s (A) 一定是 2 的幂（并且 2 的任何幂都是可能的）。然而，这个结果并不扩展到环：在[DLP]中，表明存在任何规定级别的交换N-代数，或者等效地，对于任何整数n，“通用”代数An= N [Xl..... x,]/(1+ Xl z+. 9 9+ x 2) 的级别完全等于n。[DLP]中s (A,)= n的证明基于拓扑事实：博尔苏克-乌拉姆定理。这个证明的想法表明，球体拓扑与环平方和算术之间存在着自然而有趣的关系。为了更正式地研究这种关系，我们定义了
By definition, the level s (A) of a commutative ring A is the smallest integer n such that-1 is the sum of n squares in A.(If-1 is not a sum of squares in A, we define s (A) to be~.) By a well-known theorem of A. Pfister, if A is a field and if s (A)<~, then s (A) must be a power of 2 (and any power of 2 is possible). This result, however, does not extend to rings: in [DLP], it was shown that there exist commutative N-algebras of any prescribed level, or, equivalently, for any integer n, the" generic" algebra An= N [Xl..... x,]/(1+ Xl z+. 9 9+ x 2) has level exactly equal to n.The proof that s (A,)= n in [DLP] was based on a topological fact: the Borsuk-Ulam Theorem. The idea of this proof suggested that there is a natural and interesting relationship between the topology of spheres and the arithmetic of sums of squares in rings. To study this relationship more formally, we defined in