Quadratic forms with absolutely maximal splitting
Quadratic forms with absolutely maximal splitting
复制标题
具有绝对最大分裂的二次形式
DOI：
10.1090/conm/272/04399
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发表时间：
2010
影响因子：
1.8
通讯作者：
A. Vishik
中科院分区：
文献类型：
--
作者：
O. Izhboldin;A. Vishik
设F是域，φ是F上的二次型.φ的高阶维特指标可以递归地定义为ik+1（φ）= ik（（φan）F（φan）），其中i 0（φ）= iW（φ）是φ形式的维特指标。我们说各向异性形式φ有绝对最大分裂，如果对所有k > 1，i1（φ）> ik（φ）。本文的一个主要结果是，对于满足条件2n−1 + 2n−3 < dimφ ≤ 2n的所有各向异性形式φ，下列三个条件是等价的：（i）自然同态Hn（F，Z/2 Z）→ Hn（F（φ），Z/2 Z）的核是非平凡的，（ii）φ有绝对最大分裂，（iii）φ有最大分裂（即，i1（φ）= dimφ− 2n−1）。此外，我们证明了如果我们另外假设dimφ ≥ 2n − 7，那么这三个条件成立当且仅当φ是各向异性的n重Pfister近邻。在我们的证明中，我们使用了V. Voevodsky在证明Milnor猜想时所使用的技术。
Let F be a field and φ be a quadratic form over F . The higher Witt indices of φ are defined recursively by the rule ik+1(φ) = ik((φan)F (φan)), where i0(φ) = iW (φ) is the usual Witt index of the form φ. We say that anisotropic form φ has absolutely maximal splitting if i1(φ) > ik(φ) for all k > 1. One of the main results of this paper claims that for all anisotropic forms φ satisfying the condition 2n−1 + 2n−3 < dimφ ≤ 2n, the following three conditions are equivalent: (i) the kernel of the natural homomorphism Hn(F,Z/2Z) → Hn(F (φ),Z/2Z) is nontrivial, (ii) φ has absolutely maximal splitting, (iii) φ has maximal splitting (i.e., i1(φ) = dimφ− 2n−1). Moreover, we show that if we assume additionally that dimφ ≥ 2n − 7, then these three conditions hold if and only if φ is an anisotropic n-fold Pfister neighbor. In our proof we use the technique developed by V. Voevodsky in his proof of Milnor’s conjecture.