AlgebraicK-theory and quadratic forms
AlgebraicK-theory and quadratic forms
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代数 K 理论和二次形式
DOI：
10.1007/bf01425486
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发表时间：
1970
影响因子：
3.1
通讯作者：
J. Milnor
中科院分区：
文献类型：
--
作者：
J. Milnor
本文第一节定义并研究了分次环K。F与任意域F相联系。根据定义，K~F是从F~x…·F~到加法群的泛n-线性函数的目标群，满足当i-q-ai+~=1时al·‘xa应映射为零的条件。这里F~表示乘法群F0。第二节构造了一个与F上具有剩余类域F的离散赋值有关的同态~：K，F-，Ki~~。这些同态被用来计算有理函数域的环K，F(T)，利用John Tate的技巧。第三节涉及K。通过在特征域F上定义二次模的某些“Stiefel-Whitney不变量”来推广二次模的理论。2.该定义与Delzant[-5]密切相关。设W是F上各向异性二次模的Witt环，IcW是由偶数秩模组成的极大理想。第四节研究了相关分次环的猜想
The first section of this paper defines and studies a graded ring K . F associated to any field F. By definition, K~F is the target group of the universal n-linear function from F ~ x ... • F ~ to an additive group, satisfying the condition that al • " ' x a, should map to zero whenever a i -q-a i + ~ = 1. Here F ~ denotes the multiplicative group F 0 . Section 2 constructs a homomorphism ~: K,F---, K~__I_~ associated with a discrete valuation on F with residue class field F. These homomorphisms ~ are used to compute the ring K, F(t) of a rational function field, using a technique due to John Tate. Section 3 relates K . F to the theory of quadratic modules by defining certain " Stiefel-Whitney invariants" of a quadratic module over a field F of characteristic . 2 . The definition is closely related to Delzant [-5]. Let W be the Witt ring of anisotropic quadratic modules over F, and let I c W be the maximal ideal, consisting of modules of even rank. Section 4 studies the conjecture that the associated graded ring