【作者】 许宁；

【导师】 李克正；

【作者基本信息】 中国科学院研究生院（数学科学学院） ， 代数几何， 2008， 博士

【摘要】 赋值理论具有很长的历史，他在很多领域都有重要的应用。赋值是许多代数对象的研究工具。令K为k的有限生成扩域，tr．deg（K／k）=n<+∞，v是K的一个k-赋值。如果n=1，则K是k上曲线C的函数域，v可以由C的一个素除子来定义，我们称它为除子类型。每个除子类型的赋值都是离散的。我们把它推广到高维（即n≥2）情形，它对K-群的计算以及刚性解析空间的研究都有帮助。本论文主要研究n=2的情形，它比n=1的情形要复杂得多。我们定义了赋值的高度以及单项式ax^syⁿ，s∈Q，n∈N，a∈k的（?）-次数，其中（?）（x）=sum from i=1 to +∞a_ix^r_i,a_i∈k，r_i∈Q，0<r₁<r₂<…，（?）r_i=r<+∞为任意给定的形式级数。从而得到曲面上k-赋值的完整分类，并且给出了赋值和超越级数的关系。进一步地，我们证明了曲面上所有非平凡k-赋值都可以由爆发的无穷序列给出，具体给出了爆发的过程。一个平行的问题是算术曲面上赋值的分类，也就是对Q的超越度为1的有限生成扩域上的赋值的分类。在本论文中，我们给出了赋值的高度的定义以及大域C_p,G的定义，其中p为素数，G（?）R为包含1的加法子群。我们得出C_p,G是一个域并且C_p,Q是代数闭的。从而得到算术曲面上赋值的完整分类。进一步地，对任意m≤n∈Z，令V_m,n为n-m+1维R-向量空间，坐标指数从m到n。我们推广C_p,G的定义，使得其中p为素数，G（?）V_m,n为包含1的加法子群。我们得出如果m≤0≤n，则C_p,G是一个域。更多还原

【Abstract】 The valuation theory has a long history. It has important applications in various areas. Valuation is a researching tool for many algebraic objects. Let K be a finitely generated field extension of k with tr.deg（K/k） = n <+∞, v be a k-valuation ofK. When n = 1, it is well known that K is the function field of a curve C over k,then v can be defined by a prime divisor of C. We call it the divisor type. Each valuation of divisor type is discrete. There have been attempts to generalize this method to higher dimensional （i.e. n≥2） cases. It is helpful to the calculations of K-groups and the study of rigid analytic spaces.In this paper we study the case when n = 2, it is much more difficult than the casewhen n = 1. We define the height of a valuation and the （?） -degree of a monomialax^syⁿ, s∈Q, n∈N, a∈k , for any given formal series（?）（x） = sum from i=1 to +∞a_ix^r_i,a_i∈k,r_i∈Q, 0 < r₁ < r₂ <…, （?）r_i= r < +∞. Based on this the author obtains thecomplete classification of k -valuations on surfaces. In addition, we get the relationship between the valuation and transcendental series. Furthermore, we show that all the nontrivial k -valuations can be given by the infinite sequences of blowing-ups and give the process of blowing-ups.A parallel problem is the classification of valuations on arithmetic surfaces, i.e. theclassification of all the valuations of a finitely generated field extension of Q withtranscendental degree 1.In this paper, we give the definition of the height of a valuation and the definition ofthe big field C_p,G, where p is a prime and G（?）R is an additive subgroupcontaining 1. We conclude that C_p,G is a field and C_p,G is algebraically closed. Based on this the author obtains the complete classification of valuations on arithmetic surfaces. Furthermore, for any m≤n∈Z, let V_m,n be an R -vector spaceof dimension n-m + 1 , whose coordinates are indexed from m to n . We generalize the definition of C_p,G, where p is a prime and G（?）V_m,n is an additivesubgroup containing 1. We also conclude that C_p,G is a field if m≤0≤n.更多还原