Valuations On Surfaces And On Arithmetic Surfaces

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.