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Study on subrings of polynomial rings

多项式环的子环研究

基本信息

批准号：
16540018
负责人：
ONODA Nobuharu
金额：
$ 0.64万
依托单位：
University of Fukui
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540018/
关键词：
International collaboration India polynomial ring finite generation fiber ring valuation ring

项目摘要

The purpose of the study is to consider the following problem on subrings A of a polynomial ring in n-variables over a commutative ring R :(1) Find conditions for A to be finitely generated over R.(2) Find conditions for A to be a polynomial ring or an A^<[r]->fibration over R.For the first problem, in collaboration with Dr.Amartya K.Dutta, I investigated the case where R is a discrete valuation ring and n=1, and gave a condition for the closed fiber of A over R to be finitely generated. In connection with this result, I studied Noetherian subrings A of a polynomial ring in one variable over a unique factorization domain R, and gave a condition for A to be finitely generated over R. Furthermore I proved that, under this condition, A is a polynomial ring.For the problem (2), I investigated a faithfully flat integral domain A over a unique factorization domain R such that generic and codimension one fibers of A over R are polynomial rings in one variable. I proved that such A is a direct limit of certain algebras, and using this result I gave a condition for A to be a polynomial ring.Concerning the problem (2), I studied the following problem with Professor T.Asanuma (Toyama University) :Let R be a valuation ring with quotient field K and let V be a valuation ring of an algebraic function field K(x,y) in one variable over K such that V dominates R. Find out the algebraic structure of the residue field of V.For this problem, we investigated the case where K(x,y) is a hyperelliptic function field defined by y^2=x^n+ax+b, and proved that among the valuation rings V of K(x,y) dominating R, there exists at most one V such that the residue field of V is not a rational function field in one variable over the residue field of R. Furthermore, for such V, we determined the defining equation of the residue field of V.

本文研究交换环R上n元多项式环的子环A的下列问题：（1）求A在R上生成的条件。(2)对于第一个问题，我与Amartya K.Dutta博士合作，研究了R是离散赋值环且n=1的情况，并给出了R上A的闭纤维是n-生成的条件。结合这个结果，研究了唯一分解整环R上的一元多项式环的Noether子环A，并给出了A在R上生成的一个条件.对于问题（2），研究了唯一分解整环R上的忠实平坦整环A，使得R上A的通有和余维为1的纤维是单变量多项式环.关于问题（2），我与浅沼教授（富山大学）研究了以下问题：设R是具有商域K的赋值环，V是K上的一元代数函数域K（x，y）的赋值环，使得V支配R。对于这个问题，我们研究了K（x，y）是由y ^2 =x^n+ax+B定义的超椭圆函数域的情形，证明了在K（x，y）控制R的赋值环V中，至多存在一个V使得V的剩余域不是R的剩余域上的一元有理函数域.进一步，对这样的V，我们确定了V的剩余域的定义方程。