Research of the cancellation problem in affine algebraic geometry

仿射代数几何抵消问题的研究

• 批准号: 16540016
• 负责人: ASANUMA Teruo
• 金额: $ 1.73万
• 依托单位: University of Toyama
• 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-16540016/
• 关键词: affine algebraic geometry; algebraic curve; polynomial ring; valuation ring; cancellation problem; algebraic function field; 代数関数

From the abstracts of papers in the references :(1)We give an algebraic structure theorem of purely inseparable $k$-forms of geometrically normal affine plane curves over a field $k$ of characteristic $p>2$.(2)Let $R$ be a discrete valuation ring with quotient field $K$ and residue field $k$. For a finitely generated integrally closed domain $A$ over $R$, we give an explicit algebraic structure of the reduced $k$-algebra $(Aotimes_Rk)_{rm red}$ when the generic fibre $Aotimes_RK$ is a polynomial ring or a Laurent polynomial ring in one variable over $K$.(3)Let $V$ be a valuation ring with residue field $k$ and quotient field $K$. Let $K(x,y)$ be an algebraic function field in one variable over $K$ defined by a polynomial equation $f(x,y)=0$ and let $V_{xy}$ be a valuation ring of $K(x,y)$ with residue field $k_{xy}$. Suppose that $V_{xy}$ dominates $V$ and $k_{xy}/k$ is a transcendental field extension. Then $k_{xy}$ is an algebraic function field in one variable over $k$. We consider a $k$-algebraic structure of $k_{xy}$. In particular when $k$ is an algebraic closed field of characteristic zero, we will give a defining equation $g(z,w)$ of such $k_{xy}=k(z,w)$ explicitly when $y^2=x^m+lambda_1x+lambda_0$ for $0<m$ and $lambda_0,lambda_1in K$.

从文献摘要中得到：（1）给出了特征p>2的域k上的几何正规仿射平面曲线的纯不可分k-型的代数结构定理。（2）设R是具有商域K和剩余域K的离散赋值环。对于R上的积分闭整环A，当泛纤维Aotimes_Rk是K上的多项式环或单变量Laurent多项式环时，给出了约化k-代数（Aotimes_Rk）_{rmred}的显式代数结构.（3）设V是具有剩余域K和商域K的赋值环。令$K（x，y）$是由多项式方程$f（x，y）=0$定义的$K$上的单变量代数函数域，令$V_{xy}$是$K（x，y）$的赋值环，具有剩余域$k_{xy}$。设V_{xy}$优于V$，且k_{xy}/k$是超越域扩张.则$k_{xy}$是在$k$上的单变量代数函数域。我们考虑了k_{xy}$的一个k-代数结构。特别地，当k是特征为零的代数闭域时，当0 <m，k中的y ^2 =x^m+ x ^da_1x + x ^da_0时，我们将明确地给出k_{xy}=k（z，w）的定义方程g（z，w）。

项目成果

Generic fibrations by A^1 and A^* over discrete valuation rings

A^1 和 A^* 在离散评估环上的通用纤维

• 发表时间: 2005
• 期刊: Contemporary Mathematics 369
• 作者: T.Asanuma;N.Onoda
• 通讯作者: N.Onoda

Valuation rings of algebraic function fields in one variable

一个变量中代数函数域的评估环

• 发表时间: 2006
• 期刊: 宮西正宜教授退官記念論文集 Affine Algebraic Geometry 1(発行予定)
• 作者: 浅沼照雄;小野田信春
• 通讯作者: 小野田信春

Purely inseparable $k$-forms of affine algebraic curves

仿射代数曲线的纯粹不可分离的 $k$ 形式

• 发表时间: 2005
• 期刊: Affine algebraic geometry, Contemporary Mathematics 369
• 作者: Shiga;H.;浅沼照雄
• 通讯作者: 浅沼照雄

Generic fibrations by $A^1$ and $A^*$ over discrete valuation rings

$A^1$ 和 $A^*$ 在离散估值环上的通用纤维

• 发表时间: 2005
• 期刊: Affine Algebraic Geometry, Contemp.Math. 369
• 作者: Teruo Asanuma;S.M.Bhatwadekar;Nobuharu Onoda
• 通讯作者: Nobuharu Onoda