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 $lambda_0,lambda_1in K$.

在文献[1]的基础上，给出了特征为$p&gt；2$的域$k上几何正规仿射平面曲线的纯不可分$k$形式的代数结构定理.(2)设$R$是具有商域$K$和剩余域$k$的离散赋值环.对于$R$上有限生成的整闭域$A$，当普通纤维$AoTimes_Rk$是$K$上的多项式环或一元Laurent多项式环时，我们给出了简化的$k$-代数$(AoTimes_Rk)_{Rm red}$的显式代数结构.(3)设$V$是具有剩余域$k$和商域$K$的赋值环.设$K(x，y)$是由多项式方程$f(x，y)=0定义的$K$上的一元代数函数域，$V_{xy}$是具有剩余域$k_{xy}$的$K(x，y)$赋值环.设$V_{xy}$支配$V$，且$k_{xy}/k$是超越域扩张。则$k_{xy}$是$k$上的一元代数函数域。我们考虑$k_{xy}$的$k$-代数结构。特别地，当$k$是特征为零的代数闭域时，当$y^2=x^m+lambda_1x+lambda_0$时，我们将显式地给出这样的$k_{xy}=k(z，w)$的定义方程$g(z，w)$，其中$0&lt；m$和$lambda_0，lambda_1在K$中。

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