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鐃緒申鐃緒申鐃� II 12 鐃塾駕申鐃緒申

鐃緒申鐃緒申 12.1   $A=\mbox{${\Bbb R}$ }[X,Y,Z]$ 鐃緒申鐃緒申 $A$ 鐃舜の環緒申同鐃緒申鐃緒申鐃緒申 $\phi$ 鐃緒申

\begin{displaymath}\phi(X)=XZ,\quad \phi(Y)=(YZ),\quad \phi(Z)=Z \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申襦ｏ申鐃緒申鐃祝￥申$A$ 鐃塾ワ申鐃叔ワ申鐃緒申 $I$ 鐃緒申

\begin{displaymath}I=(X^2+Y^2-1) \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申襦ｏ申鐃緒申里箸鐃緒申鐃�
1.
$V(I)$ 鐃塾ワ申鐃緒申佞粒鐃緒申鐃緒申鐃書きなわ申鐃緒申鐃緒申
2.
$\psi=\pi\circ\phi: A \to A/I $ 鐃塾核わ申鐃緒申覆鐃緒申鐃緒申鐃� 鐃緒申鐃緒申鐃緒申 $\pi:A\to A/I$ 鐃熟種申鐃緒申鐃淑射縁申 ($p$ 鐃緒申鐃出わ申鐃銃わ申鐃塾ワ申鐃初ス $[p]$鐃緒申鐃出縁申鐃緒申鐃緒申鐃緒申鐃銃縁申鐃�)鐃叔わ申鐃暑。
3.
${}^a\psi$ 鐃緒申鐃緒申鐃塾鰹申鐃緒申鐃緒申颪�なわ申鐃緒申鐃緒申

(鐃緒申鐃緒申)

(1) 鐃緒申鐃塾件申鐃熟縁申鐃緒申鐃叔わ申鐃獣て￥申鐃淳ではなわ申鐃緒申

\includegraphics[scale=0.5]{tube.ps}

(2) $\operatorname{Ker}(\psi)$ 鐃緒申

\begin{displaymath}\operatorname{Ker}(\psi)=\{f(X,Y,Z)\in \mbox{${\Bbb R}$}[X,Y,Z]; f(XZ,YZ,Z)\in (X^2+Y^2-1)\} \end{displaymath}

鐃叔わ申鐃暑。鐃緒申鐃緒申呂鐃緒申鐃塾計誌申鐃叔わ申鐃暑が鐃緒申 鐃殉わ申 $f_0(X,Y,Z)=X^2+Y^2-Z^2$ 鐃緒申 $\operatorname{Ker}(\psi)$ 鐃緒申鐃緒申鐃暑こ鐃夙わ申鐃叔э申鐃緒申鐃緒申澆鐃緒申鐃緒申鐃� 鐃緒申鐃獣わ申鐃緒申鐃緒申

\begin{displaymath}f_0(XZ,YZ,Z)=(XZ)^2+(YZ)^2-Z^2=(X^2+Y^2-1)Z^2 \end{displaymath}

鐃緒申鐃緒申鐃緒申立鐃縦わ申鐃緒申任鐃緒申襦� 鐃緒申鐃緒申鐃緒申鐃獣て￥申 $\operatorname{Ker}(\psi)\supset (f_0)$. $\operatorname{Ker}(\psi)$ 鐃緒申 $(f_0)$ 鐃夙わ申鐃渋際わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃夙を示わ申鐃緒申鐃緒申砲蓮鐃� $f_0$ 鐃祝わ申鐃緒申蟷誌申鐃緒申鐃術わ申鐃緒申里鐃緒申茲わ申鐃� 鐃緒申鐃淑わ申鐃緒申鐃� $g\in \operatorname{Ker}(\psi)$ 鐃緒申任鐃春にとわ申函鐃� $g$ 鐃緒申 $f_0$ 鐃緒申(鐃術随申 $X$ 鐃緒申鐃緒申鐃旬わ申鐃緒申)鐃緒申襪鰹申箸砲鐃所、 $q,r\in \mbox{${\Bbb R}$ }[X,Y,Z] $ 鐃叔￥申

\begin{displaymath}g=f_0q+r, \quad \end{displaymath}

鐃緒申澆鐃緒申鐃緒申鐃� $r$ 鐃緒申 $X$ 鐃祝関わ申鐃暑次鐃緒申鐃緒申 $2$ 鐃淑駕申鐃緒申鐃夙なわ申鐃塾わ申存鐃淳わ申鐃暑。 $r$ 鐃緒申 $X$ 鐃祝関わ申鐃緒申 $2$ 鐃緒申鐃淑駕申鐃淑のわ申鐃緒申鐃初、鐃緒申鐃緒申鐃�

\begin{displaymath}r(X,Y,Z)=r_1(Y,Z)X+r_0(Y,Z) \end{displaymath}

鐃夙わ申鐃緒申鐃緒申鐃祝書いてみわ申弌鐃�

\begin{displaymath}0=\psi(g)=\psi(r)=r_1(YZ,Z)XZ+r_0(YZ,Z) \end{displaymath}

$X$ 鐃塾件申鐃緒申鐃緒申鐃緒申鐃旬わ申鐃銃￥申多鐃準式鐃夙わ申鐃緒申 $r_1=0$鐃緒申 同鐃粛にわ申鐃銃￥申多鐃準式鐃夙わ申鐃緒申 $r_0=0$鐃緒申鐃緒申鐃暑こ鐃夙わ申鐃叔わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $g=f_0q$. 鐃緒申鐃淑わ申鐃� $g\in (f_0)$ 鐃緒申鐃緒申鐃緒申鐃叔わ申鐃緒申錣縁申鐃緒申鐃�

(3) 鐃瞬居申鐃叔駕申鐃駿わ申鐃獣わ申鐃処う鐃祝わ申鐃緒申榔濘鐃緒申任鐃緒申襦�

\includegraphics[scale=0.5]{cone.ps}

鐃緒申鐃緒申 12.2   $B=\mbox{${\Bbb R}$ }[X,Y]$ 鐃緒申鐃緒申 $B$ 鐃舜の環緒申同鐃緒申鐃緒申鐃緒申 $\varphi$ 鐃緒申

\begin{displaymath}\varphi(X)=XY,\qquad \varphi(Y)=Y \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申襦ｏ申鐃緒申鐃祝￥申$B$ 鐃塾ワ申鐃叔ワ申鐃緒申 $J$ 鐃緒申

\begin{displaymath}J=(Y-X^2+1) \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申襦ｏ申鐃緒申里箸鐃緒申鐃�
1.
鐃緒申鐃緒申10.1 鐃祝なわ申辰鐃� ${}^a\varphi(S)$ 鐃薯グワ申佞暴颪�なわ申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申

\begin{displaymath}S=\{ (x,y)\in \mbox{${\Bbb R}$}^2; x \in {\mbox{${\Bbb Z}$}}\text{ or } y \in {\mbox{${\Bbb Z}$}}\} \end{displaymath}

鐃叔わ申鐃暑。
2.
$V(J)$ 鐃塾ワ申鐃緒申佞粒鐃緒申鐃緒申鐃書きなわ申鐃緒申鐃緒申
3.
$\rho=\pi\circ\varphi: B \to B/J $ 鐃塾核わ申鐃緒申覆鐃緒申鐃緒申鐃� 鐃緒申鐃緒申鐃緒申 $\pi:B\to B/J$ 鐃熟種申鐃緒申鐃淑射縁申 ($p$ 鐃緒申鐃出わ申鐃銃わ申鐃塾ワ申鐃初ス $[p]$鐃緒申鐃出縁申鐃緒申鐃緒申鐃緒申鐃銃縁申鐃�)鐃叔わ申鐃暑。
4.
${}^a\rho$ 鐃緒申鐃緒申鐃塾鰹申鐃緒申鐃緒申颪�なわ申鐃緒申鐃緒申

(鐃緒申鐃緒申)

(1) 鐃緒申略鐃緒申鐃暑が鐃緒申鐃瞬格子」わ申鐃緒申鐃淑わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃淑わ申里鐃塾になわ申覆鐃緒申鐃� 鐃瞬違い鐃淑のに居申鐃重わ申鐃緒申鐃淳わ申鐃緒申鐃緒申 直鐃緒申鐃緒申鐃緒申鐃緒申丹念鐃緒申鐃宿わ申鐃緒申鐃緒申鐃緒申匹鐃緒申里粘鐃獣縁申覆呂鐃緒申任鐃緒申襦�

(2) 鐃緒申鐃緒申呂笋居申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃淑�鐃緒申鐃叔わ申鐃暑。 $S$ 鐃緒申(2)鐃緒申同鐃緒申鐃祝書き刻申鐃緒申函鐃緒申鐃緒申里茲�鐃淑ワ申鐃緒申佞砲覆襦�

\includegraphics[scale=0.5]{parabora.ps}

(3) $T=XY,U=Y$ 鐃夙わ申鐃緒申鐃銃￥申 $X,Y$鐃塾関件申鐃緒申 $Y-X^2+1=0$ 鐃緒申鐃緒申 $T,U$ 鐃塾関件申鐃緒申鐃緒申 鐃竣誌申鐃緒申鐃緒申鐃叔醐申鐃� $T,U$ 鐃緒申 $X,Y$ 鐃緒申鐃瞬わ申鐃緒申鐃緒申鐃緒申个茲わ申鐃� 鐃緒申鐃塾常申鐃熟件申鐃緒申的鐃緒申 $X=T/U,Y=U$ 鐃夙わ申鐃緒申鐃緒申鐃祝逆に解けるか鐃緒申擇任鐃緒申襦� 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申的鐃淑計誌申鐃薯しわ申鐃緒申鐃夙にわ申鐃緒申鐃緒申12.1鐃緒申(2)鐃塾駕申鐃緒申能劼戮鐃緒申茲�鐃淑ワ申鐃緒申鐃獣ワ申鐃緒申 鐃緒申鐃銃わ申鐃緒申鐃駿わ申鐃叔わ申鐃暑。

$(X^2-Y^2-Y^3)$

(4) 鐃緒申鐃塾ワ申鐃緒申佞鐃瞬居申鐃叔書いわ申鐃緒申鐃夙わ申鐃緒申鐃暑。 鐃緒申鐃緒申鐃緒申鐃獣て詳わ申鐃緒申鐃緒申鐃緒申鐃熟常申略鐃緒申鐃暑。 (1)鐃緒申(3)鐃夙わ申同鐃緒申鐃祝書くと￥申 鐃緒申鐃塾よう鐃祝なる。 鐃緒申鐃緒申鐃緒申鐃塾ワ申鐃緒申佞鐃緒申鐃駿わ申伐鐃緒申鐃緒申鐃緒申鐃緒申辰討鐃緒申襪�分鐃緒申鐃緒申笋刻申鐃緒申鐃緒申鐃緒申鐃緒申鐃� 鐃緒申鐃緒申佞鬟好鐃緒申鐃祝書いてわ申鐃緒申鐃銃￥申 鐃緒申鐃緒申鐃緒申硫鐃緒申鐃緒申佞鐃述っわ申鐃緒申鐃瞬わ申(鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申劼佑蠅�鐃獣わ申鐃�)鐃緒申 鐃緒申鐃順う鐃宿わ申鐃塾よう鐃淑件申鐃祝なる。鐃�

\includegraphics[scale=0.5]{singular.ps}


Yoshifumi Tsuchimoto
2001-07-24