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鐃緒申鐃緒申鐃� II 鐃緒申鐃緒申 No.5

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{多項式関数・座標環}

鐃緒申鐃緒申 5.1  
1.
$k[X_1,\dots,X_n]$ 鐃塾各醐申鐃緒申 $k^n$ 鐃緒申鐃� $k$-鐃粛関随申鐃夙醐申鐃暑こ鐃夙わ申鐃叔わ申鐃暑。
2.
鐃緒申鐃緒申鐃出縁申鐃祝わ申辰董鐃渋随申犲逸申鐃渋�鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃熟鰹申鐃緒申鐃叔わ申足鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃出縁申鐃緒申鐃暑。
3.
$k$ 鐃塾醐申鐃塾個随申鐃緒申無鐃渋個なわ申鐃� $p\in k[X_1,\dots, X_n]$ 鐃熟わ申鐃緒申 $k^n$ 鐃叔わ申 鐃粛にわ申辰憧鐃緒申鐃緒申坊鐃緒申蠅居申鐃暑。

鐃緒申鐃緒申 5.2   $k^n$ 鐃緒申鐃緒申鐃重�鐃緒申分鐃緒申鐃緒申 $V$ 鐃緒申鐃縦件申鐃銃わ申鐃緒申鐃緒申鐃緒申鐃塾とわ申鐃緒申
1.
多鐃準式鐃緒申 $k[X_1,\dots,X_n]$ 鐃塾各醐申 $p$ 鐃緒申 $V$ 鐃緒申鐃� $k$-鐃粛関随申 $p\vert _V$鐃緒申鐃緒申鐃暑。
2.
鐃緒申鐃緒申鐃出縁申鐃祝わ申辰董鐃渋随申犲逸申鐃渋�鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃熟鰹申鐃緒申鐃叔わ申足鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃出縁申鐃緒申鐃暑。
3.
$p\vert _V=q\vert _V \quad {\Leftrightarrow}\quad p-q\in I(V)$

鐃緒申鐃� 5.1   $k^n$ 鐃緒申鐃緒申鐃重�鐃緒申鐃緒申 $V$ 鐃緒申鐃出わ申鐃銃￥申 $k[X_1,\dots,X_n]/I(V)$ 鐃塾わ申鐃夙わ申 $V$ 鐃塾削申標鐃縦と呼び￥申$A(V)$ 鐃夙わ申鐃緒申鐃緒申

鐃緒申鐃緒申砲鐃緒申 $A(V)$ 鐃緒申 鐃緒申$V$ 鐃緒申鐃� $k$-鐃粛関随申鐃緒申鐃塾の緒申鐃緒申廚鐃� 鐃緒申分鐃緒申鐃緒申鐃粛随申鐃緒申襪鰹申箸鐃緒申錣�鐃暑。

(鐃緒申足)

鐃緒申鐃縮に￥申$I(V)$ 鐃淑鰹申鐃緒申 $R=k[X_1,\dots,X_n]$ 鐃塾ワ申鐃叔ワ申鐃緒申 $I$ 鐃緒申鐃出わ申鐃銃も、 $R/I$ 鐃緒申鐃瞬削申標鐃縦」とみなわ申鐃緒申茲�鐃淑器申鐃緒申鐃緒申的鐃出象わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申任鐃緒申襦� 鐃緒申鐃塾よう鐃淑わ申里里鐃緒申鐃緒申鐃緒申屮櫂團鐃初ー鐃淑わ申里魯鐃緒申侫鐃緒申鐃緒申鵐好鐃緒申鐃緒申鐃� $\operatorname{Spec}(R/I)$ 鐃夙わ申个鐃緒申鐃塾で￥申鐃緒申鐃重�鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申悗任鐃緒申羶甘�鐃緒申鐃緒申鐃緒申未鐃緒申鐃緒申鐃�

$V$ 鐃塾削申標鐃縦わ申 $V$ 鐃塾常申鐃緒申鬚�なり握鐃獣てわ申鐃暑。鐃緒申鐃緒申呂鐃緒申里覆鐃緒申琉鐃縦わ申鐃緒申鐃緒申 鐃述べよう鐃緒申

鐃緒申鐃緒申 5.3   $k^n$ 鐃緒申鐃緒申鐃重�鐃緒申鐃緒申 $V$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� 鐃盾し鐃緒申 $A(V)$ 鐃緒申 $1$ 鐃淑鰹申鐃塾駈申鐃緒申鐃緒申 $e$ (鐃緒申鐃淑わ申鐃緒申鐃�$e^2=e$ 鐃夙なる元) 鐃緒申鐃銃ば￥申$V$ 鐃熟互わ申鐃祝醐申鐃緒申覆鐃緒申鐃緒申任覆鐃緒申鐃緒申的鐃緒申鐃緒申 $V_0,V_1$ 鐃緒申 鐃渋緒申鐃緒申砲覆襦� 鐃緒申鐃緒申的鐃祝は￥申

\begin{displaymath}V_0=\{x\in V; e\vert _V(x)=0\} \end{displaymath}


\begin{displaymath}V_1=\{x\in V; e\vert _V(x)=1\} \end{displaymath}

Zariski 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申廚鐃緒申鐃緒申鐃緒申鐃緒申弌鐃緒申鐃緒申鐃緒申鐃緒申麓鐃緒申里茲�鐃祝醐申鐃緒申鐃緒申鐃夙わ申任鐃緒申襦�

$A(V)$ 鐃緒申 $1$ 鐃淑鰹申鐃塾駈申鐃緒申鐃緒申鐃緒申鐃緒申討弌鐃�$V$ 鐃緒申($k^n$ 鐃緒申 Zariski 鐃緒申鐃所か鐃緒申鐃� 誘導鐃緒申鐃緒申砲弔鐃緒申鐃�)連鐃緒申任覆鐃緒申鐃�

(鐃渋際にはわ申鐃緒申命鐃緒申鐃緒申亢鐃緒申鐃緒申鐃緒申鐃緒申鐃縮ｏ申鐃緒申箸鐃巡す鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申)

鐃緒申檗鐃緒申鐃緒申鐃緒申鐃緒申5.1鐃殉わ申鐃緒申5.2鐃緒申鐃緒申鐃薯するこ鐃夙￥申 両鐃緒申鐃緒申鐃薯してわ申茲わ申鐃緒申鐃宿常申鐃緒申鐃緒申鐃緒申里呂匹鐃緒申蕕�鐃処い鐃緒申鐃塾みでわ申鐃暑。

鐃緒申鐃緒申 5.1   鐃緒申鐃塾各￥申鐃塾環の駈申鐃緒申鐃緒申($0,1$ 鐃緒申泙鐃�)鐃緒申鐃緒申蠅刻申鐃緒申鐃緒申鐃緒申鐃緒申鐃淑わ申鐃緒申鐃緒申 (鐃緒申鐃緒申鐃緒申 $X$ 鐃淑わ申鐃緒申 $X,Y$ 鐃緒申多鐃準式鐃叔緒申表鐃緒申鐃緒申鐃叔わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申磴わ申鐃塾わ申 鐃緒申鐃緒申茲�鐃緒申鐃緒申鐃熟わ申鐃暑こ鐃夙￥申 鐃殉わ申鐃緒申鐃銃￥申鐃塾駈申鐃緒申鐃緒申 $e$ 鐃祝つわ申鐃銃￥申鐃緒申鐃緒申鐃緒申 $e^2=e$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃夙わ申 鐃緒申鐃緒申鐃銃わ申鐃緒申鐃緒申鐃夙￥申 $0,1$ 鐃祝つわ申鐃銃は常申略鐃緒申鐃銃わ申茲わ申鐃�)
1.
$\mbox{${\Bbb R}$ }[X]/\langle X^3-X\rangle_{\text{ideal}}$         ($0,1$ 鐃緒申泙鐃緒申8鐃緒申)
2.
$\mbox{${\Bbb R}$ }[X,Y]/\langle X^2Y-X\rangle_{\text{ideal}}$         ($0,1$ 鐃緒申泙鐃緒申4鐃緒申)

(鐃緒申鐃緒申) 鐃緒申鐃緒申鐃緒申鐃叔は￥申鐃緒申鐃緒申鐃緒申鐃緒申鐃春わ申促鐃緒申鐃緒申鐃緒申鐃�

\begin{displaymath}\mbox{${\Bbb R}$}[X,Y]/\langle X^2Y-X\rangle_{\text{ideal}} \end{displaymath}

鐃夙わ申鐃緒申鐃緒申鐃緒申鐃緒申僂鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申咾鐃緒申啗鐃緒申鐃緒申茲�鐃祝わ申鐃緒申鐃�

\begin{displaymath}\mbox{${\Bbb R}$}[X,Y]/(X^2Y-X) \end{displaymath}

鐃夙書かわ申襪鰹申箸鐃緒申鐃緒申鐃渋随申鐃緒申鐃�

鐃緒申鐃緒申 5.2  
1.
${\Bbb C}[X,Y,Z]/(XZ-1,X-Y,X+Y)$ 鐃塾駈申鐃緒申鐃緒申鐃叔￥申$0,1$ 鐃淑鰹申鐃塾わ申里鐃緒申鐃縦わ申鐃緒申鐃処。
2.
鐃緒申鐃縮にわ申鐃緒申鐃緒申5.3鐃塾逆わ申鐃緒申鐃緒申鐃緒申鐃淑わ申鐃緒申 鐃緒申鐃緒申鐃緒申鬚△鐃緒申茵�


Yoshifumi Tsuchimoto
2001-05-14