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鐃緒申鐃緒申鐃� II 鐃緒申鐃緒申 No.9

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{関数環の間の準同型と代数的集合の間の射}

鐃緒申鐃� 9.1 (鐃緒申鐃緒申)   鐃緒申 $R$ 鐃緒申鐃緒申鐃� $S$ 鐃舜の種申鐃緒申 $\phi$ 鐃緒申(鐃縦とわ申鐃銃わ申)鐃緒申同鐃緒申鐃緒申鐃緒申鐃叔わ申鐃緒申箸蓮鐃� $\phi$ 鐃緒申鐃緒申鐃塾常申鐃緒申鐃緒申足鐃緒申鐃緒申箸鐃緒申砲鐃緒申鐃緒申鐃�
1.
$\phi(x+y)=\phi(x)+\phi(y)\qquad (\forall x,y\in R)$
2.
$\phi(xy)=\phi(x)\phi(y)\qquad (\forall x,y\in R)$
3.
$\phi(1)=1$
$\phi$ 鐃緒申鐃緒申両鐃緒申鵬辰鐃緒申鐃緒申鐃獣縁申佑任發�鐃緒申箸鐃緒申砲鐃� $\phi$ 鐃塾逆種申鐃緒申鐃緒申 鐃緒申同鐃緒申鐃緒申鐃緒申鐃祝なるこ鐃夙わ申鐃銃易わ申分鐃緒申鐃暑。鐃緒申鐃塾よう鐃緒申(鐃緒申単鐃粛緒申同鐃緒申鐃叔わ申鐃緒申茲�鐃緒申) $\phi$ 鐃塾わ申鐃夙わ申(鐃縦とわ申鐃銃わ申)同鐃緒申鐃緒申鐃緒申鐃夙わ申鐃緒申鐃緒申

鐃緒申鐃� 9.2 (鐃緒申鐃緒申)   鐃緒申 $k$ 鐃緒申泙鐃緒申 $R$ 鐃緒申鐃緒申鐃� $S$ 鐃舜の緒申同鐃緒申 $\phi$ 鐃緒申 $k$-鐃緒申同鐃緒申鐃緒申鐃緒申鐃叔わ申鐃緒申箸蓮鐃� $\phi$ 鐃緒申 $k$ 鐃塾醐申鐃緒申鐃楯つとわ申鐃緒申 鐃緒申鐃淑わ申鐃緒申鐃�

\begin{displaymath}\phi(c)=c \qquad \forall c\in k \end{displaymath}

鐃緒申鐃緒申鐃緒申立鐃縦とわ申鐃祝醐申鐃緒申鐃緒申

(鐃緒申鐃�1 鐃熟￥申$\phi$ 鐃緒申 $(R,+)$ 鐃緒申鐃緒申 $(S,+)$ 鐃舜の加件申鐃夙わ申鐃銃わ申 鐃緒申同鐃緒申鐃緒申鐃緒申鐃叔わ申鐃暑こ鐃夙わ申鐃緒申辰討鐃緒申襦�)

鐃緒申鐃緒申 9.1   $R,S$ 鐃緒申鐃緒申 $k$ 鐃緒申泙鐃縦とわ申鐃暑。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] $ 鐃緒申鐃緒申 $S$ 鐃舜わ申 $k$-鐃緒申同鐃緒申 $\phi$ 鐃緒申 $X_1,\dots,X_n$ 鐃塾行わ申鐃緒申 $\phi(X_1),\dots,\phi(X_n)$ 鐃祝わ申辰鐃� 鐃緒申佞坊鐃殉る。
2.
鐃春に￥申$S$ 鐃塾醐申 $s_1,\dots,s_n$ 鐃緒申任鐃春わ申与鐃緒申鐃緒申鐃夙わ申鐃緒申

\begin{displaymath}\phi(X_1)=s_1,\dots,\phi(X_n)=s_n \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申 $k[X_1,\dots,X_n] $ 鐃緒申鐃緒申 $S$ 鐃舜わ申 $k$-鐃緒申同鐃緒申鐃緒申唯鐃緒申鐃渋醐申澆鐃緒申襦�
3.
$R$ 鐃塾ワ申鐃叔ワ申鐃緒申 $I$ 鐃緒申与鐃緒申鐃緒申譴随申箸鐃� 鐃緒申$R$ 鐃緒申鐃緒申 $S$ 鐃舜わ申 $k$-鐃緒申同鐃緒申 $\psi$ 鐃叔￥申$\psi\vert I=0$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申痢鐃� 鐃緒申 鐃緒申$R/I$ 鐃緒申鐃緒申 $S$ 鐃舜わ申 $k$-鐃緒申同鐃緒申鐃竣とは逸申鐃出逸申鐃緒申弍鐃緒申鐃緒申襦�

鐃緒申鐃緒申 9.2   鐃緒申鐃重�鐃緒申鐃緒申 $V \subset k^n$ 鐃緒申 $W\subset k^m$ 鐃夙わ申与鐃緒申鐃緒申譴随申箸鐃緒申鐃� 鐃緒申鐃緒申鐃塾ワ申鐃春ワ申鐃緒申弔鬚修譴常申鐃� $A(V)=k[X_1,\dots,X_n]/I(V)$, $A(W)=k[Y_1,\dots,Y_m]/I(W)$鐃夙書くと￥申
1.
$V$ 鐃緒申鐃緒申鐃緒申 $A(V)$ 鐃緒申鐃緒申 $k$ 鐃舜わ申 $k$-鐃緒申同鐃緒申鐃夙逸申鐃出逸申鐃緒申弍鐃緒申鐃緒申襦�
2.
$A(V)$ 鐃緒申鐃緒申 $A(W)$ 鐃舜わ申 $k$-鐃緒申同鐃緒申 $\phi$ 鐃緒申与鐃緒申鐃緒申鐃緒申函鐃� $W$ 鐃緒申鐃緒申 $V$ 鐃舜の種申鐃緒申 ${}^a\phi$ 鐃緒申鐃緒申

\begin{displaymath}{}^a\phi(y_1,\dots,y_m)= (\phi(X_1)(y_1,\dots,y_m), \phi(X_2)(y_1,\dots,y_m), \dots \phi(X_n)(y_1,\dots,y_m)) \end{displaymath}

鐃祝わ申辰鐃緒申鐃殉る。

鐃緒申鐃� 9.3  
1.
$A(V)$ 鐃緒申鐃緒申 $A(W)$ 鐃舜の環緒申同鐃緒申 $\phi$ 鐃祝わ申辰鴇鐃緒申鐃緒申鐃緒申里茲�鐃祝わ申鐃緒申鐃緒申泙鐃� $W$ 鐃緒申鐃緒申 $V$ 鐃舜の種申鐃緒申鐃緒申 $W$ 鐃緒申鐃緒申 $V$ 鐃舜の種申(鐃緒申鐃暑い鐃緒申多鐃準式鐃緒申鐃緒申)鐃夙呼わ申
2.
鐃夙わ申鐃祝￥申鐃盾し $\phi$ 鐃緒申同鐃緒申鐃淑わ申鐃� ${}^a\phi$ 鐃緒申鐃緒申単鐃粛でわ申鐃所、鐃緒申鐃塾逆種申鐃緒申鐃緒申 多鐃準式鐃緒申鐃緒申鐃緒申与鐃緒申鐃緒申鐃暑。鐃緒申鐃塾よう鐃祝環わ申同鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃殉わ申 $W$ 鐃緒申鐃緒申 $V$ 鐃舜わ申鐃緒申単鐃粛のわ申鐃夙わ申鐃緒申鐃重�鐃緒申鐃緒申鐃銃縁申鐃緒申佑箸鐃瞬￥申

鐃緒申 9.1   $k=\mbox{${\Bbb R}$ }$ 鐃夙わ申鐃緒申 $V=\mbox{${\Bbb R}$ }^1$, $W=V(Y^2-X^2-X^3) \subset \mbox{${\Bbb R}$ }^2 $ 鐃夙わ申鐃緒申鐃緒申鐃緒申鐃塾とわ申鐃緒申
1.
$A(V)=\mbox{${\Bbb R}$ }[T]$, $A(W)=\mbox{${\Bbb R}$ }[X,Y]/(Y^2-X^2-X^3)$ 鐃叔わ申鐃暑。
2.
$A(W)$ 鐃叔わ申 $X,Y$ 鐃塾ワ申鐃初ス鐃薯それぞ鐃緒申 $[X],[Y]$ 鐃夙書くわ申鐃夙にわ申鐃緒申函鐃�

\begin{displaymath}[Y]^2-[X]^2-[X]^3=0 \end{displaymath}

鐃緒申鐃淑りた鐃縦￥申
3.
(鐃緒申鐃緒申鐃緒申鐃術わ申鐃緒申鐃�) $W$ 鐃塾常申暴鐃緒申鐃叔わ申鐃緒申佑砲箸辰討呂鐃緒申鐃� $X$ 鐃緒申標鐃緒申 $Y$ 鐃緒申標鐃薯それぞ鐃緒申表鐃緒申鐃舜随申鐃祝は常申里茲�鐃淑関件申鐃緒申鐃緒申鐃暑。
4.
$A(W)$ 鐃緒申鐃緒申 $A(V)$ 鐃舜わ申 $\mbox{${\Bbb R}$ }$-鐃緒申同鐃緒申 $\phi$ 鐃緒申

\begin{displaymath}\phi([X])=T^2-1, \phi([Y])=T(T^2-1) \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申襪鰹申箸鐃緒申任鐃緒申襦�
5.
${}^a\phi$ 鐃熟種申鐃塾よう鐃淑種申鐃緒申鐃叔わ申鐃暑。

\begin{displaymath}{}^a\phi(t)=(t^2-1,t(t^2-1)) \end{displaymath}

鐃緒申鐃緒申 9.1   $R=\mbox{${\Bbb R}$ }[X,Y]$ 鐃塾ワ申鐃叔ワ申鐃緒申 $I,J$ 鐃緒申$I=(Y-X^3)$, $J=(Y^2-X)$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃� $R$ 鐃塾醐申 $p$ 鐃緒申 $R/I,R/J $ 鐃叔のワ申鐃初ス鐃薯それぞ鐃緒申 $[p],\overline{p}$ 鐃夙書くわ申鐃夙にわ申鐃暑。 $R/I$ 鐃緒申鐃緒申 $R/J$ 鐃舜の種申鐃緒申 $\phi$ 鐃緒申

\begin{displaymath}\phi([p(X,Y)])=\overline{p(X^4,X^{12})} \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申鐃夙わ申鐃緒申鐃緒申鐃塾鰹申鐃緒申鐃緒申鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�
1.
$\phi([X^2+Y])$ 鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�
2.
${}^a\phi(1,1)$, ${}^a\phi(4,2)$, ${}^a\phi(9,3)$ 鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 9.2   $R={\Bbb C}[X,Y]/((Y-X^2)(Y-1))$ 鐃緒申鐃緒申 $R$ 鐃舜の環とわ申鐃銃わ申 同鐃緒申鐃緒申鐃緒申 $\phi$ 鐃叔￥申

\begin{displaymath}{}^a\phi(0,0)=(0,1) \end{displaymath}

鐃緒申澆鐃緒申鐃緒申鐃塾わ申鐃縦醐申鐃縦わ申鐃淑わ申鐃緒申鐃緒申

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2001-06-28