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\begin{center}{\bf M953, Homework 4, due Friday, February 22, 2013}\end{center}
\noindent Instructions: Do any three problems.
\begin{itemize}
\item[(1)] Let $k$ be a field and let
$J_1,J_2\subseteq k[x_1,\ldots,x_m]$ be ideals.
\begin{itemize}
\item[(a)] Show that $V(J_1)=V(J_2)$ in ${\bf A}^m(k)$ if $\sqrt{J_1}=\sqrt{J_2}$.
\item[(b)] If $k$ is algebraically closed, show that $\sqrt{J_1}=\sqrt{J_2}$
if $V(J_1)=V(J_2)$ in ${\bf A}^m(k)$.
\item[(c)] For any field $k$, show that $\sqrt{J_1}=\sqrt{J_2}$
if $V(J_1)=V(J_2)$, if for any ideal $J\subseteq k[x_1,\ldots,x_m]$
we take $V(J)$ to mean all $K$-points of $V(J)$ in ${\bf A}^m(K)$ for
every field extension $K$ of $k$. (Recall that a point $p\in {\bf A}^m(K)$ is a $K$-point
of $V(J)$ if $f(p)=0$ for all $f\in J$, where we regard $J$ as a subset of
$K[x_1,\ldots,x_m]$ via the inclusion $k[x_1,\ldots,x_m]\subseteq K[x_1,\ldots,x_m]$.)
You may assume the fact from commutative algebra that
$\sqrt{J'}\cap k[x_1,\ldots,x_m]=\sqrt{J}$, whenever $J\subseteq k[x_1,\ldots,x_m]$
is an ideal and $J'$ is the ideal in $K[x_1,\ldots,x_m]$ generated by $J$.
\end{itemize}
\vskip\baselineskip
\item[(2)] Let $k$ be a field and let $S\subseteq {\bf A}^m(k)$ be any subset. Show that
the closure of $S$ in the Zariski topology is $V(I(S))$. (Recall that the closure of a set
$S$ is the intersection of all closed sets containing $S$.)
\vskip\baselineskip
\item[(3)] Let $S\subset {\bf A}^2({\bf R})$ be the set $(a,b)$ with $ab=1$ and $a>0$.
Show that the Zariski closure of $S$ is $V(xy-1)$, where ${\bf R}$ is the field of reals and
we take the coordinate ring of ${\bf A}^2({\bf R})$ to be ${\bf R}[x,y]$.
\vskip\baselineskip
\item[(4)] Let $C=V(y-x^2)\subset{\bf A}^2(k)$, where $k={\bf Q}$ is the field of rationals.
Let $K$ be the reals. Thus $\pi=3.14159265\ldots\in K$. Show that $p=(\pi,\pi^2)$ is
a $K$-point of $C$, and that $f(p)=0$ for $f\in k[x,y]$ if and only if $g(x,y)=y-x^2$ divides $f$.
Conclude that $J\cap k[x,y]=(y-x^2)$, where the intersection takes place in $K[x,y]$ and
$J\subset K[x,y]$ is the ideal generated by
$x-\pi$ and $y-\pi^2$. (You may assume that $\pi$ is transcendental over $k$. Hint:
apply Problem 5 on Homework 1.)
\vskip\baselineskip
\item[(5)] Let $k$ be a field and let
$C_1,\ldots,C_r$ and $D_1,\ldots, D_s$ be irreducible algebraic varieties in ${\bf A}^m(k)$.
Assume that $C_i\subseteq C_j$ implies $i=j$ and $D_i\subseteq D_j$ implies $i=j$ (i.e,
none of the $C$'s contains any of the others, and likewise for the $D$'s).
If $C_1\cup \cdots \cup C_r=D_1\cup \cdots \cup D_s$, show that
$r=s$ and, after reordering if need be, that $C_i=D_i$ for each $i$.
\vskip\baselineskip
\item[(6)] Let $C=V(J)\subset {\bf A^3}$
for the ideal $J=(xy^2-xy,xyz-xz,y^3-y^2,y^2z-yz,y^2z-y^2-yz+y,yz^2-yz-z^2+z)
\subset k[x,y,z]$, where $k$ is an algebraically closed field. Find the irreducible components of $C$.
\end{itemize}
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