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\begin{center}\begin{large}\textbf{Math 137 - Problem Set 2\\
Due Wednesday, Feb 12}\end{large}\end{center}



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All rings are commutative, and $k$ is an algebraically closed field.

\begin{enumerate} 

\item
Let $I$ be an ideal in a ring $R$. Show that there is a one-to-one correspondence between radical ideals in $R$ containing $I$ and radical ideals in $R/I$. (Convince yourself that the same holds for prime ideals. You don't need to write up that part.)

\item
\begin{enumerate}
\item  Let $I \subset k[x_1, \ldots, x_n]$ be an ideal. Show that $I$ is radical if and only if it is equal to the intersection of all the maximal ideals containing it.
\item Show that the radical of the ideal $I = (x^2-2xy^4 +y^6, y^3 -y)\subset \C[x, y]$ is the intersection of three maximal ideals.
\end{enumerate}

\item 
Let $X = V(x^2-yz, xz-x) \subset \A_\C^3$. Find the irreducible components of $X$ and their corresponding prime ideals. Make sure you justify your solution.

\item Let $a_1, a_2, \ldots a_n \in k$. Show that $(x_1 - a_1, \ldots, x_n - a_n) \subset k[x_1, \ldots, x_n]$ is a maximal ideal. (Hint: reduce to the case where the $a_i$ are all 0.)

\item Let $X \subset \A^n$ be a set (not necessarily algebraic). The \textbf{\textit{Zariski closure}} of $X$, denoted $\overline{X}$, is the intersection of all Zariski closed sets containing $X$. Show that $V(I(X))  = \overline{X}$.


\item A subset of affine space $U \subset \A^n$ is called \textbf{\textit{compact}} (in the Zariski topology) if for every collection $\{U_i\}_{i\in J}$ (where $J$ is some indexing set) of Zariski open sets such that if $$U \subset \bigcup_{i \in J} U_i,$$ then $U$ is also contained in some finite union of the $U_i$. That is, there is some finite set $L \subset J$ such that  $$U \subset \bigcup_{i \in L} U_i.$$ More concisely, $U$ is compact if every open cover has a finite subcover.
Show that if $X \subset \A_k^n$ is an algebraic set, $X$ is compact in the Zariski topology.
\end{enumerate}

\noindent \textbf{Bonus topology questions} (Extra credit)

\begin{enumerate}
  \setcounter{enumi}{6}
  \item Identify $\A^1 \times \A^1$ with $\A^2$ in the natural way. Show that the Zariski topology on $\A^2$ is not the product topology induced by the Zariski topology on $\A^1$.

\item For each $f \in k[x_1, \ldots, x_n]$ define $U_f $ to be the set of points $P \in \A^n$ such that $f(P) \neq 0$. Prove that the collection of all such sets $U_f$ forms a basis for the Zariski topology on $\A^n$.

\end{enumerate}

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