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\begin{center}\begin{large}\textbf{Math 137 - Problem Set 4\\
Due Wednesday, Feb 26}\end{large}\end{center}



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

\begin{enumerate} 

\item Verify the following claims/assumptions from class.
\begin{enumerate}
\item Let $R$ be a reduced finitely generated $k$-algebra (reduced means $(0)$ is radical). Then $R$ is isomorphic to the coordinate ring of some algebraic set.
\item Let $\phi, \psi: X \to Y$ be morphisms of algebraic sets. If $\phi^* = \psi^*$, then $\phi = \psi$.
\end{enumerate}



\item Let $\phi: \A^2 \to \A^2$ be the regular map given by $\phi(x,y) = (x,xy)$. 
\begin{enumerate}
\item Show that the image of $\phi$ is neither open nor closed. 
\item What is the closure of the image of $\phi$?
\end{enumerate}

\item Let $V = V(x^3- y^2) \subset \A^2$. Consider the morphism $\phi: \A^1 \to V$ given by $\phi(t) = (t^2, t^3)$. 
\begin{enumerate}
\item Show that $\phi$ is a bijection but not an isomorphism.
\item Show that $V$ is not isomorphic to $\A^1$. (Note that this is not implied by (a).)
\end{enumerate}

\item Let $V = (x^2 - y, x^3 - z) \subset \A^3.$ Define $\phi : \A^1 \to V$ by $\phi(t) = (t,t^2, t^3)$. Show that $\phi$ is an isomorphism.

\item For the morphisms defined in the previous three problems, explicitly describe the induced map of coordinate rings.


\item Let $\phi: X \to Y$ be a morphism of algebraic sets. Show that if $X$ is irreducible, the Zariski closure of $\phi(X)$ is irreducible.

\item Let $$V = \left\{(x,y,z,w) \in \A^4  \mid \text{rank} \begin{pmatrix} y & z & w \\ x & y & z \end{pmatrix} \leq 1\right\}.$$ Show that $V$ is an irreducible variety. \textit{(Hint: Can you find a map from a certain 2-dimensional irreducible variety that has $V$ as its image?)}

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 \noindent \textbf{Bonus question} (Extra credit)

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  \item Let $X \subset \A^n$ and $Y \subset \A^m$ be algebraic sets. Show that the coordinate ring of $X \times Y \subset \A^{m+n}$ is isomorphic to $\Gamma(X) \otimes_k \Gamma(Y)$.


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