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\begin{center}\begin{large}\textbf{Math 137 - Problem Set 3\\
Due Wednesday, Feb 19}\end{large}\end{center}



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

\begin{enumerate} 

\item
Let $R \subset S \subset T$ be integral domains.
\begin{enumerate}
\item If $S$ is module-finite over $R$ and $T$ is module-finite over $S$, prove that $T$ is module-finite over $R$.
\item If $S$ is integral over $R$ and $T$ is integral over $S$, prove that $T$ is integral over $R$.
\end{enumerate}

\item Let $X = V(f)$ be an irreducible hypersurface in $\A^n$. Show that if $Y \subset \A^n$ is an affine variety containing $X$, then $Y =\A^n$ or $Y =X$.


\item Let $R = \C[x,y]$. For each ideal $I \subset R$, find $V(I)$, and compute $\dim_k(R/I)$.
\begin{enumerate}
\item $I = (y^2-x^3, y-x^2)$
\item $I = (y^2-x^2, y^2+x^2)$.
\end{enumerate}

\item Let $V \subset \A^n$ be a nonempty variety. Recall that the \textit{\textbf{ring of regular functions}} on $V$, denoted $\Gamma(V)$, is the set of functions $f: V \to k$ such that $f = F|_V$, where $F \in k[x_1, \ldots, x_n]$, and $F|_V$ denotes the restriction of $F$ to $V$. Let $\phi: k[x_1, \ldots, x_n] \to \Gamma(V)$ be the restriction map $F \mapsto F|_V$.

\begin{enumerate}
\item Check that $\phi$ is a $k$-algebra homomorphism (i.e. a ring homomorphism that acts as the identity on $k$).
\item Show that $\ker(\phi) = I(V)$.
\item Conclude that $\Gamma(V) \cong k[x_1, \ldots, x_n]/I(V)$.
\end{enumerate}

\item Let $V \subset \A^n$ be a nonempty variety. Show that the following are equivalent.

\begin{enumerate}[(i)]
\item $V$ is a single point;
\item $\dim_k \Gamma(V) < \infty$;
\item $\Gamma(V) = k$.
\end{enumerate}



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