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\begin{center}\begin{large}\textbf{Math 137 - Problem Set 5\\
Due Wednesday, Mar 4}\end{large}\end{center}



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

\begin{enumerate} 

\item Let $X \subset \A^n$ be a nonempty linear subvariety (i.e. its corresponding ideal is generated by nonzero polynomials of degree 1). 

\begin{enumerate}
\item Show that there is some change of coordinates $T: \A^n \to \A^n$ such that $T(X) = V(x_{m+1}, \ldots, x_n)$, for some $m < n$. 

\item Conclude that $X$ is isomorphic to $\A^m$.
\end{enumerate}

\item Let $\mathcal{O}_P(V )$ be the local ring of a variety $V$ at a point $P.$ Show that there is a natural one-to-one correspondence between the prime ideals in $\mathcal{O}_P(V)$ and the subvarieties of $V$ that pass through $P.$


\item Let $T: \A^n \to \A^n$ be an affine change of coordinates. Let $V$ be a subvariety of $\A^n$, and $P \in V$ a point and $Q = T(P).$
\begin{enumerate}
\item Show that the map  $\mathcal{O}_Q(\A^n) \to \mathcal{O}_P(\A^n)$ induced by $T^*$ is an isomorphism\footnote{We described this map in class.}. [Hint: Use the fact that $T$ is an isomorphism. Your solution should be pretty short.]
\item Show that $T(V)$ is a closed subvariety of $\A^n$.
\item Show that the map $\mathcal{O}_Q(T(V)) \to \mathcal{O}_P(V)$ induced by $T^*$ is an isomorphism.
\end{enumerate}


\item Let $f$ be a rational function on an affine variety $V$. Let $$U =\{P \in  V \mid f \text{ is defined at } P\}.$$ Recall that $U$ is open, since we showed in class that its complement is closed. Then $f$ defines a function from $U$ to $k$. Show that this function determines $f$ uniquely.


\item Let $X \subset \A^n$ be a variety. You may (but don't need to) assume $\Gamma(X)$ is a UFD. Let $U \subset X$ be a Zariski open set in $X$ (i.e. the complement of some algebraic subset of $X$). Define $\mathcal{O}_X(U) \subset k(X)$ to be the set of rational functions that are regular (i.e. defined) at every point of $U$. 

\begin{enumerate}
\item Show that $\mathcal{O}_X(U)$ is a ring (In fact, a $k$-algebra).
\item For $f \in \Gamma(X)$, define the open set $$U_f := X - V(f).$$ Describe explicitly the ring $\mathcal{O}_X(U_f)$. 
\item If $V \subset U$ is an open set contained in $U$, describe the natural ring homomorphism $$ \phi_{U,V}: \mathcal{O}_X(U) \to \mathcal{O}_X(V).$$
\end{enumerate}

\item Let $X = \A^2$. 
\begin{enumerate}
\item Describe the ring $\mathcal{O}_X(U)$, where $U \subset \A^2$ is the complement of the $x$-axis.
\item Describe the ring $\mathcal{O}_X(V)$, where $V$ is the complement of the origin. (Do you see why your answer makes sense?)
\end{enumerate}

\end{enumerate}

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

\begin{enumerate}
  \setcounter{enumi}{6}
  \item Look up the definition of a sheaf. Show that $\mathcal{O}_X$ described in problem 5 is a sheaf of rings\footnote{The sheaf $\mathcal{O}_X$ is called the \textit{\textbf{structure sheaf}} of $X$. } (in fact, of $k$-algebras) on $X$. (You've already done a lot of the work above.) 

\item Look up the definition of the stalk of a sheaf. Show that the stalk of $\mathcal{O}_X$ at a point $P \in X$ is the local ring\footnote{The local ring is often denoted $\mathcal{O}_{P,X}$ to avoid notation confusion.} $\mathcal{O}_P(X)$. This gives $X$ the structure of a \textit{\textbf{locally ringed space}}.

\end{enumerate}

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