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\begin{document}

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\centerline{\sc Problem Set Two: Due Thursday, February 1}

See the course website for policy on collaboration. 
All references to FOAG refer to the July 31, 2023 edition of \href{http://math.stanford.edu/~vakil/216blog/FOAGjul3123public.pdf}{\emph{Foundations of Algebraic Geometry}}, by Ravi Vakil.

\textbf{Problems from the textbook:}

\textbf{14.2.B} This exercise is ``essential", but it is also straightforward. It tells you that pullbacks of vector bundles are computed in a completely straightforward way.\\
\textbf{14.2.D} This exercise shows you how to represent dual vector bundles in coordinates. It is a good example to make you do a concrete computation.\\
\textbf{14.2.F} Tensoring with a locally free sheaf preserves exactness. This comes up all the time.\\
\textbf{14.2.H} The definition of the Picard group, an important object in algebraic geometry.\\
\textbf{14.3.G} This exercise shows how to write down a short exact sequence of vector bundles in coordinates. You'll get more practice with it in Problem~\ref{EulerSequence}.


\begin{Problem} \label{HomsInCoords}
As described in Section 14.2 of \emph{FOAG}, to give a vector bundle of rank $r$ on a scheme $X$ is to give an open cover $U_i$ of $X$ and transition matrices $T_{ij} \in \GL_r(\cO(U_i \cap U_j))$ obeying\footnote{If I were writing the book, I would switch the order of the subscripts in $T_{ij}$, but I will follow the book.} $T_{ik}|_{U_i \cap U_j \cap U_k} = T_{jk}|_{U_i \cap U_j \cap U_k} \circ T_{ij}|_{U_i \cap U_j \cap U_k}$. 
Let $\cE$ and $\cF$ be vector bundles of ranks $p$ and $q$ on $X$. Let $U_i$ be an open cover of $X$ on which both $\cE$ and $\cF$ are trivial,  with transition matrices $P_{ij}$ and $Q_{ij}$. 

\begin{enumerate}
\item Show that $\Hom(\cE, \cF)$ is isomorphic to the set of collections of matrices $H_i \in \Mat_{q \times p}(\cO(U_i))$ such that $H_j|_{U_i \cap U_j} \circ P_{ij} = Q_{ij} \circ H_i|_{U_i \cap U_j}$. 
\item Show that $\cE \cong \cF$ if and only if $p=q$ and there are invertible matrices $G_i \in \GL_p(\cO(U_i))$ such that $Q_{ij} = G_j|_{U_i \cap U_j} \circ P_{ij} \circ G^{-1}_i|_{U_i \cap U_j}$. 
\end{enumerate}
\end{Problem}

\begin{Problem} \label{EulerSequence}
In this problem, we work with some basic vector bundles on $\PP^1$. Let $k$ be a field. Cover $\PP^1$ with two open charts: $U = \Spec k[t]$ and $V = \Spec k[t^{-1}]$, so $U \cap V = \Spec k[t, t^{-1}]$.  The line bundle $\cO(n)$ is formed by taking a section $\sigma$ of $\cO_U$ and gluing it $t^{-n} \sigma$ in $\cO_V$. So the global sections of $\cO(n)$ are ordered pairs $(f,g)$ with $f \in k[t]$ and $g \in k[t^{-1}]$ such that $g = t^{-n} f$. Let $x_1$ and $x_2$ be the sections $(1, t^{-1})$ and $(t, 1)$ of $\cO(1)$.
\begin{enumerate}
\item Define $\beta: \cO(0)^{\oplus 2}$ to $\cO(1)$ by $(f_1, f_2) \mapsto f_1 x_1 + f_2 x_2$. Show that $\beta$ is a surjective map of sheaves.
\item Show that the kernel of $\beta$ is isomorphic to $\cO(-1)$. So we have a short exact sequence $0 \to \cO(-1) \overset{\alpha}{\longrightarrow} \cO^{\oplus 2} \overset{\beta}{\longrightarrow} \cO(1) \to 0$. 
\item Encode the vector bundles $\cO(-1)$, $\cO^{\oplus 2}$ and $\cO(1)$ with the matrices $\begin{sbm} t \end{sbm}$, $\begin{sbm} 1 & 0 \\ 0& 1 \\ \end{sbm}$ and $\begin{sbm} t^{-1} \\ \end{sbm}$ respectively. Give the matrices in $\Mat_{1 \times 2}(k[t])$ and $\Mat_{1 \times 2}(k[t^{-1}])$ representing $\beta$ and the matrices in $\Mat_{2 \times 1}(k[t])$ and $\Mat_{2 \times 1}(k[t^{-1}])$ representing $\alpha$. 
\item From \textbf{14.3.G}, there must be a matrix of the form $\begin{sbm} t^{-1} & \ast \\ 0 & t \end{sbm}$ representing $\cO^{\oplus 2}$. From Problem~\ref{HomsInCoords}, we should be able to factor 
\[ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = G_1 \begin{bmatrix} t^{-1} & \ast \\ 0 & t \end{bmatrix} G_2^{-1} \]
for $G_1 \in \GL_2(k[t])$ and $G_2 \in \GL_2(k[t^{-1}])$. Give the matrices $\begin{sbm} t^{-1} & \ast \\ 0 & t \end{sbm}$, $G_1$ and $G_2$.
\end{enumerate}
\end{Problem}

\end{document}


