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\begin{document}

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\centerline{\sc Problem Set One: Due Thursday, January 25}

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{Exercise 4.1.D}: The construction of a sheaf $\widetilde{M}$ from a module $M$. Feel free to just verify that $\widetilde{M}$ is a sheaf on the affine base, without spelling out why that implies that it gives a sheaf on $\Spec A$ and without checking that $\widetilde{M}$ is an $\cO_{\Spec A}$-module.\\
\textbf{Exercise 4.1.G}: The full subcategory of quasi-coherent sheaves is equivalent to the category of modules on an affine scheme.\\
\textbf{Exercise 6.3.A}: Exact sequences of  quasi-coherent sheaves are exact on every affine open. Contrast this result with Problem~\ref{NotExactSections}.


\begin{Problem}
Let $k$ be a field and let $\AA^n = \Spec k[x_1, x_2, \ldots, x_n]$. Conventionally, we think of $\AA^n$ as ``$n$-dimensional space". In each part of this problem, I'll describe a geometric map involving $\AA^n$ in informal terms and ask you to give the corresponding map of rings.
\begin{enumerate}
\item ``Vector addition" is a map $\AA^n \times \AA^n \to \AA^n$. What is the corresponding map $k[x_1, \ldots, x_n] \to k[y_1, \ldots, y_n] \otimes_k k[z_1, \ldots, z_n]$?
\item ``Scalar multiplication" is a map $\AA^1 \times \AA^n \to \AA^n$. What is the corresponding map $k[x_1, \ldots, x_n] \to k[t] \otimes_k k[y_1, \ldots, y_n]$?
\end{enumerate}
\end{Problem}



\begin{Problem} \label{NotExactSections}
Let $k$ be a field and let $\AA^2$ be $\Spec k[x,y]$. Let $(0,0)$ be the point of $\AA^2$ corresponding to the maximal ideal $\langle x,y \rangle$ of $k[x,y]$ and let $U$ be the open set $\AA^2 \setminus \{ (0,0) \}$.

Let $\phi$ be the map $k[x,y]^{\oplus 2} \to k[x,y]$ given by $\phi\left( \begin{sbm} f\\g \end{sbm} \right) = xf+yg$; let $\widetilde{\phi}$ be the induced map of sheaves $\cO_{\AA^2}^{\oplus 2} \to \cO_{\AA^2}$ on $\AA^2$.
\begin{enumerate}
\item Is the map of modules $\phi: k[x,y]^{\oplus 2} \to k[x,y]$ surjective?
\item Is the map of sheaves $\widetilde{\phi}: \cO_{\AA^2}^{\oplus 2} \to \cO_{\AA^2}$ surjective on $\AA^2$?
\item Is the map of sheaves $\widetilde{\phi}: \cO_U^{\oplus 2} \to \cO_U$ surjective on $U$? This is the restriction of $\widetilde{\phi}$ to the open set $U$.
\item Is the map of global sections, $\widetilde{\phi}(U) : \cO(U)^{\oplus 2} \to \cO(U)$, surjective?
\end{enumerate}
\end{Problem}





\begin{Problem}
Let $R$ be an integral domain and let $J$ be an ideal of $R$. The ideal $J$ is called \newword{principal} if $J = fR$ for some nonzero $f \in R$. The ideal $J$ is called locally principal if there are localizations $R_{f_1}$, $R_{f_2}$, \dots, $R_{f_k}$ with $\Spec R = \bigcup_j \Spec R_{f_j}$ such that $J_{f_i}$ is principal in $R_{f_i}$ for each $i$. In this exercise, we'll see an example of a locally principal, but non-principal, ideal, which is surprisingly geometric.

Let $R = \RR[x,y]/\langle x^2+y^2-1 \rangle$. Let $J$ be the ideal $\langle x-1, y \rangle$.

\begin{enumerate}
\item Verify that $J$ is locally principal.
\item Verify that $J$ is not principal. There are several ways to do this, but one way is to think about how the function $f(x,y)$ could behave on the circle $x^2+y^2=1$, if we had $J=f(x,y)R$.
\end{enumerate}
\end{Problem}
\end{document}


