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\begin{document}

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\centerline{\sc Problem Set Four: Due Thursday, February 15}

See the course website for policy on collaboration. 
All references to FOAG refer to the February 6, 2024 edition of \href{http://math.stanford.edu/~vakil/216blog/FOAGfeb0624public.pdf}{\emph{Foundations of Algebraic Geometry}}, by Ravi Vakil.

\textbf{Problems from the textbook:} \\
\textbf{15.2.E and 15.2.G}: I'm not sure why they are in this section of the book, but these are useful concrete facts to know about projective space, which are often helpful in computations.\\
\textbf{15.4.A}: This may be labeled ``easy", but it is also important. I would do it by first thinking, for some divisor $D \subset \PP^n$, about what the intersection $D \cap \AA^n$ could look like.\\
\textbf{15.4.H}: \sout{We finally learn that there are no line bundles on $\PP^n$ other than $\cO(d)$.} Removed because done in class. \\
\textbf{15.5.C}: This verifies that there is a section of $\cO(D)$ which, indeed, has zero set $D$. It is ``tricky" because there are a lot of parts to keep track of, not because the argument is long or surprising. \\

\begin{Problem}
Let $R$ be the ring $\RR[x,y]/(x^2+y^2-1)$. In this problem, we'll compute the class group of $R$ by hand. You may take for granted that $R$ is a normal, one dimensional, Noetherian integral domain -- also known as a Dedekind domain.
\begin{enumerate}
\item Let $\fm$ be a maximal ideal of $R$. Show that $R/\fm$ is either isomorphic to $\RR$ or to $\CC$. We'll call $\fm$ ``real" or ``complex" accordingly.
\item Let $\fm_1$ and $\fm_2$ be two real maximal ideals of $R$. Show that $\fm_1 \fm_2$ is principal. Remember to cover the case $\fm_1 = \fm_2$.
\item Let $\fm$ be a complex maximal ideal of $R$. Show that there are real scalars $a$, $b$ and $c$, not all $0$, such that $ax+by+c \in \fm$. Show furthermore that $\fm = \langle ax+by+c \rangle$.
\item Argue that $\text{Pic}(\Spec\ R)$ is cyclic of order $2$. If the route sketched above doesn't work for you, you can also prove this by some other means; I tried to give the most direct attack I could.
\end{enumerate}
\end{Problem}

\end{document}


