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\textbf{18.782 Introduction to Arithmetic Geometry\hspace{118pt}Fall~2013}\\\vspace{4pt}
\textbf{Problem Set \#7\hspace{243pt}Due: 11/5/2013}
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These problems are related to the material covered in Lectures 13-16.
I have made every effort to proof-read them, but some errors may remain.
The first person to spot each error will receive 1-5 points of extra credit.

The problem set is due by the start of class on 11/5/2013 and should be submitted electronically as a pdf-file e-mailed to  \url{drew@math.mit.edu}.
You can use the latex source for this problem set as a template for writing up your solutions;
be sure to include your name in your solutions and to identify collaborators and any sources not listed in the syllabus.

\subsection*{Problem 1. Short answer questions (25 points)}
Please justify each of the following assertions in 25 words or less.
They all have short easy proofs and are meant to check your understanding, not to challenge you.

\begin{enumerate}
\item For any ideals $I,J\subseteq\kbar[x_1,\ldots,x_n]$ the affine algebraic sets $Z_{I\cup J}$ and $Z_{I+J}$ are equal.
\item For any ideals $I,J\subseteq\kbar[x_1,\ldots,x_n]$ the affine algebraic sets $Z_{I\cap J}$ and $Z_{IJ}$ are equal.
\item Every nonempty open set in $\A^n$ is dense.
\item Every affine variety is connected.
\item The closure of the image of a morphism of affine varieties is a variety.
\item A morphism of affine varieties is dominant if and only if the corresponding morphism of coordinate rings is injective.
\item $\A^n$ and $\P^n$ are birationally equivalent (prove this explicitly by writing down the rational maps; indicate in each case whether your rational map is a morphism or not).
\end{enumerate}

\subsection*{Problem 2. Sheaves (25 points)}

In this problem and the next all rings are commutative with identity.
This includes the zero ring, which is the unique ring with exactly one element $0=1$.

\begin{definition}
A \emph{presheaf} (of rings) $\mathcal{F}$ on a topological space $X$ assigns to each open set $U$ a ring $\mathcal{F}(U)$,
and to each nested pair of open sets $U\subseteq V$ a ring homomorphism
\[
\rho_{VU}\colon\mathcal{F}(V)\to\mathcal{F}(U),
\]
(note the direction)
such that $\rho_{UU}$ is the identity map and for all open sets $U\subseteq V\subseteq W$,
\[
\rho_{VU}\circ\rho_{WV}=\rho_{WU}.
\]
\end{definition}

In the motivating examples for presheaves, each $\mathcal{F}(U)$ is a ring of functions defined on~$U$, and the maps $\rho_{VU}$ simply restrict the domains of these functions from $V$ to $U$.
For this reason the maps $\rho_{VU}$ are called \emph{restriction maps}, and for each $f\in\mathcal{F}(U)$ the element $\rho_{VU}(f)\in\mathcal{F}(V)$ is called the \emph{restriction} of $f$ to $V$.\footnote{One can define presheaves for any category: just replace ``ring" with your favorite type of object and ``ring homomorphism" with the appropriate morphism; in general a presheaf is just a contravariant functor from the category of open sets of $X$ (whose morphisms are inclusion maps) to any category you like.}

\pagebreak
Now let $X$ be an affine variety over $k=\kbar$ with function field $k(X)$.
\begin{enumerate}
\item[\textbf{(a)}]
Show that for any open set $U\subseteq X$ the set
\[
\O(U)=\{f\in k(X): U\subseteq\dom(f)\}
\]
is a subring of $k(X)$ that contains $k$ (hence is a $k$-algebra), and has fraction field $k(X)$.
Prove that $\O(X)$ is equal to the coordinate ring $k[X]$.

\item[\textbf{(b)}] For open sets $U$ in $X$, let $\mathcal{F}(U)=\O(U)$, and for nested open sets $U\subseteq V$ in $X$, let $\rho_{VU}$ be the inclusion map $\O(V)\subseteq \O(U)$.
Show that $\mathcal{F}$ is a presheaf.
\end{enumerate}

\begin{definition}
An \emph{open cover} of an open set $U$ is a collection of open sets $\{U_\alpha\}_{\alpha\in A}$ whose union is $U$.
A presheaf $\mathcal{F}$ is a \emph{sheaf} if it satisfies the \emph{sheaf axiom}.
\medskip

\noindent
\textbf{Sheaf axiom:} For every open cover $\{U_a\}_{a\in A}$ of $U=\bigcup_a U_a$ and every collection $\{f_a\}$ with $f_a\in \mathcal{F}(U_\alpha)$ such that for all $a,b\in A$ the restrictions of $f_a$ and $f_b$ to $U_a\cap U_b$ are equal, there is a \emph{unique} $f\in\mathcal{F}(U)$ whose restriction to $U_a$ is $f_a$ for all $a\in A$.
\end{definition}

\begin{enumerate}
\item[\textbf{(c)}] Prove that in any sheaf of rings $\mathcal{F}$, the ring $\mathcal{F}(\emptyset)$ must be the zero ring.\footnote{Some authors include this restriction in the definition of a presheaf, but this is unnecessary.}

\item[\textbf{(d)}] Modify the presheaf $\mathcal{F}$ defined in part (b) so that $\mathcal{F}(\emptyset)$ is the zero ring and $\rho_{U\emptyset}$ is the unique homomorphism to the zero ring (for each open set $U$).
The resulting presheaf~$\O_X$ is called the \emph{structure sheaf} of $X$.
Prove that $\O_X$ is indeed a sheaf.\footnote{While this modification may seem artificial, it can actually be justified by noting that if you restrict the domain of a set of functions with the same codomain to the empty set they become a single function (there is exactly one function from the empty set to a given set $S$, it is the function whose graph is the empty set).}
\end{enumerate}

\begin{definition}
A pair $(X,\mathcal{O}_X)$ where $X$ is a topological space and $\O_X$ is a sheaf of rings on $X$ is called a \emph{ringed space}.
\end{definition}

\subsection*{Problem 3. Affine schemes (50 points)}
As in Problem 2, all rings in this problem are commutative rings with identity.

Let $R$ be a ring whose elements $f$ we shall call \emph{regular functions}.
Let 
\[
X=\spec R
\]
be the set of prime ideals of $R$.
We shall call the elements $P$ of $X$ \emph{points}.
For any point $P\in X$,  the \emph{residue field} of $P$ is the quotient field of the integral domain $R/P$.
For any regular function $f\in R$ and any point $P\in X$ we define $f(P)$ to be the image of $f$ in the residue field of $P$
(this means we reduce $f$ mod $P$ to get an element $\bar{f}$ of $R/P$ that we then regard as the element $\bar{f}/1$ of the fraction field).
Notice that $f(P)$ is defined at every point $P\in X$, which is why we call $f$ a regular function.
The field in which $f(P)$ lies will vary with $P$, but in any case we can meaningfully ask whether $f(P)=0$ or not.
We now define an \emph{algebraic set} $Z$ as the zero locus of a set of regular functions $S\subseteq R$:
\[
Z_S=\{P\in X: f(P)=0\text{ for all $f\in S$}\}.
\]
We now endow $X$ with the Zariski topology by letting the closed sets be the algebraic sets.

\begin{enumerate}
\item[\textbf{(a)}] Prove that this actually defines a topology by showing that the empty set and $X$ are both closed, and that arbitrary intersections and finite unions are closed.
\item[\textbf{(b)}] Show that for $S\subseteq R$ the algebraic set $Z_S$ is precisely the set of prime ideals of $R$ (points of $X$) that contain~$S$, and that $Z_S=Z_I$, where $I$ is the ideal generated by $S$.
\item[\textbf{(c)}] We call a point $P\in X$ a \emph{closed point} if the set $\{P\}$ is closed in the Zariski topology.
Prove that the closed points in $X$ are precisely the maximal ideals of $R$.
\item[\textbf{(d)}] Let $R$ be the polynomial ring $k[x_1,x_2]$ for some field $k=\kbar$.
Give a bijection between closed points $P$ in $ X$ and elements $(a_1,a_2)$ of $\A^2(k)$, and show that
under this bijection we have $f(P)=f(a_1,a_2)$ for any $f\in R$ and closed point $P$.
Describe the points of $X$ that are not closed and evaluate $3x_1^2+2x_1x_2-5x_2^2$ at two such points.
\item[\textbf{(e)}] Let $R=\Z$.  Describe the opens sets in $X$ and the residue fields at points $P\in X$.
Evaluate $15\in R$ at the points $(0), (2), (3), (5)\in X$, and describe the zero locus of $15$.
\end{enumerate}

Now let $V\subseteq\A^n$ be an affine variety of dimension 1 over a field $k=\kbar$, let $R=k[V]$ be its coordinate ring, let $k(V)$ be its function field, and let $X=\spec R$.

\begin{enumerate}
\item[\textbf{(f)}] Prove that every point in $V$ corresponds to a maximal ideal of $k[x_1,\ldots,x_n]$ that contains $I(V)$, which in turn corresponds to a maximal ideal of $R$, hence a closed point in $X$.  We may therefore identify $V$ with the set of closed points in $X$.
\item[\textbf{(g)}] Prove that the only point in $X$ that is not in $V$ is the zero ideal and that this point is dense in $X$ and contained in every nonempty open set in $X$.
\item[\textbf{(h)}] Let us say that an element $f=g/h$ of the function field $k(V)$ is \emph{regular} at $P\in X$ if $h(P)\ne 0$ (where $h(P)$ is defined as above).  Argue that for points $P\in V$ this definition agrees with the corresponding definition given in Lecture 15.  Then show that every $f\in k(V)$ is regular at the point $P=(0)$ and that we may thus add $(0)$ to $\dom(f)$ for every $f\in k(V)$.
\end{enumerate}

\begin{definition}
An \emph{affine scheme} is a ringed space $(X,\O_X)$ where $X=\spec R$ for a ring~$R$.
\end{definition}

\begin{enumerate}
\item[\textbf{(i)}] Use (h) to define a structure sheaf $\O_X$ for $X=\spec k[V]$, as in part (d) of Problem~2, thereby constructing an affine scheme corresponding to $V$.

\item[\textbf{(j)}] Use (e) to define a structure sheaf $\O_X$ for $X=\spec \Z$.
\end{enumerate}

\noindent
Just in case you are interested, here is the general definition of a scheme.

\begin{definition}
A \emph{scheme} is a ringed space $(X,\O_X)$ for which every point has a neighborhood~$U$ for which $(U,\O_U)$ is an affine scheme, where $\O_U$ is $\O_X$ restricted to $U$.
\end{definition}



\subsection*{Problem 4. Survey}
Complete the following survey by rating each problem on a scale of 1 to~10 according to how interesting you found the problem (1 = ``mind-numbing," 10 = ``mind-blowing"), and how difficult you found the problem (1 = ``trivial," 10 = ``brutal").  Also estimate the amount of time you spent on each problem.

\begin{center}
\begin{tabular}{l|r|r|r|}
& Interest & Difficulty & Time Spent\\\hline
Problem 1 & & & \\\hline
Problem 2 & & & \\\hline
Problem 3 & & & \\\hline
\end{tabular}
\end{center}
\noindent
Please rate each of the following lectures that you attended, according to the quality of the material (1=``useless", 10=``fascinating"), the quality of the presentation (1=``epic fail", 10=``perfection"), the pace (1=``way too slow", 10=``way too fast"), and the novelty of the material (1=``old hat", 10=``all new").

\begin{center}
\begin{tabular}{l|l|r|r|r|r|}
Date & Lecture Topic & Material & Presentation & Pace & Novelty\\\hline
10/24 & Morphisms and rational maps & & & & \\\hline 
10/29 & Function fields & & & &\\\hline 
\end{tabular}
\end{center}

\noindent
Feel free to record any additional comments you have on the problem sets or lectures; in particular, how you think they might be improved.

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