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\large
\textbf{18.782 Introduction to Arithmetic Geometry\hspace{118pt}Fall~2013}\\\vspace{4pt}
\textbf{Problem Set \#8\hspace{237pt}Due: 11/13/2013}
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These problems are related to the material covered in Lectures 16-17.
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 11/12/2013 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. Products of varieties and completeness (50 points)}

For topological spaces $X$ and $Y$ we write $X\toptimes Y$ to denote their product as a topological space; if $\pi_X$ and $\pi_Y$ are the projection maps, the sets of the form $\pi_X^{-1}(S)$ and $\pi_Y^{-1}(S)$ with~$S$ closed generate the closed sets in $X\toptimes Y$ (under intersection and finite unions).
For varieties $X$ and $Y$ we write $X\times Y$ to denote the product variety with the Zariski topology, as defined in Lecture 16.  In this problem $k$ denotes an algebraically closed field.

\begin{enumerate}
\item Prove that every closed set in $\A^m\toptimes \A^n$ is closed in $\A^m\times \A^n$, but that the converse is not true for any $m$ and $n$.
\item Prove that $\A^m\times \A^n$ is isomorphic to $\A^{m+n}$ (for any $m$ and $n$), but that $\P^1\times \P^1$ is not isomorphic to $\P^2$, nor to any subsvariety of $\P^2$.
\item Define the map $\phi\colon \P^1\times\P^1\to \P^3$ by
\[
\phi((x_0:x_1),(y_0:y_1))=[x_0y_0:x_0y_1:x_1y_0:x_1y_1].
\]
Prove that $\phi$ is a morphism whose image $V$ is a variety (specify $I(V)$ explicitly).
Then prove that $\P^1\times\P^1\simeq V$ by giving an inverse morphism.
\item Let $x_0,\ldots,x_m$ and $y_0,\ldots,y_n$ be homogeneous coordinates for $\P^m$ and $\P^n$ respectively, and for $0\le i\le m$ and $0\le j\le n$ let $z_{ij}$ be homogeneous coordinates for $\P^N$, where $N=(m+1)(n+1)-1$.
Prove that the \emph{Segre morphism} from $\varphi\colon \P^m\times\P^n\to \P^N$ defined by $z_{ij}=x_iy_j$ is an isomorphism to a projective variety $V\subseteq \P^N$.
\item The field $\C$ is an affine $\R$-algebra (an integral domain finitely generated over $\R$).
Show that the tensor product of $\R$-algebras $\C\otimes_\R\C$ is not an integral domain, hence not an affine $\R$-algebra.
Thus Lemma 16.4, which states that a tensor product of affine algebras is again an affine algebra, depends on the assumption that $k$ is algebraically closed.  Explain exactly where in the proof this assumption is used.
\item Let $X$ be a variety.  Chevalley's criterion states that if for every variety $Z\subseteq X$ and every valuation ring $R$ of $k(Z)/k$ there is a point $P\in Z$ such that $\O_{P,Z}\subseteq R$, then $X$ is complete.
We proved in Lecture 16 that affine varieties of positive dimension are not complete, hence such varieties cannot satisfy Chevalley's criterion.  Write down an explicit affine plane curve $X$ and exhibit $Z$ and $R$ for which~$X$ does not satisfy Chevalley's criterion.  Then indicate which point or points in the projective closure of~$X$ address this deficiency.
\end{enumerate}

\subsection*{Problem 2. Valuation rings (50 points)}
An \emph{ordered abelian group} is an abelian group $\Gamma$ with a total order $\le$ that is compatible with the group operation.
This means that for all $a,b,c\in \Gamma$ the following hold:
\[
\begin{matrix}
a\le b\le a &\implies &a=b &\text{(antisymmetry)}\\
a\le b\le c &\implies & a\le c &\text{(transitivity)}\\
a\not\le b &\implies & b\le a &\text{(totality)}\\
a\le b &\implies &a+c\le b+c &\text{(compatibility)}
\end{matrix}
\]
Note that totality implies reflexivity $(a\le a)$ .  Given an ordered abelian group $\Gamma$, we define the relations $\ge,<,>$ and the sets $\Gamma_{\le 0}, \Gamma_{\ge 0}, \Gamma_{<0}$, and $\Gamma_{>0}$ in the obvious way.

A \emph{valuation} $v$ on a field $K$ is a surjective homomorphism $v\colon K^\times\to \Gamma$ to an ordered abelian group $\Gamma$ that satisfies $v(x+y)\ge \min (v(x),v(y))$ for all $x,y\in K^\times$.
The group $\Gamma$ is called the \emph{value group} of $v$, and when $\Gamma=\{0\}$ we say that $v$ is the \emph{trivial valuation}.

\begin{enumerate}
\item 
Let $R$ be a valuation ring with fraction field $F$, and let $v\colon F^\times\to F^\times/R^\times= \Gamma$ be the quotient map.
Show that the relation $\le$ on $\Gamma$ defined by
\[
v(x)\le v(y) \Longleftrightarrow y/x\in R,
\]
makes $\Gamma$ an ordered abelian group and that $v$ is a valuation on $F$.
\item  Let $F$ be a field and let $v\colon F^\times\to \Gamma$ be a non-trivial valuation.  Prove that
the set
\[
R_v:=v^{-1}(\Gamma_{\ge 0})\cup \{0\}
\]
is a valuation ring of $F$ and that $v(x)\le v(y) \Longleftrightarrow y/x\in R_v$
(note that this includes proving that $R_v$ is actually a ring).


\item
Let $\Gamma$ an ordered abelian group, let $k$ be a field, and let $A$ be the $k$-algebra generated by the set of formal symbols $\{x^a\colon a\in \Gamma_{\ge 0}\}$, with multiplication defined by $x^ax^b=x^{a+b}$.
This means that $A$ consists of all sums of the form
\[
\sum_{a\in\Gamma_{\ge 0}} c_a x^a
\]
with $c_a\in k$ and only finitely many nonzero.
Let $F$ be the fraction field of $A$ and define the function $v\colon F^\times\to \Gamma$ by
\[
v\left(\frac{\sum c_a x^a}{\sum d_a x^a}\right) = \min\{ a: c_a\ne 0\} - \min\{ a:d_a\ne 0\}.
\]
Prove that $v$ is a valuation of $F$ with value group $\Gamma$.

\item
Let $v\colon F^\times\to \Gamma_v$ and $w\colon F^\times\to \Gamma_w$ be two valuations on a field $F$, and let $R_v$ and $R_w$ be the corresponding valuation rings.
Prove that $R_v=R_w$ if and only if there is an order preserving isomorphism $\rho\colon \Gamma_v\to \Gamma_w$ for which $\rho\circ v = w$ (in which case we say that $v$ and $w$ are equivalent).
Thus there is a one-to-one correspondence between valuation rings and equivalence classes of valuations.

\item Let $D$ be an integral domain that is properly contained in its fraction field $F$, and let~$\mathcal{R}$ be the set of local rings that contain $D$ and are properly contained in $F$.
Partially order $\mathcal{R}$ by writing $R_1\le R_2$ if $R_1\subseteq R_2$ and the maximal ideal of $R_1$ is contained in the maximal ideal of $R_2$.\footnote{This is known as the \emph{dominance} ordering.}
Zorn's lemma states that any partially ordered set in which every chain has an upper bound contains at least one maximal element.
Use this to prove that $\mathcal{R}$ contains a maximal element $R$, and then show that any such $R$ is a valuation ring (note: the hypothesis of Zorn's lemma is understood to include the empty chain, so you must prove that $\mathcal{R}$ is nonempty).

\item Prove that every valuation ring $R$ is \emph{integrally closed}.  This means that every element of the fraction field of $R$ that is the root of a monic polynomial in $R[x]$ lies in $R$.
\end{enumerate}


\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
\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
10/31 & Products of varieties, Chevalley's criterion & & & & \\\hline 
11/5 & Complete varieties, tangent spaces & & & &\\\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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