\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage{listings}
\usepackage{courier}
\usepackage{enumerate}
\usepackage{color}
\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{hyperref}
\hypersetup{colorlinks=true,urlcolor=MidnightBlue,citecolor=PineGreen,linkcolor=BrickRed}

\lstset{
	basicstyle=\small\ttfamily,
	keywordstyle=\color{blue},
	language=python,
	xleftmargin=16pt,
}

% to include source code, uses the listings package as follows:
% \begin{lstlisting}
% Put your code here.
% \end{lstlisting}

\textwidth=6in
\textheight=9in
\topmargin=-0.5in
\headheight=0in
\headsep=.5in
\hoffset  -.4in
\pagestyle{empty}

% Feel free to add new macros here

\newcommand{\kbar}{\overline{k}}
\newcommand{\A}{\mathbb{A}}
\renewcommand{\P}{\mathbb{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Qp}{\Q_p}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}_{\rm div}}
\newcommand{\Zp}{\Z_p}
\newcommand{\F}{\mathbb{F}}
\newcommand{\Fp}{\F_p}
\newcommand{\Fq}{\F_q}
\newcommand{\Fqbar}{\overline{\F}_q}
\newcommand{\Fqm}{\F_{q^m}}
\newcommand{\Fqn}{\F_{q^n}}
\newcommand{\Aut}{{\rm Aut}}
\newcommand{\Gal}{{\rm Gal}}

\newtheorem{theorem}{Theorem}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}

\begin{document}
\setlength{\unitlength}{1in}
\begin{center}
\large
\textbf{18.782 Introduction to Arithmetic Geometry\hfill Fall~2020}\\\vspace{4pt}
\textbf{Problem Set \#6\hfill Due: 10/16/2020}
\normalsize
\begin{picture}(6,.1) 
\put(0,0) {\line(1,0){6}}         
\end{picture}
\end{center}

These problems are related to the material covered in Lectures 14-16.
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 5pm on 10/16/2020 and should be submitted electronically as a pdf-file e-mailed to \url{zzyzhang@mit.edu} and \url{roed@mit.edu} (please include ``18.782'' in the subject of the email).
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.
\smallskip

\subsection*{Problem 1. Homogeneous ideals and projective algebraic sets (50 points)}
Let $k$ be a perfect field and fix an algebraic closure $\kbar$.
Fix a positive integer $n$, let $R$ be the polynomial ring $\kbar[x_0,\ldots,x_n]$, and let $R^h$ be the set of all homogeneous polynomials in $R$.
Recall from lecture that (1) a homogenous ideal in $R$ is an ideal generated by elements of $R^h$,
(2) for any subset $S\subset R$ the algebraic set $Z_S$ is the zero locus in $\P^n$ of $S\cap R^h$,
and (3) for any subset $Z\subset\P^n$ the ideal $I(Z)$ is the homogeneous ideal generated by the polynomials in $R^h$ that vanish at every point in $Z$.

\begin{enumerate}
\item[\textbf{(a)}] Prove the ``homogeneous \emph{Nullstellensatz}," which says that if $I\subseteq R$ is a homogeneous ideal and $f\in R^h$ is a nonconstant polynomial that vanishes at every point in $Z_I$, then $f^r\in I$ for some $r>0$ (hint: re-interpret the problem in $\A^{n+1}$ and use the usual \emph{Nullstellensatz}).

\item[\textbf{(b)}] Let $R_+$ denote the set of all polynomials in $R$ with no constant term.
Prove that for any homogeneous ideal $I\subseteq R$ the following are equivalent:
\begin{enumerate}[(i)]
\item $Z_I$ is the empty set;
\item Either $\sqrt{I}=R$ or $\sqrt{I}=R_+$.
\item $I$ contains every polynomial in $R^h$ of degree $d$, for some $d>0$.
\end{enumerate}

\item[\textbf{(c)}] Prove the following:
\begin{enumerate}[(i)]
\item If $S\subseteq T$ are subsets of $R^h$ then $Z_T\subseteq Z_S$.
\item If $Y\subseteq Z$ are subsets of $\P^n$ then $I(Z)\subseteq I(Y)$.
\item For any two subsets $Y,Z$ in $\P^n$ we have $I(Y\cup Z)=I(Y)\cap I(Z)$.
\item For any algebraic set $Z\subseteq \P^n$ we have $Z_{I(Z)}=Z$.
\item For any homogeneous radical ideal $I$ for which $Z_I\ne\emptyset$ we have $I(Z_I)=I$.
\end{enumerate}

\item[\textbf{(d)}] Conclude from (a), (b), and (c) that there is a one-to-one inclusion-reversing correspondence between algebraic sets in $\P^n$ and homogeneous radical ideals in $R$ that are not equal to $R_+$, given by the maps $Z\to I(Z)$ and $I\to Z_I$.\footnote{The ideal $R_+$ that does not appear in this correspondence is called the \emph{irrelevant} ideal.}

\item[\textbf{(e)}] Prove that an algebraic set $Z\in\P^n$ is irreducible if and only if $I(Z)$ is a prime ideal.
\end{enumerate}

\subsection*{Problem 2. 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
\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.

%\begin{thebibliography}{9}
%\end{thebibliography}

\end{document}