\documentclass[12pt,letterpaper]{article}
\usepackage{multicol}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{mathrsfs}
\usepackage{enumerate}
\usepackage{hyperref}
%\usepackage{pstricks, pst-node, pst-plot}

\newcommand{\ds}{\displaystyle}
\newcommand{\R}{\mathbb{R}}
\newcommand{\A}{\mathbb{A}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\str}{\mathcal{O}}
\renewcommand{\P}{\mathbb{P}}

\DeclareMathOperator{\Spec}{Spec}

\begin{document}


\begin{center}\begin{large}\textbf{Math 523 - Problem Set 2\\
Due Wednesday, Feb 16}\end{large}\end{center}



\pagestyle{empty}



\begin{enumerate}

\item Let $\phi: R \to S$ be a homomorphism of rings. Verify the claims made in class regarding the relationships between expansion and contraction of ideals:
\begin{enumerate}
\item $(J \cap R) S \subseteq J$ for all ideals $J \subseteq S$. 
\item $I \subseteq (IS) \cap R$ for all ideals $I \subseteq R.$
\item For each (a) and (b), give an example in which the inclusion is proper.
\end{enumerate} 

\item Let $U \subseteq R$ be a multiplicative set.
\begin{enumerate} 
\item Give an example to show that the natural map $R \to U^{-1}R$ is not always injective. 
\item Give necessary and sufficient conditions for $R$ to be a subring of $U^{-1}R$ (via the natural map).
\end{enumerate}

\item Show that localization at $U$ is a indeed a functor from $R$-modules to $U^{-1}R$-modules as follows:
\begin{enumerate}
\item If $\phi: M \to N$ is an $R$-module homomorphism, describe the natural map $$U^{-1}\Phi: U^{-1}M \to U^{-1}N$$ and check that it is a well-defined $U^{-1}R$-module homomorphism.
\item If $M$ is an $R$-module, show that $U^{-1}id_M = id_{U^{-1}M}$.
\item For $R$-module homomorphisms $\phi: M \to N$ and $\psi: N \to L,$ show that $$U^{-1}(\psi \circ \phi) = (U^{-1}\psi) \circ (U^{-1}\phi).$$
\end{enumerate}


\item Show that the natural map $$f: \Spec(U^{-1}R) \to \Spec R$$ is a homeomorphism onto its image $Y =\{P \mid U \cap P = \emptyset\}$.\footnote{The closed sets of $Y$ are those of the form $C \cap Y$, where $C$ is closed in $\Spec R$.}

\item If $M$ is an $R$-module, use the universal property to verify that $R\otimes_R M \cong M$.

\item \begin{enumerate} 
\item Finish the proof from class of the claim that the tensor product is right exact.
\item Give an example to show that it is not necessarily left exact.\footnote{Hint: Take $M =R/I$ and consider a sequence $0 \to R \to R \to R/(f)$, where $f\ \in I$ is chosen appropriately.}
\end{enumerate}

\item \begin{enumerate}
\item If $I$ and $J$ are ideals of $R$, show that $R/I \otimes_R R/J \cong R/(I+J)$.\footnote{Hint: Use right-exactness of tensor product.}
\item Describe $\Z/(m) \otimes \Z/(n)$ for each pair $m,n\in \Z$. 
\end{enumerate}

\item Let $\phi: \C[t] \to \C[x,y,t]/(xy-t)$ be the inclusion map. Describe as explicitly as possible the fibers of the induced map of Spec. 

\end{enumerate}

\end{document}