\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amscd}
\usepackage{verbatim}
%\usepackage[arrow,matrix,cmtip]{xy}
\usepackage{fullpage}

\def\cprime{$'$}
\newcommand{\Num}{\operatorname{Num}}
\newcommand{\num}{\operatorname{Num}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\eps}{\epsilon}
\newcommand{\bpr}{\begin{proof}}
\newcommand{\epr}{\end{proof}}
\newcommand{\Ab}{\operatorname{\textbf{Ab}}}
\newcommand{\dep}{\operatorname{depth}}
\newcommand{\TS}{\textstyle}
\newcommand{\ol}{\overline}
\newcommand{\reg}{\operatorname{reg}}
\newcommand{\dlim}{\underset{\lra} \lim}
\newcommand{\lm}{\lim_{n \ra \infty}}
\newcommand{\cat}{\operatorname{-dist}}
\newcommand{\Cat}{\operatorname{-Dist}}
\newcommand{\sat}{\operatorname{sat}}
\newcommand{\fg } {f.g.\ }
\newcommand{\spec} {\mbox{Spec}\ }
\newcommand{\ext} {\operatorname{Ext}}
\newcommand{\caphom} {\operatorname{Hom}}
\newcommand{\ann} {\operatorname{ann}}
\newcommand{\Llra}{\Longleftrightarrow}
\newcommand{\lra} {\longrightarrow}
\newcommand{\lla}{\longleftarrow}
\newcommand{\da} {\downarrow}
\newcommand{\adj}{\operatorname{adj}}
\newcommand{\ra}{\rightarrow}
\newcommand{\st}{s.t.\ }
\newcommand{\D}{\displaystyle}
\newcommand{\mc}{\mathcal}
\newcommand{\mf}{\mathfrak}
\newcommand{\cha}{\operatorname{char}}
\newcommand{\mb}{\mathbb}
\newcommand{\hra}{\hookrightarrow}
\newcommand{\pgl}{\operatorname{PGL}}
\newcommand{\gl}{\operatorname{GL}}
\newcommand{\m}{\mf{m}}
\newcommand{\p}{\mf{p}}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\uext}{\underline{\operatorname{Ext}}}
\newcommand{\E}{\underline{\operatorname{Ext}}}
\newcommand{\uhom}{\underline{\operatorname{Hom}}}
\renewcommand{\hom}{\operatorname{Hom}}
\renewcommand{\H}{\underline{\operatorname{Hom}}}
\newcommand{\GK}{\operatorname{GK}}
\newcommand{\Kdim}{\operatorname{K.dim}}
\newcommand{\wt}{\widetilde}
\newcommand{\ndo}{\operatorname{End}}
\newcommand{\coH}{\operatorname{H}}
\newcommand{\ucoH}{\underline{\operatorname{H}}}
\newcommand{\ugext}{\operatorname{ug.Ext}}
\newcommand{\ughom}{\operatorname{ug.Hom}}
\newcommand{\tor}{\operatorname{Tor}}
\newcommand{\utor}{\underline{\operatorname{Tor}}}
\newcommand{\T}{\underline{\operatorname{Tor}}}
\newcommand{\ugtor}{\operatorname{ug.Tor}}
\newcommand{\ClKdim}{\operatorname{Cl.K.dim}}
\newcommand{\im}{\operatorname{Im}}
\newcommand{\cd}{\operatorname{cd}}
\newcommand{\ucd}{\underline{\operatorname{cd}}}
\newcommand{\Proj}{\operatorname{-Proj}}
\newcommand{\proj}{\operatorname{-proj}}
\newcommand{\cproj}{\operatorname{proj}}
\newcommand{\rproj}{\operatorname{proj-}}
\newcommand{\rProj}{\operatorname{Proj-}}
\newcommand{\coh}{\operatorname{coh}}
\newcommand{\catmod}{\operatorname{-mod}}
\newcommand{\catMod}{\operatorname{-Mod}}
\newcommand{\rcatmod}{\operatorname{mod-}}
\newcommand{\rcatMod}{\operatorname{Mod-}}
\newcommand{\gldim}{\operatorname{gl.dim}}
\newcommand{\gd}{\operatorname{gd}}
\newcommand{\Gr}{\operatorname{-Gr}}
\newcommand{\gr}{\operatorname{-gr}}
\newcommand{\rQch}{\operatorname{Qch}}
\newcommand{\Qgr}{\operatorname{-Qgr}}
\newcommand{\qgr}{\operatorname{-qgr}}
\newcommand{\rGr}{\operatorname{Gr-}}
\newcommand{\rgr}{\operatorname{gr-}}
\newcommand{\rQgr}{\operatorname{Qgr-}}
\newcommand{\rqgr}{\operatorname{qgr-}}
\newcommand{\Tors}{\operatorname{-Tors}}
\newcommand{\tors}{\operatorname{-tors}}
\newcommand{\rTors}{\operatorname{Tors-}}
\newcommand{\und}{\underline}
\newcommand{\id}{\operatorname{id}}
\newcommand{\aut}{\operatorname{Aut}}
\newcommand{\quasiopen}{quasi-open}
\newcommand{\ideal}[1]{\mf{m}_{#1}}
%\newcommand{\todo}[1]{\vspace{5 mm}\par \noindent \marginpar{\textsc{ToDo}}
%\framebox{\begin{minipage}[c]{0.95 \textwidth} \tt #1
%\end{minipage}}\vspace{5 mm}\par}
\newcommand{\mult}{\star}
%\newcommand{todo}[1]{$\bigstar$ \textbf{#1}}
\newcommand{\segre}{\overset{s}{\otimes}}
\DeclareMathOperator{\pr}{pr}
\newcommand{\name}{subtwisted algebra}
\newcommand{\regmult}{\circ}
\newcommand{\twistmult}{\star}
\newcommand{\Tmod}[1]{T^{#1}}
\newcommand{\pointmod}[1]{P({#1})}
\newcommand{\specmod}[1]{M^{#1}}
\newcommand{\twistmod}[2]{{#1}^{#2}}
\DeclareMathOperator{\Hom}{Hom}
\newcommand{\invlim}{\underleftarrow{\lim}}
\DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\HB}{H}
\DeclareMathOperator{\HS}{h}
\newcommand{\GL}{\operatorname{GL}}

\DeclareMathOperator{\lann}{l.ann} \DeclareMathOperator{\rann}{r.ann}
\newcommand{\Tor}{\operatorname{Tor}}

\newcommand{\rings}{\mathbf{Rings}}
\newcommand{\sets}{\mathbf{Sets}}
\newcommand{\kalg}{\mathbf{alg(k)}}
\newcommand{\kschemes}{\mathbf{Schemes(k)}}
\newcommand{\on}{\operatorname}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{subclaim}[theorem]{Subclaim}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{question}[theorem]{Question}

%\theoremstyle{definition}
\newtheorem{hypothesis}[theorem]{Hypothesis}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{example}[theorem]{Example}
\newtheorem{construction}[theorem]{Construction}
\newtheorem{remark}[theorem]{Remark}

\numberwithin{equation}{section}

\makeatletter              % This sequence of commands will
\let\c@equation\c@theorem  % incorporate equation numbering
                           % into theorem numbering scheme
\makeatother




\begin{document}
\title{Math 200b Winter 2021 Homework 2}
\date{Due 1/22/2020 by midnight on Gradescope}
\maketitle


1.  Let $R$ be an integral domain which is noetherian (that is, such that every ideal of $R$ is finitely generated as an ideal).   

\medskip

(a). Show that if every finitely generated torsionfree $R$-module is free, then $R$ is a PID.

\medskip

(b).  Show that if every torsionfree $R$-module is free, then $R$ is a field.

\bigskip

2.  Let $R = F[x]$ where $F$ is a field.  Suppose that 
\[
M = R/(x^2 -1)  \oplus R/(x^2 + 1) \oplus R/(x^2 - 2x +1) \oplus R/(x^3 +1).
\]

\medskip

(a).  Assuming that $F = \mb{Q}$ is the rational numbers, find the elementary divisors and invariant factors of $M$ as an $F[x]$-module.

\medskip

(b).  Assuming that $F = \mb{C}$ is the complex numbers, find the elementary divisors and invariant factors of $M$ as an $F[x]$-module.

\bigskip

3.  Let $M$ be a finitely generated torsion module over the PID $R$.  
Suppose that $M$ has $s$ elementary divisors and $t$ invariant factors.  
Define 
\[
S = \{ n \in \mb{N} | M\ \text{is isomorphic to a direct sum of}\ n\ \text{cyclic modules} \}.   
\]

\medskip

(a) Show that $s = \max(S)$.

\medskip

(b) Show that $t = \min(S)$.

\bigskip





4.  In class we proved that a finitely generated module over a PID is the direct sum of its  torsion submodule and a torsionfree (in fact, free) module.  This exercise shows that the same is not true of infinitely generated modules in general.

\medskip

Let $M = \prod_p \mb{Z}/p\mb{Z}$, where the product runs over all distinct (positive) prime numbers $p$.  Consider $M$ as an abelian group (i.e. $\mb{Z}$-module).  

\medskip

(a) Show that the torsion submodule of $M$ is $\on{Tors}(M) = \bigoplus_p \mb{Z}/p \mb{Z}$.

\medskip

(b) Show that $N = M/\on{Tors}(M)$ is a \emph{divisible} $\mb{Z}$-module.  This means for any $x \in N$ and prime number $p$, there is $y \in N$ such that $p y = x$.

\medskip

(c) Show that $M$ has no nonzero divisible submodules over $\mb{Z}$.  

\medskip

(d) Conclude that $M \not \cong \on{Tors}(M)  \oplus (M/\on{Tors}(M))$.

















 


















\end{document}
