\documentclass[12pt]{amsart}

\usepackage[margin=1in]{geometry}

\newcommand{\bZ}{\mathbf{Z}}
\newcommand{\Hom}{\mathrm{Hom}}

\title{Math 490 exercises}

\begin{document}

\maketitle

Here is a list of things to think about based on topics that arise in lecture. You don't need to turn these in. However, to gain familiarity with the subjects, you'll want to think about how to solve these.

\section{January 17}

\begin{enumerate}
\item Verify that $\bZ/n$ is a commutative ring.

\item Show that $\bZ/p$ is a field when $p$ is a prime. More specifically, show that if $a \not\equiv 0 \pmod p$, then there exists $x$ such that $ax \equiv 1 \pmod p$. 

Hint: use the Euclidean algorithm.

\item Let $R$ be a commutative ring and let $M$ and $N$ be $R$-modules. Recall that $\Hom_R(M,N)$ is the set of homomorphisms $f \colon M \to N$. For $r \in R$, $f,g \in \Hom_R(M,N)$, define $f+g$ by $(f+g)(m) = f(m) + g(m)$ and $rf$ by $(rf)(m) = rf(m)$. Verify that $\Hom_R(M,N)$ is an $R$-module.
\end{enumerate}

\section{January 19}

\begin{enumerate}
\item Let $R$ be a commutative ring. Let $f \colon M \to N$ be a homomorphism of $R$-modules.
  \begin{enumerate}
  \item Verify that ${\rm im}(f)$ is a submodule of $N$.
  \item Construct a short exact sequence
\[
0 \to \ker f \to M \to {\rm im}(f) \to 0.
\]
  \end{enumerate}
\item Let $R$ be a field. 
  \begin{enumerate}
  \item Let
\[
0 \to M_2 \xrightarrow{d_1} M_1 \xrightarrow{d_0} M_0 \to 0
\]
be a short exact sequence of vector spaces. Show that 
\[
\dim M_0 - \dim M_1 + \dim M_2 = 0.
\]
Hint: Show that $M_2 \cong {\rm im}(d_1)$ and that $M_0 \cong M_1 / {\rm im}(d_1)$.

\item Consider a longer complex
\[
0 \to M_3 \xrightarrow{d_2} M_2 \xrightarrow{d_1} M_1 \xrightarrow{d_0} M_0 \to 0
\]
which is exact everywhere. Show that 
\[
\dim M_0 - \dim M_1 + \dim M_2 - \dim M_3 = 0.
\]

Hint: Split this up into two short exact sequences by applying 1(b) to $d_1$ and $d_0$ and then apply 2(a) to both.
\item Formulate and prove a general statement for longer exact complexes.
  \end{enumerate}
\end{enumerate}

\section{January 24}

\begin{enumerate}
\item We defined a bijection $\Hom_R(R^n, M) \to M^n$ which sends a homomorphism $f$ to $(f(e_1), f(e_2), \dots, f(e_n))$. Show that this function is in fact an $R$-module homomorphism.
\item We explained how to turn a linear map $f \colon R^n \to R^m$ into an $m \times n$ matrix $M(f)$. Verify that given $g \colon R^m \to R^p$, we have $M(g \circ f) = M(g) M(f)$, i.e., that composition turns into matrix multiplication.
\end{enumerate}

\end{document}