\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}

\usepackage[margin=1in]{geometry}
\usepackage{enumitem}
\usepackage{amsmath, amssymb, amsthm, mathrsfs}
\usepackage{tikz, tikz-cd}
\usetikzlibrary{decorations.markings}
\usepackage{xcolor}
\usepackage{verbatim}
\usepackage{graphicx}
\usepackage{hyperref}
\hypersetup{
    colorlinks=true,
    linkcolor=blue,
    filecolor=magenta,      
    urlcolor=blue,
    pdftitle={Knot Theory},
    pdfpagemode=FullScreen,
    }

% THEOREM ENVIRONMENTS
\theoremstyle{definition}
\newtheorem*{theorem}{{Theorem}}
\newtheorem*{lemma}{{Lemma}}
\newtheorem*{proposition}{{Proposition}}
\newtheorem*{definition}{{Definition}}
\newtheorem*{corollary}{{Corollary}}
\newtheorem*{example}{{Example}}
\newtheorem*{remark}{{Remark}}
\newtheorem*{claim}{{Claim}}
\newtheorem*{question}{{Question}}

\theoremstyle{plain}
\newtheorem*{reminder}{{\textit{Reminder}}}
\newtheorem*{hint}{{\textit{Hint}}}



% COMMANDS
\newcommand{\solution}{{\color{blue} \textsc{\ \\ Solution.\ \ }}}
\newcommand{\Z}{\mathbb{Z}} % the integers
\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\R}{\mathbb{R}} % real numbers
\newcommand{\C}{\mathbb{C}} % complex numbers
\newcommand{\Q}{\mathbb{Q}} % rational numbers
% \newcommand{\st}{ \ : \ } % such that
\newcommand{\st}{ \ | \ } % such that
\newcommand{\inv}{^{-1}} % inverse
\newcommand{\into}{\hookrightarrow}  % injection
\newcommand{\onto}{\twoheadrightarrow} % surjection
\newcommand{\map}[1]{\xrightarrow{#1}} % named map
\newcommand{\id}{\mathrm{id}} % identity map
\newcommand{\Frac}{\mathrm{Frac}} % fraction field
\newcommand{\normal}{\unlhd} % normal subgroup of
\newcommand{\F}{\mathbb{F}} % random field
\newcommand{\Isom}{\mathrm{Isom}} % isometries group

\DeclareMathOperator{\coker}{coker} % cokernel
\DeclareMathOperator{\img}{im} % image
\DeclareMathOperator{\sgn}{sgn} % sign of permutation
\DeclareMathOperator{\ord}{ord} % order of group element
\DeclareMathOperator{\lcm}{lcm} % least common multiple
% \DeclareMathOperator{\gcd}{gcd} % greatest common divisor
\newcommand{\op}{^{\mathrm{op}}} % opposite thing

\DeclareMathOperator{\Mod}{\mathbf{Mod}} % module category
\newcommand{\RMod}{_R\Mod} % left R mods
\newcommand{\ModR}{\Mod_R} % right R mods
\DeclareMathOperator{\Grp}{\mathbf{Grp}} % cat of groups
\newcommand{\SC}{\mathscr{C}} % category C
\newcommand{\CC}{\mathcal{C}} % category C (Rotman) or chain complex
\DeclareMathOperator{\Hom}{\mathrm{Hom}} % module category
\DeclareMathOperator{\Ab}{\mathbf{Ab}} % cat of abelian groups






% SOME ENVIRONMENTS
\newcommand{\exheading}[1]{\section*{Exercise #1}}
\newcounter{exnum}
% \setcounter{exnum}{1} % default 0 start
\newcommand{\exercise}{
    \stepcounter{exnum} 
    \exheading{\theexnum}
    }
\newcommand{\bea}{\begin{enumerate}[label={(\alph*)}]}
\newcommand{\ee}{\end{enumerate}}
% round matrix
\newcommand{\bpmat}{\begin{pmatrix}}
\newcommand{\epmat}{\end{pmatrix}}
% square matrix
\newcommand{\bbmat}{\begin{bmatrix}}
\newcommand{\ebmat}{\end{bmatrix}} 





\title{MAT 250B HW02}
\author{[add your name here]}
\date{Due Friday, 1/19/24 at 11:59 pm on Gradescope}

\begin{document}

\maketitle

\begin{reminder} 
Your homework submission \textbf{\large must be typed} (TeX'ed) up in full sentences, with proper mathematical formatting. 

The following resources may be useful as you learn to use TeX and Overleaf:
\begin{itemize}
    \item Overleaf's introduction to LaTeX: \\ \url{https://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes}
    \item Detexify: \\ \url{https://detexify.kirelabs.org/classify.html}
\end{itemize}

\end{reminder}


\paragraph{Grading} Most problems will be graded for completion out of 2 points: 1 if you wrote something meaningful, 2 if your solution is basically correct.
A few select problems will be graded more carefully.

\paragraph{Solutions} I will not be posting full solutions, but might post some abridged solutions for selected problems. You are responsible for looking at the solutions and determining whether your solutions are correct! This is a very important skill to have when you transition to doing research; you \textit{must} learn how to check your own work. So, if you have any questions about whether your solution is valid, come ask me or the TA (after checking my abridged solutions). 

\paragraph{Regarding figures} You are not required to \texttt{tikz} your diagrams, but you're welcome to. Feel free to hand-draw your diagrams and insert them using a \texttt{figure} environment and/or 
\begin{verbatim}
    \includegraphics[]{}
\end{verbatim}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%-----------
\exercise
(Review) 

\bea
\item Let $M,N \in \RMod$, and let $f: M \to N$ be a module map. Prove that there is an exact sequence 
\[
    0 \to \ker f \to M \map{f} N \to \coker f \to 0.
\]
\item Let $A \map{f} B \to C \map{h} D$ be an exact sequence. Prove that $f$ is surjective if and only if $h$ is injective. 
\ee

\solution
% add your solution here



%-----------
\exercise
(Review)
A module $M \in \RMod$ is called \textbf{irreducible} if $M \neq 0$ and if $0$ and $M$ are the only submodules of $M$. 

\bea
\item Show that $M$ is irreducible if and only if $M \neq 0$ and $M$ is a cyclic module with any nonzero element as a generator.
\item Determine all the irreducible $\Z$-modules.
\ee




%-----------
\exercise
\bea 
\item Let $k$ be a field, and let $G$ be a \textbf{finite} group. Prove that $(kG)\op \cong kG$. 
\item Let $\mathbb{H}$ be the division ring of \href{https://en.wikipedia.org/wiki/Quaternion}{real quaterions}. Use part (a) to deduce that $\mathbb{H}\op \cong \mathbb{H}$.
\ee


\solution
% add your solution here





%----------
\exercise
Prove that products (a) exist and (b) are unique up to unique isomorphism in $\RMod$. 

\begin{hint}
    We covered the proof for coproducts in class, and verbally discussed the proof of uniqueness. Both these proofs are also in the book; use a similar diagram-chasing argument.
\end{hint}

\solution

% add your solution here



%----------
\exercise
What are the notions of (a) product and (b) coproduct in the category of groups $\Grp$? 

\begin{hint}
    Let $A,B \in \Grp$ and think about different ways you know how to build a new (bigger) group from $A$ and $B$. Once you've chosen your candidate constructions, prove that they satisfy the universal property defining products / coproducts.
\end{hint}

\solution

% add your solution here



%---------
\exercise 
Let $\SC$ be a category, let $A,B \in \SC$ be objects, and let $f \in \Hom(A,B)$ be an \textbf{isomorphism}. Prove that its inverse $f\inv \in \Hom(B,A)$ is unique.

\solution

% add your solution here




%-----------
\exercise
We now generalize products and coproducts to infinite families of objects. 
Let $\SC$ be a category, and let $A_i \in \SC$ for all $i \in I$, where $I$ is an arbitrary indexing set. 

\begin{definition}
The \textbf{coproduct} of the family of objects $\{A_i\}_{i \in I}$ is the pair $(C, \{\alpha_i: A_i \to C \}$ such that for any $X \in \SC$ with and set of maps $\{f_i: A_i \to X\}$, there exists a unique morphism $\theta: C \to X$ such that the following diagram commutes for all $i$:
\begin{center}
\begin{tikzcd}
    A_i \arrow{dr}{\alpha_i} \arrow[bend right=30, swap]{ddr}{f_i} & \\
    & C \arrow[dashed]{d}{\exists ! \  \theta} \\
    & X \\
\end{tikzcd}
\end{center}
\end{definition}

Such a coproduct, if it exists, is unique up to unique isomorphism, and is usually denoted $\coprod_{i \in I} A_i$. The \textbf{product} $P = \prod_{i \in I} A_i$ is also defined similarly:
\begin{center}
\begin{tikzcd}
    X \arrow[dashed, swap]{d}{\exists ! \  \theta}
        \arrow[bend left=30]{ddr}{f_i} & \\
    P \arrow[swap]{dr}{\alpha_i} & \\
    & A_i \\
\end{tikzcd}
\end{center}
Once again, if the product exists, is unique up to unique isomorphism. 


\bea
\item In $\RMod$, the coproduct and product are denoted $\bigoplus_i A_i$ and $\prod_i A_i$. Prove that if the indexing set $I$ is finite, then $\bigoplus_i A_i \cong \prod_i A_i$.

\begin{hint}
    First prove this for the family $\{A,B\}$. Then use induction.
\end{hint}

\item Let $\Ab$ denote the category of $\Z$-modules (abelian groups). Prove that when $I = \N$, the coproduct and product are \textbf{not} in general isomorphic. 

\begin{hint}
    Exhibit a counterexample. There's a hint in the lecture notes.
\end{hint}

\ee







\end{document}
