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% THEOREM ENVIRONMENTS
\theoremstyle{definition}
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\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{\cat}{\SC} % 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
\DeclareMathOperator{\Set}{\mathbf{Set}} % cat of sets





% SOME ENVIRONMENTS
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% round matrix
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% square matrix
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\title{MAT 250B HW04}
\author{[add your name here]}
\date{Due Friday, 2/2/24 at 11:59 pm on Gradescope}

\begin{document}

\maketitle

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

\paragraph{Notes} 
\begin{itemize}
    \item We will continue to write ``$R$-module'' as shorthand for ``left $R$-module''. 
    \item We talked about pullbacks in class, but we didn't get to pushouts. You can read about this dual construction in the book or in the posted lecture notes.
\end{itemize}


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


%-----------
\exercise


Let $R$ be a ring and let $M \in \RMod$. 

\bea

\item Prove that $\Hom_{\Z}(R,M)$ is a left $R$-module under the action $(r\varphi)(x) = \varphi(xr)$ where $r,x \in R$. 

\item Let $0 \to A \map{\alpha} B$ is an exact sequence of $R$-modules. Prove that if every $R$-map $f: A \to M$ lifts to an $R$-map $\tilde f: B \to M$ where $f = \tilde f \circ \alpha$, then every $R$-map $F: A \to \Hom_{\Z}(R,M)$ lifts to an $R$-map $\tilde F: B \to \Hom_{\Z}(R,M)$. 

\textit{Hint:} Given $F$, show that  $f(a) = F(a)(1_R)$ defines an $R$-map from $A$ to $M$. 
Then show that $\tilde F(b)(r) = \tilde f(rb)$ defines an $R$-map from $B$ to $\Hom_{\Z}(R,M)$. Finally, check that $\tilde F$ lifts $F$. 

\item Prove that if $Q$ is an injective $R$-module, then $\Hom_{\Z}(R,Q)$ is also an injective $R$-module. 

\ee






%-----------
\exercise
Let $R$ be a ring.
In this exercise you will prove the following theorem. 

\begin{theorem}
    Every $R$-module $M$ is contained in an injective $R$-module. 
\end{theorem}

\bea

\item Show that $M$ (viewed as a $\Z$-module) is contained in an injective $\Z$-module $Q$. 

\item Prove that $\Hom_R(R,M) \subseteq \Hom_{\Z}(R,M) \subseteq \Hom_{\Z}(R,Q)$.

\item Prove that $M \cong \Hom_R(R,M)$ as $R$-modules, and then conclude that $M$ is contained in an injective $R$-module.

\ee





%-----------
\exercise
Let $G$ be an abelian group, written additively. Its \textbf{Pontrjagin dual} is the group 
\[
    G^* = \Hom_\Z(G, \Q/\Z).
\]
\bea
\item Prove that for any nonzero $a \in G$, there exists a homomorphism $f: G \to \Q/\Z$ with $f(a) \neq 0$.
\textit{Why do we care? See remark below.}
\item Prove that if  $0 \to A \to G \to B \to 0$ is an exact sequence of abelian groups, then so is $0 \to B^* \to G^* \to A^* \to 0$.
\item If $G$ is a finite abelian group, prove that $G^* \cong G$.
\ee

\begin{remark}
    In the wild, we usually let $G$ be a locally compact abelian topological group (e.g. $\R^n, \C^\times, \Q_p,$ any torus $\mathbb{T}^n$), and the Pontrjagin dual $\widehat G = \Hom(G,\R/\Z)$ consists of all continuous homomorphisms from $G$ to the circle group $\R/\Z$. This allows us to generalize Fourier transform to any such $G$. In this exercise, we are treating $\Q$ as a topological space with the discrete topology.
\end{remark}



%-----------
\exercise

Let $\varphi: A \to B$ be an $R$-module homomorphism. Prove that $\coker \varphi$ is a pushout. 
\textit{Hint: Come up with a pushout diagram for $\coker \varphi$, and show that $\coker \varphi$ satisfies the universal property of pushouts.}


%-----------
\exercise
In this exercise, you will prove that in the the category $\RMod$, pullbacks and pushouts always exist. 
\bea
\item Let $f: A \to C$ and $g: B \to C$ be two $R$-module maps. Prove that the \textbf{pullback} exists, by considering the set  
\[
    P = \{ (a,b) \in A \times B  \st f(a) = g(b) \} \qquad ( = A \times_C B)
\]

Here is a \texttt{tikzcd} pullback diagram with some suggested map names:
\[
    \begin{tikzcd}
        P \arrow{r}{\beta} \arrow{d}[swap]{\alpha}
            & B \arrow{d}{g} \\
        A \arrow{r}[swap]{f} 
            & C \\
    \end{tikzcd}
\]
\textit{
    First, briefly verify that $P$ is actually an object in $\RMod$; one sentence is enough. Then check that $(P, \alpha, \beta)$ is a solution and that it satisfies the universal property of pullbacks.
}


\item Now let $f: C \to A$ and $g: C \to B$ be two $R$-module maps. Prove that the \textbf{pushout} exists, by defining
\[
    S = \{ (f(c), -g(c)) \in A \oplus B \st c \in C\}
\]
and defining the $R$-module 
\[
    Q = (A \oplus B) / S.
\]
\textit{
    First briefly verify that $S$ is actually a submodule of $A \oplus B$; one sentence is enough.
}
\ee


%-----------
\exercise

\begin{definition}
Let $\cat$ be a category. 
\begin{itemize}
    \item An object $A \in \cat$ is an \textbf{initial object} if, for every object $C \in \cat$, there exists a unique morphism $A \to C$.
    \item An object $\Omega \in \cat$ is a \textbf{terminal object} if, for every object $C \in \cat$, there exists a unique morphism $C \to \Omega$.  
    \item An object $Z \in \cat$ is a \textbf{zero object} if it is both initial and terminal.
\end{itemize}    
\end{definition}

\bea

\item Prove that initial and terminal objects are unique (up to unique isomorphism) if they exist.
\item Suppose $\cat$ contains a terminal object $\Omega$. Characterize products in $\cat$ as pullbacks.
\item Suppose a category contains an initial object $A$. Characterize coproducts in $\cat$ as pushouts.
\item What are the initial and terminal objects in $\Set$? Prove that $\Set$ does not have a zero object.
\item What are the initial and terminal objects in $\Ab$? What is the zero object?

\ee








\end{document}
