\documentclass[12pt]{amsart}
\title{Algebra 2. Homework 4}
\author{}

\usepackage{amsmath,amssymb}

\usepackage[margin=1in]{geometry}
\usepackage[all]{xy}
\xyoption{arc}
\newcommand{\ds}{\displaystyle}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}

\newcommand{\Ad}{\operatorname{Ad}}
\newcommand{\ol}[1]{\overline{#1}}
\newcommand{\wt}[1]{\widetilde{#1}}

\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}

\newcommand{\morph}[1]{\stackrel{#1}{\longrightarrow}}
\newcommand{\Ker}{\operatorname{Ker}}
\newcommand{\CalA}{\mathcal{A}}
\newcommand{\CalB}{\mathcal{B}}

\newcommand{\CalC}{\mathcal{C}}
\newcommand{\CalD}{\mathcal{D}}
\newcommand{\Ob}{\operatorname{Ob}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Id}{\operatorname{Id}}
\newcommand{\h}{\mathsf{h}}
\newcommand{\op}{\scriptscriptstyle{\operatorname{op}}}
\newcommand{\fd}{\scriptscriptstyle{\operatorname{fd}}}


\newcommand{\Cov}[2]{{\mathcal{F}}(#1,#2)}
\newcommand{\Contra}[2]{{\mathcal{F}}(#1^{\op},#2)}

\newcommand{\Sets}{{\mathbf{Sets}}}
\newcommand{\Gps}{{\mathbf{Gps}}}
\newcommand{\Ab}{{\mathbf{Ab}}}
\newcommand{\Vect}{{\mathbf{Vect}}}
\newcommand{\GSets}[1]{#1\mbox{-}\Sets}
\newcommand{\Rmod}[1]{#1\mbox{-}\mathbf{\operatorname{mod}}}
\newcommand{\isom}{\stackrel{\sim}{\longrightarrow}}

\newcommand{\bfI}{\mathbf{I}}

\newcommand{\Dlim}[1]{\ds \lim_{\begin{subarray}{c}
\longrightarrow \\ #1\end{subarray}}}
\newcommand{\Ilim}[1]{\ds \lim_{\begin{subarray}{c}
\longleftarrow \\ #1\end{subarray}}}

\newcommand{\PP}[2]{\noindent {\bf (#1) #2.--}}

\newtheorem*{thm}{Theorem}
\newtheorem*{prop}{Proposition}
\newtheorem*{lem}{Lemma}
\newtheorem*{cor}{Corollary}
\newenvironment{pf}{\paragraph{{\sc Proof}}}{\qed\par\medskip}
\theoremstyle{definition}
\newtheorem*{defn}{Definition}
\newtheorem*{rem}{Remark}
\newtheorem*{rems}{Remarks}
\newtheorem*{ex}{Example}

\newcommand{\problem}[1]{\noindent {\bf Problem #1.}}

\begin{document}
\maketitle

\problem{1} Let $\CalA$ be an abelian category.
\begin{enumerate}
\item Show that a sequence of morphisms $0\to A\morph{f}B\morph{g}C$
is exact if, and only if for every $X\in \CalA$, the following is
an exact sequence in $\Ab$:
\[
0\to \Hom_{\CalA}(X,A) \morph{f\circ -}\Hom_{\CalA}(X,B)\morph{g\circ -}
\Hom_{\CalA}(X,C).
\]

\item Show that a sequence of morphisms $A\morph{f}B\morph{g}C\to 0$
is exact if, and only if for every $X\in \CalA$, the following is
an exact sequence in $\Ab$:
\[
0\to \Hom_{\CalA}(C,X) \morph{-\circ g}\Hom_{\CalA}(B,X)\morph{-\circ f}
\Hom_{\CalA}(A,X).
\]

\end{enumerate}

\problem{2} Let $\CalA$ and $\CalB$ be two abelian categories.
Consider a pair of additive covariant functors $\ds \xy
(0,0)*{\CalA}; (10,0)*{\CalB}; {\ar^F (2,1); (8,1)};
{\ar^G (8,-1); (2,-1)};
\endxy$. Assume that $(F,G)$ is an adjoint pair. Prove that
$F$ is right exact, and $G$ is left exact.\\

\problem{3} Assume that $\CalC$ is an additive category in which
arbitrary direct sums and products exist. Further assume that
direct sums and products are isomorphic. Prove that $\CalC$
is trivial (that is, every object of $\CalC$ is isomorphic
to the trivial one).\\

\hrule

\vspace*{0.25in}

\begin{center}
{\em In problems 4--6 below, $\CalA=\Rmod{R}$ is the category
of left $R$--modules for a unital ring $R$.}\\
\end{center}

\problem{4} Let $J$ be a set. Prove that $\ds \bigoplus_J,\prod_J
: \CalA^J \to \CalA$ are exact functors.\\

\problem{5} Let $(I,\leq)$ be a {\em right directed} partially ordered
set. Prove that the direct limit 
$\Dlim{(I,\leq)} : \Cov{\bfI}{\CalA} \to \CalA$ is an exact functor.
{\em Recall that $\bfI$ is the category associated to the partially
ordered set $(I,\leq)$ and $\Cov{\bfI}{\CalA}$ is the category
of direct systems valued in $\CalA$.}\\

\problem{6} Let $(I,\leq)$ be a partially ordered set. Prove that
the inverse limit $\Ilim{(I,\leq)} : \Contra{\bfI}{\CalA}\to\CalA$
is left exact, but not right exact.\\

\hrule

\vspace*{0.25in}

\begin{center}
{\em In problems 7--10, $\CalA=\Rmod{A}$ for a unital
commutative ring $A$.}
\end{center}

\problem{7} Let $I$ be a set, $\{M_i\}_{i\in I}$ a set of $A$--modules.
For $N\in\Rmod{A}$, show that we have an isomorphism:
\[
\left(\bigoplus_{i\in I} M_i\right) \otimes N
\cong
\bigoplus_{i\in I} (M_i\otimes N).
\]
\ \\

\problem{8} Prove or give a counterexample to the statement of the
previous problem, if direct sum (on both sides of the equation) 
is replaced by direct product.\\

\problem{9} Let $(I,\leq)$ be a {\em right directed} poset, and
$\ds \left\{(M_i)_{i\in I}, (\psi_{ji}:M_i\to M_j)_{i\leq j}\right\}$
a directed system of $A$--modules. Prove that, for every $N\in\Rmod{A}$:
\[
\left(\Dlim{i\in I} M_i\right) \otimes N \cong
\Dlim{i\in I} (M_i\otimes N).
\]

\problem{10} Is the statement from the previous problem true for
inverse limits? 

\end{document}

