\documentclass{amsart}
\title{Algebra 2. Exam 1}
\author{}

\usepackage{amsmath,amssymb}
\usepackage[usenames,dvipsnames]{color}
\usepackage{hyperref}
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    colorlinks=true,                            % false: boxed links; true: colored links
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\newcommand{\corb}[1]{\color{blue}{#1}}
\newcommand{\corr}[1]{\color{red}{#1}}
\usepackage[all]{xy}
\xyoption{arc}
%\usepackage[margin=1.5in]{geometry}

\newcommand{\sst}[1]{\scriptstyle{#1}}

\newcommand{\problem}[1]{\noindent {\bf Problem #1.}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\B}{\mathcal{B}}

\newcommand{\Csets}{\mathbf{Sets}}
\newcommand{\Hom}{\operatorname{Hom}}

\begin{document}
\maketitle

\noindent {\em Instructions.}\\
\begin{itemize}
\item Please read the hypotheses of each problem carefully.\\

\item Please do not seek help from your friends, professors, or
the internet!\\

\item You may consult the class notes. If you are going to use
a result, that we proved in class, or homework, please write the complete
reference (format, for example: Remark 9.6, page 8; or Homework 3, problem 4).\\

\item Please hand in your solutions to Sjuvon during recitation on
Tuesday (February 13, 2018) at 11.30AM. We will not accept late solutions,
or solutions by email!\\

\end{itemize}

\hrule\hrule

\vskip 0.3in


\problem{1} Let $\C$ be a category and let $A,B_1,B_2$ be objects of $\C$ and
let $b_\ell : A\to B_{\ell}$ ($\ell=1,2$) be two morphisms in $\C$.\\

\begin{itemize}
\item[(a)] Write the covariant functor $F : \C \to \Csets$, which if representable, is
represented by the following data:\\

An object $C$ of $\C$ together with two morphisms $c_{\ell} : B_{\ell}\to C$,
$\ell = 1,2$, such that $c_1b_1 = c_2b_2$. This data is then subject to the
following (universal) condition:\\

Given an object $D$ of $\C$ together with two morphisms $d_{\ell} : B_{\ell}\to D$,
$\ell = 1,2$, such that $d_1b_1 = d_2b_2$, there exists a unique morphism
$f : C\to D$ such that $d_{\ell} = fc_{\ell}$ (again, $\ell = 1,2$).\\

\[
\xy
(0,0)*{A}="a"; (20,0)*{B_1}="b1"; (0,-20)*{B_2}="b2"; (20,-20)*{C}="c";
(30,-30)*{D}="d";
{\ar^{b_1} "a"; "b1"};
{\ar_{b_2} "a"; "b2"};
{\ar^{c_1} "b1"; "c"};
{\ar_{c_2} "b2"; "c"};
"b1"; (30,-28)*{} **\crv{(35,-12)}?(1)*\dir{>};
"b2"; (28,-30)*{} **\crv{(12,-35)}?(1)*\dir{>};
{\ar@{-->}^f "c"; "d"};
(33,-15)*{\scriptstyle{d_1}};
(15,-33)*{\scriptstyle{d_2}};
\endxy
\]

\vskip 0.3in

\item[(b)] Assume that such an object $C$ together with morphisms $c_{\ell}$
exists, and assume
that $\widetilde{B} := B_1\oplus B_2$ also exists in $\C$, and let $\iota_{\ell}
: B_{\ell}\to \widetilde{B}$ be the morphisms from the definition of the
direct sum.\\

Prove that there exists a unique morphism $\pi: \widetilde{B}\to C$ such
that the following sequence is {\em left exact} (see what it means below), 
for every object $D$ of $\C$:\\

\[
\xy
(0,0)*{\Hom_{\C}(C,D)}="a";
(30,0)*{\Hom_{\C}(\widetilde{B},D)}="b";
{\ar "a"; "b"};
(70,0)*{\Hom_{\C}(A,D)};
{\ar (40,1)*{}; (60,1)*{}};
{\ar (40,-1)*{}; (60,-1)*{}};
(15,1)*{\sst{-\circ\pi}};
(50,2)*{\sst{-\circ\iota_1b_1}};
(50,-2)*{\sst{-\circ\iota_2b_2}};
\endxy
\]

\noindent {\em Meaning.} The leftmost map is one to one; and its image
is equal to the set of elements of the middle Hom set which go to the 
same element via the two rightmost maps.\\

\end{itemize}

\problem{2} Let $\C$ be an abelian category and consider a sequence of 
morphisms in $\C$:
\[
A\rightarrow B\rightarrow C\rightarrow 0
\]
Prove that this sequence is exact if, and only if, for every $X\in \C$
the following sequence of abelian groups is exact:
\[
0\rightarrow \Hom_{\C}(C,X)\rightarrow \Hom_{\C}(B,X)\rightarrow
\Hom_{\C}(A,X)
\]

\vskip 0.3in

\problem{3} Let $\A$ and $\B$ be two abelian categories. Assume we are given
two additive covariant functors $F:\A\to \B$ and $G:\B\to\A$ such that
$(F,G)$ is an adjoint pair. Prove that $F$ is always right exact. (Also,
$G$ is always left exact, but you don't have to prove it).\\

\noindent {\bf Bonus problem.} Take $R$ to be a commutative ring with $1\neq 0$.
Let $\A$ be the category of $R$--modules. For each $M\in \A$, prove
that we have a pair of adjoint functors $(M\otimes_R -, \Hom_{\A}(M,-))$
from $\A$ to itself. {\em Hence, by Problem 2, we get another proof that $M\otimes_R -$
is a right exact functor}.
\end{document}

