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Assignment 2.  Due October 24 in class.  Feel free to work with other people in the class and use google and webpages to help to learn concepts (or ask me for definitions if you'd like); please do not ask me for hints or ask people online for help, but I will give clarification about what questions mean or explain concepts that are unclear.  I will post solutions later.
\begin{enumerate}

%\item One cannot form a category of categories, but we can create a category $\mathcal{C}$ whose objects are small categories and morphisms are functors.  (This is not a small category!)  Let $F:\mathcal{C}\to Set$, be the functor that sends a small category to the disjoint union of all of its Hom sets (and you can figure out what the functor does to a morphism).  Show that this is a representable functor.  
\item Let $R$ be a commutative ring.  Show that every $R$-module $M$ is a filtered colimit of its finitely generated submodules (Here, one takes the category of all finitely generated submodules of $M$ with maps between them being the inclusions.) 

\item Let $R$ be a commutative ring.  Show that every $R$-module is a filtered colimit of finitely presented modules (finitely presented just means that it is isomorphic to $R^n/N$ for some natural number $n\ge 1$ and some finitely generated submodule $N$ of $R^n$).
\item Let $\mathcal{B}$ be a small filtered subcategory of the category of commutative $k$-algebras ($k$ a field).  Show that if all objects of $\mathcal{B}$ are integral domains then the colimit is an integral domain.  What if we work in the category of noncommutative (but associative) $k$-algebras instead?
\item (Completion of a ring) Let $R$ be a local noetherian ring with maximal ideal $P$.  Then we have a subcategory 
of the category of rings whose objects are $R/P^n$ for $n\ge 1$ and where we have the ``reduction'' map $R/P^n\to R/P^m$ for $n\ge m$.  We define $\widehat{R}$ to be the limit of this diagram.  Show that the cone coming from the maps $R/P^n$ gives us a map $R\to \widehat{R}$ and that this map is injective. (Hint: you probably need Nakayama's lemma.)
\item Let $R$ and $\widehat{R}$ be as in the preceding question.  Suppose that $R$ is a $K$-algebra that is a discrete valuation ring.  Show that if the composition $K\to R\to R/P$ is an isomorphism then $\widehat{R}\cong K[[t]]$, the ring of power series in one variable.  
\item Show that if $p$ is prime and $R=\{a/b\colon p\not | b\}\subseteq \mathbb{Q}$ then $R$ is a local ring and the completion is isomorphic to $\mathbb{Z}_p$.  Why does this not contradict the preceding exercise?
\item Let $R$ be a ring and let $M$ be an $R$-module.  Show that if $M$ is a submodule of $Q$ then there is a maximal submodule $E$ of $Q$ such that $M\cap N$ is nonzero for every nonzero submodule $N$ of $E$.   
\item if $\mathcal{C}$ is a category and $\mathcal{H}$ is a class of morphisms in $\mathcal{C}$ then an object $I$ of $\mathcal{C}$ is $\mathcal{H}$-injective whenever $f:A\to I$ is a morphism and $h:A\to B$ is a morphism in $\mathcal{H}$ we have $f=g\circ h$ for some $g:B\to I$.  If $\mathcal{H}$ is not specified, one takes it to be the class of monomorphisms and we just say that $I$ is injective.  Let $R$, $M$, $E$, and $Q$ be as in the preceding exercise.  Show that if $Q$ is an injective $R$-module then so is $E$.
\item Show that an injective abelian group $A$ is divisible; that is, if $x\in A$ is nonzero and $n$ is a nonzero integer then there is some $y\in A$ such that $x=ny$.  

\item We say that an injective module is decomposable if we have $I\cong I'\oplus I''$ with $I'$ and $I''$ nonzero---otherwise, it is indecomposable.  Show that any indecomposable injective abelian group is isomorphic to either $\mathbb{Q}$ or to the subgroup of $\mathbb{C}^*$ given by
$\{\exp(2\pi i j/p^n)\colon n\ge 1, j\ge 0\}$ with $p$ a prime number.  


\end{enumerate}
\end{document}
\item Let $\mathcal{C}$ be a category with a zero object.  Show that if $C$ and $D$ are objects 
\item A category $\mathcal{A}$ is called an \emph{abelian category} if it has the following properties:
\begin{enumerate}
\item[a]
it has a zero object $0$ (i.e., $0$ is an initial and final object);
\item[b] if $A,B\in \mathcal{A}$ then $A\prod B$ and $A\coprod B$ exist;
\item[c] if $f:A\to B$ then the equalizer of $(f,0_{A\to B})$ (kernel) and the coequalizer of $(f,0_{A\to B})$ (cokernel) exist, where $0_{A\to B}$ is the map from $A\to 0\to B$;
\item[d] if $f:A\to B$ is a monomorphism (that is if $f\circ g=f\circ h$ then $g=h$ then $f$ is the kernel of some morphism (what does this mean?  ;
\item[e] if $f:A\to B$ is an epimorphism (that is if $g\circ f=h\circ f$ then $g=h$) then $f$ is the cokernel of some morphism.
\end{enumerate}
Show that if $\mathcal{A}$ is an abelian category then $\mathcal{A}^{\rm op}$ is an abelian category and that  if $A$ and $B$ are objects in $\mathcal{A}$ then ${\rm Hom}(A,B)$ can be given the structure of an abelian group.
\item Let $\mathcal{A}$ be an abelian category.  Show that the representable functor $h_A : \mathcal{A}\to Set$ can in fact be viewed as a functor from
$\mathcal{A}\to Ab$, where $Ab$ is the category of abelian groups.  In fact show that the functor $h_A$ is an \emph{additive functor}; that is, if $C$ and $D$ are objects in $\mathcal{A}$ then $h_A$ gives a group homomorphism from ${\rm Hom}(C,D)$ to ${\rm Hom}(h_A(C),h_A(D))$.
\item Use Yoneda's lemma to show that if $\mathcal{A}$ is a small abelian category then $\mathcal{A}$ embeds as a subcategory of $Ab$.  (If $\mathcal{A}$ is simply a locally small abelian category then the Freyd-Mitchell theorem says that an abelian category embeds in the category of left $R$-modules of some ring $R$, but this is much harder to prove.)
\item (Localization of categories) Let $\mathcal{C}$ be a category and let $S$ be a collection of morphisms.  A localization of $\mathcal{C}$ at $S$ is a category $\mathcal{C}[S^{-1}]$ equipped with a functor $F: \mathcal{C}\to \mathcal{C}[S^{-1}]$ such that:
\begin{enumerate}
\item[a] for all $f\in S$, $Q(f)$ is an isomorphism;
\item[b] (universality) for any category $\mathcal{A}$ and any functor $G:\mathcal{C}\to \mathcal{A}$ such that $G(f)$ is an isomorphism for every $f\in S$, there is a functor $G_S: C[S^{-1}]\to \mathcal{A}$ and a natural isomorphism $G\cong G_S \circ F$;
\item[c] If $\mathcal{A}$ is a category Then map $H\mapsto H\circ F$ from ${\rm Funct}(\mathcal{C}[S^{-1}],\mathcal{A})$ to ${\rm Funct}(\mathcal{C},\mathcal{A})$ is full and faithful (that is we can view the one functor category as a full subcategory of the other) for every category $\mathcal{A}$.  
\end{enumerate}
Show that if $\mathcal{C}$ is a category with one object, given by a ring $R$, and morphisms $r:R\to R$ given by left multiplication by $r$, then if $S$ is a multiplicatively closed subset of $R$ (which we think of as morphisms) then $\mathcal{C}[S^{-1}]$ is equivalent to the category with object $S^{-1}R$ and morphisms given by left multiplication by elements $s^{-1}r$.
\item Sheafification
