\documentclass[12pt]{amsart}
\title{Algebra 2. Homework 3}
\author{}

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\begin{document}
\maketitle

\begin{center}
{\em $(I,\leq)$ is a partially ordered set, and $\bfI$ is the
category associated to it, as in \S 6.2. Recall that, for a
category $\CalC$, $\Cov{\bfI}{\CalC}$ is the category
of direct systems valued in $\CalC$ and $\Contra{\bfI}{\CalC}$
is the category of inverse systems valued in $\CalC$.}\\
\end{center}

\hrule

\vspace*{0.25in}

\problem{1} Assume that $(I,\leq)$ has a unique maximal element $i_0\in I$
(that is, $i\leq i_0$ for every $i\in I$). Prove that
for any direct system $\{(X_i),(\psi_{ji}:X_i\to X_j)\}$,
we have an isomorphism $\Dlim{(I,\leq)} X_i \isom X_{i_0}$. \\

\problem{2} With the same hypothesis as in Problem 1, prove that
for any inverse system $\{(Y_i), (\varphi_{ij}:Y_j\to Y_i)\}$,
we have an isomorphism $Y_{i_0}\isom \Ilim{(I,\leq)} Y_i$.\\

\problem{3} Let $\CalC$ be a category such that for every
$\mathfrak{X}\in\Cov{\bfI}{\CalC}$, its direct limit exists.

\begin{enumerate}
\item Prove that this gives a functor:
\[
\lim_{\longrightarrow} : \Cov{\bfI}{\CalC} \to \CalC,
\]
sending a direct system to its limit.\\

\item Consider the constant functor $C : \CalC \to \Cov{\bfI}{\CalC}$,
which associates to each $X\in\CalC$, the direct system
$C(X)=\{(X_i)_{i\in I}, (\psi_{ji}:X_i\to X_j)_{i\leq j}\}$ with
$X_i=X$ for every $i\in I$ and $\psi_{ji}=\Id_X$ for every $i\leq j$.
Prove that $\ds \left(\lim_{\longrightarrow},C\right)$ is an
adjoint pair of functors.\\
\end{enumerate}

\problem{4} Let $\CalC$ be a category such that for every
$\mathfrak{Y}\in\Contra{\bfI}{\CalC}$, its inverse limit exists.

\begin{enumerate}
\item Prove that this gives a functor:
\[
\lim_{\longleftarrow} : \Contra{\bfI}{\CalC} \to \CalC,
\]
sending an inverse system to its limit.\\

\item Again, consider the constant functor $C : \CalC \to \Contra{\bfI}{\CalC}$,
which associates to each $Y\in\CalC$, the inverse system
$C(Y)=\{(Y_i)_{i\in I}, (\varphi_{ij}:Y_j\to Y_i)_{i\leq j}\}$ with
$Y_i=Y$ for every $i\in I$ and $\varphi_{ij}=\Id_Y$ for every $i\leq j$.
Prove that $\ds \left(C,\lim_{\longleftarrow}\right)$ is an
adjoint pair of functors.\\
\end{enumerate}

\problem{5} (Same hypothesis on $\CalC$ as in Problem 3 above).
Assume that we have two direct systems $\mathfrak{X}_1,\mathfrak{X}_2\in
\Cov{\bfI}{\CalC}$:
\[
\mathfrak{X}_{\ell} = \left\{ \left( X^{(\ell)}_i\right)_{i\in I},
\left(\psi^{(\ell)}_{ji} : X^{(\ell)}_i\to X^{(\ell)}_j\right)_{i\leq j}
\right\},\ \qquad \ell=1,2.
\]
Let $(f_i)_{i\in I}$ be a morphism in
$\Cov{\bfI}{\CalC}$ from $\mathfrak{X}_1$ to $\mathfrak{X}_2$,
such that $f_i$ is surjective, for every $i\in I$. Prove
that $\ds \lim_{\longrightarrow} f_i : \lim_{\longrightarrow} X^{(1)}_i
\to \lim_{\longrightarrow} X^{(2)}_i$ is surjective.\\

\problem{6} (Same hypothesis on $\CalC$ as in Problem 4 above).
Prove that inverse limit of injective morphisms is injective
(see Problem 5, where this statement is written in detail for
direct limits).\\

\problem{7} Assume $(I,\leq)$ is right--directed (i.e, for every
$i,j\in I$, there exists $k\in I$ such that $i\leq k$ and $j\leq k$).
Give a construction of the direct limit over $(I,\leq)$ in
the category of groups $\Gps$.\\

\problem{8} Let $G\in\Ab$ be an abelian group. Consider the
direct system in $\Ab$, whose elements are {\em finitely generated}
subgroups of $G$, and morphisms are natural inclusions.
Thus, the partially ordered set labels finitely generated
subgroups of $G$: $\{G_i\}_{i\in I}$, and $i\leq j$ means
$G_i\subset G_j$. Verify that this is a right--directed poset,
and prove that $\Dlim{(I,\leq)} G_i \isom G$.\\

\problem{9} Let $(I,\leq)=\N$ with its usual total order. Consider
the following two inverse systems, valued in $\Ab$:
\begin{itemize}
\item[(InvSys1)] $X_n=\Z$ for every $n\in\N$, and for every $m<n$,
$\varphi_{mn} : \Z \to \Z$ is multiplication by $3^{n-m}$.\\

\item[(InvSys2)] $X_n = \Z/2\Z$ for every $n\in \N$, and
all the morphisms are identities.\\
\end{itemize}

\begin{enumerate}
\item Prove that the inverse limit of (InvSys1) is $\{0\}$.
\item Prove that the canonical surjection $\Z\to\Z/2\Z$ is a morphism
from (InvSys1) to (InvSys2), whose inverse limit is $\{0\}\to \Z/2\Z$.
\end{enumerate}
({\em Hence, inverse limit of surjective maps need not be surjective.})\\

\problem{10} Again take $(I,\leq)=\N$. For each $n\in\N$,
let $A_n = \C[x]/(x^{n+1})$ and for each $m<n$, $\pi_{mn}:A_n\to A_m$
the natural projection (see \S 7.3).
\begin{enumerate}
\item Prove that in the
category of commutative rings, $\C[\![x]\!]\isom \Ilim{n\in\N} A_n$.\\

\item Consider the category of {\em $\N$--graded} commutative rings.
An object of this category is a (unital) commutative ring $R$,
together with abelian subgroups $R[k] (k\in \N)$ such that
\begin{itemize}
\item $\ds R=\bigoplus_{k=0}^{\infty} R[k]$ as an abelian group.
\item $x\in R[k]$ and $y\in R[{\ell}]$ implies $xy\in R[k+\ell]$.
\end{itemize}
Prove that, in this category $\ds \C[x]\isom \Ilim{n\in\N} A_n$.
\end{enumerate}
\end{document}

