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\begin{document}

\title{Homework 6}

\maketitle

The goal of the first problem is to prove the Artin-Rees Lemma, which was
stated without proof in {\textit{Dimension II}} and applied to the Krull
Dimension Theorem.

\bigbreak\noindent
{\textbf{Problem 1.}} Let $A$ be a Noetherian commutative ring,
$\mathfrak{a}$ an ideal, and $M$ a finitely-generated $A$-module. Let $R = A
[x]$ be the polynomial ring and $M [x]$ the $R$-module of polynomials $\sum_i
m_i x^i$ with $m_i \in M$.
\[ R' = \left\{ \sum_{k = 0}^m a_k x^k \in R|a_k \in \mathfrak{a}^k \right\},
   \qquad M' = \left\{ \sum_{k = 0}^n m_k x^k |m_k \in \mathfrak{a}^k M
   \right\} \]
(i) Prove that $R'$ is a Noetherian ring and $M'$ a finitely generated
$R'$-module.

\smallskip\noindent
(ii) Let $N$ be a submodule of $M$. Let $N' = \left\{ \sum_{k = 0}^n m_k x^k
|m_k \in N \cap \mathfrak{a}^k M \right\}$. Show that $N'$ is a finitely
generated $R'$-module, and deduce the Artin-Rees Lemma.

\bigbreak

Now let $A$ be a commutative ring and $M$ an $A$-module. Define a topology on
$M$ in which a subset $U \subseteq M$ is open if and only if for every $x \in
U$, the set $U$ contains $x +\mathfrak{a}^n M$ for some $n$. Thus a sequence $x_i
\longrightarrow 0$ if and only if for every $n$, $x_i \in \mathfrak{a}$ for
sufficiently large $i$. This is called the $\mathfrak{a}$--adic topology on
$M$. This leads to the {\textit{$\mathfrak{a}$-adic completion}} $\hat{M}$ whose
use is a powerful technique for studying $M$.

\bigbreak\noindent
{\textbf{Problem 2.}} Assume that $A$ is Noetherian, and let $M$ a finitely
generated $\mathfrak{a}$-module with the $\mathfrak{a}$-adic topology. Let $N$
be a submodule of $M$. We have two topologies on $N$: its own
$\mathfrak{a}$-adic topology, and its subspace topology as a submodule of $M$.
Show that these two topologies are the same.

\medbreak\noindent
{\textbf{Hint:}} Use the Artin-Rees Lemma.

\subsection*{Representations}

If $M$ is a module for the group algebra $\mathbb{C} [G]$ of a finite group,
and $g \in G$, we define an endomorphism $\pi (g) : M \longrightarrow M$ by
$\pi (g) v = g \cdot v$. This is a representation of $G$. Conversely, given a
representation $\pi : G \longrightarrow \GL (M)$, where $M$ is a complex
vector space, then $M$ admits the structure of a $\mathbb{C} [G]$-module by
\[ \left( \sum_{g \in G} a_g g \right) \cdot v = \sum a_g \pi (g) (v), \qquad
   v \in M. \]
In these notes we will always assume that the module $M$ is \textit{finite-dimensional}.
The {\textit{character}} of $M$ is the character of the corresponding
representation. So
\[ \chi (g) = \tmop{tr} \; \pi (g) . \]
If $V$ is a vector space, a map $p : V \longrightarrow V$ is called a
{\textit{projection}} if $p^2 = p$.

\bigbreak\noindent
{\textbf{Problem 3}}. Let $G$ be a finite group, $R =\mathbb{C} [G]$ its
group algebra. If $M$ is an $R$-module, let
\[ M^G = \left\{ x \in M| \text{$g x = x$ for all $g \in G$} \right\} \]
be the module of invariants. Define the endomorphism $p : M \longrightarrow M$
by
\begin{equation}
  \label{pvdef} p (v) = \frac{1}{| G |} \sum_{g \in G} g \cdot v.
\end{equation}
Prove that $p$ is a projection map whose image is $M^G$ and deduce that
\[ \dim (M^G) = \frac{1}{| G |} \sum_{g \in G} \chi (g) . \]

\bigbreak\noindent
{\textbf{Problem 4.}} Let $M$ and $N$ be $\mathbb{C} [G]$-modules. Let $W =
\tmop{Hom}_{\mathbb{C}} (M, N)$ be the space of linear transformations $T : M
\longrightarrow N$. Show that $W$ has the structure of a $\mathbb{C}
[G]$-module such that
\[ (g \phi) (x) = g \cdot \phi (g^{- 1} \cdot
   x), \qquad x \in M. \]
Show that the character of $W$ is $\chi_M (g^{-1}) \chi_N (g)$, where
$\chi_M$ and $\chi_N$ are the characters of $M$ and $N$.

\bigbreak\noindent
{\textbf{Problem 5.}} Let $F$ be a field of characteristic $p$, and let $G
= \langle \sigma \rangle$ a cyclic group of order $p$ generated by $\sigma \in
G$. Let $R$ be the group algebra $F [G]$. Show that $R$ has elements $\tau_1,
\cdots, \tau_p$ such that
\begin{eqnarray*}
  \sigma \tau_1 & = & \tau_1,\\
  \sigma \tau_i & = & \tau_i - \tau_{i - 1} \quad (i > 1) .
\end{eqnarray*}
Describe all ideals of $R$ and deduce that $R$ is not a semisimple ring.

\end{document}
