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\begin{document}

\noindent Math 742, Spring 2016\\
Homework 7\\
Due: March 14 (MONDAY)

\section{Exercises}

\begin{enumerate}[(1)]

\item Let $k$ be a field and $R=k[x_1,\dots,x_n]$. Let $I$ be a monomial ideal.
  \begin{enumerate}
  \item Let $m$ be a monomial which is a minimal generator of $I$ and suppose we have a factorization into other monomials $m = m'm''$. Show that $I = (I + (m')) \cap (I + (m''))$.
  \item Use (a) to describe a primary decomposition of a monomial ideal into primary monomial ideals.
  \item In the case $R=k[x,y,z]$, find a primary decomposition of $(x^2, xz^3, yz)$. Identify the minimal and embedded associated primes.
  \end{enumerate}

\bigskip

\item Let $k$ be a field and let $R$ be a finitely generated $k$-algebra. Show that $R$ is artinian if and only if $R$ is a finite-dimensional vector space over $k$.

\bigskip

\item The results of Chapter 19 show that any artinian commutative ring is a direct product of artinian local rings. In particular, any finite commutative ring is a direct product of finite local rings. We will see that this is false in the non-commutative setting. Let $R$ be a nonzero ring (not necessarily commutative).

Some terminology: $x \in R$ is {\bf left-invertible} if there exists $y \in R$ such that $yx = 1$. {\bf Right-invertible} is defined in a similar way. An element is a {\bf unit} if it is both left- and right-invertible. A {\bf left ideal} $I$ is a subset closed under addition such that $x \in R$ and $y \in I$ implies $xy \in I$.  A left ideal $I$ is {\bf maximal} if it is not the whole ring and any other left ideal that contains $I$ is either the whole ring or equal to $I$. The notion of {\bf right ideal} and maximal right ideals is similar.

\begin{enumerate}
\item Assume that $xy = 1$ and $yx \ne 1$. Use the identity $yx(1-yx)=(1-yx)yx=0$ to show that $yx$ and $1-yx$ are not left-invertible and not right-invertible.

\item Show that if $R$ has a unique maximal left ideal, then this ideal is also a right ideal and, in fact, is the unique maximal right ideal. Also, show that this is the set of non-units of $R$. In this case we say that $R$ is {\bf local}. Conclude that in a local ring, left-invertibility and right-invertibility are equivalent.


\item $e \in R$ is a {\bf central idempotent} if $e^2 = e$ and $ex = xe$ for all $x \in R$, and it is nontrivial if $e \ne 0$ and $e \ne 1$. Show that $R$ is isomorphic to a direct product of two nonzero rings if and only if there exists a nontrivial central idempotent $e \in R$.

\item Let $k$ be a commutative field. Let 
\[
R = \left\{\left.\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \right| a,b,c \in k\right\}
\]
be the ring of $2 \times 2$ upper triangular matrices. Show that $R$ is not local and that $R$ has no nontrivial central idempotents. In particular, if $k$ is finite, this gives the desired example.
\end{enumerate}

\bigskip

\item Let $k$ be a field. Consider the polynomial ring $R = k[x,y,z,w]$ as a graded ring in the usual way. Compute the Hilbert polynomial and Hilbert series of the quotient ring $R/I$ where $I = (x,y) \cap (z,w)$.

\bigskip

\item Corollary 20.8 says that if $R$ is a polynomial ring where each variable has degree $1$, then the Hilbert function of a finitely generated module $M$ is eventually a polynomial function. Here's the general setting.

A {\bf quasi-polynomial} is a function $f \colon \bZ \to \bQ$ which is of the form $n \mapsto \sum_{i=0}^d c_i(n) n^i$ where $c_i(n) \colon \bZ \to \bQ$ is a periodic function for each $i$, i.e., there exists $p_i$ such that $c_i(n) = c_i(n + p_i)$ for all $n$; if $p_i$ is as small as possible, this is the minimal period of $c_i$. 

In the notation of Theorem 20.7, show that there exists a quasi-polynomial $h(M,n)$ such that the Hilbert function of $M$ is given by $h(M,n)$ for sufficiently large $n$, and the minimal period of each coefficient function divides $k_1 k_2 \cdots k_r$.
\end{enumerate}

\section{Suggested exercises (don't submit)}

From Altman--Kleiman:
\begin{itemize}
\item Chapter 18: 17, 26, 27
\item Chapter 19: 2, 5, 13, 16
\item Chapter 20: 5, 6, 9
\end{itemize}

\end{document}
