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\centerline{\Kopfont Commutative Algebra}

\smallskip
\centerline{Excercises 9}

\bigskip\noindent
A Noetherian local ring $A$ is said to be {\it Cohen-Macauley}
of dimension $d$ if it has dimension $d$ and there exists a system
of parameters $x_1,\ldots,x_d$ for $A$ such that $x_i$ is a nonzero
divisor in $A/(x_1,\ldots,x_{i-1})$ for $i=1,\ldots,d$.

\medskip\noindent
{\bf 1.} Cohen-Macauley rings of dimension 1. Part I: Theory.
\item{\bf (a)} Let $(A, \gm)$ be a local Noetherian with $\dim A=1$.
Show that if $x\in \gm$ is not a zero divisor then
\itemitem{\bf (1)} $\dim A/xA=0$, in other words $A/xA$ is Artinian,
in other words $\{x\}$ is a system of parameters for $A$.
\itemitem{\bf (2)} $A$ is has no embedded prime.
\item{\bf (b)} Conversely, let $(A,\gm)$ be a local Noetherian ring of
dimension $1$. Show that if $A$ has no embedded prime then there exists
a nonzero divisor in $\gm$.

\medskip\noindent
{\bf 2.} Cohen-Macauley rings of dimension 1. Part II: Examples.
\item{\bf b.} Let $A$ be the local ring at $(x,y)$ of $k[x,y]/(x^2,xy)$.
\itemitem{\bf (1)} Show that $A$ has dimension 1.
\itemitem{\bf (2)} Prove that every element of $\gm\subset A$ is a
zero divisor.
\itemitem{\bf (3)} Find $z\in \gm$ such that $\dim A/zA=0$
(no proof required).
\item{\bf c.} Let $A$ be the local ring at $(x,y)$ of $k[x,y]/(x^2)$.
Find a nonzero divisor in $\gm$ (no proof required).

\medskip\noindent
{\bf 3.} Local rings of embedding dimension $1$.
Suppose that $(A,\gm,k)$ is a Noetherian local ring
of embedding dimension $1$, i.e.,
$$
\dim_k \gm/\gm^2 = 1.
$$
Show that the function $f(n) = \dim_k \gm^n/\gm^{n+1}$
is either constant with value $1$, or its values are
$$
1,1,\ldots,1,0,0,0,0,0,\ldots
$$

\medskip\noindent
{\bf 4.} Regular local rings of dimension $1$.
Suppose that $(A,\gm,k)$ is a regular Noetherian local ring of
dimension $1$. Recall that this means that $A$ has dimension $1$
and embedding dimension $1$, i.e.,
$$
\dim_k \gm/\gm^2 = 1.
$$
Let $x\in\gm$ be any element whose class in $\gm/\gm^2$ is not zero.
\item{\bf (a)} Show that for every element $y$
of $\gm$ there exists an integer $n$ such that $y$ can be written as
$y=ux^n$ with $u\in A^\ast$ a unit.
\item{\bf (b)} Show that $x$ is a nonzero divisor in $A$.
\item{\bf (c)} Conclude that $A$ is a domain.

\medskip\noindent
{\bf 5.} Let $(A,\gm,k)$ be a Noetherian local ring with associated
graded $Gr_\gm(A)$. 
\item{\bf (a)} Suppose that $x\in \gm^d$ maps to a nonzero
divisor $\bar x \in \gm^d/\gm^{d+1}$ in degree $d$ of $Gr_\gm(A)$.
Show that $x$ is a nonzero divisor.
\item{\bf (b)} Suppose the depth of $A$ is at least $1$.
Namely, suppose that there exists a nonzero divisor $y \in \gm$.
In this case we can do better: assume just that $x\in \gm^d$ maps to
the element $\bar x \in \gm^d/\gm^{d+1}$ in degree $d$ of $Gr_\gm(A)$
which is a nonzero divisor on sufficiently high degrees: $\exists N$ such
that for all $n \geq N$ the map
$$
\gm^n/\gm^{n+1} \mapright{\bar x \cdot} 
\gm^{n+d}/\gm^{n+d+1}
$$
is injective. Then show that $x$ is a nonzero divisor.

\medskip\noindent
{\bf 6.} Suppose that $(A,\gm,k)$ is a Noetherian local ring of
dimension $1$. Assume also that the embedding dimension of $A$ is
$2$, i.e., assume that
$$
\dim_k \gm/\gm^2 = 2.	
$$
Notation: $f(n) = \dim_k \gm^n/\gm^{n+1}$. Pick generators $x,y \in \gm$ 
and write $Gr_\gm(A) = k[\bar x,\bar y]/I$ for some homogenous ideal $I$.
\item{\bf (a)} Show that there exists a homogenous element
$F\in k[\bar x, \bar y]$ such that $I \subset (F)$ with equality
in all sufficiently high degrees.
\item{\bf (b)} Show that $f(n) \leq n+1$.
\item{\bf (c)} Show that if $f(n) < n+1$ then $n \geq \deg(F)$.
\item{\bf (d)} Show that if $f(n) < n+1$, then $f(n+1) \leq f(n)$.
\item{\bf (e)} Show that $f(n) = \deg(F)$ for all $n >> 0$.

\medskip\noindent
{\bf 7.} Cohen-Macauley rings of dimension 1 and embedding dimension 2.
Suppose that $(A,\gm,k)$ is a Noetherian local ring which is Cohen-Macauley
of dimension $1$. Assume also that the embedding dimension of $A$ is
$2$, i.e., assume that
$$
\dim_k \gm/\gm^2 = 2.	
$$
Notations: $f$, $F$, $x,y\in \gm$, $I$ as in Ex.\ 6 above. Please
use any results from the problems above.
\item{\bf (a)} Suppose that $z\in \gm$ is an element whose class
in $\gm/\gm^2$ is a linear form
$\alpha \bar x + \beta \bar y \in k[\bar x, \bar y]$
which is coprime with $f$.
\itemitem{\bf (1)} Show that $z$ is a nonzero divisor on $A$.
\itemitem{\bf (2)} Let $d = \deg(F)$. Show that $\gm^n = z^{n+1-d}\gm^{d-1}$
for all sufficiently large $n$. (Hint: First show
$z^{n+1-d}\gm^{d-1} \to \gm^n/\gm^{n+1}$ is surjective by what you know
about $Gr_\gm(A)$. Then use NAK.)
\item{\bf (b)} What condition on $k$ garantees the existence of 
such a $z$? (No proof required; it's too easy.)

\noindent
Now we are going to assume there exists a $z$ as above. This turns out
to be a harmless assumption (in the sense that you can reduce to
the situation where it holds in order to obtain the results in
parts (d) and (e) below).
\item{\bf (c)} Now show that $\gm^\ell = z^{\ell-d+1} \gm^{d-1}$
for all $\ell \geq d$.
\item{\bf (d)} Conclude that $I = (F)$.
\item{\bf (e)} Conclude that the function $f$ has values
$$
2,3,4,\ldots,d-1,d,d,d,d,d,d,d,\ldots
$$

\medskip\noindent
This suggests that a local Noetherian Cohen-Macauley ring of dimension 1
and embedding dimension 2 is of the form $B/FB$, where $B$ is a 2-dimensional
regular local ring. This is more or less true (under suitable ``niceness''
properties of the ring).


\end
