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\begin{document}

\noindent Math 746, Spring 2016\\
Homework 2\\
Due: February 26

\section{Exercises}

\begin{enumerate}[(1)]

\item Eisenbud 18.12(a,b)

\bigskip

\item Cohen--Macaulayness of graded rings can be used in combinatorics to prove positivity statements. Here's one simple positivity statement that can be proven without much machinery.

Let $R = \bigoplus_{n \ge 0} R_n$ be a noetherian graded algebra which is generated by $R_1$ over the field $k=R_0$. Assume that $R$ has depth $d$ with respect to the homogeneous ideal $R_+ = \bigoplus_{n > 0} R_n$.
  \begin{enumerate}
  \item Show that if $k$ is infinite, then there is a regular sequence of length $d$ consisting of elements in $R_1$.
  \item The Hilbert series of $R$ can be written as
\[
\sum_{n \ge 0} (\dim_k R_n) t^n = \frac{h_0 + h_1t + \cdots + h_r t^r}{(1-t)^{\dim R}}.
\]
If $R$ is Cohen--Macaulay, show that $h_i \ge 0$ for $i=0,\dots,r$. (Reduce to the case where $k$ is infinite and use (a)).
  \end{enumerate}

\bigskip

\item Let $k$ be a field and let $n$ be a positive integer. Let $R = k[x_{ij} \mid 1 \le i,j \le n]$ be the polynomial ring in $n^2$ variables, which we think of as the polynomial functions on the set of $n \times n$ matrices $X = (x_{ij})$. Let $\cN$ be the set of nilpotent matrices and let $I$ be the ideal of all polynomial functions which vanish on $\cN$. 

  \begin{enumerate}
  \item Show that $\cN$ is an irreducible algebraic set. (Depending on your background, you might not have enough to do this -- so feel free to skip this part and use it for the remaining parts.)

  \item Find $n$ equations $f_1, \dots, f_n$ with $\deg(f_i)=i$ so that a matrix is nilpotent if and only if each $f_i$ vanishes on it. In particular, $\sqrt{(f_1, \dots ,f_n)} = I$. For example, when $n=2$, $f_1 = {\rm trace}(X) = x_{1,1} + x_{2,2}$ and $f_2 = \det(X) = x_{1,1} x_{2,2} - x_{1,2} x_{2,1}$.

  \item Show that the Jacobian of $f_1, \dots, f_n$ evaluated on a generic nilpotent matrix (i.e., having one Jordan block) has full rank $n$. 

  \item Conclude that $(f_1, \dots, f_n) = I$ and hence $R/I$ is Cohen--Macaulay.

  \item Show that $R/I$ satisfies $({\rm R}_1)$\footnote{Geometrically: the singular locus of $\cN$ has codimension $\ge 2$ in $\cN$} and hence is normal. (Again, depending on background, you might not have enough to do this, so feel free to skip.)
  \end{enumerate}

\end{enumerate}

\section{Further reading}

\begin{itemize}
\item Eisenbud 18.14 says that the ring of invariants $R^G$ of a Cohen--Macaulay ring $R$ is also Cohen--Macaulay in characteristic $0$. For a proof, see Corollary 6.4.6 of Bruns, Herzog, {\it Cohen--Macaulay Rings}.

\item Hochster's ICM talk on Cohen--Macaulay rings may also be of interest:
\url{http://www.mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0291.0298.ocr.pdf}.
\end{itemize}

\end{document}
