\documentclass[12pt]{amsart}

\usepackage[margin=1in]{geometry}
\usepackage{enumerate}
\usepackage{eucal, hyperref}

%%defines mathcal, mathbb, and mathfrak capital letters
\def\makeCal#1{%
\expandafter\newcommand\csname c#1\endcsname{\mathcal{#1}}}
\def\makeBf#1{%
\expandafter\newcommand\csname b#1\endcsname{\mathbf{#1}}}
\def\makeFrak#1{%
\expandafter\newcommand\csname f#1\endcsname{\mathfrak{#1}}}

\makeatletter
\count@=0
\loop
\advance\count@ 1
\edef\y{\@Alph\count@}%
\expandafter\makeCal\y
\expandafter\makeBf\y
\expandafter\makeFrak\y
\ifnum\count@<26
\repeat
\makeatother

\newcommand{\Set}{\mathsf{Set}}
\DeclareMathOperator{\Ann}{Ann}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Supp}{Supp}
\newcommand{\op}{\mathrm{op}}

\usepackage{xypic}
\renewcommand{\cD}{\mathcal{D}}
\newcommand{\fm}{\mathfrak{m}}

\begin{document}

\noindent Math 746, Spring 2016\\
Homework 3\\
Due: March 18

\section{Exercises}

\begin{enumerate}[(1)]

\item Eisenbud 19.10

\item Eisenbud 19.15

\bigskip

\item Let $\phi$ be a skew-symmetric matrix (i.e., $\phi_{ii} = 0$ and $\phi_{ij} = -\phi_{ji}$) of odd size $2n+1$ with entries in $R$. Let $f_j$ be the Pfaffian\footnote{Recall that if $X$ is a skew-symmetric matrix of even size, there is a function, the Pfaffian, such that ${\rm Pf}(X)^2 = \det(X)$. This has a Laplace expansion formula similar to the determinant case; the wikipedia page has many basic properties, or see any introductory algebra text.} of the skew-symmetric submatrix obtained by deleting the $j$th column and $j$th row of $\phi$. Let $J(\phi)$ be the ideal generated by $f_1,\dots,f_{2n+1}$. Define 
\[
\bF \colon 0 \to R \xrightarrow{\phi_3} R^{2n+1} \xrightarrow{\phi_2} R^{2n+1} \xrightarrow{\phi_1} R
\]
as follows:
\begin{itemize}
\item $\phi_1 = \begin{pmatrix} f_1 & -f_2 & \cdots & -f_{2n} & f_{2n+1} \end{pmatrix}$,
\item $\phi_2 = \phi$,
\item $\phi_3 = \phi_1^T$, where $T$ denotes transpose.
\end{itemize}

\begin{enumerate}
\item Verify that $\bF_\bullet$ is a complex.
\item Consider the universal case $R = \bZ[x_{ij} \mid 1 \le i < j \le 2n+1]$ and 
\[
\phi_{ij} = \begin{cases} 0 & \text{if $i=j$}\\ x_{ij} & \text{if $i<j$} \\ -x_{ij} & \text{if $i>j$} \end{cases}.
\]
Show that ${\rm depth}\ J(\phi) = 3$ and that $\bF_\bullet$ is exact.
\item Go back to the general case of a noetherian ring $R$ and assume that $J(\phi) \ne R$. Use the generic perfection theorem to show that ${\rm depth}\ J(\phi) \le 3$ and that $\bF_\bullet$ is exact if and only if ${\rm depth}\ J(\phi) = 3$.
\end{enumerate}

\bigskip

\item Let $k$ be a field and let $I \subset k[x,y,z]$ be the ideal of polynomials vanishing on a finite set of points in $\bP^2_k$.\footnote{Algebraically, this means that $I$ is radical and $R/I$ is equidimensional of dimension $1$.} Use the Hilbert--Burch theorem to show that if the points lie on a curve of degree $d$, then $I$ can be generated by $d+1$ elements.

\end{enumerate}

\section{Further reading}

The complex in \#3 was studied in Buchsbaum, Eisenbud, ``Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$'' and plays a role similar to the Hilbert--Burch complex. Namely, assuming that the ideal $J(\phi)$ has depth $3$, the ring $R/J(\phi)$ is Gorenstein, which we will learn about later in Chapter 21. Conversely, a depth $3$ ideal whose quotient is Gorenstein and which has a finite free resolution of length 3 must be of the form $J(\phi)$ for some skew-symmetric matrix $\phi$ (up to a choice of basis, $\phi$ is the second map in the free resolution) -- the existence of the resolution is automatic if $R$ is a regular (graded) local ring.

\end{document}
