%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Hyperplane Arrangement Cohomology and Monomials in
%            the Exterior Algebra
%
% David Eisenbud, Sorin Popescu, Sergey Yuzvinsky
%
% 01/07/2000
%%%% PlainTeX (requires diagrams.tex)
%------------------------------
%=========================================================================%
% load begin.tex only once, but keep count to match \bye commands 
%=========================================================================% 
 
\ifx\begin\undefined\else\global\advance\srcdepth by 
1\expandafter\endinput\fi 
 
\def\begin{} 
\newcount\srcdepth 
\srcdepth=1 
 
\outer\def\bye{\global\advance\srcdepth by -1 
  \ifnum\srcdepth=0 
    \def\endcmd{\vfill\eject\nopagenumbers\par\vfill\supereject\end} 
  \else\def\endcmd{}\fi 
  \endcmd 
} 
 
%=========================================================================% 
% initialize TeX 
%=========================================================================% 
 
\magnification=\magstephalf 
\baselineskip=13pt 
\hsize = 5.5truein 
\hoffset = 0.5truein 
\vsize = 8.5truein 
\voffset = 0.2truein 
\emergencystretch = 0.05\hsize 
 
\newif\ifblackboardbold 
 
% comment out the following line if AMS msbm fonts aren't available 
\blackboardboldtrue 
 
%=========================================================================% 
% select fonts 
%=========================================================================% 
 
\font\titlefont=cmbx12 scaled\magstephalf 
\font\sectionfont=cmbx12 
\font\netfont=cmtt9 
 
% Establish AMS blackboard bold fonts without using amssym.def, amssym.tex 
 
\newfam\bboldfam 
\ifblackboardbold 
\font\tenbbold=msbm10 
\font\sevenbbold=msbm7 
\font\fivebbold=msbm5 
\textfont\bboldfam=\tenbbold 
\scriptfont\bboldfam=\sevenbbold 
\scriptscriptfont\bboldfam=\fivebbold 
\def\bbold{\fam\bboldfam\tenbbold} 
\else 
\def\bbold{\bf} 
\fi 
 
%=========================================================================% 
% font size-changing command ("A Beginner's Book of TeX" p35, p275) 
%=========================================================================% 
 
\font\Arm=cmr8 
\font\Ai=cmmi8 
\font\Asy=cmsy8 
\font\Abf=cmbx8 
\font\Brm=cmr6 
\font\Bi=cmmi6 
\font\Bsy=cmsy6 
\font\Bbf=cmbx6 
\font\Crm=cmr5 
\font\Ci=cmmi5 
\font\Csy=cmsy5 
\font\Cbf=cmbx5 
 
\ifblackboardbold 
\font\Abbold=msbm10 at 8pt 
\font\Bbbold=msbm7 at 6pt 
\font\Cbbold=msbm5 
\fi 
 
\def\smallmath{% 
\textfont0=\Arm \scriptfont0=\Brm \scriptscriptfont0=\Crm 
\textfont1=\Ai \scriptfont1=\Bi \scriptscriptfont1=\Ci 
\textfont2=\Asy \scriptfont2=\Bsy \scriptscriptfont2=\Csy 
\textfont\bffam=\Abf \scriptfont\bffam=\Bbf \scriptscriptfont\bffam=\Cbf 
\def\rm{\fam0\Arm}\def\mit{\fam1}\def\oldstyle{\fam1\Ai}% 
\def\bf{\fam\bffam\Abf}% 
\ifblackboardbold 
\textfont\bboldfam=\Abbold 
\scriptfont\bboldfam=\Bbbold 
\scriptscriptfont\bboldfam=\Cbbold 
\def\bbold{\fam\bboldfam\Abbold}% 
\fi 
} 
 
%=========================================================================% 
% single-pass symbolic theorem labeling 
%=========================================================================% 
 
% Because this is a single-pass mechanism with no .aux file, forward 
% references need to be declared in advance: 
 
%   \forward{thm:main}{Theorem}{1.1} 
 
% This is also the mechanism for "timely" declaration of labels, which 
% will usually be buried within the corresponding theorem macros. 
% A warning is issued if a label redeclaration is inconsistent, allowing 
% forward references to be manually fixed. 
 
%   \ref{thm:main} produces "Theorem~1.1" 
%   \refs{thm:main} produces "Theorems~1.1" 
%   \refn{thm:main} produces "1.1" 
 
% Some TeX adapted from "The Advanced TeXbook" by David Salomon, chapter 9. 
 
% Implementers: The code for \forward is subtle. Its second argument must 
% be provided literally, e.g. "Theorem" rather that "\capitalize{theorem}". 
% Its third argument must either be literal or a macro that expands 
% directly to a literal, e.g. "\edef\numtoks{\number\proccount}". 
% This use of \edef cannot be replaced by \def, which defers expansion. 
% Failure to follow these rules will cause spurious warnings that forward 
% references are inconsistent, when they are in fact consistent after 
% expansion. Note the "Towers of Palo Alto" recreational math problem 
% involving the iterated use of \expandafter to expand the first argument 
% to \forwardsub before calling it. 
 
\newlinechar=`@ 
\def\forwardmsg#1#2#3{\immediate\write16{@*!*!*!* forward reference should 
be: @\noexpand\forward{#1}{#2}{#3}@}} 
\def\nodefmsg#1{\immediate\write16{@*!*!*!* #1 is an undefined reference@}} 
 
\def\forwardsub#1#2{\def\newref{{#2}{#1}}} 
 
\def\forward#1#2#3{% 
\expandafter\expandafter\expandafter\forwardsub\expandafter{#3}{#2} 
\expandafter\ifx\csname#1\endcsname\relax\else% 
\expandafter\ifx\csname#1\endcsname\newref\else% 
\forwardmsg{#1}{#2}{#3}\fi\fi% 
\expandafter\let\csname#1\endcsname\newref} 
 
\def\firstarg#1{\expandafter\argone #1}\def\argone#1#2{#1} 
\def\secondarg#1{\expandafter\argtwo #1}\def\argtwo#1#2{#2} 
 
\def\ref#1{\expandafter\ifx\csname#1\endcsname\relax 
  {\nodefmsg{#1}\bf`#1'}\else 
  \expandafter\firstarg\csname#1\endcsname 
  ~\expandafter\secondarg\csname#1\endcsname\fi} 
 
\def\refs#1{\expandafter\ifx\csname#1\endcsname\relax 
  {\nodefmsg{#1}\bf`#1'}\else 
  \expandafter\firstarg\csname #1\endcsname 
  s~\expandafter\secondarg\csname#1\endcsname\fi} 
 
\def\refn#1{\expandafter\ifx\csname#1\endcsname\relax 
  {\nodefmsg{#1}\bf`#1'}\else 
  \expandafter\secondarg\csname #1\endcsname\fi} 
 
%=========================================================================% 
% widow control 
%=========================================================================% 
 
% usage: 
% \widow{.2} % start new page if <.2 page left 
 
\def\widow#1{\vskip 0pt plus#1\vsize\goodbreak\vskip 0pt plus-#1\vsize} 
 
%=========================================================================% 
% sections and theorems 
%=========================================================================% 
 
% use \showlabels or \showlabelsabove to display section and theorem labels 
 
\def\marginlabel#1{} 
 
\def\showlabels{ 
\font\labelfont=cmss10 at 8pt 
\def\marginlabel##1{\llap{\labelfont##1\quad}} 
} 
 
\def\showlabelsabove{ 
\font\labelfont=cmss10 at 6pt 
\def\marginlabel##1{\rlap{\smash{\raise 10pt\hbox{\labelfont##1}}}} 
} 
 
\newcount\seccount 
\newcount\proccount 
\seccount=0 
\proccount=0 
 
\def\stdskip{\vskip 9pt plus3pt minus 3pt} 
\def\stdbreak{\par\removelastskip\penalty-100\stdskip} 
 
\def\proof{\stdbreak\noindent{\sl Proof. }} 
 
\def\qed{\vrule height 1.2ex width .9ex depth .1ex} 
 
\def\Box{ 
  \ifmmode\eqno\qed 
  \else\ifvmode\removelastskip\line{\hfil\qed} 
  \else\unskip\quad\hskip-\hsize 
    \hbox{}\hskip\hsize minus 1em\qed\par 
  \fi\stdbreak\fi} 
 
\def\references{ 
  \removelastskip 
  \widow{.05} 
  \vskip 24pt plus 6pt minus 6 pt 
  \leftline{\sectionfont References} 
  \nobreak\stdskip\noindent} 
 
\def\ifempty#1#2\endB{\ifx#1\endA} 
\def\makeref#1#2#3{\ifempty#1\endA\endB\else\forward{#1}{#2}{#3}\fi} 
 
\outer\def\section#1 #2\par{ 
  \removelastskip 
  \global\advance\seccount by 1 
  \global\proccount=0\relax 
                \edef\numtoks{\number\seccount} 
  \makeref{#1}{Section}{\numtoks} 
  \widow{.05} 
  \vskip 24pt plus 6pt minus 6 pt 
  \message{#2} 
  \leftline{\marginlabel{#1}\sectionfont\numtoks\quad #2} 
  \nobreak\stdskip} 
 
\def\proclamation#1#2{ 
  \outer\expandafter\def\csname#1\endcsname##1 ##2\par{ 
  \stdbreak 
  \advance\proccount by 1 
  \edef\numtoks{\number\seccount.\number\proccount} 
  \makeref{##1}{#2}{\numtoks} 
  \noindent{\marginlabel{##1}\bf #2 \numtoks\enspace} 
  {\sl##2\par} 
  \stdbreak}} 
 
\def\othernumbered#1#2{ 
  \outer\expandafter\def\csname#1\endcsname##1{ 
  \stdbreak 
  \advance\proccount by 1 
  \edef\numtoks{\number\seccount.\number\proccount} 
  \makeref{##1}{#2}{\numtoks} 
  \noindent{\marginlabel{##1}\bf #2 \numtoks\enspace}}} 
 
\proclamation{definition}{Definition} 
\proclamation{lemma}{Lemma} 
\proclamation{proposition}{Proposition} 
\proclamation{theorem}{Theorem} 
\proclamation{corollary}{Corollary} 
\proclamation{conjecture}{Conjecture} 
 
\othernumbered{example}{Example} 
\othernumbered{remark}{Remark} 
\othernumbered{construction}{Construction} 
\othernumbered{problem}{Problem} 
%=========================================================================% 
% enable postscript illustrations using epsf.tex 
%=========================================================================% 
 
% Usage: 
% \draw{70}{fig}{} % draw fig.eps at 70% scale 
% \draw{999}{fig}{} % draw fig.eps scaled to width of page 
 
% Optional third argument can be multiple calls to \figtext; see below. 
% More generally, the third argument is read in vertical mode, with the 
% reference point at the lower left corner of the eps picture, whose 
% dimensions are contained in the dimen registers \drawx and \drawy. 
% This enables using TeX to generate the text that goes with the picture. 
% To request that the picture be widened to respect the added text,  
% examine and modify the dimen registers \ngap, \egap, \sgap, \wgap. 
% This is done automatically by the \figtext macro. 
 
% These macros rely on "epsf.tex" which is the lowest level interface 
% available for including encapsulated Postscript files in TeX documents. 
% Rather that manually reading the .eps file to compute the nominal size, 
% the \epsfbox macro is called twice, and two of its internal registers 
% are examined after the first call. A major change to epsf.tex (unlikely) 
% will require changes here.  
 
% \input epsf 
 
\newcount\figcount 
\figcount=0 
\newbox\drawing 
\newcount\drawbp 
\newdimen\drawx 
\newdimen\drawy 
\newdimen\ngap 
\newdimen\sgap 
\newdimen\wgap 
\newdimen\egap 
 
\def\drawbox#1#2#3{\vbox{ 
  \setbox\drawing=\vbox{\offinterlineskip\epsfbox{#2.eps}\kern 0pt} 
  \drawbp=\epsfurx 
  \advance\drawbp by-\epsfllx\relax 
  \multiply\drawbp by #1 
  \divide\drawbp by 100 
  \drawx=\drawbp truebp 
  \ifdim\drawx>\hsize\drawx=\hsize\fi 
  \epsfxsize=\drawx 
  \setbox\drawing=\vbox{\offinterlineskip\epsfbox{#2.eps}\kern 0pt} 
  \drawx=\wd\drawing 
  \drawy=\ht\drawing 
  \ngap=0pt \sgap=0pt \wgap=0pt \egap=0pt  
  \setbox0=\vbox{\offinterlineskip 
    \box\drawing \ifgridlines\drawgrid\drawx\drawy\fi #3} 
  \kern\ngap\hbox{\kern\wgap\box0\kern\egap}\kern\sgap}} 
 
\def\draw#1#2#3{ 
  \setbox\drawing=\drawbox{#1}{#2}{#3} 
  \advance\figcount by 1 
  \goodbreak 
  \midinsert 
  \centerline{\ifgridlines\boxgrid\drawing\fi\box\drawing} 
  \smallskip 
  \vbox{\offinterlineskip 
    \centerline{Figure~\number\figcount} 
    \smash{\marginlabel{#2}}} 
  \endinsert} 
 
\def\prevfigtoks{\edef\numtoks{\number\figcount}} 
 
\def\nextfigtoks{% 
  \advance\figcount by 1% 
  \edef\numtoks{\number\figcount}% 
  \advance\figcount by -1} 
   
\def\prevfig{\prevfigtoks Figure~\numtoks} 
\def\nextfig{\nextfigtoks Figure~\numtoks} 
 
\def\prevfiglabel#1{\prevfigtoks\forward{#1}{Figure}{\numtoks}} 
\def\nextfiglabel#1{\nextfigtoks\forward{#1}{Figure}{\numtoks}} 
  
%=========================================================================% 
% figure text macros 
%=========================================================================% 
 
% \figtext{x1}{y1}{x2}{y2}{text} places "text" in a box, and overlays this 
% text onto an eps picture, when called as part of the third argument to 
% \draw, above. (x1,y1) are eps picture coordinates, from 0 to 1 
% and relative to its lower left corner. (x2,y2) are text box 
% coordinates, from 0 to 1 and relative to its lower left corner. 
% Any of these coordinates may take values outside of the range 0 to 1. 
% These two points are attached, so text annotations survive changes of 
% scale of the eps picture. If text is placed extending outside the 
% eps picture, the box containing the eps picture is enlarged without 
% disturbing this coordinate system. 
 
% As an aid to placement, call the \gridlines macro to provide grid lines. 
% Crosshairs mark the reference point on each text box; minor ticks are 
% .002 units, and major ticks are .01 units. 
 
% To understand the motivation for the remaining macros, consider that 
% there are three separate units of measure impinging on a placement 
% of text over a picture: picture units, text units, and absolute units. 
% To survive picture rescaling and overall changes in the magnification of 
% the document, it is important to work out idioms that record placement 
% decisions in the correct units. 
 
% The macros \swtext, \nwtext, \setext, \netext, \wtext, \stext, \etext, 
% and \ntext refer to compass points on the perimeter of the text box, 
% replacing the (x2,y2) arguments to \figtext. Similarly, \ctext sets 
% (x2,y2) to the center of the text box. 
 
% The macros \swpad, \nwpad, \sepad, \nepad, \wpad, \spad, \epad, 
% and \npad place space in absolute TeX units around each box. The macro 
% \mimic gives the height of the first argument to its second 
% argument, so placements with north reference points come out aligned. 
% Thus, one idiom for using these macros would be: 
 
%   \draw{70}{twistedfiber}{ 
%     \setext{.594}{.288}{\sepad{3pt}{3pt}{$b^3c$}} 
%     \ntext{.326}{0}{\npad{3pt}{\mimic{$b$}{$a$}}} 
%     \ntext{.722}{0}{\npad{3pt}{$b$}} 
%   } 
 
% As these macros operate in vertical mode, extra spaces are permitted. 
 
\newif\ifgridlines 
\newbox\figtbox 
\newbox\figgbox 
\newdimen\figtx 
\newdimen\figty 
 
\def\gridlines{\gridlinestrue} 
 
\newdimen\bwd 
\bwd=2sp % 2sp (1/32768") is smallest visible width for Textures 
 
\def\hline#1{\vbox{\smash{\hbox to #1{\leaders\hrule height \bwd\hfil}}}} 
 
\def\vline#1{\hbox to 0pt{% 
  \hss\vbox to #1{\leaders\vrule width \bwd\vfil}\hss}} 
 
\def\clap#1{\hbox to 0pt{\hss#1\hss}} 
\def\vclap#1{\vbox to 0pt{\offinterlineskip\vss#1\vss}} 
 
\def\hstutter#1#2{\hbox{% 
  \setbox0=\hbox{#1}% 
  \hbox to #2\wd0{\leaders\box0\hfil}}} 
 
\def\vstutter#1#2{\vbox{ 
  \setbox0=\vbox{\offinterlineskip #1} 
  \dp0=0pt 
  \vbox to #2\ht0{\leaders\box0\vfil}}} 
 
\def\crosshairs#1#2{ 
  \dimen1=.002\drawx 
  \dimen2=.002\drawy 
  \ifdim\dimen1<\dimen2\dimen3\dimen1\else\dimen3\dimen2\fi 
  \setbox1=\vclap{\vline{2\dimen3}} 
  \setbox2=\clap{\hline{2\dimen3}} 
  \setbox3=\hstutter{\kern\dimen1\box1}{4} 
  \setbox4=\vstutter{\kern\dimen2\box2}{4} 
  \setbox1=\vclap{\vline{4\dimen3}} 
  \setbox2=\clap{\hline{4\dimen3}} 
  \setbox5=\clap{\copy1\hstutter{\box3\kern\dimen1\box1}{6}} 
  \setbox6=\vclap{\copy2\vstutter{\box4\kern\dimen2\box2}{6}} 
  \setbox1=\vbox{\offinterlineskip\box5\box6} 
  \smash{\vbox to #2{\hbox to #1{\hss\box1}\vss}}} 
 
\def\boxgrid#1{\rlap{\vbox{\offinterlineskip 
  \setbox0=\hline{\wd#1} 
  \setbox1=\vline{\ht#1} 
  \smash{\vbox to \ht#1{\offinterlineskip\copy0\vfil\box0}} 
  \smash{\vbox{\hbox to \wd#1{\copy1\hfil\box1}}}}}} 
 
\def\drawgrid#1#2{\vbox{\offinterlineskip 
  \dimen0=\drawx 
  \dimen1=\drawy 
  \divide\dimen0 by 10 
  \divide\dimen1 by 10 
  \setbox0=\hline\drawx 
  \setbox1=\vline\drawy 
  \smash{\vbox{\offinterlineskip 
    \copy0\vstutter{\kern\dimen1\box0}{10}}} 
  \smash{\hbox{\copy1\hstutter{\kern\dimen0\box1}{10}}}}} 
 
\def\figtext#1#2#3#4#5{ 
  \setbox\figtbox=\hbox{#5} 
  \dp\figtbox=0pt 
  \figtx=-#3\wd\figtbox \figty=-#4\ht\figtbox 
  \advance\figtx by #1\drawx \advance\figty by #2\drawy 
  \dimen0=\figtx \advance\dimen0 by\wd\figtbox \advance\dimen0 by-\drawx 
  \ifdim\dimen0>\egap\global\egap=\dimen0\fi 
  \dimen0=\figty \advance\dimen0 by\ht\figtbox \advance\dimen0 by-\drawy 
  \ifdim\dimen0>\ngap\global\ngap=\dimen0\fi 
  \dimen0=-\figtx 
  \ifdim\dimen0>\wgap\global\wgap=\dimen0\fi 
  \dimen0=-\figty 
  \ifdim\dimen0>\sgap\global\sgap=\dimen0\fi 
  \smash{\rlap{\vbox{\offinterlineskip 
    \hbox{\hbox to \figtx{}\ifgridlines\boxgrid\figtbox\fi\box\figtbox} 
    \vbox to \figty{} 
    \ifgridlines\crosshairs{#1\drawx}{#2\drawy}\fi 
    \kern 0pt}}}} 
 
\def\nwtext#1#2#3{\figtext{#1}{#2}01{#3}} 
\def\netext#1#2#3{\figtext{#1}{#2}11{#3}} 
\def\swtext#1#2#3{\figtext{#1}{#2}00{#3}} 
\def\setext#1#2#3{\figtext{#1}{#2}10{#3}} 
 
\def\wtext#1#2#3{\figtext{#1}{#2}0{.5}{#3}} 
\def\etext#1#2#3{\figtext{#1}{#2}1{.5}{#3}} 
\def\ntext#1#2#3{\figtext{#1}{#2}{.5}1{#3}} 
\def\stext#1#2#3{\figtext{#1}{#2}{.5}0{#3}} 
\def\ctext#1#2#3{\figtext{#1}{#2}{.5}{.5}{#3}} 
 
% macros to add space to text on specified sides 
 
\def\hpad#1#2#3{\hbox{\kern #1\hbox{#3}\kern #2}} 
\def\vpad#1#2#3{\setbox0=\hbox{#3}\dp0=0pt\vbox{\kern #1\box0\kern #2}} 
 
\def\wpad#1#2{\hpad{#1}{0pt}{#2}} 
\def\epad#1#2{\hpad{0pt}{#1}{#2}} 
\def\npad#1#2{\vpad{#1}{0pt}{#2}} 
\def\spad#1#2{\vpad{0pt}{#1}{#2}} 
 
\def\nwpad#1#2#3{\npad{#1}{\wpad{#2}{#3}}} 
\def\nepad#1#2#3{\npad{#1}{\epad{#2}{#3}}} 
\def\swpad#1#2#3{\spad{#1}{\wpad{#2}{#3}}} 
\def\sepad#1#2#3{\spad{#1}{\epad{#2}{#3}}} 
 
% macro to give one text string the apparent height of another 
 
\def\mimic#1#2{\setbox1=\hbox{#1}\setbox2=\hbox{#2}\ht2=\ht1\box2} 
 
% macro to center one text string over another 
 
\def\stack#1#2#3{\vbox{\offinterlineskip 
  \setbox2=\hbox{#2} 
  \setbox3=\hbox{#3} 
  \dimen0=\ifdim\wd2>\wd3\wd2\else\wd3\fi 
  \hbox to \dimen0{\hss\box2\hss} 
  \kern #1 
  \hbox to \dimen0{\hss\box3\hss}}} 
 
% macros to hide size of trailing exponents 
 
\def\sexp#1{\rlap{${}^{#1}$}} 
\def\hexp#1{% 
  \setbox0=\hbox{${}^{#1}$}% 
  \hbox to .5\wd0{\box0\hss}} 
 
%=========================================================================% 
% macros for matrices and arrows 
%=========================================================================% 
 
% typical usage: 
%   \rightarrowmat{2pt}{4pt}{d & bd \cr \!-c & 0 \cr 0 & -ac \cr} 
 
\def\bmatrix#1#2{{\smallmath\left[\vcenter{\halign 
  {&\kern#1\hfil$##\mathstrut$\kern#1\cr#2}}\right]}} 
 
\def\rightarrowmat#1#2#3{ 
  \setbox1=\hbox{\kern#2$\bmatrix{#1}{#3}$\kern#2} 
  \,\vbox{\offinterlineskip\hbox to\wd1{\hfil\copy1\hfil} 
    \kern 3pt\hbox to\wd1{\rightarrowfill}}\,} 
 
\def\leftarrowmat#1#2#3{ 
  \setbox1=\hbox{\kern#2$\bmatrix{#1}{#3}$\kern#2} 
  \,\vbox{\offinterlineskip\hbox to\wd1{\hfil\copy1\hfil} 
    \kern 3pt\hbox to\wd1{\leftarrowfill}}\,} 
 
\def\rightarrowbox#1#2{ 
  \setbox1=\hbox{\kern#1\hbox{\smallmath #2}\kern#1} 
  \,\vbox{\offinterlineskip\hbox to\wd1{\hfil\copy1\hfil} 
    \kern 3pt\hbox to\wd1{\rightarrowfill}}\,} 
 
\def\leftarrowbox#1#2{ 
  \setbox1=\hbox{\kern#1\hbox{\smallmath #2}\kern#1} 
  \,\vbox{\offinterlineskip\hbox to\wd1{\hfil\copy1\hfil} 
    \kern 3pt\hbox to\wd1{\leftarrowfill}}\,} 
 
%=========================================================================% 
% quire macros for preview mode and making booklets 
%=========================================================================% 
 
% \legalbooklet{20} makes a booklet from legal paper in landscape 
% orientation, where "20" is the page count. To preview, give a negative 
% pagecount. Either print using the legal duplex option on a modern laser 
% printer, or struggle to simulate this effect manually. Bind using a long 
% reach stapler. 
 
% \preview squeezes two pages side by side in landscape orientation. It 
% is not suitable for printing, but ideal for previewing on a two page 
% monitor. 
 
% \twoup squeezes two pages onto letter paper in landscape mode, 
% suitable for printing. 
 
% Each of these macros calls the file "quire.tex" 
 
\def\bookletdims{ 
  \hsize=5.25truein 
  \vsize=7truein 
} 
 
\def\legalbooklet#1{ 
  \input quire 
  \bookletdims 
  \htotal=7.0truein 
  \vtotal=8.5truein 
  % below computed from above 
  \hoffset=\htotal 
  \advance\hoffset by -\hsize 
  \divide\hoffset by 2 
  \voffset=\vtotal 
  \advance\voffset by -\vsize 
  \divide\voffset by 2 
  \advance\voffset by -.0625truein 
  \shhtotal=2\htotal 
  % below doesn't need to change 
  \horigin=0.0truein 
  \vorigin=0.0truein 
  \shstaplewidth=0.01pt 
  \shstaplelength=0.66truein 
  \shthickness=0pt 
  \shoutline=0pt 
  \shcrop=0pt 
  \shvoffset=-1.0truein 
  \ifnum#1>0\quire{#1}\else\qtwopages\fi 
} 
 
\def\preview{ 
  \input quire 
  \bookletdims 
  \hoffset=0.1truein 
  \vtotal=8.5truein 
  \shhtotal=14truein 
  % below computed from above 
  \voffset=\vtotal 
  \advance\voffset by -\vsize 
  \divide\voffset by 2 
  \advance\voffset by -.0625truein 
  \htotal=2\hoffset 
  \advance\htotal by \hsize 
  % below doesn't need to change 
  \horigin=0.0truein 
  \vorigin=0.0truein 
  \shstaplewidth=0.5pt 
  \shstaplelength=0.5\vtotal 
  \shthickness=0pt 
  \shoutline=0pt 
  \shcrop=0pt 
  \shvoffset=-1.0truein 
  \qtwopages 
} 
 
\def\twoup{ 
  \input quire 
  \hsize=4.79452truein % 5.25/1.095 
  \vsize=7truein 
  \vtotal=8.5truein 
  \shhtotal=11truein 
  % below computed from above 
  \hoffset=-2\hsize 
  \advance\hoffset by \shhtotal 
  \divide\hoffset by 6 
  \voffset=\vtotal 
  \advance\voffset by -\vsize 
  \divide\voffset by 2 
  \advance\voffset by -12truept 
  \htotal=2\hoffset 
  \advance\htotal by \hsize 
  % below doesn't need to change 
  \horigin=0.0truein 
  \vorigin=0.0truein 
  \shstaplewidth=0.01pt 
  \shstaplelength=0pt 
  \shthickness=0pt 
  \shoutline=0pt 
  \shcrop=0pt 
  \shvoffset=-1.0truein 
  \qtwopages 
} 
 
%=========================================================================% 
% timestamp (adapted from eplain.tex) 
%=========================================================================% 
 
\newcount\countA 
\newcount\countB 
\newcount\countC 
 
\def\monthname{\begingroup 
  \ifcase\number\month 
    \or January\or February\or March\or April\or May\or June\or 
    July\or August\or September\or October\or November\or December\fi 
\endgroup} 
 
\def\dayname{\begingroup 
  \countA=\number\day 
  \countB=\number\year 
  \advance\countA by 0 % adjust after each leap day 
  \advance\countA by \ifcase\month\or 
    0\or 31\or 59\or 90\or 120\or 151\or 
    181\or 212\or 243\or 273\or 304\or 334\fi 
  \advance\countB by -1995 
  \multiply\countB by 365 
  \advance\countA by \countB 
  \countB=\countA 
  \divide\countB by 7 
  \multiply\countB by 7 
  \advance\countA by -\countB 
  \advance\countA by 1 
  \ifcase\countA\or Sunday\or Monday\or Tuesday\or Wednesday\or 
    Thursday\or Friday\or Saturday\fi 
\endgroup} 
 
\def\timename{\begingroup 
   \countA = \time 
   \divide\countA by 60 
   \countB = \countA 
   \countC = \time 
   \multiply\countA by 60 
   \advance\countC by -\countA 
   \ifnum\countC<10\toks1={0}\else\toks1={}\fi 
   \ifnum\countB<12 \toks0={\sevenrm AM} 
     \else\toks0={\sevenrm PM}\advance\countB by -12\fi 
   \relax\ifnum\countB=0\countB=12\fi 
   \hbox{\the\countB:\the\toks1 \the\countC \thinspace \the\toks0} 
\endgroup} 
 
\def\timestamp{\dayname, \the\day\ \monthname\ \the\year, \timename} 
 
%========================================================================== 
% macros (specific to this paper) 
%========================================================================== 
 
% surround with $ $ if not already in math mode 
\def\enma#1{{\ifmmode#1\else$#1$\fi}} 
\def\th{{^{\rm th}}} 
 
\def\mathbb#1{{\bbold #1}} 
\def\mathbf#1{{\bf #1}} 
 
% blackboard bold symbols 
\def\NN{\enma{\mathbb{N}}} 
\def\ZZ{\enma{\mathbb{Z}}} 
 
 
% bold symbols 
\def\aa{\enma{\mathbf{a}}} 
\def\bb{\enma{\mathbf{b}}} 
\def\cc{\enma{\mathbf{c}}} 
\def\ee{\enma{\mathbf{e}}} 
\def\xx{\enma{\mathbf{x}}} 
\def\mm{\enma{\mathbf{m}}} 
\def\uu{\enma{\mathbf{u}}} 
\def\kVect{\enma{{k{\mathbf{Vect}}}}} 
 
 
\def\set#1{\enma{\{#1\}}} 
\def\setdef#1#2{\enma{\{\;#1\;\,|\allowbreak 
  \;\,#2\;\}}} 
 
\def\sf{\mathop{\rm sf}\nolimits} 
 
\def\depth{\mathop{\rm depth}\nolimits} 
\def\reg{\mathop{\rm reg}\nolimits} 
 
\def\Ext{\mathop{\rm Ext}\nolimits} 
\def\Tor{\mathop{\rm Tor}\nolimits} 
\def\link{\mathop{\rm lk}\nolimits} 
\def\star{\mathop{\rm star}\nolimits} 
\def\core{\mathop{\rm core}\nolimits} 
\def\abs#1{\enma{\left| #1 \right|}} 
\def\supp{\mathop{\rm supp}\nolimits} 
 
 
\def\mtext#1{\;\,\allowbreak\hbox{#1}\allowbreak\;\,} 
 
\def\A{\enma{\cal A}} 
\def\B{\enma{\cal B}} 
\def\E{\enma{\cal E}} 
\def\Sym{\mathop{\rm Sym}\nolimits} 
 
\input diagrams 
%\showlabels 
 
%========================================================================== 
% macros (specific to this paper) 
%========================================================================== 
 
% surround with $ $ if not already in math mode 
\def\enma#1{{\ifmmode#1\else$#1$\fi}} 
\def\th{{^{\rm th}}} 
\def\st{{^{\rm st}}} 
 
\def\mathbb#1{{\bbold #1}} 
\def\mathbf#1{{\bf #1}} 
 
% blackboard bold symbols 
\def\N{\enma{\mathbb{N}}} 
\def\NN{\enma{\mathbb{N}}} 
\def\CC{\enma{\mathbb{C}}} 
\def\Z{\enma{\mathbb{Z}}} 
\def\P{\enma{\mathbb{P}}} 
 
% caligraphic symbols 
\def\A{\enma{{\cal A}}} 
 
% bold symbols 
\def\aa{\enma{\mathbf{a}}} 
\def\bb{\enma{\mathbf{b}}} 
\def\cc{\enma{\mathbf{c}}} 
\def\ee{\enma{\mathbf{e}}} 
\def\FF{\enma{\mathbf{F}}} 
\def\LL{\enma{\mathbf{L}}} 
\def\RR{\enma{\mathbf{R}}} 
\def\xx{\enma{\mathbf{x}}} 
\def\mm{\enma{\mathbf{m}}} 
\def\uu{\enma{\mathbf{u}}} 
\def\kVect{\enma{{k{\mathbf{Vect}}}}} 
 
% 
\font\abst=cmr9 
% 
%gothic symbols 
% Gothic fonts from AMSTeX  
\font\tengoth=eufm10  \font\fivegoth=eufm5 
\font\sevengoth=eufm7 
\newfam\gothfam  \scriptscriptfont\gothfam=\fivegoth  
\textfont\gothfam=\tengoth \scriptfont\gothfam=\sevengoth 
\def\goth{\fam\gothfam\tengoth} 
% 
\def \gm {{\goth m}} 
 
\def\set#1{\enma{\{#1\}}} 
\def\setdef#1#2{\enma{\{\;#1\;\,|\allowbreak 
  \;\,#2\;\}}} 
 
\def\sf{\mathop{\rm sf}\nolimits} 
 
\def\depth{\mathop{\rm depth}\nolimits} 
\def\cx{\mathop{\rm cx}\nolimits} 
\def\reg{\mathop{\rm reg}\nolimits} 
\def\ann{\mathop{\rm Ann}\nolimits} 
\def\Hom{\mathop{\rm Hom}\nolimits} 
\def\Ext{\mathop{\rm Ext}\nolimits} 
\def\ext{\mathop{\rm Ext}\nolimits} 
\def\Tor{\mathop{\rm Tor}\nolimits} 
\def\tor{\mathop{\rm Tor}\nolimits} 
%\def\H{\mathop{\rm H}\nolimits} 
\def\H{{\rm H}} 
\def\Tor{\mathop{\rm Tor}\nolimits} 
\def\ker{\mathop{\rm ker}\nolimits} 
\def\link{\mathop{\rm lk}\nolimits} 
\def\star{\mathop{\rm star}\nolimits} 
\def\core{\mathop{\rm core}\nolimits} 
\def\abs#1{\enma{\left| #1 \right|}} 
\def\supp{\mathop{\rm supp}\nolimits} 
\def\initial{\mathop{\rm in}\nolimits} 
 
\def\A{\enma{\cal A}} 
\def\B{\enma{\cal B}} 
\def\E{\enma{\cal E}} 
\def\Sym{\mathop{\rm Sym}\nolimits} 
 
\def\mtext#1{\;\,\allowbreak\hbox{#1}\allowbreak\;\,} 
\def\fix#1{{\bf ((**** #1 ****))}} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% Forward references 
% 
\forward{coho of hyp arr}{Section}{1} 
\forward{OSrank-var}{Section}{2} 
\forward{S-module}{Section}{3} 
\forward{Appendix}{Section}{4} 
% 
\forward {coho res}{Theorem}{1.1} 
\forward {codim of rank var}{Corollary}{2.4} 
\forward{generic example continued}{Example}{3.3} 
\forward {characterization of generic}{Corollary}{3.6} 
\forward {sf bettis exterior}{Proposition}{4.3} 
\forward {betti}{Corollary}{4.7} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 

\font\smc=cmcsc10
\font\smallsmc=cmcsc8
\font\smallrm=cmcsc8

\headline={\ifodd\pageno \ifnum\pageno>1 \smallrm \hfil
Hyperplane Arrangement Cohomology and Monomials in the Exterior Algebra%
\hfil\folio \else\hfill\fi \else \smallrm \folio \hfill
David Eisenbud, Sorin Popescu, and Sergey Yuzvinsky%
\hfill\fi} \footline={\hss}   % footline is blank


\hbox{} 
%\rightline{01/07/2000}  
 
\bigskip 
\centerline{\titlefont Hyperplane Arrangement Cohomology} 
\smallskip 
\centerline{\titlefont  and} 
\smallskip 
\centerline{\titlefont Monomials in the Exterior Algebra 
\footnote{$^{*}$}{\rm Mathematics Subject Classification (MSC 2000) numbers:  
Primary 15A75, 52C35, 55N45; secondary 55N99, 14Q99.}}  
\bigskip\smallskip 
\centerline{David Eisenbud, Sorin Popescu, and Sergey Yuzvinsky 
\footnote{$^{**}$}{\rm The first two authors are grateful to the NSF for  
support during the preparation of this work. The authors would like  
to thank the Mathematical Sciences Research Institute in Berkeley  
for its support while part of this paper was being written.}} 
 
\bigskip\bigskip 
{\narrower 
\noindent{\bf Abstract:} \abst  
We show that if $X$ is the complement of a complex hyperplane 
arrangement, then the homology of $X$ has linear free resolution as 
a module over the exterior algebra on the first cohomology of $X$. 
We study invariants of $X$ that can be deduced from this resolution. 
A key ingredient is a result of Aramova, Avramov, and Herzog [1999] on 
resolutions of monomial ideals in the exterior algebra. We 
give a new conceptual proof of this result. 
 
\bigskip} 
 
\noindent Let $X$ be the complement of a complex hyperplane arrangement $\A$. 
In this paper we study the singular  
homology $\H_*(X)$ as a module 
over the exterior algebra $E$ on the first singular cohomology 
$V:=\H^1(X)$ always with coefficients in a fixed field $K$.  Our main 
result (\ref{coho of hyp arr}) asserts that $\H_*(X)$ is generated in 
a single degree and has a linear free resolution; this amounts to an 
infinite sequence of statements asserting the nontriviality of the 
multiplication in the Orlik-Solomon algebra $\H^*(X)$. We also 
analyze some other topological examples from the  
point of view of resolutions over the exterior algebra. 
 
In \ref{OSrank-var} we study an invariant of an $E$-module $M$ called 
the {\it singular variety\/}, the algebraic subset of $V$ consisting of 
those elements $x$ whose annihilator in $M$ is not equal to $xM$.  The 
singular variety is the same for $M$ and for $M^*$, and thus for the 
homology and cohomology of $X$.  Aramova, Avramov and Herzog [1999] 
show that the codimension of the singular variety gives the rate of growth 
of the free resolution of $M$.  We compute the singular variety of 
$\H^*(M)$ and show that its codimension is the number of central  
arrangements in an 
expression of $\A$ as a product of irreducible arrangements. 
 
Using the Bernstein-Gel'fand-Gel'fand correspondence on the linear 
resolution of $\H_*(X)$, we define in \ref{S-module} an invariant of 
$X$ (or really of the intersection poset of the arrangement) which 
is a module $F(\A)$ over the symmetric algebra of $\H_1(X)$, supported on 
the singular variety. We compute some homological invariants of $F(\A)$ and we 
use its properties to show that the cones over a generic hyperplane 
arrangement may be characterized as the ones for which the defining 
ideal (the Orlik Solomon ideal) of $\H^*(X)$ also has a linear free 
resolution. 
 
A key ingredient in the proof of our main theorem is the theorem of 
Aramova, Avramov, and Herzog [1999] (later improved by R\"omer [1999]) 
relating the resolutions of square-free monomial ideals (and some more 
general modules) over symmetric and exterior algebras. This allows us to 
apply the results on resolutions and Alexander duality due to Eagon and 
Reiner [1998].   The proof given by Aramova, Avramov and Herzog depends 
on an intricate computation. In \ref{Appendix} we offer a conceptual 
description of the relationship which leads to a transparent proof. 
 
We are glad to acknowledge the essential role of the computer algebra 
system written  
by Grayson and Stillman [Macaulay2] in the genesis of 
this paper: It was only through ``playing" with this program that we were 
lead to  guess at the main result (\ref{coho res}) and most of the 
lesser result were carefully checked for plausibility before 
we looked for proofs. 
 
\bigskip 
\noindent{\bf Notation:} 
Throughout this paper, $\A$ will denote an essential  affine complex 
hyperplane arrangement, that is, a set of $n$ affine hyperplanes in 
$\CC^\ell$ whose intersection poset has rank $\ell$. We will denote the 
complement of the union of the hyperplanes in $\A$ by  $X$. We denote 
with $K$  an arbitrary field. 
 
We use notation as in Orlik-Terao [1992].  In particular we write 
$A:=A(\A)$ for the Orlik-Solomon algebra of $\A$, isomorphic to the 
singular cohomology of $X$ with coefficients in $K$.  By Orlik-Solomon 
[1980], the vector space $E_1=V=\H^1(X)$ has basis $e_1,\ldots,e_n$ 
corresponding to the hyperplanes of $\A$, and we may write 
$\H^*(X)=E/I$, where $E$ is the exterior algebra on $E_1$ and 
$I\subset E$ is the {\it Orlik-Solomon ideal\/} generated by the 
elements 
$$ 
\partial(e_{i_1}\wedge\cdots\wedge e_{i_t})= 
\sum_j (-1)^j 
e_{i_1}\wedge\cdots e_{i_{j-1}}\wedge \widehat {e_{i_j}} 
                   \wedge e_{i_{j+1}}\cdots \wedge e_{i_t} 
$$ 
where  
$\{ H_{i_1},\dots,H_{i_t}\}$  
is a minimal linearly dependent set of hyperplanes of $\A$, 
and the monomials $e_{i_1}\wedge\cdots\wedge e_{i_t}$ where 
$\{ H_{i_1},\dots,H_{i_t}\}$  
have empty intersection. 
 
We grade $E$ by taking the elements of $V$ to have degree $1$ 
(this is the opposite convention from that of  Eisenbud and 
Schreyer [2000]). The homology module $\H_*(X)$ is dual to $E/I$, and 
thus is graded in negative degrees. 
 
We will write 
$\chi(\A,-)$ for the characteristic polynomial  
of the arrangement $\A$. 
For our purposes $\chi$ may be defined by the relation 
$$ 
\chi(\A,t)=t^{\ell}\pi(A,-1/t), 
$$  
where $\pi$ is the Poincar\'e polynomial polynomial of $X$, that is 
$$ 
\pi(t)=\sum_j\dim_K\H^j(X)t^j; 
$$  
see Orlik-Terao [1992, Definition 2.52 and Theorem 3.68]. 
 
If $A$ is a skew commutative algebra, we write $A\langle e\rangle$ 
and $A[t]$  to denote the skew-commutative algebra obtained by 
adjoining a variable of degree 1 or 2 respectively; thus $A[t]$ 
is an ordinary polynomial ring on one commuting variable over 
$A$, while if $E$ is the exterior algebra of $V$ then  
$E\langle e\rangle$ is the exterior algebra of $V\oplus Ke$. 
 
\section{coho of hyp arr} The Cohomology of Hyperplane Arrangements 
 
\theorem{coho res} The minimal free resolution of $\H_*(X)$, 
regarded as a module over the exterior algebra $E=\wedge(\H^1(X))$ 
by means of the cap product, has the form 
$$ 
\FF:\qquad   
\dots\to  
E^{\beta_2}(\ell-2)\to  
E^{\beta_1}(\ell-1)\to  
E^{\beta_0}(\ell)\to  
\H_*(X)\to 0. 
$$   
The ranks $\beta_i$ may be computed from the formula 
$$ 
\sum_{i=0}^\infty \beta_it^i={(-1)}^{\ell}{\chi(\A,t)\over {(1-t)}^n} 
\ \ . 
$$ 
 
In general we will say that a graded $E$-module $M$ has a {\it linear 
resolution\/} if $M$ is generated in a single degree $s$ and has 
resolution of the form given in the theorem, with $d^\th$ syzygy 
module generated in degree $s+d$; the theorem asserts that $H_*(X)$ 
has a linear resolution with $s=-\ell$.   
 
We can interpret  
the statement that a 
module of the form $E^{\beta_0}(\ell)$ can map onto $\H_*(X)$ in more 
familiar language: 
 
\corollary{socle} An element $c\in \H^*(X)$ is annihilated by 
the (cup) product with every element of $\H^1(X)$ if and only if 
$c\in \H^\ell(X)$. 
 
\noindent{\sl Proof of \ref{socle}.\/}  
Because $E^{\beta_0}(\ell)$ maps onto $\H_*(X)$, we see that $\H_*(X)$ 
is generated as an $E$-module by $\H_\ell(X)$. In particular we 
recover the well-known fact that $\H^j(X)= (\H_j(X))^*= 0$ for 
$j>\ell$, so that every element of $\H^\ell(X)$ is annihilated by 
$\H^1(X)$. 
 
Conversely, let $c\in\H^*(X)$ be annihilated by $\H^1(X)$.  The 
Orlik-Solomon description shows that $\H^*(X)$ is generated as an 
algebra by $\H^1(X)$, so $c$ is annihilated by $\H^+$, the ideal of 
elements of positive degree in $\H^*(X)$. In particular, $c \cdot 
(\H^+\cdot \H_*(X))=0$.  Because $\H_*(X)$ is generated by 
$\H_\ell(X)$ we have $\H^+\cdot \H_*(X)=\sum_{j<\ell}\H_j(X)$.  It 
follows that $c\in \H^\ell(X)$.\Box 
 
 
For the proof of \ref{coho res} it is convenient to reduce to 
the central case. Recall that an arrangement is {\it central\/}  
if the intersection of its hyperplanes is nonempty. 
Given a (not necessarily central) arrangement  
$\A$ of $n$ hyperplanes in $\CC^\ell$, we can projectivize and add the 
hyperplane at infinity, to get an arrangement in ${\P}^\ell_\CC$; 
the affine cone over this arrangement is a central arrangement  
$\B=c\A$ of $n+1$ hyperplanes in $\CC^{\ell+1}$, 
called the {\it cone over} $\A$. Conversely, given a central 
arrangement $\B$ of $n+1$ hyperplanes and a chosen hyperplane $H$ in it,  
we may form the corresponding arrangement of $n+1$ hyperplanes 
in projective $\ell$-space. Removing $H$, we get a (not necessarily 
central) arrangement $\A=d\B$ of $n$ hyperplanes in  
$\CC^\ell$, which we call the {\it deconing\/} of $\B$ 
with respect to $H$. It is clear that $\A$ is the deconing 
of $c\A$ with respect to the ``new'' hyperplane. 
 
The basic result connecting the Orlik-Solomon algebras 
is motivated as follows: The complement of 
the projective arrangement associated to $\B$ is the same as the 
complement of the arrangement associated to any of the deconings of $\B$; 
thus the complement of $\B$ is an $S^1$-bundle over the  
complement of any of the deconings of $\B$. It follows 
that the cohomology algebra of any deconing is 
canonically isomorphic to the cohomology algebra of the 
complement of $\B$ modulo a degree $1$ form. 
The following result gives this identification 
algebraically. For 
this it is convenient to factor the Orlik-Solomon 
relations as products of linear forms: 
 
\proposition{deconing} Suppose $\B=\{H_0,\dots,H_n\}$  
is a central hyperplane arrangement,  
with Orlik Solomon ideal $I$ in the exterior algebra 
$E=K\langle e_0,\dots,e_n\rangle$ whose generators 
$e_i$ correspond to the hyperplanes $H_i$.  
Let $E'$ be the subalgebra generated by the differences 
$e_i-e_j$. Let $I'\subset E'$ be  
ideal generated by 
$$ 
%\eqalign{ 
\{ (e_{i_1}-e_{i_2})(e_{i_2}-e_{i_3}) 
%& 
\cdots(e_{i_s-1}-e_{i_s})\mid 
%\cr 
                %& 
H_{i_1},\dots,H_{i_s}\hbox{\ are linearly dependent}\}. 
%} 
$$ 
The Orlik-Solomon ideal of $\B$ is $I=I'E$, and 
$E/I\cong (E'/I')\langle e_j\rangle$ for any $j$. Furthermore, 
if $\A$ is the deconing of $\B$ with respect to 
$H_j$, then the Orlik-Solomon algebra of $\A$ is  
$E/(I+(e_j))\cong E'/I'$. 
 
\proof One checks directly that 
$ 
(e_{i_1}-e_{i_2})\cdots(e_{i_s-1}-e_{i_s})=  
\partial(e_{i_1}\wedge\cdots\wedge e_{i_s}). 
$ 
It follows that $I=I'E$. The rest 
of the statements are consequences.\Box 
    
\noindent{\sl Proof of \ref{coho res}.\/} To prove 
that the resolution of $\H_*(X)$ is linear, we first 
reduce to the central case. By \ref{deconing}, 
the Orlik-Solomon algebra of $c\A$ is  
$A\langle e_0\rangle=A\otimes_KK\langle e_0\rangle$ 
as skew-commutative algebras, and it follows that the 
free resolution of the homology of the complement of $c\A$ 
is deduced from that of $\A$ by tensoring over $K$ with 
$K\langle e_0\rangle$. In particular, one is linear if and only if the 
other one is, and we may assume that $\A$ is central to begin with. 
 
With respect to 
the lexicographic order on the monomials of $E$, taking 
$e_i<e_j$ if $i<j$, the 
initial (largest) terms of the generators for the Orlik-Solomon ideal (as 
given in the introduction) are 
$$\eqalign{ 
\{  
e_{i_1}\wedge\cdots\wedge e_{i_s} &\mid i_1<\dots <i_s, \hbox{and}\cr 
&\hbox{there exists $i_0<i_1$ such that }\cr 
                     &\{H_{i_0},\dots,H_{i_s}\} 
                        \hbox{ is a dependent set of hyperplanes} 
\}. 
}$$ 
The subsets 
that appear in this expression are exactly the broken circuits 
of $c\A$. By Bj\"orner [1982], the monomials that are not 
divisible by broken circuits are a basis for $A$. It follows that the 
generators of $I$ given in the introduction form a Gr\"obner basis. 
Consequently the initial ideal of $I$ is the ideal $I_0$ generated by the 
monomials in the display. $I_0$ 
is the {\it broken circuit ideal\/} of  
the matroid defined by the dependence relations among the hyperplanes 
of $\A$.  
 
From the general theory of Gr\"obner bases (as for example 
in Eisenbud [1995] where the completely parallel theory is treated for 
ideals in a polynomial ring) we see that $I_0$ is a flat degeneration 
of $I$. More formally, there is an ideal  
$I_t\subset K[t]\otimes_K E$  
such that the algebra  
$ K[t]\otimes_K E/I_t$ is free (and thus flat) over $ K[t]$, and 
$I_0:=(I+(t))/(t)\subset  K[t]\otimes_K E/(t)=E$ 
is the initial ideal $\initial(I)$, while for $0\neq a\in  K$ we have 
$I_a:=(I+(t-a))/(t-a)\subset  K[t]\otimes_K E/(t-a)=E$ 
is conjugate to $I$ by a linear automorphism of $E$. 
 
The module structure on $\H_*(X)$ comes 
from the identification $\H_*(X)=\Hom_K(\H^*(X), K))$, 
so $\H_*(X)$ degenerates flatly to 
$M_0=\Hom_K(E/I_0,  K)$. More formally, the module 
$M_t=\Hom_{ K[t]}(E/I_t,  K[t])$ is free (and thus flat) 
over $ K[t]$, and has special fiber $M/(t)M\cong M_0$, 
whereas for $a\neq 0$ the fiber $M_a:=M/(t-a)M$ is conjugate to  
$\H_*(X)$ by an automorphism of $E$. 
 
The first statement of \ref{coho res} amounts to saying that the $k^\th$ graded 
component, $\Tor_j^E(M, K)_k,$ vanishes for all $j>0$ and  
$k\neq \ell-j$. The vanishing of any one of these vector spaces 
is an open condition in flat families, so it suffices to show that  
$M_0=\Hom_K(E/I_0,  K)$ satisfies the conditions of  
\ref{coho res}.  
 
The algebra $E$ is Gorenstein (injective as a module over itself) with 
socle in degree $n$, so 
$M_0=\Hom_K(E/I_0,  K)=\Hom_E(E/I_0, E)(n)$ as $E$-modules.  
On the other hand $\Hom_E(E/I_0, E)$ may be identified with the 
annihilator $J_0$ of $I_0$, and we see that it suffices to show 
that $J_0$ has free resolution of the form 
$$ 
\FF(-n):\qquad   
\dots\to  
E^{\beta_2}(\ell-2-n)\to  
E^{\beta_1}(\ell-1-n)\to  
E^{\beta_0}(\ell-n)\to  
J_0\to 0. 
$$ 
 
Since $I_0$ is generated by monomials, so is the ideal $J_0$. 
Following Aramova, Avramov, and Herzog [1999] (see also  
\ref{Appendix} below for more details) we let $_SI_0$ and $_SJ_0$ 
be the ideals of $S= K[e_1,\ldots,e_n]$ generated by the monomials 
corresponding to the generators of $I_0$ and $J_0$, respectively, so 
that $_SI_0$ and $_SJ_0$ are square-free monomial ideals of $S$. 
Aramova, Avramov, and Herzog [1999] show that $J_0$ has a free 
resolution as above with $d^\th$ syzygies generated in degree 
$d+\ell-n$ if and only if $_SJ_0\subset S$ has a resolution with this 
same property; for another proof, see \ref{Appendix}, below. 
 
Any square-free monomial ideal $J$ corresponds to a simplicial 
complex $\Delta(J)$. Since $I_0$ and $J_0$ are 
annihilators of one another in $E$, the simplicial complex 
$\Delta(I_0)$ is the Alexander dual of $\Delta(J_0)$; that is, the 
faces of $\Delta(J_0)$ are the complements of the nonfaces of 
$\Delta(I_0)$. By Eagon-Reiner [1998], $_SJ_0$ has a (linear) resolution  
as above if and only if $_SI_0$ has codimension $n-\ell$ and $S/(_SI_0)$ is 
Cohen-Macaulay, or in combinatorial terms, that the simplicial complex 
$\Delta(I_0)$ is Cohen-Macaulay of dimension $\ell-1$. (See also the 
later papers of Terai [1997], Bayer-Charalambous-Popescu [1999], 
Musta\c t\v a [1999], and Yanagawa [1998] for 
more sophisticated versions of this result.) 
 
It was observed by Hochster [1972] and Stanley [1975] 
that the Cohen-Macaulay property of 
a simplicial complex follows from a simpler geometric property called {\it 
shellability\/}; see also Stanley [1996, Theorem 2.5],   
Bruns-Herzog [1993, Theorem 5.1.13].  It is known that the 
simplicial complex corresponding to the broken circuits of a matroid 
of rank $\ell$ is shellable of dimension $\ell-1$ (Provan [1977]; 
see Bj\"orner [1992, 7.4.2(ii) and 7.4.3] and his reference 
Billera and Provan [1980]), 
concluding the proof of the first statement. 
 
In order to prove the second statement we note that, from the 
given resolution,  
$$\eqalign{ 
\pi(\H_*(X), t)&=\sum_i (-1)^i\pi(E^{\beta_i}(\ell-i),t)\cr 
                  &=\sum_i (-1)^i\beta_it^{-\ell+i}(1+t)^n. 
}$$                   
On the other hand, since homology and cohomology are dual, 
$ 
\pi(\H_*(X), t)=\pi(A,1/t)=(-1)^\ell\chi(A,-t)/t^\ell, 
$ 
whence the desired formula.\Box 
 
In general we do not know how to write the free resolution of the 
Orlik-Solomon ideal explicitly; this seems an interesting problem.  
 
\remark{surfaces} Here are a few other topological examples 
treated from the point of view of resolutions over 
the exterior algebra: \hfill\smallskip 
% 
$a)$ Perhaps the most familiar topological spaces with 
cohomology generated in degree one are compact orientable 
surfaces. If $Y$ is an 
orientable compact connected surface of genus 
$g>0$, then the homology $\H_*(Y)$ {\it does not\/} 
satisfy \ref{coho res}: By Poincar\'e duality the homology 
$\H_*(Y)$ is isomorphic as a module over $E=\wedge \H^1(Y)$ 
to $\H^*(Y)$, which 
has relations of degree $>1$.  
However, if we write $\H^*(Y)=E/I$ then one can check that 
(with respect 
to any monomial order on $E$) the initial ideal of $I$ 
is the square-free 
stable ideal consisting of all but the last monomial of degree  
$2$ in $E$. 
By Aramova, Herzog, Hibi [1998, Corollary 
2.5], the initial ideal, and with it 
$I$ itself, has linear resolution.  It follows that the  minimal free 
resolution of the homology module has the form 
$$ 
\FF:\qquad   
\dots\to  
E^{\beta_3}(-2)\to  
E^{\beta_2}(-1)\to  
E^{{2g\choose 2}-1}\to  
E(2)\to  
\H_*(Y)\to 0. 
$$ 
\smallskip  
% 
$b)$ A result analogous to 
\ref{coho res} holds for the homology module of an essential 
arrangement of real subspaces of codimension two in ${\mathbb R}^{2n}$ 
with even dimensional intersections. In this case the cohomology ring of 
the complement has again the shape of an Orlik-Solomon algebra, however 
in contrast with the complex case it is not determined merely by the 
intersection lattice,  but requires the knowledge of extra information 
on sign patterns (computed as determinants of linear relations, or as 
linking numbers in the sense of knot theory); see Bj\"orner-Ziegler 
[1992] and Ziegler [1993] for details.\hfill\smallskip   
% 
$c)$ The complements of codimension two subspace arrangements in  
${\mathbb R}^{4}$  
are equivalent to the link complements obtained 
by intersecting them with the three-sphere $S^3$. 
More generally, consider 
the case of an arbitrary tame link $L=\cup_{i=1}^n L_i$ in $S^3$, 
and let $X$ be the compact manifold with boundary that 
is the complement of a tubular neighborhood of $L$. 
Alexander duality gives 
$\dim 
\H^1(X)=n$, 
$\dim \H^2(X)=n-1$ and $\H^{\ge 3}(X)=0$. 
More explicitly, let $e_i\in\H^1(X)$ be the dual of the meridian 
of the $i^\th$ boundary component, and let 
$f_{i,j}\in\H^2(X)$ be the Alexander dual of the (relative) 
homology class of an arc $\gamma_{i,j}$ connecting 
the $i^\th$ and $j^\th$ components of the boundary. 
The elements $e_i$ form a basis of $\H^1(X)$ and 
(with the conventions $f_{i,i}=0$ and $f_{i,j}=-f_{j,i}$) 
the $f_{i,j}$ generate $\H^2(X)$. 
 
A Mayer-Vietoris argument shows 
that the cohomology ring of $X$ has a presentation 
$$ 
\H^*(X) = \wedge V'/(e_i\wedge e_j - l_{i,j} f_{i,j},\, 
f_{i,j} + f_{j,k} + f_{k,i},\, e_k\wedge f_{i,j},\, f_{i,j}\wedge f_{k,l}), 
$$ 
where $V'=\H^1(X)\oplus \H^2(X)$, 
the numbers $i$ and $j$ run from $1$ to $n$, 
 and  
$l_{i,j}:=\link(L_i,L_j)$  is the linking number of $L_i$ and 
$L_j$. In particular, the cohomology algebra 
$\H^*(X)$ depends only on the linking numbers  
(for most of this, see Milnor [1957]). 
 
Let $G$ be the graph whose vertices are 
the components $L_i$, $i=1,\ldots, n$, and where 
two vertices $L_i$ and $L_j$ are connected by an edge if their 
linking number $l_{i,j}$ is non-zero. Assume that $G$ 
is connected and the ground field has characteristic $0$.  
The given relations then suffice to eliminate all the $f_{i,j}$, 
and it 
follows that 
$\H^*(X)$ is generated in degree $1$ (see also 
Massey-Traldi [1986, Theorem 1 and Proposition 4.1],
or Matei-Suciu [1998]). 
 
Under these hypotheses, the cohomology ring behaves very 
nicely: 
 
\theorem{links} Both the homology module 
$\H_*(X)$ and the presentation ideal $I$ of the cohomology ring  
$\H^*(X)$ have  
linear free resolutions over the exterior algebra $E=\wedge\H^1(X)$. 
 
\noindent{\sl Proof Sketch.} 
With these hypotheses the presentation ideal $I\subset E$ is  
generated by the monomials  
$e_i\wedge e_j$, where $i$ and $j$ are vertices not connected by an  
edge in $G$, together with elements 
$\sum_k (1/l_{i_k,i_{k+1}})e_{i_k}\wedge e_{i_{k+1}}$, where the 
sums is over a cycle in the graph $G$. In particular, $E/I$ is a 
quotient ring of the  (exterior algebra) Stanley-Reisner ring of the 
graph $G$, regarded as $1$-dimensional simplicial complex on the vertex 
set $\{e_1,\ldots,e_n\}$. 
   
Now suppose we have chosen $T$ a spanning tree of the connected graph 
$G$,  and a total order on the edges of $G$. Recall that an edge $e\in 
G\setminus T$ is called {\it externally active} in $T$ if it is the 
largest edge in the unique cycle $C_e$ contained in $T\cup \{e\}$. It is 
a standard fact that for each enumeration of the edges of $G$ (say 
corresponding to  the choice of a monomial order in $E$) there exists a 
spanning tree $T_0$ of $G$ such that every edge of $G$ not in $T_0$ is 
externally active in $T_0$ (see Bollob\'as [1998, proof of Theorem 10, 
p. 351 and Exercise 8, p. 372]). Since the  cycles $C_e$ 
form a basis of the cycle space of $G$ (see for 
example Bollob\'as [1998, proof of Theorem 9, p. 53]), it follows that   
the ideal $I$ has an initial ideal $I_0$, which is the Stanley-Reisner  
ideal of the chosen spanning tree $T_0$ in $G$.  
 
The fact that the Stanley-Reisner ideal 
$I_0$ has a linear resolution follows from Hochster's 
formula for the Betti numbers of a square-free monomial ideal (see 
Hochster [1977] or, for an exposition, Stanley [1996]) since any 
subcomplex of a tree is a forest, which is acyclic in all positive 
homological degrees. The  linearity of the injective resolution of 
$\H^*(X)$ follows from the fact that  
$T$ is a Cohen-Macaulay simplicial complex as in the proof of 
\ref{coho res}.\Box 
 
 
 
\section{OSrank-var} The singular variety of an Orlik-Solomon algebra 
 
An element $x\in V= E_1$ is said to be singular on a module $M$ if the 
set of elements of $M$ annihilated by $x$ is not the same as $xM$. 
The set $V(M)$ of singular elements is an algebraic subset of $V$ 
called the {\it singular variety\/} of $M$; see Aramova, Avramov, and 
Herzog [1999] for a discussion. These authors prove, among other 
things, that the dimension of $V(M)$ is 
the {\it complexity\/} of $M$, defined as the exponent of growth of 
the betti numbers of $M$. This complexity plays, for 
modules over an exterior algebra, a role analogous to that of the 
projective dimension for modules over a polynomial ring. In this 
section we will compute the singular variety of the Orlik-Solomon algebra $A$ 
of an arrangement $\A$. It follows at once from the definition that 
the singular variety of a module $M$ is the same as that of $\Hom_K(M, 
K)$, so this also gives the singular variety of $\H_*(X)$. 
 
Before describing the singular variety explicitly, we note that  
in the case of $A$, \ref{coho 
res} gives a particularly simple criterion for an element to be 
singular, extending Theorem 4.1 (i) of Yuzvinsky [1995].  
 
\corollary{highesthom} 
An element $e\in V$ is singular 
for $A$ (or equivalently for $\H_*(X)$)  
if and only if 
there is a nonzero element of $\H_\ell(X)$  
annihilated by $e$. 
 
\proof We have $\H_{\ell+1}(X)=0$; for example 
this follows from the statement that $\H_*(X)$ 
is generated by $\H_\ell(X)$. Thus if 
$e$ annihilates a nonzero element of $\H_\ell(X)$  
it follows that $e$ is singular.  
 
The converse is a nontrivial fact that holds more generally 
for $E$-modules 
with linear free resolution; see  Eisenbud and Schreyer 
[2000, Corollary 4.4]. \Box 
 
Recall that the product $\A_1\times\A_2$ of 
arrangements $\A_i$ in $ \CC^{\ell_i}$ is the arrangement $\A$ 
in $\CC^{\ell_1+\ell_2}$ consisting of the hyperplanes 
$H\times \CC^{\ell_2}$ for $H\in \A_1$ and the hyperplanes 
$\CC^{\ell_1}\times H$ for $H\in \A_2$. Any arrangement can 
be expressed uniquely as the product of irreducible arrangements. 
The following well-known remark shows that to compute the singular variety 
of the Orlik-Solomon algebra as a module over $E$, it suffices to 
treat the irreducible case: 
 
\proposition{tensor product} The Orlik-Solomon algebra of a product 
$\A=\A_1\times\A_2$ of two arrangements is given by 
$A(\A)=A(\A_1) \otimes_K A(\A_2)$, the tensor product 
in the category of graded skew-commutative $K$-algebras. Thus 
$V(A(\A))=V(A(\A_1))\times V(A(\A_2))$. 
 
\proof A minimal dependent set of hyperplanes 
in $\A$, or a minimal set with empty intersection, comes from 
a similar set either in $\A_1$ or in $\A_2$, 
proving the first statement. The second follows because  
$A(\A)_1$ is the direct sum of the corresponding spaces for 
$\A_1$ and $\A_2$. A linear form $x=(x_1,x_2)$ is singular for 
$A(\A)$ if $x_i$ is singular on $A(\A_i)$ for both $i=1,2$.\Box 
 
The main result of this section is: 
 
\theorem{rank of irr} Let $\A$ be an irreducible complex 
hyperplane arrangement with Orlik-Solomon algebra 
$A$ and elements $e_i\in V:=A_1$ corresponding to the hyperplanes 
of $\A$.  
\item{$a)$} If $\A$ is noncentral then the singular variety of 
$A$ is $V$. 
\item{$b)$} If $\A$ is central then the singular variety of 
$A$ is the hyperplane spanned by the elements $e_i-e_j$. 
 
\proof 
If the singular variety of the Orlik-Solomon algebra $A$ of 
an arrangement $\A$ does not contain an element $e\in V$, 
then $A$ is a free module over the subring $K[e]/e^2$. It 
follows that the Poincar\'e polynomial $1+t$ 
of $K[e]/e^2$ divides the Poincar\'e 
polynomial 
$\pi(A,t)$  of $A$. 
 
On the other hand,  
Crapo [1967]  
(see also Schechtman-Terao-Varchenko [1995, Sect.~2]) 
shows that if $\B$ is an irreducible central arrangement with 
deconing 
$\A$, then 
$$ 
{\pi(A(\B),t)/(1+t)}_{\mid_{t=-1}}\not=0. 
$$  
It follows that in this case the singular variety of $A(\A)$  contains 
everything of degree $1$. In particular, if $\A$ is 
an irreducible noncentral arrangement, we may  
apply this remark to $\B=c\A$. Part $a)$ now follows from 
\ref{deconing}. 
 
If now $\B$ is an irreducible central arrangement, then 
the formula $A(\B)=A(d\B)[e]$ from  \ref{deconing} implies that 
the singular variety of $A(\B)$ is equal to the singular variety 
of $E'/I'\cong A(d\B)$; that is, it consists of precisely the elements 
of $V'$ as required.\Box 
 
From \ref{rank of irr} and \ref{tensor product} we get the general 
case: 
 
\corollary{codim of rank var} 
The singular variety of the Orlik-Solomon 
algebra $A$ of any arrangement $\A$ is a linear 
space of codimension equal to  
the number of central factors in an irreducible  
decomposition of $\A$.\Box 
 
\example{generic example} 
A central arrangement $\A$  
in $\CC^\ell$ is called {\it generic\/} 
if no set of $\ell$ or fewer hyperplanes of $\A$ is dependent. 
Analogously, a noncentral arrangement is called generic  
if every set of $\ell+1$ or fewer hyperplanes meet transversely 
(in particular, they don't meet if the number of hyperplanes 
is $\ell+1$).  
In the generic noncentral case it follows immediately 
from the definition that the Orlik-Solomon ideal $I$ is the  
$(\ell+1)^\st$ power $\gm^\ell$ of the maximal ideal $\gm$ 
of $E$. From \ref{deconing} it follows from this that in the  
generic central case the 
Orlik-Solomon ideal is the $\ell^\th$ power of the maximal ideal of the 
subalgebra $E'$ generated by the differences $e_i-e_j$ of the 
generators of $E$. 
 
The homology module $\H_*(X)$ 
is, as for every arrangement, given by 
$$ 
\H_*(X)=\Hom_K(E/I, K)\cong \Hom_E(E/I,E(n))=(0:_EI)(n), 
$$ 
the $n^\th$ twist of the annihilator of $I$. 
If $I=\gm^\ell$, then $(0:_EI)(n)=\gm^{n-\ell+1}$. 
An explicit computation of the resolution of this ideal is given in 
terms of Schur functors in  Eisenbud and Schreyer [2000, 
Corollary 3.3]; in particular the resolution is linear. 
 
 
\section{S-module} The module $F(\A)$  
 
Let $W=V^*=\H_1(X)$ be the dual vector space to $V$, and let $S=\Sym(W)$ 
be the symmetric algebra of $W$, a polynomial ring over $ K$. 
 
As usually stated, the Bernstein-Gel'fand-Gel'fand  
correspondence (BGG) is an isomorphism between the derived category 
of bounded complexes of coherent sheaves on ${\P}(V^*$) and the 
derived category of bounded complexes of finitely generated graded 
modules over $E=\wedge V$.  But if one examines the proof one can 
extract a functor ${\RR}$ from the category of graded modules 
over $S$ and the category of linear 
free complexes over $E$, and also a functor $\LL$ 
from the category of graded $E$-modules to the category 
of linear free complexes over $S$. These functors are equivalences of 
categories; see Eisenbud and Schreyer [2000, Proposition 2.1]. 
 
Starting with a graded $E$-module $P$ the corresponding complex $\LL(P)$ 
over $S$ is 
$$ 
\cdots \rTo S\otimes P_i\rTo S\otimes P_{i+1} \rTo \cdots 
$$ 
with differential $1\otimes p\mapsto\sum x_i\otimes e_ip$, where 
$x_i$ and $e_i$ are dual bases of $W$ and $V$. Starting with 
a graded $S$-module $M$ the corresponding complex $\RR(M)$ over $E$ is 
$$ 
\cdots \rTo \Hom_K(E, M_i)\rTo \Hom_K(E, M_{i+1})\rTo \cdots , 
$$ 
with differential defined similarly. 
 
Starting from a hyperplane arrangement $\A$, we 
consider the injective resolution of $A$ as an $E$-module. 
Recall that since $E$ is 
Gorenstein, injective resolutions over $E$ are simply 
the duals (with respect to $E$ or to $ K$)  
of free resolutions. Thus the 
injective resolution of $A$ is the $ K$-dual of the free resolution of 
$\H_*(X)$. By \ref{coho res}, this free resolution, and with it the  
injective resolution of $A$, is linear.  
 
Thus we may 
define 
$F(\A)$  to be the graded $S$-module that is mapped by 
$\RR$ to the injective resolution of $A$ as an $E$-module. 
The reason for choosing the injective 
resolution over the free resolution in the definition of $F(\A)$ 
is to make 
$F(\A)$ finitely generated.  
 
The following result, which is Corollary 6.2 of Eisenbud and Schreyer 
[2000] allows us to derive some basic properties of $F(\A)$: 
 
\theorem{reciprocity} 
If $M$ is a 
graded $S$-module and $P$ is a finitely generated graded 
$E$-module, then $\LL(P)$ is a free resolution of $M$  
if and only if $\RR(M)$ is an injective resolution of $P$.  
\Box 
 
\corollary{F properties}  
$F(\A)$ is generated over $S$ in degree $\ell$ and 
has linear free resolution equal to $\LL(A)$. In particular, 
\item{$a)$} 
$F(\A)$ has projective dimension $\ell$ and $\Ext^\ell_S(F(\A), S)= K$. 
\item{$b)$} 
 The support of $F(\A)$ is a linear space whose codimension is the  
number of central arrangements in an irreducible decomposition of $\A$. 
\item{$c)$} 
The Hilbert function of $F(\A)$ is  
$$ 
\sum_{i=0}^{\infty}\dim_K(F(\A)_i)t^i= 
(-1)^{\ell}{{\chi(\A,t)}\over{(1-t)^n}} 
$$ 
 
\proof By 
\ref{coho res} the injective resolution of $A$ over $E$, which 
is dual to the free resolution of $H_*(X)$, is linear.  
By \ref{reciprocity}, 
$$ 
\LL(A):\quad  
0\rTo S\otimes_K A_0\rTo\cdots\rTo  
S\otimes_K A_\ell \rTo F(\A)\rTo 0 
$$  
is a (linear) free resolution of $F(\A)$, proving the first 
statement and computing the projective dimension. 
 
$a)$: The degree 0 and 1 parts of $A$ coincide with those 
of $E$; thus the left-hand terms of the resolution above 
are the same as those in $\LL(E)$, the Koszul complex. This 
allows us to compute the $\Ext$ in part a). 
 
$b)$: Aramova, Avramov and Herzog [1999] show in general that 
the singular variety of an $E$-module $P$  
is the support of the $S=\Ext_E^*( K, K)$-module 
$\Ext_E^*(P, K)$, which is the same (since $E$ is 
Gorenstein) as the support of the module $\Ext_E^*( K,P)$. 
By  Eisenbud and Schreyer [2000, Proposition 5.2] 
this is the module $F(\A)$. 
 
$c)$: Knowing the free resolution of $F(\A)$ allows us to 
compute its Hilbert series, just as in  
the proof of \ref{coho res}.\Box 
 
\example{generic example continued} 
If $\A$ is a generic noncentral arrangement of $n$ hyperplanes in 
$ K^\ell$, then $A$ is $E/\gm^{\ell+1}$, so the 
free resolution of $F(\A)$ is a truncation of the Koszul complex, 
and $F(\A)$ is isomorphic to the $(v-\ell)^\th$ syzygy module 
of the trivial $S$-module $ K$.\Box 
 
We have already seen that if $\A$ is a generic noncentral arrangement 
then the Orlik-Solomon ideal of $\A$ is a power of the maximal 
ideal of $E$, and thus has a linear free resolution. We will 
show that this property characterizes generic arrangements and 
their cones. We begin with a general result characterizing 
deformations of powers of the maximal ideal: 
 
 
\theorem{characterization of powers} 
Let $I\subset E$ 
be an ideal in the exterior algebra. Both $I$ and $(E/I)^*$ 
admit linear free resolutions if and only if $I$ reduces to 
a power of the maximal ideal modulo some (respectively any) 
maximal $E/I$ regular sequence of linear forms of $E$. 
 
\proof If $f_1,\dots,f_s\in E_1$ is a regular sequence on 
$E/I$ then $I$ and $(E/I)^*=\Hom_K(E/I,K)$ are also free 
over $K\langle f_1,\dots,f_s\rangle$. The freeness of 
$E/I$ over $K\langle f_1,\dots,f_s\rangle$ implies that,  
the image of $I$ in $E/(f_1,\dots,f_s)$ is 
isomorphic to $I/(f_1,\dots,f_s)I$, and also that the dual 
of $E/(I+(f_1,\dots,f_s))$ is $(E/I)^*\otimes_E E/(f_1,\dots,f_s)$. 
Thus the minimal  free 
resolutions of $I$ and $(E/I)^*$ are linear if and 
only if the minimal resolutions of  
$I/(f_1,\dots,f_s)I$ and $(E/(I+(f_1,\dots,f_s)))^*$ 
are linear, and it follows from  Eisenbud and  
Schreyer [2000, Section 3] that if the image of 
$I$ in $E/(f_1,\dots,f_s)$ is a power of the maximal ideal, 
then the minimal free resolutions of $I$ and $(E/I)^*$ are 
linear. 
 
To prove the converse, the argument given above 
reduces us to showing, in the case where the singular variety 
of $E/I$ is $V$, that if the resolutions of $I$ and $(E/I)^*$ 
are linear, then $I$ is itself a power of the maximal ideal. 
 
Our hypothesis implies in particular that  module $(E/I)^*$ is generated 
in a single degree. It follows by Nakayama's Lemma and duality  
that the socle of $E/I$ (the annihilator in $E/I$ of $\gm$) 
is generated in a single degree, say degree $s$. Thus  
$I_j=E_j$ for $j>s$, 
and it suffices to show that $I_j=0$ for $j\leq s$. 
 
By \ref{reciprocity}, both $\LL(E/I)$ and  
$\LL(I^*)$ are free resolutions; let $F$ be the 
module whose resolution is $\LL(E/I)$.  
By Aramova, Avramov, and Herzog [1999] its support is 
the singular variety of $E/I$, that is, $V$. 
 
Duality (into $ K$) over the  
exterior algebra gives an exact sequence 
$0\to (E/I)^*\to E^*\to I^*\to 0$. Taking duals commutes 
with the functor $\LL$ (up to shifts),  
so we get an exact sequence 
of complexes 
$0\to \LL(E/I)^*\to \LL(E)^*\to \LL(I^*)\to 0$, where 
now the duals denote $\Hom_S(-, S)$. The homology 
of $\LL(E/I)^*$ at $S\otimes_K ((E/I)_s)^*$ 
is $\Ext_S^0(F,S)$, which is nonzero because $F$ 
has support $V$. It follows from the exact sequence that  
$\LL(I^*)$ has nonzero homology at the term 
$S\otimes (I_{s+1})^*$. Since $\LL(I^*)$ is a resolution, 
this must be the last term of the complex---that is, 
$I_j=0$ for $j\leq s$, as required. 
\Box 
 
\example{nontrivial defo} The ideals characterized in 
\ref{characterization of powers} include  powers of the maximal ideal  
in subalgebras generated by linear forms (this will be the 
case for cones over hyperplane arrangements) but also 
many that are {\it not\/} of this form. Here is 
the simplest concrete example: Let  
$$ 
I:=(ab+cd, ac, bc)\ \subset \ E:=K\langle a,b,c,d\rangle . 
$$ 
It is easy to check that the three given quadrics form 
a Gr\"obner basis with respect to any order with $ab>cd$. 
Since $d$ is a regular element on $E$ modulo the initial 
ideal $(ab,ac,bc)$, it follows that $d$ is regular on $E/I$. 
It is evident that $I$ reduces modulo $d$ to the square of the maximal  
ideal. To see that $I$ is not the square of the maximal ideal 
of any exterior subalgebra on 3 variables, note that the quadrics 
in 3 variables are all of rank 2, where as $I$ contains an element 
of rank 4 (here the rank is defined via the identification between 
elements of $E_2$ and skew-symmetric $4\times 4$ matrices.) 
 
 
\corollary{characterization of generic}  
The Orlik-Solomon 
ideal of $\A$ admits a linear free resolution over $E$ if and only if 
$\A$ is obtained by successively coning a generic noncentral  
arrangement. 
 
\proof We have already seen that the property holds for generic 
noncentral arrangements. If $I$ is the Orlik-Solomon ideal 
of $\A$, then the Orlik-Solomon ideal of $c\A$ in $E[e_0]$ 
is $IE[e_0]=I\otimes_EE[e_0]$, which has free resolution obtained 
from that of $I$ by tensoring with $E[e_0]$; in particular, 
the linearity is not affected.  
 
Deconing $\A$ as many times as possible,  it now suffices to show  
that if $\A$ is noncentral and $I$ has a linear resolution then 
$\A$ is generic. Since $\A$ is noncentral it can have 
no central factors in its irreducible decomposition, and 
thus the singular variety of $\A$ is the whole of the vector space 
$V$ of linear forms.  
 
The theorem now follows from a more general result. Recall  
from Aramova, Avramov, and Herzog [1999] that a sequence 
of elements $f_1,\dots,f_s\in E_1$ is called a regular sequence on an 
$E$-module 
$M$ if $M$ is free over $K\langle f_1,\dots,f_s\rangle$, or 
equivalently, if the annihilator of $f_i$ in $M/(f_1,\dots,f_s)M$ 
is $f_iM/(f_1,\dots,f_s)M$ for every $i$. In this case 
the minimal free resolution of $M/(f_1,\dots,f_s)M$ over 
$E/(f_1,\dots,f_s)$ is obtained by reducing the minimal  
$E$-free resolution of $M$ modulo $(f_1,\dots,f_s)$. The 
length of any maximal regular sequence on $M$ is equal 
to the codimension of the singular variety of $M$ in $V$. 
 
\remark{horrocks} \ref{characterization of powers} is  
actually equivalent to  
the Theorem of Horrocks that characterizes the bundle 
$\Omega_{\P(W)}^i(i)$ as the unique indecomposable sheaf ${\cal F}$  
such that the only nonzero intermediate 
cohomology of any twist of ${\cal F}$ is $\H^i({\cal F})=K$. 
To see this one uses the correspondence between powers of 
the maximal ideal of $E$ and the twisted exterior powers of 
the cotangent sheaf $\Omega_{\P(W)}^i(i)$, as well as the relation 
between resolutions over $E$ and cohomology of sheaves on 
$\P(W)$, all explained in  Eisenbud and Schreyer [2000]. 
 
 
\section{Appendix} 
Syzygies of Monomial Ideals in the Exterior Algebra 
 
In this section we give a conceptual description and proof of the 
correspondence between free resolutions of certain modules over exterior 
and symmetric algebras first proved by Aramova, Avramov, and Herzog 
[1998] and R\"omer [1999].  The main idea is an isomorphism between 
certain subcategories of the categories of modules over these two 
algebras. Our approach provides a simple explanation for the shape of the 
formula relating the corresponding multigraded Betti numbers  
(\ref{betti}). 
 
Let $V$ be an $n$-dimensional vector space over the field $K$, with 
basis $x_1,\ldots,x_n$. We will denote by $S=\Sym(V)$  
the symmetric algebra over $V$,  which we identify with  
the ring of polynomials over $K$ in the $n$ variables $x_1,\ldots,x_n$, 
and by $E=\Lambda(V)$ the exterior algebra of the vector space  
$V$. Both these algebras have a natural $\Z^n$ grading in which each 
monomial (product of the $x_i$) generates a homogeneous component. 
(Note that in earlier sections we wrote $S=\Sym(W)$, where 
$W$ was the dual of $V$. Since we have explicitly chosen 
a basis of $V$ we may identify $V$ with $W$.) 
 
We say that a $\Z^n$-graded module $M$ over $E$ or $S$ is 
{\it square-free\/} if it admits a free presentation  
$F\rTo G\rTo M\rTo 0$ where each generator of $F$ and $G$ has the  
degree of a square-free monomial. Note that the presentation map 
$F\rTo G$ is represented by a matrix whose entries are scalars 
times monomials. Examples include the  
Stanley-Reisner rings $S/I$ where $I$ is an ideal generated by 
square-free monomials, but also such things as the cokernel of  
the matrix 
$$\pmatrix{ 
x_0 &0 \cr 
-x_1 &x_1 \cr 
0 & -x_2}, 
$$ 
the canonical module of the cone over 3 points in the plane. 
 
There is a 1-1 correspondence between square-free modules over $S$ and 
over $E$ obtained by interpreting the presentations as matrices over 
$S$ or over $E$; we will write $_SM$ and $_EM$ for the two.  
 
We can describe the correspondence of resolutions in a simple way 
as follows: 
 
Start from a free resolution of a square-free module $_SM$. Replace 
each free module in the resolution by a module made from the sum of 
the vector spaces of its multihomogeneous elements of square-free 
degree. It turns out---this is the main point---that this complex 
of vector spaces has the structure both of a complex of 
$S$-modules and a complex of $E$-modules. The modules in this 
complex are not free, but they have simple and functorial free 
resolutions. The free resolutions of the 
$E$-modules in the complex fit together to make a double complex, 
whose total complex is the minimal free resolution of $_EM$. A 
similar procedure allows us to pass in the opposite direction.  
 
The correspondence described above works, 
with appropriate definitions, in a more general setting, 
in which $E$ is replaced by one  
of the algebras 
$$ 
R_q:={K\{\,x_1,\ldots , x_n\,\}\over 
({\langle  
x_jx_i-qx_ix_j \mid 1\le i<j\le n\rangle 
+ 
\langle (1-q)x_i^2\mid 1\le i\le n\rangle})} 
$$  
where $K\{\,x_1,\ldots , x_n\,\}$ denotes the free 
 $K$-algebra on 
$x_1,\ldots , x_n$, and $q\ne 0$. We leave the details of this 
generalization to the interested reader. 
 
All modules and free resolutions considered will 
be assumed $\Z^n$-graded. We identify $\N^n\subset\Z^n$ with the set 
of monomials of $S$. By the support of a monomial in either $E$ or $S$, 
we will mean the collection of variables present in it. 
A square-free monomial (or multidegree) is an element 
$\aa\in\{0,1\}^n\subset\NN^n$, so $\supp(\aa)=\setdef 
{x_j} {a_j\ne 0}$.   
 
\medskip\noindent 
{\bf Modules With Square-free Presentation}. 
The following result is due Bruns and Herzog [1995, Theorem 3.1 a)]: 
 
\proposition{sf bettis} Let $\Gamma$ be any set of monomials of $S$ 
closed under taking least common multiples. If $M$ is an $S$-module 
with generators and relations having degrees in $\Gamma$, then all the 
free modules in a minimal free resolution of $M$ have degrees in 
$\Gamma$. 
 
\proof  
We give a new proof using Gr\"obner bases, which will 
easily extend to give \ref{sf bettis exterior} as well. 
Let  
$F\rTo^\phi G\rTo M\rTo 0$ 
be a $\Z^n$-graded free presentation with degrees of $F$ and $G$  
in $\Gamma$.  
We may replace $F\rTo^\phi G$ by a map $F'\rTo^{\phi'} G$  
so that the generators 
of $F'$ map to a Gr\"obner basis of $\ker(G\rTo M)$ by using the Buchberger 
algorithm; this involves adding free generators whose 
degrees are the least common multiples of pairs of generators 
already present, and thus still in $\Gamma$. 
Schreyer's theorem  
(Eisenbud [1995, Theorem 15.10]) shows that in the symmetric 
case the 
kernel of $\phi'$ is generated by elements of degrees equal 
to the least common multiples of pairs of degrees of generators of $F'$. 
%In the exterior case we must also add the 
 
It follows as in Eisenbud [1995, Theorem 20.2], 
that the minimal presentation of $M$  has also  
degrees in $\Gamma$, and iterating this process  
we see that the same is true for 
the whole syzygy chain. 
\Box 
 
If $M$ is a square-free module in the sense above, then 
we say that the {\it square-free part\/} of $M$ is the  
module obtained by factoring out all the homogeneous elements 
of $M$ with non square-free degrees. Thus for example  
the square-free part of $S$ itself is the factor ring 
$R:=S/(x_1^2,\ldots,x_n^2)$.  More generally, if $\aa$ is 
any square-free monomial, then 
$S(-\aa)$ has square-free part $R/\supp(\aa)(-\aa)$.  
 
\corollary{sf parts} If $M$ is a square-free module over $S$ 
then the square-free part of $M$  
admits a resolution by direct sums of modules of the 
form $R/\supp(\aa)(-\aa)$.  
\Box 
 
An analogous result also holds over $E$: 
 
\proposition{sf bettis exterior} If $M$ is a module over $E$ whose 
generators and relations have square-free  degrees,  
then the square-free part of 
$M$ admits a finite resolution by modules of the form  
$E_\aa:= E/\supp(\aa)(-\aa)$. 
 
\proof Because the generators and relations of $M$ have square-free degrees, 
we may write $M$ as the cokernel of a map (always $\Z^n$-homogeneous) 
between finite direct sums of modules of the form $E_\aa$, and it thus 
suffices to show that the kernel of such a map is generated in 
square-free degrees. Using Gr\"obner bases we may reduce as above to 
the monomial case. Exactly as in Eisenbud [1995, Lemma 15.1], one 
shows that all the relations among monomials are generated by those 
determined by the fact that any monomial $\aa$ is annihilated by the 
variables in the support of $\aa$, and the two-at-a time relations 
coming from the least common multiples (``divided Koszul 
relations''). The desired result follows.\Box 
 
\medskip\noindent 
{\bf The Common Subcategory}.  
The category of modules 
over $E$ and the category of modules over $R=S/(x_1^2,\dots,x_n^2)$  
have much in common. 
We make one such connection precise as follows: 
 
Let $\aa$ and $\bb$ be two monomials in $E$ such that 
$\supp(\aa)\subseteq\supp(\bb)$, and let $E_\aa$ and $E_\bb$ 
be the cyclic $E$-submodules generated by these monomials. 
The natural inclusion $E_\bb\subseteq E_\aa\subseteq E$ induces 
a functorial commutative diagram 
\newarrow{Equals}===== 
$$\diagram[midshaft,small] 
E/\supp(\aa)(-\aa)&\rTo^\cong& E_\aa&\rIntoA&E\\  
\uTo^{\cdot \bb\aa^{-1}} &&\uIntoB&&\uEquals\\ 
E/\supp(\bb)(-\bb)&\rTo^\cong& E_\bb&\rIntoA&E\\  
\enddiagram 
$$ 
where the horizontal isomorphisms are defined by sending 1 
to the distinguished generator, and the upper left 
monomorphism is induced by right multiplication in $E$ 
with $\bb\aa^{-1}$, the signed exterior monomial such that 
$(\bb\aa^{-1})\aa=\bb$. 
 
The same commutative diagram holds if we replace 
$E$ by $R$, and in fact 
identifying square-free monomials in $E$ with the corresponding 
monomials in $R$ defines an equivalence of categories. More precisely: 
 
\proposition{equivER} Let $\A$ denote the $K$-additive extension of the 
category of $\Z^n$-graded submodules of $R$, with morphisms given 
by inclusions, let $\B$ denote the $K$-additive extension of the 
category of $\Z^n$-graded submodules of $E$, also with morphisms given 
by inclusions, and let $\kVect$ be the category of $K$-vector spaces. 
\hfill\break  
The above identification of square-free monomials in $E$ with those of $R$  
induces an equivalence $\Psi$ of categories  
$$ 
\diagram[midshaft,small] 
\A&&\rTo^\Psi&&\B\\ 
&\rdTo&&\ldTo\\  
&&\kVect&&\\ 
\enddiagram 
$$ 
whose restriction (via the natural forgetful functors)  
to the underlying $K$-vector spaces is the 
identity functor.  
In particular, the functor $\Psi$ preserves acyclic complexes.\Box 
 
Notice that if $\aa$ is a square-free monomial, then the square 
free parts of  
$S(-\aa)$ is $R/\supp(\aa)(-\aa)=(\aa)R$, an object of $\A$. 
Similarly, the square-free part $E_\aa=E/\supp(\aa)(-\aa)=(\aa)E$ 
of $E(-\aa)$ is an object of $\B$. 
 
 
\medskip\noindent 
{\bf Resolutions over $S$ and $E$}.  
We let $\A_0$ and $\B_0$ be the 
additive subcategories generated by these modules and the inclusion 
morphisms $(\aa)R\subset(\bb)R$ and $(\aa)E\subset(\bb)E$ when 
$\aa|\bb$ as monomials in $S$. 
 
Certain free complexes over $S$ and $E$ correspond to complexes  
in the categories $\A_0\cong\B_0$. We describe the connection 
with $S$ first: 
 
To a given $\Z^n$-graded complex ${\mathbf F}_\bullet$ of free $S$-modules 
$$ 
{\mathbf F}_\bullet: \quad 0  \rTo F_r  
\rTo \ldots \rTo F_1 \rTo  F_0,  
$$  
with generators in square-free degrees, 
we associate a complex $\sf({\mathbf F}_\bullet)$ of 
$R$-modules, that we may regard as a complex in $\A_0$. Namely 
we define $\sf({\mathbf F}_\bullet)$ as the complex of 
square-free degrees of ${\mathbf F}_\bullet$, that is 
$$ 
{\sf({\mathbf F}_\bullet)}_i= 
\oplus_{\bb\in\set{0,1}^n}{({\mathbf F}_i)}_\bb, 
$$ 
for all $i$, and where the differentials are induced by the 
differentials of the original complex ${\mathbf F}_\bullet$. 
It is easy to see that $\sf$  
defines a functor from the category of $\Z^n$-graded complexes 
of free $S$-modules to the category of complexes in $\A_0$. 
 
It follows from \ref{sf bettis} 
that ${\mathbf F}_\bullet$ is square-free acyclic (that is it has 
no homology in square-free multidegrees) if and only if the complex 
$\sf({\mathbf F}_\bullet)$ is acyclic. It is also clear that 
${\mathbf F}_\bullet$ is minimal if and only if 
$\sf({\mathbf F}_\bullet)$ is minimal. 
 
We have proven: 
 
\proposition{S to A} The functor $\sf$ is an equivalence between the category 
of square-free complexes of free $S$-modules and the category of 
complexes in $\A_0$. It preserves minimality and acyclicity. 
 
Now we turn to complexes over $E$. For functorial constructions, 
we will use the divided power algebra. 
If $U$ is a finitely dimensional graded vector space, we write  
$D_l(U)$ for the $l\th$-divided power of $U$.  It is convenient to define 
$D_l(U)$ as the dual of the $l\th$-symmetric power of the dual space, 
that is $D_l(U) = (\Sym_l(U^*))^*$.  The divided powers $D_l(U)$ have 
``diagonal'' maps  
$$D_{l+1}(U) \rTo D_l(U)\otimes U$$  
which are the monomorphisms dual to the surjective  
natural multiplication map in the symmetric algebra 
$$\Sym_l (U^*)\otimes U^* \rTo \Sym_{l+1} (U^*).$$  
 
We can now go from complexes in the category $\B_0$ to  
free complexes over $E$ using the 
Cartan Resolution. 
 
\proposition{equivEE} There exists a functor $\Phi$ from the 
category of complexes in $\B_0$ to the category of complexes of free 
modules over $E$ whose inverse is obtained by taking 
square-free parts. $\Phi$ preserves acyclicity and minimality. 
Applied to an acyclic complex in $\B_0$ with homology 
$M$, the functor $\Phi$ provides an $E$-free resolution of $M$. 
 
\proof  We first define $\Phi$ on modules in $\B_0$. 
It associates to a cyclic module  
$E_\aa\cong E/\supp(\aa)(-\aa)$  
the (resolution)  
$\Phi(E_\aa):=D(L_\aa)\otimes E$,  
where  
$L_\aa:=\oplus_{x_i\in\supp(\aa)}k(-\ee_i)$  
is the $\Z^n$-graded subspace of $V$ spanned by  
$\supp(\aa)$, and whose  differentials are induced by  
the diagonals followed by multiplication in $E$.  
 
More precisely $\Phi(E_\aa)$ is the complex 
$$\Phi(E_\aa):\qquad 
\ldots\rTo D_2(L_\aa)\otimes E(-\aa)\rTo L_\aa\otimes E(-\aa)\rTo 
E(-\aa),$$ 
which is a minimal free resolution of  the cyclic module $E_\aa$. 
We see at 
once that $E_\aa$ is the square-free part of $\Phi(E_\aa)$. 
 
If $\aa$ and $\bb$ are two monomials in $E$ such that 
$\supp(\aa)\subseteq\supp(\bb)$, then  
$$D(L_\bb)\otimes E(-\bb)\rTo^{\pi\otimes(\cdot \bb\aa^{-1})} 
D(L_\aa)\otimes E(-\aa),$$  
where $\pi$ is the map induced to divided powers by the  
canonical projection $\pi: L_\bb\rTo L_\aa$, is a morphism of 
chain complexes lifting the inclusion 
$E_\bb\subseteq E_\aa\subseteq E$. 
 
Given a complex ${\mathbf F}_\bullet$ in $\B$, we may apply $\Phi$ 
to obtain a double complex of free $E$-modules, and we 
set $\Phi({\mathbf F}_\bullet)$ to be the total complex of 
this double complex. Because of the way $\Phi$ is defined on each 
object of $\B_0$, this functor preserves minimality. The spectral 
sequences of the double complex shows that it also preserves  
acyclicity.\Box 
 
As Aramova, Avramov, Herzog [1999] and R\"omer [1999] 
observe, the existence of such a construction 
shows that if an $S$-module $M$ 
has a linear free resolution over $S$ if and only if the 
corresponding $E$ module has a linear free resolution over $E$. 
Our version of the construction also ``explains'' these authors' 
formula for betti numbers: 
 
\corollary{betti} The  following equality holds among  Poincar\'e  
series: 
$$\sum_{i=0}^\infty\sum_{\aa\in\NN^n}\beta_{i,\aa}^E(_EM)t^i\uu^\aa= 
\sum_{i=0}^\infty\sum_{\aa\in\NN^n}\beta_{i,\aa}^S(_SM) 
{{t^i\uu^\aa}\over{\prod_{j\in\supp(\aa)}(1-tu_j)}}$$ 
where $\beta_{i,\aa}^E(_EM)$ denotes the dimension of the  
degree $\aa$ part of $\tor^E_i(M,K)$, and similarly for 
$\beta_{i,\aa}^S(_SM)$. 
 
 
\references  
\parindent=0pt 
\frenchspacing 
\item{} A.~Aramova, L.A.~Avramov, J.~Herzog:  
Resolutions of monomial ideals and cohomology over exterior algebras,  
{\sl Trans. Amer. Math. Soc.} (1999) (to appear). 
\medskip 
 
\item{} A.~Aramova, J.~Herzog, T.~Hibi:  
Squarefree lexsegment ideals,  
{\sl Math. Z.} {\bf 228}, (1998), 353--378. 
\medskip 
 
\item{}  D.~Eisenbud and F.-O.~Schreyer:  
Free Resolutions over Exterior Algebras,  preprint 2000.  
\medskip 
 
\item{} D.~Bayer, H.~Charalambous, S.~Popescu: 
Extremal Betti Numbers and  Applications to Monomial Ideals, 
{\sl J. Algebra} {\bf 221}, (1999), 497--512. 
\medskip 
 
\item{}  L.~J.~Billera, J.~S.~Provan:  Decompositions of simplicial  
complexes related to diameters of convex polyhedra,  
{\sl Math. Oper. Res.} {\bf 5}, (1980), 576--594. 
\medskip 
 
\item{} A.~Bj\"orner: On the homology of geometric lattices,  
{\sl   Algebra Univ.}  {\bf 14}, (1982), 107--128. 
\medskip 
 
\item{} A.~Bj\"orner: The homology and shellability of matroids and 
  geometric lattices, Chapter 7 of {\it Matroid Applications}, ed. 
  Neil White, 226--283, Encyclopedia Math. Appl., {\bf 40}, Cambridge 
  Univ. Press, Cambridge, 1992.   
\medskip 
   
\item{} A.~Bj\"orner, G.~Ziegler:  
Combinatorial stratification of complex arrangements'',  
{\sl J. Amer. Math. Soc.} {\bf 5}, (1992), no. 1, 105--149.  
\medskip 
 
\item{} 
B.~Bollob\'as: {\it Modern Graph Theory}, Graduate Texts 
in Mathematics {\bf 184}, Springer, New York, 1998. 
\medskip 
 
\item{} W.~Bruns, J.~Herzog: {\it Cohen-Macaulay Rings}, Cambridge 
  Studies in advanced mathematics, {\bf 39}, Cambridge University 
  Press 1993.   
\medskip 
   
\item{} W.~Bruns, J.~Herzog: On multigraded resolutions,  
  {\sl Math. Proc. Camb. Phil. Soc.}, {\bf 118}, (1995), 245--257.   
\medskip 
 
\item{} H.~Crapo: A higher invariant for matroids, {\sl J. of 
    Combinatorial Theory} {\bf 2}, (1967), 406--417.   
\medskip 
 
 
\item{} J.~Eagon, V.~Reiner: Resolutions of Stanley-Reisner rings and  
Alexander duality, {\sl J. Pure Appl. Algebra} {\bf 130}, (1998), no. 
3, 265--275.  
\medskip 
 
\item{} D.~Eisenbud: 
{\sl Commutative Algebra with a View Toward Algebraic Geometry}, 
Springer, New York, 1995. 
\medskip 
 
\item{} D.~Grayson, M.~Stillman: {\it Macaulay2}, a 
software system devoted to supporting research in algebraic geometry 
and commutative algebra.  Contact the authors, or download from 
{\tt http://www.math.uiuc.edu/Macaulay2}. 
\medskip 
 
\item{}  M.~Hochster: Rings of invariants of tori, Cohen-Macaulay 
rings generated by monomials, and polytopes, {\sl Ann. of Math.} 
{\bf 96}, (1972), 318--337. 
\medskip 
 
\item{} M.~Hochster: Cohen-Macaulay rings, combinatorics and simplicial 
complexes, in Ring theory II,  McDonald B.R., Morris, R. A. (eds), 
{\it Lecture Notes in Pure and Appl. Math.} {\bf 26}, M. Dekker 1977. 
\medskip 
 
\item{} D.~Matei, A.~Suciu: Cohomology rings and nilpotent 
quotients of real and complex arrangements, to appear in 
{\it Singularities and Arrangements, Sapporo-Tokyo 1998}, 
Advanced Studies in Pure Mathematics , preprint 
{\tt math.GT/9812087}.  
\medskip 
 
\item{} W.~Massey, L.~Traldi: 
On a conjecture of K. Murasugi, 
{\sl Pacific J. Math.} {\bf 124} (1986), 
no. 1, 193--213. 
\medskip 
 
\item{} J.~Milnor: 
Isotopy of links, in  {\it Algebraic geometry and topology. 
A symposium in honor of S. Lefschetz}, pp. 280--306. 
Princeton University Press, Princeton, N. J., 1957. 
\medskip 
 
\item{} M.~Musta\c t\v a: Local Cohomology at Monomial Ideals, preprint 1998. 
 \medskip 
 
\item{} P.~Orlik, L.~Solomon: Combinatorics and topology of 
complements of hyperplanes, {\sl Invent. Math.} {\bf 56}, (1980), 167--189. 
\medskip 
 
\item{} P.~Orlik, H.~Terao: {\it Arrangements of hyperplanes},  
Grundlehren der Mathematischen Wissenschaften {\bf 300},  
Springer-Verlag, Berlin, 1992.   
\medskip 
 
\item{} J.~S.~Provan: Decompositions, shellings, and diameters of  
simplicial complexes and convex polyhedra, Thesis Cornell Univ. 1977. 
\medskip 
 
\item{} T.~R\"omer: Generalized Alexander Duality and Applications,  
Preprint 1999. To appear in {\sl Osaka J. Math.}\medskip 
\medskip 
 
\item{} V.~Schechtman, H.~Terao, A.~Varchenko: Local systems over 
complements of hyperplanes and the Kac-Kazhdan conditions for singular 
vectors,  {\sl J. Pure Appl. Algebra} {\bf 100}, (1995), 93--102. 
\medskip 
 
\item{} R.~Stanley: {\it Combinatorics and Commutative Algebra}, Second 
edition, Progress in Math. {\bf 41}, Birkh\"auser, 1996. 
\medskip 
 
\item{} R.~Stanley: Cohen-Macaulay rings and constructible 
polytopes, {\sl Bull. Amer. Math. Soc.} {\bf 81}, (1975), 133-135.  
\medskip 
 
\item{} N.~Terai: Generalization of Eagon-Reiner theorem 
and $h$-vectors of graded rings, preprint 1997.      
\medskip 
 
\item{} K.~Yanagawa: Alexander duality for Stanley-Reisner rings and 
   square-free ${\bf N}^n$-graded modules, preprint 1998. 
\medskip 
 
\item{} S.~Yuzvinsky: Cohomology of the Brieskorn-Orlik-Solomon algebras,  
{\sl Comm. Algebra} {\bf 23}, (1995), 5339--5354. 
\medskip 
 
\item{} G.~Ziegler: On the difference  between real and  
complex arrangements,  {\sl Math. Z.} {\bf 212} (1993),  
no. 1,  1--11.  
 
 
\bigskip\bigskip 
\vbox{\noindent Author Addresses: 
\smallskip 
\noindent{David Eisenbud}\par 
\noindent{Department of Mathematics, University of California, Berkeley, 
Berkeley CA 94720}\par 
\noindent{de@msri.org} 
\smallskip 
\noindent{Sorin Popescu}\par 
\noindent{Department of Mathematics, Columbia  University, 
New York, NY 10027, and}\par 
\noindent{Department of Mathematics, SUNY at Stony Brook,  
Stony Brook, NY 11794}\par 
\noindent{psorin@math.columbia.edu}\par 
\smallskip 
\noindent{Sergey Yuzvinsky}\par 
\noindent{Department of Mathematics, University of Oregon, 
Eugene, OR 97403}\par 
\noindent{yuz@math.uoregon.edu}\par 
} 
 
\bye 
