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\setlength{\headheight}{15pt}
\lhead{{\sf Math 9850, Algebraic Systems Biology}: Homework 6}
\chead{}\rhead{\thepage}
\lfoot{Due Friday, November 1 2024}
\cfoot{}\rfoot{Written by M.~Macauley}
\pagestyle{fancy}

\begin{document}

$\;$ \vspace{-4mm}

\begin{enumerate}

\item \textbf{Alexander duality}.  Consider the simplicial complex $\Delta$ over $X=\{a,b,c,d,e\}$ shown below:
  \[
  \begin{tikzpicture}[scale=1.25, xscale=1.25,yscale=.7,v/.style={draw,Red,fill=Red, circle, inner sep=1pt},baseline=-1ex]
    \draw[Red,fill=Red!30] (0,0) -- (.707,1) -- (1.41,0) -- cycle;
    \node[v, label=left:{$a$}] (a) at (.707,2) {};
    \node[v, label=below:{$b$}] (b) at (.707,1) {};
    \node[v, label=above:{$\!\!c$}] (c) at (0,0) {};
    \node[v, label=above:{$d\!\!$}] (d) at (1.41,0) {};
    \node[v, label=above:{$e$}] (e) at (2.41,0) {};
    \draw [Red] (a)--(b); \draw [Red] (a)--(c); \draw [Red] (a)--(d);
    \draw [Red] (d)--(e);
  \end{tikzpicture}
  \]
  The 5-dimensional Boolean lattice $2^X$ also appears below.
  \begin{enumerate}[(i)]
  \item In the Boolean lattice, color the faces red, and the nonfaces blue.
  \item Find the maximal faces, and the minimal generators of the
    Stanley-Reisner ideal $I_{\Delta^c}$.
  \item Compute the primary decomposition of $I_{\Delta^c}$.
  \end{enumerate}
  \[
  \begin{tikzpicture}[scale=1.7]
    \tikzstyle{B} = [Blue]
    \tikzstyle{R} = [Red]
    \tikzstyle{F} = [faded,dashed]
    \tikzstyle{f} = [faded]
    \tikzstyle{every node}=[font=\footnotesize]
    %% Change V within \node[...] to....
    %% Up for light blue
    %% UP for dark blue
    %% DOWN for dark red
    %% Down for light red
    \node[anchor=south,V] (abcde) at (0,0) {$abcde$};
    \node[anchor=south,V] (abcd) at (-4,-1) {$abcd$};
    \node[anchor=south,V] (abce) at (-2,-1) {$abce$};
    \node[anchor=south,V] (abde) at (0,-1) {$abde$};
    \node[anchor=south,V] (acde) at (2,-1) {$acde$};
    \node[anchor=south,V] (bcde) at (4,-1) {$bcde$};
    \node[anchor=south,V] (abc) at (-4.5,-2) {$abc$};
    \node[anchor=south,V] (abd) at (-3.5,-2) {$abd$};
    \node[anchor=south,V] (abe) at (-2.5,-2) {$abe$};
    \node[anchor=south,V] (acd) at (-1.5,-2) {$acd$};
    \node[anchor=south,V] (ace) at (-.5,-2) {$ace$};
    \node[anchor=south,V] (ade) at (.5,-2) {$ade$};
    \node[anchor=south,V] (bcd) at (1.5,-2) {$bcd$};
    \node[anchor=south,V] (bce) at (2.5,-2) {$bce$};
    \node[anchor=south,V] (bde) at (3.5,-2) {$bde$};
    \node[anchor=south,V] (cde) at (4.5,-2) {$cde$};
    %%
    \node[anchor=south,V] (ab) at (-4.5,-3) {$ab$};
    \node[anchor=south,V] (ac) at (-3.5,-3) {$ac$};
    \node[anchor=south,V] (ad) at (-2.5,-3) {$ad$};
    \node[anchor=south,V] (ae) at (-1.5,-3) {$ae$};
    \node[anchor=south,V] (bc) at (-.5,-3) {$bc$};
    \node[anchor=south,V] (bd) at (.5,-3) {$bd$};
    \node[anchor=south,V] (be) at (1.5,-3) {$be$};
    \node[anchor=south,V] (cd) at (2.5,-3) {$cd$};
    \node[anchor=south,V] (ce) at (3.5,-3) {$ce$};
    \node[anchor=south,V] (de) at (4.5,-3) {$de$};
    %%
    \node[anchor=south,V] (a) at (-4,-4) {$a$};
    \node[anchor=south,V] (b) at (-2,-4) {$b$};
    \node[anchor=south,V] (c) at (0,-4) {$c$};
    \node[anchor=south,V] (d) at (2,-4) {$d$};
    \node[anchor=south,V] (e) at (4,-4) {$e$};
    \node[anchor=south,V] (empty) at (0,-5) {$\emptyset$};
    %
    %% Change [f] to:
    %% [B] for blue
    %% [R] for red
    %% [F] for dashed grey
    %%
    \draw[f] (abcde) -- (abcd); \draw[f] (abcde) -- (abce); 
    \draw[f] (abcde) -- (abde);
    \draw[f] (abcde) -- (acde); \draw[f] (abcde) -- (bcde);
    %
    \draw[f] (abcd) -- (abc); \draw[f] (abcd) -- (abd); 
    \draw[f] (abcd) -- (acd); \draw[f] (abcd) -- (bcd);
    %
    \draw[f] (abce) -- (abc); \draw[f] (abce) -- (abe); 
    \draw[f] (abce) -- (ace); \draw[f] (abce) -- (bce);
    %
    \draw[f] (abde) -- (abd); \draw[f] (abde) -- (abe); 
    \draw[f] (abde) -- (ade); \draw[f] (abde) -- (bde);
    % 
    \draw[f] (acde) -- (acd); \draw[f] (acde) -- (ace); 
    \draw[f] (acde) -- (ade); \draw[f] (cde) -- (acde); 
    %
    \draw[f] (bcde) -- (bcd); \draw[f] (bcde) -- (bce); 
    \draw[f] (bcde) -- (bde); \draw[f] (cde) -- (bcde);
    %
    \draw[f] (abc) -- (ab); \draw[f] (abc) -- (ac); \draw[f] (abc) -- (bc); 
    \draw[f] (abd) -- (ab); \draw[f] (abd) -- (ad); \draw[f] (abd) -- (bd);
    \draw[f] (abe) -- (ab); \draw[f] (abe) -- (ae); \draw[f] (abe) -- (be);
    \draw[f] (acd) -- (ac); \draw[f] (acd) -- (ad); \draw[f] (acd) -- (cd); 
    \draw[f] (ace) -- (ac); \draw[f] (ace) -- (ae); \draw[f] (ace) -- (ce); 
    \draw[f] (ade) -- (ad); \draw[f] (ade) -- (ae); \draw[f] (ade) -- (de); 
    \draw[f] (bcd) -- (bc); \draw[f] (bcd) -- (bd); \draw[f] (bcd) -- (cd); 
    \draw[f] (bce) -- (bc); \draw[f] (bce) -- (be); \draw[f] (bce) -- (ce); 
    \draw[f] (bde) -- (bd); \draw[f] (bde) -- (be); \draw[f] (bde) -- (de); 
    \draw[f] (cde) -- (cd); \draw[f] (cde) -- (ce); \draw[f] (cde) -- (de); 
    % 
    \draw[f] (ab) -- (a); \draw[f] (ab) -- (b); 
    \draw[f] (ac) -- (a); \draw[f] (ac) -- (c); 
    \draw[f] (ad) -- (a); \draw[f] (ad) -- (d); 
    \draw[f] (ae) -- (a); \draw[f] (ae) -- (e); 
    \draw[f] (bc) -- (b); \draw[f] (bc) -- (c);
    \draw[f] (bd) -- (b); \draw[f] (bd) -- (d);
    \draw[f] (be) -- (b); \draw[f] (be) -- (e);
    \draw[f] (cd) -- (c); \draw[f] (cd) -- (d);
    \draw[f] (ce) -- (c); \draw[f] (ce) -- (e);
    \draw[f] (de) -- (d); \draw[f] (de) -- (e);
    %%
    \draw[f] (a) -- (empty); \draw[f] (b) -- (empty); \draw[f] (c) -- (empty);
    \draw[f] (d) -- (empty); \draw[f] (e) -- (empty);
  \end{tikzpicture}
  \]
  
  
\end{enumerate}

\end{document}
  
%%====================================================================
  
  

