<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>dual(SimplicialComplex) -- the Alexander dual of a simplicial complex</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_faces.html">next</a> | <a href="_dim_lp__Simplicial__Complex_rp.html">previous</a> | <a href="_faces.html">forward</a> | <a href="_dim_lp__Simplicial__Complex_rp.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>dual(SimplicialComplex) -- the Alexander dual of a simplicial complex</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>dual D</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="../../Macaulay2Doc/html/_dual.html" title="dual module or map">dual</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>D</tt>, <span>a <a href="___Simplicial__Complex.html">simplicial complex</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Simplicial__Complex.html">simplicial complex</a></span>, the Alexander dual of <tt>D</tt></span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>The Alexander dual of <tt>D</tt> is the simplicial complex whose faces are the complements of the nonfaces of <tt>D</tt>. The Alexander dual of a square is the disjoint union of two edges.<table class="examples"><tr><td><pre>i1 : R = ZZ[a..d];</pre>
</td></tr>
<tr><td><pre>i2 : D = simplicialComplex {a*b,b*c,c*d,d*a}
o2 = | cd ad bc ab |
o2 : SimplicialComplex</pre>
</td></tr>
<tr><td><pre>i3 : dual D
o3 = | bd ac |
o3 : SimplicialComplex</pre>
</td></tr>
</table>
<p/>
The Alexander dual is homotopy equivalent to the complement of <tt>D</tt> in the sphere generated by all of the variables in the <a href="_ring_lp__Simplicial__Complex_rp.html">ring</a> of <tt>D</tt>. In particular, it depends on the number of variables.<table class="examples"><tr><td><pre>i4 : R = ZZ[a..e]
o4 = R
o4 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i5 : E = simplicialComplex {a*b,b*c,c*d,d*a}
o5 = | cd ad bc ab |
o5 : SimplicialComplex</pre>
</td></tr>
<tr><td><pre>i6 : dual E
o6 = | bde ace abcd |
o6 : SimplicialComplex</pre>
</td></tr>
</table>
The projective dimension of the face ring of D equals the regularity of the face ideal of the Alexander dual of D see e.g., Corollary 5.59 of Miller-Sturmfels, Combinatorial Commutative Algebra.<table class="examples"><tr><td><pre>i7 : R = QQ[a..f];</pre>
</td></tr>
<tr><td><pre>i8 : D = simplicialComplex monomialIdeal(a*b*c,a*b*f,a*c*e,a*d*e,a*d*f,b*c*d,b*d*e,b*e*f,c*d*f,c*e*f)
o8 = | def aef bdf bcf acf cde bce abe acd abd |
o8 : SimplicialComplex</pre>
</td></tr>
<tr><td><pre>i9 : A = dual D
o9 = | def aef bdf bcf acf cde bce abe acd abd |
o9 : SimplicialComplex</pre>
</td></tr>
<tr><td><pre>i10 : pdim (R^1/(ideal D))
o10 = 3</pre>
</td></tr>
<tr><td><pre>i11 : regularity ideal A
o11 = 3</pre>
</td></tr>
</table>
<p/>
Alexander duality interchanges extremal betti numbers of the face ideals. Following example 3.2 in Bayer-Charalambous-Popescu, <em>Extremal betti numbers and applications to monomial ideals</em>, we have <table class="examples"><tr><td><pre>i12 : R = QQ[x0,x1,x2,x3,x4,x5,x6];</pre>
</td></tr>
<tr><td><pre>i13 : D = simplicialComplex {x0*x1*x3, x1*x3*x4, x1*x2*x4, x2*x4*x5,
x2*x3*x5, x3*x5*x6, x3*x4*x6, x0*x4*x6,
x0*x4*x5, x0*x1*x5, x1*x5*x6, x1*x2*x6,
x0*x2*x6, x0*x2*x3}
o13 = | x3x5x6 x1x5x6 x3x4x6 x0x4x6 x1x2x6 x0x2x6 x2x4x5 x0x4x5 x2x3x5 x0x1x5 x1x3x4 x1x2x4 x0x2x3 x0x1x3 |
o13 : SimplicialComplex</pre>
</td></tr>
<tr><td><pre>i14 : I = ideal D
o14 = ideal (x0*x1*x2, x1*x2*x3, x0*x1*x4, x0*x2*x4, x0*x3*x4, x2*x3*x4,
-----------------------------------------------------------------------
x0*x2*x5, x1*x2*x5, x0*x3*x5, x1*x3*x5, x1*x4*x5, x3*x4*x5, x0*x1*x6,
-----------------------------------------------------------------------
x0*x3*x6, x1*x3*x6, x2*x3*x6, x1*x4*x6, x2*x4*x6, x0*x5*x6, x2*x5*x6,
-----------------------------------------------------------------------
x4*x5*x6)
o14 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i15 : J = ideal dual D
o15 = ideal (x0*x1*x2*x4, x0*x2*x3*x4, x0*x1*x2*x5, x1*x2*x3*x5, x0*x3*x4*x5,
-----------------------------------------------------------------------
x1*x3*x4*x5, x0*x1*x3*x6, x1*x2*x3*x6, x0*x1*x4*x6, x2*x3*x4*x6,
-----------------------------------------------------------------------
x0*x2*x5*x6, x0*x3*x5*x6, x1*x4*x5*x6, x2*x4*x5*x6)
o15 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i16 : betti res I
0 1 2 3 4 5
o16 = total: 1 21 49 42 15 2
0: 1 . . . . .
1: . . . . . .
2: . 21 49 42 14 2
3: . . . . 1 .
o16 : BettiTally</pre>
</td></tr>
<tr><td><pre>i17 : betti res J
0 1 2 3 4
o17 = total: 1 14 21 9 1
0: 1 . . . .
1: . . . . .
2: . . . . .
3: . 14 21 7 1
4: . . . 2 .
o17 : BettiTally</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="index.html" title="simplicial complexes">SimplicialComplexes</a> -- simplicial complexes</span></li>
<li><span><a href="../../Macaulay2Doc/html/_dual_lp__Monomial__Ideal_rp.html" title="the Alexander dual of a monomial ideal">dual(MonomialIdeal)</a> -- the Alexander dual of a monomial ideal</span></li>
</ul>
</div>
</div>
</body>
</html>
