<?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>coComplex -- Make a co-complex.</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_co__Complex__To__Ideal.html">next</a> | <a href="___Co__Complex.html">previous</a> | <a href="_co__Complex__To__Ideal.html">forward</a> | <a href="___Co__Complex.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>coComplex -- Make a co-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>coComplex(R,facelist)</tt><br/><tt>coComplex(R,facelist,facetlist)</tt><br/><tt>coComplex(R,facelist,Rdual)</tt><br/><tt>coComplex(R,facelist,facetlist,Rdual)</tt></div> </dd></dl> </div> </li> <li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span></span></li> <li><span><tt>facelist</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, of the faces, sorted by dimension</span></li> <li><span><tt>facetlist</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, of the facets, sorted by dimension</span></li> <li><span><tt>Rdual</tt>, <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>an <a href="___Co__Complex.html">embedded co-complex</a></span></span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div><p>Make a co-complex from a list of faces and/or facets.</p> <p>This is mostly used internally but may be occasionally useful for the end user.</p> <div/> <table class="examples"><tr><td><pre>i1 : R=QQ[x_0..x_5] o1 = R o1 : PolynomialRing</pre> </td></tr> <tr><td><pre>i2 : C=boundaryCyclicPolytope(3,R) o2 = 2: x x x x x x x x x x x x x x x x x x x x x x x x 0 1 2 0 2 3 0 3 4 0 1 5 1 2 5 2 3 5 0 4 5 3 4 5 o2 : complex of dim 2 embedded in dim 5 (printing facets) equidimensional, simplicial, F-vector {1, 6, 12, 8, 0, 0, 0}, Euler = 1</pre> </td></tr> <tr><td><pre>i3 : grading R o3 = | -1 -1 -1 -1 -1 | | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 1 0 0 | | 0 0 0 1 0 | | 0 0 0 0 1 | 6 5 o3 : Matrix ZZ <--- ZZ</pre> </td></tr> <tr><td><pre>i4 : dC=dualize C o4 = 2: v v v v v v v v v v v v v v v v v v v v v v v v 0 1 2 0 1 4 0 3 4 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 o4 : co-complex of dim 2 embedded in dim 5 (printing facets) equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1</pre> </td></tr> <tr><td><pre>i5 : fdC=fc dC o5 = {{}, {}, {}, {v v v , v v v , v v v , v v v , v v v , v v v , v v v , 0 1 2 0 1 4 0 3 4 1 2 3 1 2 5 1 4 5 2 3 4 ------------------------------------------------------------------------ v v v }, {v v v v , v v v v , v v v v , v v v v , v v v v , v v v v , 3 4 5 0 1 2 3 0 1 2 4 0 1 2 5 0 1 3 4 0 1 4 5 0 2 3 4 ------------------------------------------------------------------------ v v v v , v v v v , v v v v , v v v v , v v v v , v v v v }, 0 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 ------------------------------------------------------------------------ {v v v v v , v v v v v , v v v v v , v v v v v , v v v v v , 0 1 2 3 4 0 1 2 3 5 0 1 2 4 5 0 1 3 4 5 0 2 3 4 5 ------------------------------------------------------------------------ v v v v v }, {v v v v v v }} 1 2 3 4 5 0 1 2 3 4 5 o5 : List</pre> </td></tr> <tr><td><pre>i6 : Rdual=simplexRing dC o6 = Rdual o6 : PolynomialRing</pre> </td></tr> <tr><td><pre>i7 : grading Rdual o7 = | -1 -1 -1 -1 5 | | -1 -1 -1 5 -1 | | -1 -1 5 -1 -1 | | -1 5 -1 -1 -1 | | 5 -1 -1 -1 -1 | | -1 -1 -1 -1 -1 | 6 5 o7 : Matrix QQ <--- QQ</pre> </td></tr> <tr><td><pre>i8 : dC1=coComplex(Rdual,fdC) o8 = 2: v v v v v v v v v v v v v v v v v v v v v v v v 0 1 2 0 1 4 0 3 4 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 o8 : co-complex of dim 2 embedded in dim 5 (printing facets) equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1</pre> </td></tr> <tr><td><pre>i9 : dC==dC1 o9 = true</pre> </td></tr> </table> </div> </div> <div class="single"><h2>Caveat</h2> <div><div>If both the list of faces and facets is specified there is no consistency check.</div> </div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="___Face.html" title="The class of all faces of complexes or co-complexes.">Face</a> -- The class of all faces of complexes or co-complexes.</span></li> </ul> </div> <div class="waystouse"><h2>Ways to use <tt>coComplex</tt> :</h2> <ul><li>coComplex(PolynomialRing,List)</li> <li>coComplex(PolynomialRing,List,List)</li> <li>coComplex(PolynomialRing,List,List,PolynomialRing)</li> <li>coComplex(PolynomialRing,List,PolynomialRing)</li> </ul> </div> </div> </body> </html>