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<head><title>CoComplex -- The class of all embedded co-complexes.</title>
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<div><h1>CoComplex -- The class of all embedded co-complexes.</h1>
<div class="single"><h2>Description</h2>
<div><p>The class of all embedded co-complexes, not necessarily simplicial.</p>
<p><b>Creating co-complexes:</b></p>
<p>The following functions return co-complexes:</p>
<p><a href="_ideal__To__Co__Complex.html" title="The co-complex associated to a reduced monomial ideal.">idealToCoComplex</a> -- The co-complex associated to a reduced monomial ideal</p>
<p><a href="_dualize.html" title="The dual of a face or complex.">dualize</a> -- The dual of a complex.</p>
<p><a href="../../Macaulay2Doc/html/_complement_lp__Matrix_rp.html" title="find the minimal generators for cokernel of a matrix (low level form)">complement</a> -- The complement of a complex.</p>
<p><a href="_co__Complex.html" title="Make a co-complex.">coComplex</a> -- Make a co-complex from a list of faces</p>
<p>For further examples see the documentation of these functions.</p>
<p></p>
<p><b>The data stored in a co-complex C:</b></p>
<p><i>C.simplexRing</i>, the polynomial ring of vertices of C (note these are only faces of C if C is a polytope).</p>
<p><i>C.grading</i>, is C.simplexRing.grading, a matrix with the coordinates of the vertices of C in its rows.</p>
<p><i>C.facets</i>, a list with the facets of C sorted into lists by dimension.</p>
<p><i>C.edim</i>, the embedding dimension of C, i.e., <a href="../../Macaulay2Doc/html/_rank.html" title="compute the rank">rank</a> <a href="../../Macaulay2Doc/html/_source.html" title="source of a map">source</a> C.grading.</p>
<p><i>C.dim</i>, the dimension of C, i.e., the minimal dimension of the faces.</p>
<p><i>C.isSimplicial</i>, a <a href="../../Macaulay2Doc/html/___Boolean.html" title="the class of Boolean values">Boolean</a> indicating whether C is simplicial.</p>
<p><i>C.isEquidimensional</i>, a <a href="../../Macaulay2Doc/html/___Boolean.html" title="the class of Boolean values">Boolean</a> indicating whether C is equidimensional.</p>
<p><i>C.fc</i>, a <a href="../../Macaulay2Doc/html/___Scripted__Functor.html" title="the class of all scripted functors">ScriptedFunctor</a> with the faces of C sorted and indexed by dimension.</p>
<p><i>C.fvector</i>, a <a href="../../Macaulay2Doc/html/___List.html" title="the class of all lists -- {...}">List</a> with the F-vector of C.</p>
<p>The following may be present (if known due to creation of C or due to calling some function):</p>
<p><i>C.dualComplex</i>, the dual complex of C in the sense of dual faces of a polytope. See <a href="_dualize.html" title="The dual of a face or complex.">dualize</a>.</p>
<p><i>C.isPolytope</i>, a <a href="../../Macaulay2Doc/html/___Boolean.html" title="the class of Boolean values">Boolean</a> indicating whether C is a polytope.</p>
<p><i>C.polytopalFacets</i>, a <a href="../../Macaulay2Doc/html/___List.html" title="the class of all lists -- {...}">List</a> with the boundary faces of the polytope C.</p>
<p><i>C.complementComplex</i>, the complement complex of C (if C is a subcocomplex of a simplex). See <a href="../../Macaulay2Doc/html/_complement_lp__Matrix_rp.html" title="find the minimal generators for cokernel of a matrix (low level form)">complement</a>.</p>
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<table class="examples"><tr><td><pre>i1 : R=QQ[x_0..x_5]
o1 = R
o1 : PolynomialRing</pre>
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<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>
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<tr><td><pre>i3 : C.simplexRing
o3 = R
o3 : PolynomialRing</pre>
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<tr><td><pre>i4 : C.grading
o4 = | -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
o4 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i5 : C.fc_2
o5 = {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
o5 : List</pre>
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<tr><td><pre>i6 : C.facets
o6 = {{}, {}, {}, {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
------------------------------------------------------------------------
x x x }, {}, {}, {}}
3 4 5
o6 : List</pre>
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<tr><td><pre>i7 : dC=dualize C
o7 = 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
o7 : 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>
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<tr><td><pre>i8 : cC=complement C
o8 = 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
3 4 5 1 4 5 2 1 5 2 3 4 0 3 4 1 0 4 2 1 3 2 1 0
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>
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<tr><td><pre>i9 : dualize cC
o9 = 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
3 4 5 2 3 5 1 2 5 0 4 5 0 3 4 0 2 3 0 1 5 0 1 2
o9 : complex of dim 2 embedded in dim 5 (printing facets)
equidimensional, simplicial, F-vector {1, 6, 12, 8, 0, 0, 0}, Euler = 1</pre>
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<div class="single"><h2>Caveat</h2>
<div><div>So far a co-complex is of class complex and the methods checks of which type it really is. At some point both will have a common ancestor.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Complex.html" title="The class of all embedded complexes.">Complex</a> -- The class of all embedded complexes.</span></li>
<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>
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<div class="waystouse"><h2>Methods that use an embedded co-complex :</h2>
<ul><li><span>coComplexToIdeal(CoComplex), see <span><a href="_co__Complex__To__Ideal.html" title="The monomial ideal associated to a CoComplex.">coComplexToIdeal</a> -- The monomial ideal associated to a CoComplex.</span></span></li>
</ul>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Co__Complex.html" title="The class of all embedded co-complexes.">CoComplex</a> is <span>a <a href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a href="___Complex.html" title="The class of all embedded complexes.">Complex</a> < <a href="../../Macaulay2Doc/html/___Mutable__Hash__Table.html" title="the class of all mutable hash tables">MutableHashTable</a> < <a href="../../Macaulay2Doc/html/___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="../../Macaulay2Doc/html/___Thing.html" title="the class of all things">Thing</a>.</p>
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