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<head><title>simplicialComplex -- create a simplicial complex</title>
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<div><h1>simplicialComplex -- create 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>simplicialComplex I</tt><br/><tt>simplicialComplex L</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>a <a href="../../Macaulay2Doc/html/___Monomial__Ideal.html">monomial ideal</a></span>, the ideal of minimal nonfaces (Stanley-Reisner ideal)</span></li>
<li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of monomials representing the facets</span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Simplicial__Complex.html">simplicial complex</a></span>, the simplicial complex determined by the nonfaces <tt>I</tt> or generated by the given faces <tt>L</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>A simplicial complex on a set of vertices is a collection of subsets <tt>D</tt> of these vertices, such that if <tt>F</tt> is in <tt>D</tt>, then every subset of <tt>F</tt> is also in <tt>D</tt>. In Macaulay2, the vertices are variables in a polynomial ring, and each subset is represented as a product of the corresponding variables.<p/>
A simplicial complex is determined either by its nonfaces or by its faces. The monomials corresponding to the nonfaces are a basis of an ideal, called the Stanley-Reisner ideal, and it suffices to specify the minimal nonfaces, which generate the ideal. The monomials corresponding to the faces do not form the basis of an ideal, but it suffices to specify the maximal faces, which are called <em>facets</em>. The function <a href="_simplicial__Complex.html" title="create a simplicial complex">simplicialComplex</a> accepts either the ideal of nonfaces or the list of facets as input.<p/>
In our first example we construct the octahedron by specfying its ideal of nonfaces.<table class="examples"><tr><td><pre>i1 : R = ZZ[a..f];</pre>
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<tr><td><pre>i2 : I = monomialIdeal(a*f, b*d, c*e);
o2 : MonomialIdeal of R</pre>
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<tr><td><pre>i3 : Octahedron = simplicialComplex I
o3 = | def bef cdf bcf ade abe acd abc |
o3 : SimplicialComplex</pre>
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Note that <span>a <a href="___Simplicial__Complex.html">simplicial complex</a></span> is displayed by showing its facets. We see that there are eight facets to the octahedron. Alternatively, we could have defined the octahedron by this list of facets.<table class="examples"><tr><td><pre>i4 : L = {d*e*f, b*e*f, c*d*f, b*c*f,
a*d*e, a*b*e, a*c*d, a*b*c}
o4 = {d*e*f, b*e*f, c*d*f, b*c*f, a*d*e, a*b*e, a*c*d, a*b*c}
o4 : List</pre>
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<tr><td><pre>i5 : Octahedron' = simplicialComplex L
o5 = | def bef cdf bcf ade abe acd abc |
o5 : SimplicialComplex</pre>
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<tr><td><pre>i6 : Octahedron == Octahedron'
o6 = true</pre>
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<tr><td><pre>i7 : fVector Octahedron
o7 = HashTable{-1 => 1}
0 => 6
1 => 12
2 => 8
o7 : HashTable</pre>
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There are two "trivial" simplicial complexes: the void complex and the irrelevant complex. The void complex has no faces. This complex cannot be constructed from its facets, since it has none.<table class="examples"><tr><td><pre>i8 : void = simplicialComplex monomialIdeal 1_R
o8 = 0
o8 : SimplicialComplex</pre>
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<tr><td><pre>i9 : fVector void
o9 = HashTable{-1 => 0}
o9 : HashTable</pre>
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<tr><td><pre>i10 : dim void
o10 = -infinity
o10 : InfiniteNumber</pre>
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The irrelevant complex, which should be distinguished from the void complex, has a unique face of dimension -1, the empty set.<table class="examples"><tr><td><pre>i11 : irrelevant = simplicialComplex monomialIdeal gens R
o11 = | 1 |
o11 : SimplicialComplex</pre>
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<tr><td><pre>i12 : fVector irrelevant
o12 = HashTable{-1 => 1}
o12 : HashTable</pre>
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<tr><td><pre>i13 : dim irrelevant
o13 = -1</pre>
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<tr><td><pre>i14 : irrelevant' = simplicialComplex {1_R}
o14 = | 1 |
o14 : SimplicialComplex</pre>
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<tr><td><pre>i15 : irrelevant' == irrelevant
o15 = true</pre>
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As in Miller-Sturmfels, Combinatorial Commutative Algebra, we would avoid making such a big deal about the difference between these complexes if it did not come up so much. Many formulas for betti numbers, dimensions of local cohomology, etc., depend on this distinction.</div>
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<div class="single"><h2>Caveat</h2>
<div><ul><li>The ring of <tt>I</tt> or <tt>L</tt> must be a polynomial ring.</li>
<li>The <a href="../../Macaulay2Doc/html/_coefficient__Ring.html">coefficient ring</a> matters: for instance, it is used when computing the corresponding chain complex.</li>
<li>This construction routine does some computation, as it computes both the facets and the ideal of non-faces.</li>
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<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="_f__Vector.html" title="the f-vector of a simplicial complex">fVector</a> -- the f-vector of a simplicial complex</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>simplicialComplex</tt> :</h2>
<ul><li>simplicialComplex(List)</li>
<li>simplicialComplex(MonomialIdeal)</li>
</ul>
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