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<head><title>boundary(ZZ,SimplicialComplex) -- the boundary map from i-faces to (i-1)-faces</title>
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<div><h1>boundary(ZZ,SimplicialComplex) -- the boundary map from i-faces to (i-1)-faces</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>M = boundary(i,D)</tt></div>
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<li><span>Function: <a href="_boundary.html" title="boundary operator">boundary</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>i</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span></span></li>
<li><span><tt>D</tt>, <span>a <a href="___Simplicial__Complex.html">simplicial complex</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, the boundary map from <tt>i</tt>-faces to <tt>(i-1)</tt>-faces of <tt>D</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>The columns of the matrix <tt>M</tt> are indexed by the <tt>i</tt>-faces of <tt>D</tt>, and the rows are indexed by the <tt>(i-1)</tt>-faces, in the order given by <a href="_faces.html" title="the i-faces of a simplicial complex ">faces</a>. <tt>M</tt> is defined over the <a href="_coefficient__Ring_lp__Simplicial__Complex_rp.html">coefficient ring</a> of <tt>D</tt>.<table class="examples"><tr><td><pre>i1 : loadPackage "SimplicialComplexes";</pre>
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The boundary maps for the standard 3-simplex, defined over <tt>ZZ</tt>.<table class="examples"><tr><td><pre>i2 : R = ZZ[a..d];</pre>
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<tr><td><pre>i3 : D = simplicialComplex {a*b*c*d}
o3 = | abcd |
o3 : SimplicialComplex</pre>
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<tr><td><pre>i4 : boundary(0,D)
o4 = | 1 1 1 1 |
1 4
o4 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i5 : faces(0,D)
o5 = | a b c d |
1 4
o5 : Matrix R <--- R</pre>
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<tr><td><pre>i6 : boundary(1,D)
o6 = | -1 -1 -1 0 0 0 |
| 1 0 0 -1 -1 0 |
| 0 1 0 1 0 -1 |
| 0 0 1 0 1 1 |
4 6
o6 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i7 : faces(1,D)
o7 = | ab ac ad bc bd cd |
1 6
o7 : Matrix R <--- R</pre>
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<tr><td><pre>i8 : boundary(2,D)
o8 = | 1 1 0 0 |
| -1 0 1 0 |
| 0 -1 -1 0 |
| 1 0 0 1 |
| 0 1 0 -1 |
| 0 0 1 1 |
6 4
o8 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i9 : faces(2,D)
o9 = | abc abd acd bcd |
1 4
o9 : Matrix R <--- R</pre>
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<tr><td><pre>i10 : boundary(3,D)
o10 = | -1 |
| 1 |
| -1 |
| 1 |
4 1
o10 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i11 : faces(3,D)
o11 = | abcd |
1 1
o11 : Matrix R <--- R</pre>
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<tr><td><pre>i12 : boundary(4,D)
o12 = 0
1
o12 : Matrix ZZ <--- 0</pre>
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The boundary maps depend on the <a href="_coefficient__Ring_lp__Simplicial__Complex_rp.html">coefficient ring</a> as the following examples illustrate.<table class="examples"><tr><td><pre>i13 : R = QQ[a..f];</pre>
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<tr><td><pre>i14 : 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);</pre>
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<tr><td><pre>i15 : boundary(1,D)
o15 = | -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 |
| 1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 |
| 0 1 0 0 0 1 0 0 0 -1 -1 -1 0 0 0 |
| 0 0 1 0 0 0 1 0 0 1 0 0 -1 -1 0 |
| 0 0 0 1 0 0 0 1 0 0 1 0 1 0 -1 |
| 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 |
6 15
o15 : Matrix QQ <--- QQ</pre>
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<tr><td><pre>i16 : R' = ZZ/2[a..f];</pre>
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<tr><td><pre>i17 : 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);</pre>
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<tr><td><pre>i18 : boundary(1,D')
o18 = | 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 |
| 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 |
| 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 |
| 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 |
| 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 |
| 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 |
ZZ 6 ZZ 15
o18 : Matrix (--) <--- (--)
2 2</pre>
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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><tt>chainComplex(SimplicialComplex)</tt> (missing documentation<!-- tag: (chainComplex,SimplicialComplex) -->)</span></li>
<li><span><a href="_faces.html" title="the i-faces of a simplicial complex ">faces</a> -- the i-faces of a simplicial complex </span></li>
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