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<head><title>orlikSolomon -- defining ideal for the Orlik-Solomon algebra</title>
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<div><h1>orlikSolomon -- defining ideal for the Orlik-Solomon algebra</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>orlikSolomon(A) or orlikSolomon(A,E) or orlikSolomon(A,e)</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>A</tt>, <span>a <a href="___Arrangement.html">hyperplane arrangement</a></span>, an arrangement</span></li>
<li><span><tt>E</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, a skew-commutative polynomial ring with one variable for each hyperplane</span></li>
<li><span><tt>e</tt>, <span>a <a href="../../Macaulay2Doc/html/___Symbol.html">symbol</a></span>, a name for an indexed variable</span></li>
</ul>
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</li>
<li><div class="single">Consequences:<ul><li>the list of <a href="_circuits.html" title="list the circuits of an arrangement">circuits</a> of <tt>A</tt> is cached.</li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the defining ideal of the Orlik-Solomon algebra of <tt>A</tt></span></li>
</ul>
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</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_orlik__Solomon_lp..._cm_sp__Hyp__At__Infinity_sp_eq_gt_sp..._rp.html">HypAtInfinity => ...</a>, -- hyperplane at infinity</span></li>
<li><span><a href="_orlik__Solomon_lp..._cm_sp__Projective_sp_eq_gt_sp..._rp.html">Projective => ...</a>, -- specify projective complement</span></li>
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<div class="single"><h2>Description</h2>
<div>The Orlik-Solomon algebra is the cohomology ring of the complement of the hyperplanes, either in complex projective or affine space. The optional Boolean argument <tt>Projective</tt> specifies which. The code for this method was written by Sorin Popescu.<table class="examples"><tr><td><pre>i1 : A = typeA(3)
o1 = {x - x , x - x , x - x , x - x , x - x , x - x }
1 2 1 3 1 4 2 3 2 4 3 4
o1 : Hyperplane Arrangement </pre>
</td></tr>
<tr><td><pre>i2 : I = orlikSolomon(A,e)
o2 = ideal (e e - e e + e e , e e - e e + e e , e e - e e + e e , e e
4 5 4 6 5 6 2 3 2 6 3 6 1 3 1 5 3 5 1 2
------------------------------------------------------------------------
- e e + e e )
1 4 2 4
o2 : Ideal of QQ[e , e , e , e , e , e ]
1 2 3 4 5 6</pre>
</td></tr>
<tr><td><pre>i3 : reduceHilbert hilbertSeries I
2 3
1 + 6T + 11T + 6T
o3 = -------------------
1
o3 : Expression of class Divide</pre>
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<tr><td><pre>i4 : I' = orlikSolomon(A,Projective=>true,HypAtInfinity=>2)
o4 = ideal (e e - e e + e e , e e - e e + e e , e e - e e + e e , e e
4 5 4 6 5 6 2 3 2 6 3 6 1 3 1 5 3 5 1 2
------------------------------------------------------------------------
- e e + e e , e )
1 4 2 4 3
o4 : Ideal of QQ[e , e , e , e , e , e ]
1 2 3 4 5 6</pre>
</td></tr>
<tr><td><pre>i5 : reduceHilbert hilbertSeries I'
2
1 + 5T + 6T
o5 = ------------
1
o5 : Expression of class Divide</pre>
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The code for <tt>orlikSolomon</tt> was contributed by Sorin Popescu.</div>
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<div class="waystouse"><h2>Ways to use <tt>orlikSolomon</tt> :</h2>
<ul><li>orlikSolomon(Arrangement)</li>
<li>orlikSolomon(Arrangement,Ring)</li>
<li>orlikSolomon(Arrangement,Symbol)</li>
<li><span><tt>orlikSolomon(Arrangement,PolynomialRing)</tt> (missing documentation<!-- tag: (orlikSolomon,Arrangement,PolynomialRing) -->)</span></li>
</ul>
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