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<head><title>EPY -- compute the Eisenbud-Popescu-Yuzvinsky module of an arrangement</title>
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<div><h1>EPY -- compute the Eisenbud-Popescu-Yuzvinsky module of an arrangement</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>EPY(A) or EPY(A,S) or EPY(I) or EPY(I,S)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>A</tt>, <span>a <a href="___Arrangement.html">hyperplane arrangement</a></span>, an arrangement of n hyperplanes</span></li>
<li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, an ideal of the exterior algebra, the quotient by which has a linear, injective resolution</span></li>
<li><span><tt>S</tt>, <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span>, an optional polynomial ring in n variables</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, The Eisenbud-Popescu-Yuzvinsky module (see below) of <tt>I</tt> or, if an arrangement is given, of its Orlik-Solomon ideal.</span></li>
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<div class="single"><h2>Description</h2>
<div>Let <tt>OS</tt> denote the <a href="_orlik__Solomon.html">Orlik-Solomon algebra</a> of the arrangement <tt>A</tt>, regarded as a quotient of an exterior algebra <tt>E</tt>. The module <tt>EPY(A)</tt> is, by definition, the <tt>S</tt>-module which is BGG-dual to the linear, injective resolution of <tt>OS</tt> as an <tt>E</tt>-module.<p/>
Equivalently, <tt>EPY(A)</tt> is the single nonzero cohomology module in the Aomoto complex of <tt>A</tt>. For details, see Eisenbud-Popescu-Yuzvinsky, [TAMS 355 (2003), no 11, 4365--4383].<table class="examples"><tr><td><pre>i1 : R = QQ[x,y];</pre>
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<tr><td><pre>i2 : FA = EPY arrangement {x,y,x-y}
o2 = cokernel | -X_1 X_1+X_2+X_3 X_1 |
| X_2 0 X_1+X_3 |
2
o2 : QQ[X , X , X ]-module, quotient of (QQ[X , X , X ])
1 2 3 1 2 3</pre>
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<tr><td><pre>i3 : betti res FA
0 1 2
o3 = total: 2 3 1
0: 2 3 1
o3 : BettiTally</pre>
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In particular, <tt>EPY(A)</tt> has a linear free resolution over the polynomial ring, namely the Aomoto complex of <tt>A</tt>.<table class="examples"><tr><td><pre>i4 : A = typeA(4)
o4 = {x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x }
1 2 1 3 1 4 1 5 2 3 2 4 2 5 3 4 3 5 4 5
o4 : Hyperplane Arrangement </pre>
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<tr><td><pre>i5 : factor poincare A
o5 = (1 + T)(1 + 2T)(1 + 3T)(1 + 4T)
o5 : Expression of class Product</pre>
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<tr><td><pre>i6 : betti res EPY A
0 1 2 3 4
o6 = total: 24 50 35 10 1
0: 24 50 35 10 1
o6 : BettiTally</pre>
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<div class="waystouse"><h2>Ways to use <tt>EPY</tt> :</h2>
<ul><li>EPY(Arrangement)</li>
<li>EPY(Arrangement,PolynomialRing)</li>
<li>EPY(Ideal)</li>
<li>EPY(Ideal,PolynomialRing)</li>
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