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<head><title>koszul(ZZ,Matrix) -- a differential in a Koszul complex</title>
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<div><h1>koszul(ZZ,Matrix) -- a differential in a Koszul 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>g = koszul(i,f)</tt></div>
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<li><span>Function: <a href="_koszul.html" title="Koszul complex or specific matrix in the Koszul complex">koszul</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>i</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
<li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, a <tt>1</tt> by <tt>n</tt> matrix</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>g</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, the <tt>i</tt>-th differential in the Koszul complex of the matrix <tt>f</tt></span></li>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : R = QQ[x_1..x_4];</pre>
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<tr><td><pre>i2 : f = matrix{{x_1..x_4}}
o2 = | x_1 x_2 x_3 x_4 |
1 4
o2 : Matrix R <--- R</pre>
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To see the second differential in the Koszul complex of the matrix <tt>f</tt> look at:<table class="examples"><tr><td><pre>i3 : koszul(2,f)
o3 = {1} | -x_2 -x_3 0 -x_4 0 0 |
{1} | x_1 0 -x_3 0 -x_4 0 |
{1} | 0 x_1 x_2 0 0 -x_4 |
{1} | 0 0 0 x_1 x_2 x_3 |
4 6
o3 : Matrix R <--- R</pre>
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