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<head><title>symExt -- the first differential of the complex R(M)</title>
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<div><h1>symExt -- the first differential of the complex R(M)</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>symExt(m,E)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>m</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, a presentation matrix for a positively graded module M over a polynomial ring</span></li>
<li><span><tt>E</tt>, <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span>, exterior algebra</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, a matrix representing the map <tt>M_1 ** omega_E <-- M_0 ** omega_E</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>This function takes as input a matrix <tt>m</tt> with linear entries, which we think of as a presentation matrix for a positively graded <tt>S</tt>-module <tt>M</tt> matrix representing the map <tt>M_1 ** omega_E <-- M_0 ** omega_E</tt> which is the first differential of the complex <tt>R(M)</tt>.<table class="examples"><tr><td><pre>i1 : S = ZZ/32003[x_0..x_2]; </pre>
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<tr><td><pre>i2 : E = ZZ/32003[e_0..e_2, SkewCommutative=>true];</pre>
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<tr><td><pre>i3 : M = coker matrix {{x_0^2, x_1^2}};</pre>
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<tr><td><pre>i4 : m = presentation truncate(regularity M,M);
4 8
o4 : Matrix S <--- S</pre>
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<tr><td><pre>i5 : symExt(m,E)
o5 = {-1} | e_2 e_1 e_0 0 |
{-1} | 0 e_2 0 e_0 |
{-1} | 0 0 e_2 e_1 |
{-1} | 0 0 0 e_2 |
4 4
o5 : Matrix E <--- E</pre>
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<div class="single"><h2>Caveat</h2>
<div>This function is a quick-and-dirty tool which requires little computation. However if it is called on two successive truncations of a module, then the maps it produces may NOT compose to zero because the choice of bases is not consistent.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_bgg.html" title="the ith differential of the complex R(M)">bgg</a> -- the ith differential of the complex R(M)</span></li>
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<div class="waystouse"><h2>Ways to use <tt>symExt</tt> :</h2>
<ul><li>symExt(Matrix,PolynomialRing)</li>
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