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<head><title>janetResolution -- construct a free resolution for a given ideal or module using Janet bases</title>
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<div><h1>janetResolution -- construct a free resolution for a given ideal or module using Janet bases</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>C = janetResolution M</tt></div>
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<li><div class="single">Inputs:<ul><li><span>M, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span> or <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span> or <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>C</tt>, <span>a <a href="../../Macaulay2Doc/html/___Chain__Complex.html">chain complex</a></span>, a (non-minimal) free resolution of (the module generated by) M</span></li>
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<div class="single"><h2>Description</h2>
<div><p>The computed Janet basis for each homological degree can be extracted with <a href="_janet__Basis.html" title="compute Janet basis for an ideal or a submodule of a free module">janetBasis</a>.</p>
<p>The sets of multiplicative variables can also be extracted from the Janet basis in each homological degree with <a href="_mult__Var.html" title="extract the sets of multiplicative variables for each generator (in several contexts)">multVar</a>.</p>
<p>Note that janetResolution can be combined with <a href="../../Macaulay2Doc/html/_resolution.html" title="projective resolution">resolution</a>: when providing the option 'Strategy => Involutive' to <a href="../../Macaulay2Doc/html/_resolution.html" title="projective resolution">resolution</a>, janetResolution constructs the resolution.</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : M = matrix {{x,y,z}};
1 3
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : C = janetResolution M
1 3 3 1
o3 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o3 : ChainComplex</pre>
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<tr><td><pre>i4 : janetBasis(C, 2)
+----------+---------+
o4 = |{1} | -y ||{z, y, x}|
|{1} | x || |
|{1} | 0 || |
+----------+---------+
|{1} | -z ||{z, y, x}|
|{1} | 0 || |
|{1} | x || |
+----------+---------+
|{1} | 0 ||{z, y} |
|{1} | -z || |
|{1} | y || |
+----------+---------+
o4 : InvolutiveBasis</pre>
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<tr><td><pre>i5 : multVar(C, 2)
o5 = {set {x, y, z}, set {x, y, z}, set {y, z}}
o5 : List</pre>
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<table class="examples"><tr><td><pre>i6 : R = QQ[x,y,z];</pre>
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<tr><td><pre>i7 : I = ideal(x,y,z);
o7 : Ideal of R</pre>
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<tr><td><pre>i8 : res(I, Strategy => Involutive)
1 3 3 1
o8 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o8 : ChainComplex</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_janet__Basis.html" title="compute Janet basis for an ideal or a submodule of a free module">janetBasis</a> -- compute Janet basis for an ideal or a submodule of a free module</span></li>
<li><span><a href="_mult__Var.html" title="extract the sets of multiplicative variables for each generator (in several contexts)">multVar</a> -- extract the sets of multiplicative variables for each generator (in several contexts)</span></li>
<li><span><a href="_inv__Syzygies.html" title="compute involutive basis of syzygies">invSyzygies</a> -- compute involutive basis of syzygies</span></li>
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<div class="waystouse"><h2>Ways to use <tt>janetResolution</tt> :</h2>
<ul><li>janetResolution(Ideal)</li>
<li>janetResolution(InvolutiveBasis)</li>
<li>janetResolution(Matrix)</li>
<li>janetResolution(Module)</li>
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