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<head><title>computing syzygies</title>
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<div><h1>computing syzygies</h1>
<div>A syzygy among the columns of a matrix is, by definition, an element of the kernel of the corresponding map between free modules, and the easiest way to compute the syzygies applying the function <a href="_kernel.html" title="kernel of a ringmap, matrix, or chain complex">kernel</a>.<table class="examples"><tr><td><pre>i1 : R = QQ[x..z];</pre>
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<tr><td><pre>i2 : f = vars R
o2 = | x y z |
1 3
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : K = kernel f
o3 = image {1} | -y 0 -z |
{1} | x -z 0 |
{1} | 0 y x |
3
o3 : R-module, submodule of R</pre>
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The answer is provided as a submodule of the source of <tt>f</tt>. The function <a href="_super.html" title="get the ambient module">super</a> can be used to produce the module that <tt>K</tt> is a submodule of; indeed, this works for any module.<table class="examples"><tr><td><pre>i4 : L = super K
3
o4 = R
o4 : R-module, free, degrees {1, 1, 1}</pre>
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<tr><td><pre>i5 : L == source f
o5 = true</pre>
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The matrix whose columns are the generators of <tt>K</tt>, lifted to the ambient free module of <tt>L</tt> if necessary, can be obtained with the function <a href="_generators.html" title="provide matrix or list of generators">generators</a>, an abbreviation for which is <tt>gens</tt>.<table class="examples"><tr><td><pre>i6 : g = generators K
o6 = {1} | -y 0 -z |
{1} | x -z 0 |
{1} | 0 y x |
3 3
o6 : Matrix R <--- R</pre>
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We can check at least that the columns of <tt>g</tt> are syzygies of the columns of <tt>f</tt> by checking that <tt>f*g</tt> is zero.<table class="examples"><tr><td><pre>i7 : f*g
o7 = 0
1 3
o7 : Matrix R <--- R</pre>
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<tr><td><pre>i8 : f*g == 0
o8 = true</pre>
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Use the function <a href="_syz.html" title="the syzygy matrix">syz</a> if you need detailed control over the extent of the computation.</div>
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