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<head><title>brunsIdeal -- Returns an ideal generated by three elements whose 2nd syzygy module agrees with the given ideal</title>
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<div><h1>brunsIdeal -- Returns an ideal generated by three elements whose 2nd syzygy module agrees with the given ideal</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>j = brunsIdeal i</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>i</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, a homogeneous ideal</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>j</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, a homogeneous ideal generated by three elements whose second syzygy module is isomorphic the second syzygy module of the ideal i.</span></li>
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<div class="single"><h2>Description</h2>
<div><div>This function is a special case of the function <a href="_bruns.html" title="Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module">bruns</a>. Given an ideal, the user can find another ideal which is 3-generated, and furthermore, the second syzygy modules of both ideals are isomorphic. Although one can use <a href="_bruns.html" title="Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module">bruns</a> to do this procedure, this function cuts out some of the steps.</div>
<table class="examples"><tr><td><pre>i1 : kk=ZZ/32003
o1 = kk
o1 : QuotientRing</pre>
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<tr><td><pre>i2 : S=kk[a..d]
o2 = S
o2 : PolynomialRing</pre>
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<tr><td><pre>i3 : i=ideal(a^2,b^2,c^2, d^2)
2 2 2 2
o3 = ideal (a , b , c , d )
o3 : Ideal of S</pre>
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<tr><td><pre>i4 : betti (F=res i)
0 1 2 3 4
o4 = total: 1 4 6 4 1
0: 1 . . . .
1: . 4 . . .
2: . . 6 . .
3: . . . 4 .
4: . . . . 1
o4 : BettiTally</pre>
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<tr><td><pre>i5 : M = image F.dd_3
o5 = image {4} | c2 d2 0 0 |
{4} | -b2 0 d2 0 |
{4} | a2 0 0 d2 |
{4} | 0 -b2 -c2 0 |
{4} | 0 a2 0 -c2 |
{4} | 0 0 a2 b2 |
6
o5 : S-module, submodule of S</pre>
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<tr><td><pre>i6 : j1 = bruns M
4 2 2 2 2 2 2 4 2 2
o6 = ideal (-9350d , - 8444b c - a d + 7477b d , 8444b + 15572b d )
o6 : Ideal of S</pre>
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<tr><td><pre>i7 : betti res j1
0 1 2 3 4
o7 = total: 1 3 5 4 1
0: 1 . . . .
1: . . . . .
2: . . . . .
3: . 3 . . .
4: . . . . .
5: . . . . .
6: . . 5 . .
7: . . . 4 .
8: . . . . 1
o7 : BettiTally</pre>
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<tr><td><pre>i8 : j2=brunsIdeal i
4 2 2 2 2 2 2 4 2 2
o8 = ideal (-9815d , - 5142b c - a d + 9132b d , 5142b + 8380b d )
o8 : Ideal of S</pre>
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<tr><td><pre>i9 : betti res j2
0 1 2 3 4
o9 = total: 1 3 5 4 1
0: 1 . . . .
1: . . . . .
2: . . . . .
3: . 3 . . .
4: . . . . .
5: . . . . .
6: . . 5 . .
7: . . . 4 .
8: . . . . 1
o9 : BettiTally</pre>
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<tr><td><pre>i10 : (betti res j1) == (betti res j2)
o10 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_bruns.html" title="Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module">bruns</a> -- Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module</span></li>
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<div class="waystouse"><h2>Ways to use <tt>brunsIdeal</tt> :</h2>
<ul><li>brunsIdeal(Ideal)</li>
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