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<head><title>whichGm -- largest Gm satisfied by an ideal</title>
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<div><h1>whichGm -- largest Gm satisfied by an 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>whichGm 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></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, what it does</span></li>
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<div class="single"><h2>Description</h2>
<div><p>An ideal <i>I</i> in a ring <i>S</i> is said to satisfy the condition <i>G<sub>m</sub></i> if, for every prime ideal <i>P</i> of codimension <i>0<k<m</i>, the ideal <i>I<sub>P</sub></i> in <i>S<sub>P</sub></i> can be generated by at most <i>k</i> elements.</p>
<p>The call <tt>whichGm I</tt> returns the largest <i>m</i> such that <i>I</i> satisfies <i>G<sub>m</sub></i>, or infinity if <i>I</i> satisfies <i>G<sub>m</sub></i> for every <i>m</i>.</p>
<div>This condition arises frequently in work of Vasconcelos and Ulrich and their schools on Rees algebras and powers of ideals. See for example Morey, Susan; Ulrich, Bernd: Rees algebras of ideals with low codimension. Proc. Amer. Math. Soc. 124 (1996), no. 12, 3653--3661.</div>
<table class="examples"><tr><td><pre>i1 : kk=ZZ/101;</pre>
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<tr><td><pre>i2 : S=kk[a..c];</pre>
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<tr><td><pre>i3 : m=ideal vars S
o3 = ideal (a, b, c)
o3 : Ideal of S</pre>
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<tr><td><pre>i4 : i=(ideal"a,b")*m+ideal"c3"
2 2 3
o4 = ideal (a , a*b, a*c, a*b, b , b*c, c )
o4 : Ideal of S</pre>
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<tr><td><pre>i5 : whichGm i
o5 = 3</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_analytic__Spread.html" title="compute the analytic spread of a module or ideal">analyticSpread</a> -- compute the analytic spread of a module or ideal</span></li>
<li><span><a href="_minimal__Reduction.html" title="minimal reduction of an ideal">minimalReduction</a> -- minimal reduction of an ideal</span></li>
<li><span><a href="_reduction__Number.html" title="reduction number of one ideal with respect to another">reductionNumber</a> -- reduction number of one ideal with respect to another</span></li>
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<div class="waystouse"><h2>Ways to use <tt>whichGm</tt> :</h2>
<ul><li>whichGm(Ideal)</li>
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