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<head><title>saturate -- saturation of ideal or submodule</title>
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<div><h1>saturate -- saturation of ideal or submodule</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>saturate(I,J)</tt><br/><tt>saturate I</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="___Ideal.html">ideal</a></span>, <span>a <a href="___Module.html">module</a></span>, or <span>a <a href="___Monomial__Ideal.html">monomial ideal</a></span></span></li>
<li><span><tt>J</tt>, <span>an <a href="___Ideal.html">ideal</a></span> or <span>a <a href="___Ring__Element.html">ring element</a></span>. If not present, then <tt>J</tt> is the ideal generated by the variables of the ring</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Ideal.html">ideal</a></span>, <span>a <a href="___Module.html">module</a></span>, or <span>a <a href="___Monomial__Ideal.html">monomial ideal</a></span>, the saturation of <tt>I</tt> with respect to <tt>J</tt></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_saturate_lp..._cm_sp__Basis__Element__Limit_sp_eq_gt_sp..._rp.html">BasisElementLimit => ...</a>, </span></li>
<li><span><a href="_saturate_lp..._cm_sp__Degree__Limit_sp_eq_gt_sp..._rp.html">DegreeLimit => ...</a>, </span></li>
<li><span><a href="_saturate_lp..._cm_sp__Minimal__Generators_sp_eq_gt_sp..._rp.html">MinimalGenerators => ...</a>, -- whether to compute minimal generators</span></li>
<li><span><a href="_saturate_lp..._cm_sp__Pair__Limit_sp_eq_gt_sp..._rp.html">PairLimit => ...</a>, </span></li>
<li><span><a href="_saturate_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
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If I is either an ideal or a submodule of a module M, the saturation (I : J^*) is defined to be the set of elements f in the ring (first case) or in M (second case) such that J^N * f is contained in I, for some N large enough.<p/>
For example, one way to homogenize an ideal is to homogenize the generators and then saturate with respect to the homogenizing variable.<table class="examples"><tr><td><pre>i1 : R = ZZ/32003[a..d];</pre>
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<tr><td><pre>i2 : I = ideal(a^3-b, a^4-c)
3 4
o2 = ideal (a - b, a - c)
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : Ih = homogenize(I,d)
3 2 4 3
o3 = ideal (a - b*d , a - c*d )
o3 : Ideal of R</pre>
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<tr><td><pre>i4 : saturate(Ih,d)
2 2 3 2 3 2
o4 = ideal (a*b - c*d, a c - b d, b - a*c , a - b*d )
o4 : Ideal of R</pre>
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We can use this command to remove graded submodules of finite length.<table class="examples"><tr><td><pre>i5 : m = ideal vars R
o5 = ideal (a, b, c, d)
o5 : Ideal of R</pre>
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<tr><td><pre>i6 : M = R^1 / (a * m^2)
o6 = cokernel | a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 |
1
o6 : R-module, quotient of R</pre>
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<tr><td><pre>i7 : M / saturate 0_M
o7 = cokernel | a a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 |
1
o7 : R-module, quotient of R</pre>
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<p/>
If I and J are both monomial ideals, then a faster algorithm is used. If I or J is not a monomial ideal, generally Gröbner bases will be used to the compute the saturation. These will be computed as needed.<p/>
The computation is currently not stored anywhere: this means that the computation cannot be continued after an interrupt. This will be changed in a later version.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_quotient.html" title="quotient or division">quotient</a> -- quotient or division</span></li>
<li><span><a href="../../PrimaryDecomposition/html/index.html" title="">PrimaryDecomposition</a></span></li>
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<div class="waystouse"><h2>Ways to use <tt>saturate</tt> :</h2>
<ul><li>saturate(Ideal)</li>
<li>saturate(Ideal,Ideal)</li>
<li>saturate(Ideal,RingElement)</li>
<li>saturate(Module)</li>
<li>saturate(Module,Ideal)</li>
<li>saturate(Module,RingElement)</li>
<li>saturate(MonomialIdeal,MonomialIdeal)</li>
<li>saturate(MonomialIdeal,RingElement)</li>
</ul>
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