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<head><title>annihilator -- the annihilator ideal</title>
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<div><h1>annihilator -- the annihilator 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>annihilator M</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, a module, an ideal, a ring element, or a coherent sheaf</span></li>
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<li><div class="single">Outputs:<ul><li><span>the annihilator ideal, <tt>ann(M) = { f in R | fM = 0 }</tt> where <tt>R</tt> is the ring of <tt>M</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><tt>Strategy => </tt><span><span>default value Intersection</span>, <a href="___Quotient.html" title="value for an optional argument">Quotient</a> or <a href="___Intersection.html" title="value for an optional argument">Intersection</a></span></span></li>
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<div class="single"><h2>Description</h2>
<div>You may use <tt>ann</tt> as a synonym for <tt>annihilator</tt>.<p/>
As an example, we compute the annihilator of the canonical module of the rational quartic curve.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
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<tr><td><pre>i2 : J = monomialCurveIdeal(R,{1,3,4})
3 2 2 2 3 2
o2 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c)
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : M = Ext^2(R^1/J, R)
o3 = cokernel {-3} | c a 0 b 0 |
{-3} | -d -b c 0 a |
{-3} | 0 0 d c b |
3
o3 : R-module, quotient of R</pre>
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<tr><td><pre>i4 : annihilator M
3 2 2 2 3 2
o4 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c)
o4 : Ideal of R</pre>
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For another example, we compute the annihilator of an element in a quotient ring<table class="examples"><tr><td><pre>i5 : A = R/(a*b,a*c,a*d)
o5 = A
o5 : QuotientRing</pre>
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<tr><td><pre>i6 : ann a
o6 = ideal (d, c, b)
o6 : Ideal of A</pre>
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Macaulay2 uses two algorithms to compute annihilators. The default version is to compute the annihilator of each generator of the module <tt>M</tt> and to intersect these two by two. Each annihilator is done using a submodule quotient. The other algorithm computes the annihilator in one large computation and is used if <tt>Strategy => Quotient</tt> is specified.<table class="examples"><tr><td><pre>i7 : annihilator(M, Strategy=>Quotient)
3 2 2 2 3 2
o7 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c)
o7 : Ideal of R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_quotient_lp__Ideal_cm__Ideal_rp.html" title="ideal or submodule quotient">quotient(Module,Module)</a> -- ideal or submodule quotient</span></li>
<li><span><a href="___Ext.html" title="compute an Ext module">Ext</a> -- compute an Ext module</span></li>
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<div class="waystouse"><h2>Ways to use <tt>annihilator</tt> :</h2>
<ul><li>annihilator(CoherentSheaf)</li>
<li>annihilator(Ideal)</li>
<li>annihilator(Module)</li>
<li>annihilator(RingElement)</li>
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