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<head><title>analyticSpread -- compute the analytic spread of a module or ideal</title>
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<div><h1>analyticSpread -- compute the analytic spread of a module or 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>analyticSpread M</tt><br/><tt>analyticSpread(M,f)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, or <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span></span></li>
<li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, an optional element, which is a non-zerodivisor modulo <tt>M</tt> and the ring of <tt>M</tt></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span></span></li>
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<div class="single"><h2>Description</h2>
<div><div>The analytic spread of a module is the dimension of its special fiber ring. When <i>I</i> is an ideal (and more generally, with the right definitions) the analytic spread of <i>I</i> is the smallest number of generators of an ideal <i>J</i> such that <i>I</i> is integral over <i>J</i>. See for example the book Integral closure of ideals, rings, and modules. London Mathematical Society Lecture Note Series, 336. Cambridge University Press, Cambridge, 2006, by Craig Huneke and Irena Swanson.</div>
<table class="examples"><tr><td><pre>i1 : R=QQ[a,b,c,d,e,f]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : M=matrix{{a,c,e},{b,d,f}}
o2 = | a c e |
| b d f |
2 3
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : analyticSpread image M
o3 = 3</pre>
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<tr><td><pre>i4 : specialFiberIdeal image M
o4 = ideal (f, e, d, c, b, a)
o4 : Ideal of R[w , w , w ]
0 1 2</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_special__Fiber__Ideal.html" title="special fiber of a blowup">specialFiberIdeal</a> -- special fiber of a blowup</span></li>
<li><span><a href="_rees__Ideal.html" title="compute the defining ideal of the Rees Algebra">reesIdeal</a> -- compute the defining ideal of the Rees Algebra</span></li>
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<div class="waystouse"><h2>Ways to use <tt>analyticSpread</tt> :</h2>
<ul><li>analyticSpread(Ideal)</li>
<li>analyticSpread(Ideal,RingElement)</li>
<li>analyticSpread(Module)</li>
<li>analyticSpread(Module,RingElement)</li>
</ul>
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