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<head><title>isSCM -- determines if a (hyper)graph is sequentially Cohen-Macaulay</title>
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<div><h1>isSCM -- determines if a (hyper)graph is sequentially Cohen-Macaulay</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>b = isSCM H</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>H</tt>, <span>a <a href="___Hyper__Graph.html">hypergraph</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>b</tt>, <span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <tt>true</tt> if the <a href="_edge__Ideal.html" title="creates the edge ideal of a (hyper)graph">edgeIdeal</a> of <tt>H</tt> is sequentially Cohen-Macaulay</span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_is__S__C__M_lp..._cm_sp__Gins_sp_eq_gt_sp..._rp.html">Gins => ...</a>, -- use gins inside isSCM</span></li>
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<div class="single"><h2>Description</h2>
<div><p>This uses the edge ideal notion of sequential Cohen-Macaulayness; a hypergraph is called SCM if and only if its edge ideal is SCM. The algorithm is based on work of Herzog and Hibi, using the Alexander dual. <tt>H</tt> is SCM if and only if the Alexander dual of the edge ideal of <tt>H</tt> is componentwise linear.</p>
<div>There is an optional argument called <a href="___Gins.html" title="optional argument for isSCM">Gins</a> for <tt>isSCM</tt>. The default value is <tt>false</tt>, meaning that <tt>isSCM</tt> takes the Alexander dual of the edge ideal of <tt>H</tt> and checks in all relevant degrees to see if the ideal in that degree has a linear resolution. In characteristic zero with the reverse-lex order, one can test for componentwise linearity using gins, which may be faster in some cases. This approach is based on work of Aramova-Herzog-Hibi and Conca. We make no attempt to check the characteristic of the field or the monomial order, so use caution when using this method.</div>
<table class="examples"><tr><td><pre>i1 : R = QQ[a..f];</pre>
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<tr><td><pre>i2 : G = cycle(R,4)
o2 = Graph{edges => {{a, b}, {b, c}, {c, d}, {a, d}}}
ring => R
vertices => {a, b, c, d, e, f}
o2 : Graph</pre>
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<tr><td><pre>i3 : isSCM G
o3 = false</pre>
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<tr><td><pre>i4 : H = graph(monomialIdeal(a*b,b*c,c*d,a*d,a*e)); --4-cycle with whisker</pre>
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<tr><td><pre>i5 : isSCM H
o5 = true</pre>
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<tr><td><pre>i6 : isSCM(H,Gins=>true) --use Gins technique
o6 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_is__C__M.html" title="determines if a (hyper)graph is Cohen-Macaulay">isCM</a> -- determines if a (hyper)graph is Cohen-Macaulay</span></li>
<li><span><a href="_edge__Ideal.html" title="creates the edge ideal of a (hyper)graph">edgeIdeal</a> -- creates the edge ideal of a (hyper)graph</span></li>
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<div class="waystouse"><h2>Ways to use <tt>isSCM</tt> :</h2>
<ul><li>isSCM(HyperGraph)</li>
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