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<head><title>isSyzygy -- Tests if a module is a d-th syzygy</title>
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<div><h1>isSyzygy -- Tests if a module is a d-th syzygy</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 = isSyzygy(M,d)</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>, over a polynomial ring.</span></li>
<li><span><tt>d</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, positive</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>, returns <tt>true</tt> if M is a d-th syzygy, and <tt>false</tt> otherwise.</span></li>
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<div class="single"><h2>Description</h2>
<div><div>This algorithm is based upon the methods described in the book of Evans and Griffith (<em>Syzygies</em>. London Mathematical Society Lecture Note Series, 106. Cambridge University Press, Cambridge, 1985.)</div>
<table class="examples"><tr><td><pre>i1 : kk=ZZ/32003
o1 = kk
o1 : QuotientRing</pre>
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<tr><td><pre>i2 : S=kk[a..d]
o2 = S
o2 : PolynomialRing</pre>
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<tr><td><pre>i3 : F=res (ideal vars S)^2
1 10 20 15 4
o3 = S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5
o3 : ChainComplex</pre>
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<div><b>NOTE:</b> We are viewing a syzygy module as a cokernel of an appropriate map.</div>
<table class="examples"><tr><td><pre>i4 : isSyzygy(coker F.dd_3,3) -- the cokernel defined by the 3rd map is not a 3rd syzygy
o4 = false</pre>
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<tr><td><pre>i5 : isSyzygy(coker F.dd_4,3) -- the cokernel defined by the 4th map is a 3rd syzygy
o5 = true</pre>
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<p>This function is called within <a href="_evans__Griffith.html" title="Reduces the rank of a syzygy">evansGriffith</a>.</p>
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<div class="waystouse"><h2>Ways to use <tt>isSyzygy</tt> :</h2>
<ul><li>isSyzygy(Module,ZZ)</li>
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