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<head><title>isRegularSequence -- whether a list is regular over a ring or module</title>
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<div><h1>isRegularSequence -- whether a list is regular over a ring or module</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>isRegularSequence(X,A) or isRegularSequence(X) </tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>X</tt>, a <a href="../../Macaulay2Doc/html/___List.html" title="the class of all lists -- {...}">List</a> or <a href="../../Macaulay2Doc/html/___Matrix.html" title="the class of all matrices">Matrix</a></span></li>
<li><span><tt>A</tt>, a <a href="../../Macaulay2Doc/html/___Ring.html" title="the class of all rings">Ring</a> or <a href="../../Macaulay2Doc/html/___Module.html" title="the class of all modules">Module</a></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span></span></li>
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<div class="single"><h2>Description</h2>
<div>Given a list <tt>X</tt>, the function <tt>isRegularSequence</tt> tells if <tt>X</tt> forms a regular sequence. If <tt>X</tt> consists of homogeneous elements, it does this by comparing the hilbert series of <tt>A</tt> and the hilbert series of <tt>A/XA</tt>. Otherwise it checks the injectivity of the maps defined by multiplication by the elements of <tt>X</tt> and also checks if <tt>XA = A</tt>.<table class="examples"><tr><td><pre>i1 : A = ZZ/2[x, y, z];</pre>
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<tr><td><pre>i2 : X1 = {x, y*(x-1), z*(x-1)};</pre>
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<tr><td><pre>i3 : isRegularSequence X1
o3 = true</pre>
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<tr><td><pre>i4 : X2 = {z*(x-1), y*(x-1), x};</pre>
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<tr><td><pre>i5 : isRegularSequence X2
o5 = false</pre>
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<tr><td><pre>i6 : X3 = {1_A, x, y};</pre>
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<tr><td><pre>i7 : isRegularSequence X3
o7 = false</pre>
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<p>This symbol is provided by the package <a href="index.html" title="computations involving regular sequences">Depth</a>.</p>
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<div class="waystouse"><h2>Ways to use <tt>isRegularSequence</tt> :</h2>
<ul><li>isRegularSequence(List)</li>
<li>isRegularSequence(List,Module)</li>
<li>isRegularSequence(List,Ring)</li>
<li>isRegularSequence(Matrix)</li>
<li>isRegularSequence(Matrix,Module)</li>
<li>isRegularSequence(Matrix,Ring)</li>
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