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<head><title>truncate -- truncate the module at a specified degree</title>
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<div><h1>truncate -- truncate the module at a specified degree</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>truncate(i,M)</tt></div>
</dd></dl>
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<li><div class="single">Inputs:<ul><li><span><tt>i</tt>, an integer or a list of integers, representing a degree or multi-degree</span></li>
<li><span><tt>M</tt>, a module or an ideal</span></li>
</ul>
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<li><div class="single">Outputs:<ul><li><span>the ideal or submodule of M consisting of all elements of degree >= i.</span></li>
</ul>
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<div class="single"><h2>Description</h2>
<div>If <tt>i</tt> is a multi-degree, then the result is the submodule generated by all elements of degree exactly <tt>i</tt>, together with all generators of <tt>M</tt> whose first degree is higher than the first entry in <tt>i</tt>.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[a..c];</pre>
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<tr><td><pre>i2 : truncate(2,R^1)
o2 = image | a2 ab ac b2 bc c2 |
1
o2 : R-module, submodule of R</pre>
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<tr><td><pre>i3 : truncate(2, ideal(a,b,c^3)/ideal(a^2,b^2,c^4))
o3 = subquotient (| ab ac bc c3 |, | a2 b2 c4 |)
1
o3 : R-module, subquotient of R</pre>
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<tr><td><pre>i4 : truncate(2,ideal(a,b*c,c^7))
2 7
o4 = ideal (a , a*b, a*c, b*c, c )
o4 : Ideal of R</pre>
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<p/>
The base may be ZZ, or another polynomial ring. In this case, the generators may not be minimal.<table class="examples"><tr><td><pre>i5 : A = ZZ[x,y,z];</pre>
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<tr><td><pre>i6 : truncate(2,ideal(3*x,5*y,15))
2 2 2
o6 = ideal (3x , 3x*y, 3x*z, 5x*y, 5y , 5y*z, 15z )
o6 : Ideal of A</pre>
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<tr><td><pre>i7 : truncate(2,comodule ideal(3*x,5*y,15))
o7 = subquotient (| x2 xy xz y2 yz z2 |, | 3x 5y 15 |)
1
o7 : A-module, subquotient of A</pre>
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<table class="examples"><tr><td><pre>i8 : L = ZZ/691[x,y,z];</pre>
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<tr><td><pre>i9 : B = L[s,t,Join=>false];</pre>
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<tr><td><pre>i10 : truncate(2,ideal(3*x*s,5*y*t^2,s*t))
2
o10 = ideal (3x*s, 5y*t , s*t)
o10 : Ideal of B</pre>
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<tr><td><pre>i11 : truncate(2,comodule ideal(3*x,5*y,15))
o11 = subquotient (0, | 3x 5y 15 |)
1
o11 : L-module, subquotient of L</pre>
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The following includes the generator of degree {8,20}.<table class="examples"><tr><td><pre>i12 : S = ZZ/101[x,y,z,Degrees=>{{1,3},{1,4},{1,-1}}];</pre>
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<tr><td><pre>i13 : truncate({7,24}, S^1 ++ S^{{-8,-20}})
o13 = image {0, 0} | x4y3 |
{8, 20} | 0 |
2
o13 : S-module, submodule of S</pre>
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<div class="single"><h2>Caveat</h2>
<div>Bug: The answer is not correct in the example over a polynomial ring!<p/>
If the degrees of the variables are not all one, then there is currently a bug in the routine: some generators of higher degree than <tt>i</tt> may be duplicated in the generator list.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_comodule.html" title="submodule to quotient module">comodule</a> -- submodule to quotient module</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>truncate</tt> :</h2>
<ul><li>truncate(List,Ideal)</li>
<li>truncate(List,Module)</li>
<li>truncate(ZZ,Ideal)</li>
<li>truncate(ZZ,Module)</li>
</ul>
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