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<head><title>isLinearType -- is a module of linear type</title>
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<div><h1>isLinearType -- is a module of linear type</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>isLinearType M</tt><br/><tt>isLinearType(M,f)</tt></div>
</dd></dl>
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</li>
<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>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, true if <tt>M</tt> is of linear type, false otherwise</span></li>
</ul>
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</li>
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<div class="single"><h2>Description</h2>
<div><p>A module or ideal <i>M</i> is said to be “of linear type” if the natural map from the symmetric algebra of <i>M</i> to the Rees algebra of <i>M</i> is an isomorphism. It is known, for example, that any complete intersection ideal is of linear type.</p>
<div>This routine computes the <a href="_rees__Ideal.html" title="compute the defining ideal of the Rees Algebra">reesIdeal</a> of <tt>M</tt>. Giving the element <tt>f</tt> computes the <a href="_rees__Ideal.html" title="compute the defining ideal of the Rees Algebra">reesIdeal</a> in a different manner, which is sometimes faster, sometimes slower.</div>
<table class="examples"><tr><td><pre>i1 : S = QQ[x_0..x_4]
o1 = S
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : i = monomialCurveIdeal(S,{2,3,5,6})
2 3 2 2 2 2
o2 = ideal (x x - x x , x - x x , x x - x x , x - x x , x x - x x , x x
2 3 1 4 2 0 4 1 2 0 3 3 2 4 1 3 0 4 0 3
------------------------------------------------------------------------
2 2 3 2
- x x , x x - x x x , x - x x )
1 4 1 3 0 2 4 1 0 4
o2 : Ideal of S</pre>
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<tr><td><pre>i3 : isLinearType i
o3 = false</pre>
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<tr><td><pre>i4 : isLinearType(i, i_0)
o4 = false</pre>
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<tr><td><pre>i5 : I = reesIdeal i
o5 = ideal (x w - x w + x w , x w - x w - w , x w - x w + x w , x w -
2 0 3 1 4 2 1 0 3 2 5 0 0 1 1 2 2 0 4
------------------------------------------------------------------------
2
x w - x w , x w - x w - x w , x w + x w - x w , x x w + x w -
1 5 4 7 0 3 3 5 4 6 4 2 1 3 3 4 0 4 2 1 6
------------------------------------------------------------------------
2 2
x w , x w - x w + x w + x w , x w + x w - x w , x x w - x x w -
3 7 3 2 2 4 3 5 4 6 1 2 0 6 2 7 1 4 1 2 4 2
------------------------------------------------------------------------
2 2
x w + x w , x x w - x x w - x w + x w - x w , x w - x w - x w +
1 4 3 6 0 4 1 1 3 2 1 5 2 6 4 7 3 0 4 1 2 3
------------------------------------------------------------------------
2 2
x w , x w w + w w - w w , x x w - w - x w w + w w , x x w w -
4 4 4 2 5 4 6 3 7 1 4 2 6 4 1 7 4 7 3 4 0 2
------------------------------------------------------------------------
2
x w w + w - w w )
4 1 4 4 3 6
o5 : Ideal of S[w , w , w , w , w , w , w , w ]
0 1 2 3 4 5 6 7</pre>
</td></tr>
<tr><td><pre>i6 : select(I_*, f -> first degree f > 1)
2 2
o6 = {x w w + w w - w w , x x w - w - x w w + w w , x x w w - x w w +
4 2 5 4 6 3 7 1 4 2 6 4 1 7 4 7 3 4 0 2 4 1 4
------------------------------------------------------------------------
2
w - w w }
4 3 6
o6 : List</pre>
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<table class="examples"><tr><td><pre>i7 : S = ZZ/101[x,y,z]
o7 = S
o7 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i8 : for p from 1 to 5 do print isLinearType (ideal vars S)^p
true
false
false
false
false</pre>
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<div class="single"><h2>See also</h2>
<ul><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>
<li><span><a href="../../Macaulay2Doc/html/_monomial__Curve__Ideal.html" title="make the ideal of a monomial curve">monomialCurveIdeal</a> -- make the ideal of a monomial curve</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>isLinearType</tt> :</h2>
<ul><li>isLinearType(Ideal)</li>
<li>isLinearType(Ideal,RingElement)</li>
<li>isLinearType(Module)</li>
<li>isLinearType(Module,RingElement)</li>
</ul>
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