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<head><title>reesAlgebra -- compute the defining ideal of the Rees Algebra</title>
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<div><h1>reesAlgebra -- compute the defining ideal of the Rees Algebra</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>reesAlgebra M</tt><br/><tt>reesAlgebra(M,f)</tt></div>
</dd></dl>
</div>
</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> of a quotient polynomial ring <i>R</i></span></li>
<li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, any non-zero divisor modulo the ideal or module. Optional</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, defining the Rees algebra of M</span></li>
</ul>
</div>
</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_symmetric__Kernel_lp..._cm_sp__Variable_sp_eq_gt_sp..._rp.html">Variable => ...</a>, -- Choose name for variables in the created ring</span></li>
</ul>
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</li>
</ul>
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<div class="single"><h2>Description</h2>
<div><p>If <i>M</i> is an ideal or module over a ring <i>R</i>, and <i>F→M</i> is a surjection from a free module, then reesAlgebra(M) returns the ring <i>Sym(F)/J</i>, where <i>J = reesIdeal(M)</i>.</p>
<div>In the following example, we find the Rees Algebra of a monomial curve singularity. We also demonstrate the use of <a href="_rees__Ideal.html" title="compute the defining ideal of the Rees Algebra">reesIdeal</a>, <a href="_symmetric__Kernel.html" title="Compute the Rees ring of the image of a matrix">symmetricKernel</a>, <a href="_is__Linear__Type.html" title="is a module of linear type">isLinearType</a>, <a href="_normal__Cone.html" title="the normal cone of a subscheme">normalCone</a>, <a href="_associated__Graded__Ring.html" title="the associated graded ring of an ideal">associatedGradedRing</a>, <a href="_special__Fiber__Ideal.html" title="special fiber of a blowup">specialFiberIdeal</a>.</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>
</td></tr>
<tr><td><pre>i3 : time I = reesIdeal i;
-- used 0.081987 seconds
o3 : 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>i4 : reesIdeal(i, Variable=>v)
o4 = ideal (x v - x v + x v , x v - x v - v , x v - x v + x v , x v -
2 0 3 1 4 2 1 0 3 2 5 0 0 1 1 2 2 0 4
------------------------------------------------------------------------
2
x v - x v , x v - x v - x v , x v + x v - x v , x x v + x v -
1 5 4 7 0 3 3 5 4 6 4 2 1 3 3 4 0 4 2 1 6
------------------------------------------------------------------------
2 2
x v , x v - x v + x v + x v , x v + x v - x v , x x v - x x v -
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 v + x v , x x v - x x v - x v + x v - x v , x v - x v - x v +
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 v , x v v + v v - v v , x x v - v - x v v + v v , x x v v -
4 4 4 2 5 4 6 3 7 1 4 2 6 4 1 7 4 7 3 4 0 2
------------------------------------------------------------------------
2
x v v + v - v v )
4 1 4 4 3 6
o4 : Ideal of S[v , v , v , v , v , v , v , v ]
0 1 2 3 4 5 6 7</pre>
</td></tr>
<tr><td><pre>i5 : time I=reesIdeal(i,i_0);
-- used 0.405939 seconds
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 : time (J=symmetricKernel gens i);
-- used 0. seconds
o6 : 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>i7 : isLinearType(i,i_0)
o7 = false</pre>
</td></tr>
<tr><td><pre>i8 : isLinearType i
o8 = false</pre>
</td></tr>
<tr><td><pre>i9 : reesAlgebra (i,i_0)
S[w , w , w , w , w , w , w , w ]
0 1 2 3 4 5 6 7
o9 = -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 2 2 2 2 2 2 2
(x w - x w + x w , x w - x w - w , x w - x w + x w , x w - x w - x w , x w - x w - x w , x w + x w - x w , x x w + x w - x w , x w - x w + x w + x w , x w + x w - x w , x x w - x x w - x w + x w , x x w - x x w - x w + x w - x w , x w - x w - x w + x w , x w w + w w - w w , x x w - w - x w w + w w , x x w w - x w w + w - w w )
2 0 3 1 4 2 1 0 3 2 5 0 0 1 1 2 2 0 4 1 5 4 7 0 3 3 5 4 6 4 2 1 3 3 4 0 4 2 1 6 3 7 3 2 2 4 3 5 4 6 1 2 0 6 2 7 1 4 1 2 4 2 1 4 3 6 0 4 1 1 3 2 1 5 2 6 4 7 3 0 4 1 2 3 4 4 4 2 5 4 6 3 7 1 4 2 6 4 1 7 4 7 3 4 0 2 4 1 4 4 3 6
o9 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i10 : trim ideal normalCone (i, i_0)
2 3 2 2 2
o10 = ideal (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
-----------------------------------------------------------------------
2 3 2
x x - x x x , x - x x )
1 3 0 2 4 1 0 4
S[w , w , w , w , w , w , w , w ]
0 1 2 3 4 5 6 7
o10 : Ideal of -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 2 2 2 2 2 2 2
(x w - x w + x w , x w - x w - w , x w - x w + x w , x w - x w - x w , x w - x w - x w , x w + x w - x w , x x w + x w - x w , x w - x w + x w + x w , x w + x w - x w , x x w - x x w - x w + x w , x x w - x x w - x w + x w - x w , x w - x w - x w + x w , x w w + w w - w w , x x w - w - x w w + w w , x x w w - x w w + w - w w )
2 0 3 1 4 2 1 0 3 2 5 0 0 1 1 2 2 0 4 1 5 4 7 0 3 3 5 4 6 4 2 1 3 3 4 0 4 2 1 6 3 7 3 2 2 4 3 5 4 6 1 2 0 6 2 7 1 4 1 2 4 2 1 4 3 6 0 4 1 1 3 2 1 5 2 6 4 7 3 0 4 1 2 3 4 4 4 2 5 4 6 3 7 1 4 2 6 4 1 7 4 7 3 4 0 2 4 1 4 4 3 6</pre>
</td></tr>
<tr><td><pre>i11 : trim ideal associatedGradedRing (i,i_0)
2 3 2 2 2
o11 = ideal (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
-----------------------------------------------------------------------
2 3 2
x x - x x x , x - x x )
1 3 0 2 4 1 0 4
S[w , w , w , w , w , w , w , w ]
0 1 2 3 4 5 6 7
o11 : Ideal of -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 2 2 2 2 2 2 2
(x w - x w + x w , x w - x w - w , x w - x w + x w , x w - x w - x w , x w - x w - x w , x w + x w - x w , x x w + x w - x w , x w - x w + x w + x w , x w + x w - x w , x x w - x x w - x w + x w , x x w - x x w - x w + x w - x w , x w - x w - x w + x w , x w w + w w - w w , x x w - w - x w w + w w , x x w w - x w w + w - w w )
2 0 3 1 4 2 1 0 3 2 5 0 0 1 1 2 2 0 4 1 5 4 7 0 3 3 5 4 6 4 2 1 3 3 4 0 4 2 1 6 3 7 3 2 2 4 3 5 4 6 1 2 0 6 2 7 1 4 1 2 4 2 1 4 3 6 0 4 1 1 3 2 1 5 2 6 4 7 3 0 4 1 2 3 4 4 4 2 5 4 6 3 7 1 4 2 6 4 1 7 4 7 3 4 0 2 4 1 4 4 3 6</pre>
</td></tr>
<tr><td><pre>i12 : trim specialFiberIdeal (i,i_0)
2 2
o12 = ideal (x , x , x , x , x , w , w - w w , w w - w w , w - w w )
4 3 2 1 0 5 6 4 7 4 6 3 7 4 3 6
o12 : Ideal of S[w , w , w , w , w , w , w , w ]
0 1 2 3 4 5 6 7</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="_symmetric__Kernel.html" title="Compute the Rees ring of the image of a matrix">symmetricKernel</a> -- Compute the Rees ring of the image of a matrix</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>reesAlgebra</tt> :</h2>
<ul><li>reesAlgebra(Ideal)</li>
<li>reesAlgebra(Ideal,RingElement)</li>
<li>reesAlgebra(Module)</li>
<li>reesAlgebra(Module,RingElement)</li>
</ul>
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