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<head><title>universalEmbedding -- Compute the universal embedding</title>
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<div><h1>universalEmbedding -- Compute the universal embedding</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>u = universalEmbedding M</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>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>u</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, defining the universal embedding of the <tt>R</tt>-module <tt>M</tt> into a free <tt>R</tt>-module.</span></li>
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<div class="single"><h2>Description</h2>
<div><p>Suppose that M has free presentation <i>F→G</i>. universalEmbedding provides the universal map from the input module M into a free module H over the same ring, written as a map <i>u:M →H</i>, such that any map from <i>M</i> to a free <i>R</i>-module, factors uniquely through <i>u</i>. Let <i>u1</i> be the map <i>u1: G→H</i> induced by composing <i>u</i> with the surjection <i>p: G →M</i>. By definition, the Rees algebra of <i>M</i> is the image of the induced map <i>Sym(u1): Sym(G)→Sym(H)</i>, and thus can be computed with symmetricKernel(u1). The map u is computed from the dual of the first syzygy map of the dual of the presentation of <i>M</i>.</p>
<div>We first give a simple example looking at the syzygy matrix of the cube of the maximial ideal of a polynomial ring.</div>
<table class="examples"><tr><td><pre>i1 : S = ZZ/101[x,y,z];</pre>
</td></tr>
<tr><td><pre>i2 : FF=res ((ideal vars S)^3);</pre>
</td></tr>
<tr><td><pre>i3 : M=coker (FF.dd_2)
o3 = cokernel {3} | -y 0 0 -z 0 0 0 0 0 0 0 0 0 0 0 |
{3} | x -y 0 0 -z 0 0 0 0 0 0 0 0 0 0 |
{3} | 0 x -y 0 0 0 -z 0 0 0 0 0 0 0 0 |
{3} | 0 0 x 0 0 0 0 0 -z 0 0 0 0 0 0 |
{3} | 0 0 0 x y -y 0 0 0 -z 0 0 0 0 0 |
{3} | 0 0 0 0 0 x y -y 0 0 -z 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 x y 0 0 0 -z 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 x y -y 0 -z 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 x y 0 -z |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 x y |
10
o3 : S-module, quotient of S</pre>
</td></tr>
<tr><td><pre>i4 : universalEmbedding M
o4 = | x3 x2y xy2 y3 x2z xyz y2z xz2 yz2 z3 |
o4 : Matrix</pre>
</td></tr>
</table>
<div>A more complicated example.</div>
<table class="examples"><tr><td><pre>i5 : x = symbol x;</pre>
</td></tr>
<tr><td><pre>i6 : R=QQ[x_1..x_8];</pre>
</td></tr>
<tr><td><pre>i7 : m1=genericMatrix(R,x_1,2,2); m2=genericMatrix(R,x_5,2,2);
2 2
o7 : Matrix R <--- R
2 2
o8 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i9 : m=m1*m2
o9 = | x_1x_5+x_3x_6 x_1x_7+x_3x_8 |
| x_2x_5+x_4x_6 x_2x_7+x_4x_8 |
2 2
o9 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i10 : d1=minors(2,m1); d2=minors(2,m2);
o10 : Ideal of R
o11 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i12 : M=matrix{{0,d1_0,m_(0,0),m_(0,1)}, {0,0,m_(1,0),m_(1,1)}, {0,0,0,d2_0}, {0,0,0,0}}
o12 = | 0 -x_2x_3+x_1x_4 x_1x_5+x_3x_6 x_1x_7+x_3x_8 |
| 0 0 x_2x_5+x_4x_6 x_2x_7+x_4x_8 |
| 0 0 0 -x_6x_7+x_5x_8 |
| 0 0 0 0 |
4 4
o12 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i13 : M=M-(transpose M);
4 4
o13 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i14 : N= coker(res coker transpose M).dd_2
o14 = cokernel {0} | -x_2x_5-x_4x_6 -x_2x_7-x_4x_8 x_6x_7-x_5x_8 0 |
{2} | x_2x_3-x_1x_4 0 x_1x_7+x_3x_8 x_2x_7+x_4x_8 |
{2} | -x_1x_5-x_3x_6 -x_1x_7-x_3x_8 0 -x_6x_7+x_5x_8 |
{2} | 0 x_2x_3-x_1x_4 -x_1x_5-x_3x_6 -x_2x_5-x_4x_6 |
4
o14 : R-module, quotient of R</pre>
</td></tr>
<tr><td><pre>i15 : universalEmbedding(N)
o15 = {-1} | x_1 -x_6 -x_2 -x_8 |
{-1} | x_3 x_5 -x_4 x_7 |
o15 : Matrix</pre>
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<p></p>
<div>Here is an example from the paper "What is the Rees Algebra of a Module" by David Eisenbud, Craig Huneke and Bernd Ulrich, Proc. Am. Math. Soc. 131, 701--708, 2002. The example shows that one cannot, in general, define the Rees algebra of a module by using *any* embedding of that module, even when the module is isomorphic to an ideal; this is the reason for using the map provided by the routine universalEmbedding. Note that the same paper shows that such problems do not arise when the ring is torsion-free as a ZZ-module, or when one takes the natural embedding of the ideal into the ring.</div>
<table class="examples"><tr><td><pre>i16 : p = 3;</pre>
</td></tr>
<tr><td><pre>i17 : S = ZZ/p[x,y,z];</pre>
</td></tr>
<tr><td><pre>i18 : R = S/((ideal(x^p,y^p))+(ideal(x,y,z))^(p+1))
o18 = R
o18 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i19 : I = module ideal(z)
o19 = image | z |
1
o19 : R-module, submodule of R</pre>
</td></tr>
</table>
<div>As a module (or ideal), <i>Hom(I,R<sup>1</sup>)</i> is minimally generated by 3 elements, and thus the universal embedding of <i>I</i> into a free module is into <i>R<sup>3</sup></i>.</div>
<table class="examples"><tr><td><pre>i20 : betti Hom(I,R^1)
0 1
o20 = total: 3 14
0: 3 3
1: . 2
2: . 9
o20 : BettiTally</pre>
</td></tr>
<tr><td><pre>i21 : ui = universalEmbedding I
o21 = | z |
| y |
| x |
o21 : Matrix</pre>
</td></tr>
</table>
<div>it is injective:</div>
<table class="examples"><tr><td><pre>i22 : kernel ui
o22 = image 0
1
o22 : R-module, submodule of R</pre>
</td></tr>
</table>
<div>It is easy to make two other embeddings of <i>I</i> into free modules. First, the natural inclusion of <i>I</i> into <i>R</i> as an ideal is</div>
<table class="examples"><tr><td><pre>i23 : inci = map(R^1,I,matrix{{z}})
o23 = | z |
o23 : Matrix</pre>
</td></tr>
<tr><td><pre>i24 : kernel inci
o24 = image 0
1
o24 : R-module, submodule of R</pre>
</td></tr>
</table>
<div>and second, the map defined by multiplication by x and y.</div>
<table class="examples"><tr><td><pre>i25 : gi = map(R^2, I, matrix{{x},{y}})
o25 = | x |
| y |
o25 : Matrix</pre>
</td></tr>
<tr><td><pre>i26 : kernel gi
o26 = image 0
1
o26 : R-module, submodule of R</pre>
</td></tr>
</table>
<div>We can compose ui, inci and gi with a surjection R--> i to get maps u:R<sup>1</sup> --> R<sup>3</sup>, inc: R<sup>1</sup> --> R<sup>1</sup> and g:R<sup>1</sup> --> R<sup>2</sup> having image i</div>
<table class="examples"><tr><td><pre>i27 : u= map(R^3,R^{-1},ui)
o27 = | z |
| y |
| x |
3 1
o27 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i28 : inc=map(R^1, R^{-1}, matrix{{z}})
o28 = | z |
1 1
o28 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i29 : g=map(R^2, R^{-1}, matrix{{x},{y}})
o29 = | x |
| y |
2 1
o29 : Matrix R <--- R</pre>
</td></tr>
</table>
<div>We now form the symmetric kernels of these maps and compare them. Note that since symmetricKernel defines a new ring, we must bring them to the same ring to make the comparison. First the map u, which would be used by reesIdeal:</div>
<table class="examples"><tr><td><pre>i30 : A=symmetricKernel u
3 2 2 2 2 2 2
o30 = ideal (z w , y*z w , x*z w , y z*w , x*y*z*w , x z*w , x*y w , x y*w ,
0 0 0 0 0 0 0 0
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 3 3 3 4
z w , y*z*w , x*z*w , y w , x*y*w , x w , z*w , y*w , x*w , w )
0 0 0 0 0 0 0 0 0 0
o30 : Ideal of R[w ]
0</pre>
</td></tr>
</table>
<div>Next the inclusion:</div>
<table class="examples"><tr><td><pre>i31 : B1=symmetricKernel inc
3 2 2 2 2 2 2
o31 = ideal (z w , y*z w , x*z w , y z*w , x*y*z*w , x z*w , x*y w , x y*w ,
0 0 0 0 0 0 0 0
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 3 3 3 4
z w , y*z*w , x*z*w , y w , x*y*w , x w , z*w , y*w , x*w , w )
0 0 0 0 0 0 0 0 0 0
o31 : Ideal of R[w ]
0</pre>
</td></tr>
<tr><td><pre>i32 : B=(map(ring A, ring B1)) B1
3 2 2 2 2 2 2
o32 = ideal (z w , y*z w , x*z w , y z*w , x*y*z*w , x z*w , x*y w , x y*w ,
0 0 0 0 0 0 0 0
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 3 3 3 4
z w , y*z*w , x*z*w , y w , x*y*w , x w , z*w , y*w , x*w , w )
0 0 0 0 0 0 0 0 0 0
o32 : Ideal of R[w ]
0</pre>
</td></tr>
</table>
<div>Finallly, the map g1:</div>
<table class="examples"><tr><td><pre>i33 : C1 = symmetricKernel g
3 3 2 2 2 2 2
o33 = ideal (w , z w , y*z w , x*z w , y z*w , x*y*z*w , x z*w , x*y w ,
0 0 0 0 0 0 0 0
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 2
x y*w , z w , y*z*w , x*z*w , y w , x*y*w , x w )
0 0 0 0 0 0 0
o33 : Ideal of R[w ]
0</pre>
</td></tr>
<tr><td><pre>i34 : C=(map(ring A, ring C1)) C1
3 3 2 2 2 2 2
o34 = ideal (w , z w , y*z w , x*z w , y z*w , x*y*z*w , x z*w , x*y w ,
0 0 0 0 0 0 0 0
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 2
x y*w , z w , y*z*w , x*z*w , y w , x*y*w , x w )
0 0 0 0 0 0 0
o34 : Ideal of R[w ]
0</pre>
</td></tr>
</table>
<div>The following test yields “true”, as implied by the theorem of Eisenbud, Huneke and Ulrich.</div>
<table class="examples"><tr><td><pre>i35 : A==B
o35 = true</pre>
</td></tr>
</table>
<div>But the following yields “false”, showing that one must take care in general, which inclusion one uses.</div>
<table class="examples"><tr><td><pre>i36 : A==C
o36 = false</pre>
</td></tr>
</table>
</div>
</div>
<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="_rees__Algebra.html" title="compute the defining ideal of the Rees Algebra">reesAlgebra</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>universalEmbedding</tt> :</h2>
<ul><li>universalEmbedding(Ideal)</li>
<li>universalEmbedding(Module)</li>
</ul>
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