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<head><title>reesIdeal -- compute the defining ideal of the Rees Algebra</title>
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<div><h1>reesIdeal -- 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>reesIdeal M</tt><br/><tt>reesIdeal(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>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</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>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><p>The Rees algebra of a module <i>M</i> over a ring <i>R</i> is here defined, following the paper What is the Rees algebra of a module? David Eisenbud, Craig Huneke and Bernd Ulrich, Proc. Amer. Math. Soc. 131 (2003) 701--708, as follows: If <i>h:F→M</i> is a surjection from a free module, and <i>g: M→G</i> is the universal map to a free module, then the Rees algebra of <i>M</i> is the image of the induced map of <i>Sym(gh): Sym(F)→Sym(G)</i>, and thus can be computed with symmetricKernel(gh). The paper above proves that if <i>M</i> is isomorphic to an ideal with inclusion <i>g: M→R</i> (or, in characteristic zero but not in characteristic <i>>0</i> if <i>M</i> is a submodule of a free module and <i>g’: M→G</i>) is any injection), then the Rees algebra is equal to the image of <i>g’h</i>, so it is unnecessary to compute the universal embedding.</p>
<p>This package gives the user a choice between two methods for finding the defining ideal of the Rees algebra of an ideal or module <i>M</i> over a ring <i>R</i>: The call</p>
<p><tt>reesIdeal(M)</tt></p>
<p>computes the universal embedding <i>g: M→G</i> and a surjection <i>f: F→M</i> and returns the result of symmetricKernel(gf). On the other hand, if the user knows an non-zerodivisor <i>a∈R</i> such that <i>M[a<sup>-1</sup></i> is a free module (this is the case, for example, if <i>a ∈M⊂R</i> and <i>a</i> is a non-zerodivisor), then it is often much faster to call</p>
<p><tt>reesIdeal(M,a)</tt></p>
<p>which finds a surjection <i>f: F→M</i> and returns <i>(J:a<sup>∞</sup>) ⊂Sym(F)</i>, the saturation of the ideal <i>J:=(ker f)Sym(F)</i>. Note that this gives the correct answer even under the slightly weaker hypothesis that <i>M[a<sup>-1</sup>]</i> is “of linear type”. (See also <a href="_is__Linear__Type.html" title="is a module of linear type">isLinearType</a>.)</p>
<div><b>Historical Background</b>: The Rees Algebra of an ideal is the basic commutative algebra analogue of the blow up operation in algebraic geometry. It has many applications, and a great deal of modern work in commutative algebra has been devoted to it. The term "Rees Algebra" (of an ideal <i>I</i> in a ring <i>R</i>, say) is used here to refer to the ring <i>R[It]⊂R[t]</i> which is sometimes called the "blowup algebra" instead. (The origin of the name may be traced to a paper by David Rees, (On a problem of Zariski. Illinois J. Math. (1958)145âÂÂ149) where Rees used the ring <i>R[It,t<sup>-1</sup></i>, now also called the “extended Rees Algebra.”)</div>
<table class="examples"><tr><td><pre>i1 : kk = ZZ/101;</pre>
</td></tr>
<tr><td><pre>i2 : S=kk[x_0..x_4];</pre>
</td></tr>
<tr><td><pre>i3 : i=monomialCurveIdeal(S,{2,3,5,6})
2 3 2 2 2 2
o3 = 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
o3 : Ideal of S</pre>
</td></tr>
<tr><td><pre>i4 : time V1 = reesIdeal i;
-- used 0.076989 seconds
o4 : 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>i5 : time V2 = reesIdeal(i,i_0);
-- used 0.183972 seconds
o5 : Ideal of S[w , w , w , w , w , w , w , w ]
0 1 2 3 4 5 6 7</pre>
</td></tr>
</table>
<div>This example is particularly interesting upon a bit more exploration.</div>
<table class="examples"><tr><td><pre>i6 : numgens V1
o6 = 15</pre>
</td></tr>
<tr><td><pre>i7 : numgens V2
o7 = 15</pre>
</td></tr>
</table>
<div>The difference is striking and, at least in part, explains the difference in computing time. Furthermore, if we compute a Grobner basis for both and compare the two matrices, we see that we indeed got the same ideal.</div>
<table class="examples"><tr><td><pre>i8 : M1 = gens gb V1;
1 84
o8 : Matrix (S[w , w , w , w , w , w , w , w ]) <--- (S[w , w , w , w , w , w , w , w ])
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7</pre>
</td></tr>
<tr><td><pre>i9 : M2 = gens gb V2;
1 84
o9 : Matrix (S[w , w , w , w , w , w , w , w ]) <--- (S[w , w , w , w , w , w , w , w ])
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7</pre>
</td></tr>
<tr><td><pre>i10 : use ring M2
o10 = S[w , w , w , w , w , w , w , w ]
0 1 2 3 4 5 6 7
o10 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i11 : M1 = substitute(M1, ring M2);
1 84
o11 : Matrix (S[w , w , w , w , w , w , w , w ]) <--- (S[w , w , w , w , w , w , w , w ])
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7</pre>
</td></tr>
<tr><td><pre>i12 : M1 == M2
o12 = true</pre>
</td></tr>
<tr><td><pre>i13 : numgens source M2
o13 = 84</pre>
</td></tr>
</table>
<div>Another example illustrates the power and usage of the code. We also show the output in this example. While a bit messy, the user can see how we handle the degrees in both cases.</div>
<table class="examples"><tr><td><pre>i14 : S=kk[a,b,c]
o14 = S
o14 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i15 : m=matrix{{a,0},{b,a},{0,b}}
o15 = | a 0 |
| b a |
| 0 b |
3 2
o15 : Matrix S <--- S</pre>
</td></tr>
<tr><td><pre>i16 : i=minors(2,m)
2 2
o16 = ideal (a , a*b, b )
o16 : Ideal of S</pre>
</td></tr>
<tr><td><pre>i17 : time reesIdeal i
-- used 0.027996 seconds
2
o17 = ideal (b*w - a*w , b*w - a*w , w - w w )
1 2 0 1 1 0 2
o17 : Ideal of S[w , w , w ]
0 1 2</pre>
</td></tr>
<tr><td><pre>i18 : res i
1 3 2
o18 = S <-- S <-- S <-- 0
0 1 2 3
o18 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i19 : m=random(S^3,S^{4:-1})
o19 = | -10a-10b+17c -40a-5b+31c -29a+28b-18c -31a-20b+48c |
| -20a+2b-c -37a+11b+40c 20a+25b-40c 26a+14b+22c |
| -47a+36b+12c -8a-15b-2c -3a+4b+9c -2a-49b-48c |
3 4
o19 : Matrix S <--- S</pre>
</td></tr>
<tr><td><pre>i20 : i=minors(3,m)
3 2 2 3 2 2 2
o20 = ideal (- 42a + 4a b - 43a*b + 26b - 11a c - 21a*b*c + 42b c - 3a*c
-----------------------------------------------------------------------
2 3 3 2 2 3 2 2
+ 10b*c - 25c , - 48a - 5a b + 6a*b + 13b + 10a c - 22a*b*c + 9b c
-----------------------------------------------------------------------
2 2 3 3 2 2 3 2
+ 17a*c + 41b*c + 37c , - 34a - 37a b + 31a*b + 36b - 14a c +
-----------------------------------------------------------------------
2 2 3 3 2 2 3 2
33a*b*c - 23b c - b*c + 19c , - 26a + 23a b + 47a*b - 35b + 18a c +
-----------------------------------------------------------------------
2 2 2 3
23a*b*c + 41b c + 30a*c + 7b*c + 27c )
o20 : Ideal of S</pre>
</td></tr>
<tr><td><pre>i21 : time I=reesIdeal (i,i_0);
-- used 0.011998 seconds
o21 : Ideal of S[w , w , w , w ]
0 1 2 3</pre>
</td></tr>
<tr><td><pre>i22 : transpose gens I
o22 = {-1, -4} | w_0c+39w_1a+26w_1b+45w_1c-45w_2a-31w_2b+10w_2c+49w_3a-28w_
{-1, -4} | w_0b+14w_1a-25w_1b+27w_1c-32w_2a-44w_2b-46w_2c+16w_3a+27w_
{-1, -4} | w_0a-18w_1a+41w_1b-20w_1c+50w_2a-39w_2b-21w_2c-26w_3a-22w_
{-3, -9} | w_0^3+2w_0^2w_1+39w_0w_1^2+17w_1^3+16w_0^2w_2-40w_0w_1w_2-
-----------------------------------------------------------------------
3b+22w_3c
3b+44w_3c
3b+16w_3c
18w_1^2w_2-2w_0w_2^2+13w_1w_2^2-21w_2^3+23w_0^2w_3-6w_0w_1w_3+47w_1^2w_
-----------------------------------------------------------------------
3+38w_0w_2w_3+9w_1w_2w_3+11w_2^2w_3+19w_0w_3^2+29w_1w_3^2-14w_2w_3^2+
-----------------------------------------------------------------------
|
|
|
44w_3^3 |
4 1
o22 : Matrix (S[w , w , w , w ]) <--- (S[w , w , w , w ])
0 1 2 3 0 1 2 3</pre>
</td></tr>
<tr><td><pre>i23 : i=minors(2,m);
o23 : Ideal of S</pre>
</td></tr>
<tr><td><pre>i24 : time I=reesIdeal (i,i_0);
-- used 0.075989 seconds
o24 : Ideal of S[w , w , w , w , w , w , w , w , w , w , w , w , w , w , w , w , w , w ]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17</pre>
</td></tr>
</table>
<div><b>Investigating plane curve singularities</b></div>
<table class="examples"><tr><td><pre>i25 : R = ZZ/32003[x,y,z]
o25 = R
o25 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i26 : I = ideal(x,y)
o26 = ideal (x, y)
o26 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i27 : cusp = ideal(x^2*z-y^3)
3 2
o27 = ideal(- y + x z)
o27 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i28 : RI = reesIdeal(I)
o28 = ideal(y*w - x*w )
0 1
o28 : Ideal of R[w , w ]
0 1</pre>
</td></tr>
<tr><td><pre>i29 : S = ring RI
o29 = S
o29 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i30 : totalTransform = substitute(cusp, S) + RI
3 2
o30 = ideal (- y + x z, y*w - x*w )
0 1
o30 : Ideal of S</pre>
</td></tr>
<tr><td><pre>i31 : D = decompose totalTransform -- the components are the proper transform of the cuspidal curve and the exceptional curve
3 2 2 2 2
o31 = {ideal (y*w - x*w , y - x z, x*z*w - y w , z*w - y*w ), ideal (y,
0 1 0 1 0 1
-----------------------------------------------------------------------
x)}
o31 : List</pre>
</td></tr>
<tr><td><pre>i32 : totalTransform = first flattenRing totalTransform
3 2
o32 = ideal (- y + x z, w y - w x)
0 1
ZZ
o32 : Ideal of -----[w , w , x, y, z]
32003 0 1</pre>
</td></tr>
<tr><td><pre>i33 : L = primaryDecomposition totalTransform
3 2 2 2 2 2
o33 = {ideal (w y - w x, y - x z, w x*z - w y , w z - w y), ideal (y , x*y,
0 1 0 1 0 1
-----------------------------------------------------------------------
2
x , w y - w x)}
0 1
o33 : List</pre>
</td></tr>
<tr><td><pre>i34 : apply(L, i -> (degree i)/(degree radical i))
o34 = {1, 2}
o34 : List</pre>
</td></tr>
</table>
<div>The total transform of the cusp contains the exceptional with multiplicity two. The proper transform of the cusp is a smooth curve but is tangent to the exceptional curve.</div>
<table class="examples"><tr><td><pre>i35 : use ring L_0
ZZ
o35 = -----[w , w , x, y, z]
32003 0 1
o35 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i36 : singular = ideal(singularLocus(L_0));
ZZ
o36 : Ideal of -----[w , w , x, y, z]
32003 0 1</pre>
</td></tr>
<tr><td><pre>i37 : SL = saturate(singular, ideal(x,y,z));
ZZ
o37 : Ideal of -----[w , w , x, y, z]
32003 0 1</pre>
</td></tr>
<tr><td><pre>i38 : saturate(SL, ideal(w_0,w_1)) -- we get 1 so it is smooth.
o38 = ideal 1
ZZ
o38 : Ideal of -----[w , w , x, y, z]
32003 0 1</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>Caveat</h2>
<div><div/>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><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>
<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>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>reesIdeal</tt> :</h2>
<ul><li>reesIdeal(Ideal)</li>
<li>reesIdeal(Ideal,RingElement)</li>
<li>reesIdeal(Module)</li>
<li>reesIdeal(Module,RingElement)</li>
</ul>
</div>
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