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<head><title>nullhomotopy -- make a null homotopy</title>
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<div><h1>nullhomotopy -- make a null homotopy</h1>
<div class="single"><h2>Description</h2>
<div><tt>nullhomotopy f</tt> -- produce a nullhomotopy for a map f of chain complexes.<p/>
Whether f is null homotopic is not checked.<p/>
Here is part of an example provided by Luchezar Avramov. We construct a random module over a complete intersection, resolve it over the polynomial ring, and produce a null homotopy for the map that is multiplication by one of the defining equations for the complete intersection.<table class="examples"><tr><td><pre>i1 : A = ZZ/101[x,y];</pre>
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<tr><td><pre>i2 : M = cokernel random(A^3, A^{-2,-2})
o2 = cokernel | -42x2+22xy-12y2 -11x2+17xy-31y2 |
| -3x2+33xy-20y2 16x2-16xy+19y2 |
| 47x2-18xy+35y2 -8x2+24xy-41y2 |
3
o2 : A-module, quotient of A</pre>
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<tr><td><pre>i3 : R = cokernel matrix {{x^3,y^4}}
o3 = cokernel | x3 y4 |
1
o3 : A-module, quotient of A</pre>
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<tr><td><pre>i4 : N = prune (M**R)
o4 = cokernel | -31x2+32xy+3y2 -48x2+24xy-28y2 x3 x2y-2xy2-7y3 47xy2-7y3 y4 0 0 |
| x2-4xy-18y2 -6xy+50y2 0 -9xy2+40y3 38xy2-17y3 0 y4 0 |
| 18xy+37y2 x2+33xy+16y2 0 -26y3 xy2+27y3 0 0 y4 |
3
o4 : A-module, quotient of A</pre>
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<tr><td><pre>i5 : C = resolution N
3 8 5
o5 = A <-- A <-- A <-- 0
0 1 2 3
o5 : ChainComplex</pre>
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<tr><td><pre>i6 : d = C.dd
3 8
o6 = 0 : A <-------------------------------------------------------------------------- A : 1
| -31x2+32xy+3y2 -48x2+24xy-28y2 x3 x2y-2xy2-7y3 47xy2-7y3 y4 0 0 |
| x2-4xy-18y2 -6xy+50y2 0 -9xy2+40y3 38xy2-17y3 0 y4 0 |
| 18xy+37y2 x2+33xy+16y2 0 -26y3 xy2+27y3 0 0 y4 |
8 5
1 : A <-------------------------------------------------------------------------- A : 2
{2} | 14xy2-29y3 47xy2+46y3 -14y3 36y3 -3y3 |
{2} | 26xy2+44y3 -30y3 -26y3 -19y3 25y3 |
{3} | -4xy-13y2 6xy+14y2 4y2 28y2 -40y2 |
{3} | 4x2-4xy-27y2 -6x2+17xy+21y2 -4xy+17y2 -28xy-38y2 40xy+39y2 |
{3} | -26x2-45xy-46y2 38xy-40y2 26xy+y2 19xy-50y2 -25xy+34y2 |
{4} | 0 0 x+46y -44y 8y |
{4} | 0 0 -19y x+46y 37y |
{4} | 0 0 36y 41y x+9y |
5
2 : A <----- 0 : 3
0
o6 : ChainComplexMap</pre>
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<tr><td><pre>i7 : s = nullhomotopy (x^3 * id_C)
8 3
o7 = 1 : A <------------------------ A : 0
{2} | 0 x+4y 6y |
{2} | 0 -18y x-33y |
{3} | 1 31 48 |
{3} | 0 36 -8 |
{3} | 0 -20 -45 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | 7 41 0 14y -35x+20y xy+28y2 -42xy-45y2 34xy+47y2 |
{5} | -23 26 0 -17x-22y 44x-7y 9y2 xy-43y2 -38xy-25y2 |
{5} | 0 0 0 0 0 x2-46xy+8y2 44xy+17y2 -8xy+24y2 |
{5} | 0 0 0 0 0 19xy-12y2 x2-46xy+25y2 -37xy-36y2 |
{5} | 0 0 0 0 0 -36xy-11y2 -41xy-36y2 x2-9xy-33y2 |
5
3 : 0 <----- A : 2
0
o7 : ChainComplexMap</pre>
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<tr><td><pre>i8 : s*d + d*s
3 3
o8 = 0 : A <---------------- A : 0
| x3 0 0 |
| 0 x3 0 |
| 0 0 x3 |
8 8
1 : A <----------------------------------- A : 1
{2} | x3 0 0 0 0 0 0 0 |
{2} | 0 x3 0 0 0 0 0 0 |
{3} | 0 0 x3 0 0 0 0 0 |
{3} | 0 0 0 x3 0 0 0 0 |
{3} | 0 0 0 0 x3 0 0 0 |
{4} | 0 0 0 0 0 x3 0 0 |
{4} | 0 0 0 0 0 0 x3 0 |
{4} | 0 0 0 0 0 0 0 x3 |
5 5
2 : A <-------------------------- A : 2
{5} | x3 0 0 0 0 |
{5} | 0 x3 0 0 0 |
{5} | 0 0 x3 0 0 |
{5} | 0 0 0 x3 0 |
{5} | 0 0 0 0 x3 |
3 : 0 <----- 0 : 3
0
o8 : ChainComplexMap</pre>
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<tr><td><pre>i9 : s^2
5 3
o9 = 2 : A <----- A : 0
0
8
3 : 0 <----- A : 1
0
o9 : ChainComplexMap</pre>
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<div class="waystouse"><h2>Ways to use <tt>nullhomotopy</tt> :</h2>
<ul><li>nullhomotopy(ChainComplexMap)</li>
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