-- produce a nullhomotopy for a map f of chain complexes.
Whether f is null homotopic is not checked.
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.
i1 : A = ZZ/101[x,y];
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i2 : M = cokernel random(A^3, A^{-2,-2})
o2 = cokernel | -7x2+31xy-13y2 x2-28xy+14y2 |
| -46x2-46xy-17y2 -43x2+28xy-32y2 |
| -6x2+18xy-10y2 22x2+25xy+29y2 |
3
o2 : A-module, quotient of A
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i3 : R = cokernel matrix {{x^3,y^4}}
o3 = cokernel | x3 y4 |
1
o3 : A-module, quotient of A
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i4 : N = prune (M**R)
o4 = cokernel | 13x2+19xy -48x2-33xy+19y2 x3 x2y+30xy2-37y3 -6xy2-45y3 y4 0 0 |
| x2+25xy-39y2 18xy-18y2 0 -48xy2+3y3 -45xy2+10y3 0 y4 0 |
| 8xy-41y2 x2+3xy+13y2 0 15y3 xy2+28y3 0 0 y4 |
3
o4 : A-module, quotient of A
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i5 : C = resolution N
3 8 5
o5 = A <-- A <-- A <-- 0
0 1 2 3
o5 : ChainComplex
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i6 : d = C.dd
3 8
o6 = 0 : A <--------------------------------------------------------------------------- A : 1
| 13x2+19xy -48x2-33xy+19y2 x3 x2y+30xy2-37y3 -6xy2-45y3 y4 0 0 |
| x2+25xy-39y2 18xy-18y2 0 -48xy2+3y3 -45xy2+10y3 0 y4 0 |
| 8xy-41y2 x2+3xy+13y2 0 15y3 xy2+28y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------ A : 2
{2} | 27xy2+23y3 5xy2-47y3 -27y3 7y3 -49y3 |
{2} | -44xy2+4y3 -6y3 44y3 -42y3 -10y3 |
{3} | 47xy-40y2 2xy+15y2 -47y2 -26y2 -38y2 |
{3} | -47x2-42xy-19y2 -2x2-20xy+23y2 47xy-19y2 26xy-9y2 38xy-14y2 |
{3} | 44x2-9xy+42y2 -4xy+y2 -44xy+5y2 42xy+19y2 10xy-25y2 |
{4} | 0 0 x-y 7y -39y |
{4} | 0 0 49y x-40y 19y |
{4} | 0 0 -19y 32y x+41y |
5
2 : A <----- 0 : 3
0
o6 : ChainComplexMap
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i7 : s = nullhomotopy (x^3 * id_C)
8 3
o7 = 1 : A <------------------------ A : 0
{2} | 0 x-25y -18y |
{2} | 0 -8y x-3y |
{3} | 1 -13 48 |
{3} | 0 23 22 |
{3} | 0 -38 39 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <----------------------------------------------------------------------- A : 1
{5} | 33 6 0 -23y -39x-5y xy+16y2 22xy+42y2 -14xy+43y2 |
{5} | 44 -29 0 50x-10y -43x+3y 48y2 xy+y2 45xy+30y2 |
{5} | 0 0 0 0 0 x2+xy-26y2 -7xy-20y2 39xy-13y2 |
{5} | 0 0 0 0 0 -49xy-47y2 x2+40xy+26y2 -19xy+27y2 |
{5} | 0 0 0 0 0 19xy -32xy x2-41xy |
5
3 : 0 <----- A : 2
0
o7 : ChainComplexMap
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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
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i9 : s^2
5 3
o9 = 2 : A <----- A : 0
0
8
3 : 0 <----- A : 1
0
o9 : ChainComplexMap
|