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 which is multiplication by one of the defining equations for the complete intersection.
i1 = A = ZZ/101[x,y]
o1 = A
o1 : PolynomialRing
i2 = M = cokernel random(A^3, A^{-2,-2})
o2 = cokernel | 16x2-25xy-13y2 -14x2-20xy-41y2 |
| -27x2+11xy-43y2 -28x2+14xy-40y2 |
| 12x2-43xy-8y2 -46x2-2xy-16y2 |
3
A - module, quotient of A
i3 = R = cokernel matrix {{x^3,y^4}}
o3 = cokernel | x3 y4 |
1
A - module, quotient of A
i4 = N = prune (M**R)
o4 = cokernel | -47x2-19xy+34y2 -47x2+24xy-2y2 x3 x2y-26xy2+y3 -3xy2+49y3 0 0 y4 |
| 41xy+38y2 x2-21xy-32y2 0 -47xy2+13y3 4xy2+38y3 0 y4 0 |
| x2+36xy+2y2 26xy-32y2 0 6y3 xy2+48y3 y4 0 0 |
3
A - module, quotient of A
i5 = C = resolution N
3 8 5
o5 = A <-- A <-- A
0 1 2
o5 : ChainComplex
i6 = d = C.dd
3 8
o6 = 1: A <--| 26xy-32y2 x2+36xy+2y2 0 6y3 xy2+48y3 0 0 y4 |-- A
| x2-21xy-32y2 41xy+38y2 0 -47xy2+13y3 4xy2+38y3 0 y4 0 |
| -47x2+24xy-2y2 -47x2-19xy+34y2 x3 x2y-26xy2+y3 -3xy2+49y3 y4 0 0 |
8 5
2: A <--| -48xy2+36y3 20xy2+26y3 48y3 -30y3 -39y3 |-- A
| -3xy2-y3 -39y3 3y3 12y3 -39y3 |
| 33xy-3y2 49xy+9y2 -33y2 21y2 19y2 |
| -33x2-11xy+8y2 -49x2-40xy-17y2 33xy+14y2 -21xy-14y2 -19xy-3y2 |
| 3x2-3xy+2y2 15xy-21y2 -3xy+4y2 -12xy+40y2 39xy-48y2 |
| 0 0 x-14y 10y -33y |
| 0 0 -23y x-27y -24y |
| 0 0 42y 8y x+41y |
o6 : ChainComplexMap
i7 = s = nullhomotopy (x^3 * id_C)
8 3
o7 = 0: A <--| -41y x+21y 0 |-- A
| x-36y -26y 0 |
| 47 47 1 |
| 36 44 0 |
| 37 18 0 |
| 0 0 0 |
| 0 0 0 |
| 0 0 0 |
5 8
1: A <--| -6 -46 0 -37y 34x+31y xy+12y2 4xy-48y2 -13xy-22y2 |-- A
| 7 -26 0 -33x+23y 41x-29y 47y2 xy+11y2 -4xy-11y2 |
| 0 0 0 0 0 x2+14xy-6y2 -10xy+33y2 33xy-20y2 |
| 0 0 0 0 0 23xy+36y2 x2+27xy+4y2 24xy+19y2 |
| 0 0 0 0 0 -42xy+41y2 -8xy+27y2 x2-41xy+2y2 |
o7 : ChainComplexMap
i8 = s*d + d*s
3 3
o8 = 0: A <--| x3 0 0 |-- A
| 0 x3 0 |
| 0 0 x3 |
8 8
1: A <--| x3 0 0 0 0 0 0 0 |-- A
| 0 x3 0 0 0 0 0 0 |
| 0 0 x3 0 0 0 0 0 |
| 0 0 0 x3 0 0 0 0 |
| 0 0 0 0 x3 0 0 0 |
| 0 0 0 0 0 x3 0 0 |
| 0 0 0 0 0 0 x3 0 |
| 0 0 0 0 0 0 0 x3 |
5 5
2: A <--| x3 0 0 0 0 |-- A
| 0 x3 0 0 0 |
| 0 0 x3 0 0 |
| 0 0 0 x3 0 |
| 0 0 0 0 x3 |
o8 : ChainComplexMap
i9 = s^2
o9 = 0
o9 : ChainComplexMap
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