# nullhomotopy -- make a null homotopy

## Description

nullhomotopy f -- 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]; i2 : M = cokernel random(A^3, A^{-2,-2}) o2 = cokernel | 24x2-36xy-30y2 -22x2-29xy-24y2 | | -29x2+19xy+19y2 -38x2-16xy+39y2 | | -10x2-29xy-8y2 21x2+34xy+19y2 | 3 o2 : A-module, quotient of A i3 : R = cokernel matrix {{x^3,y^4}} o3 = cokernel | x3 y4 | 1 o3 : A-module, quotient of A i4 : N = prune (M**R) o4 = cokernel | -49x2-45xy-27y2 -32x2+23xy-50y2 x3 x2y-44xy2-44y3 20xy2-33y3 y4 0 0 | | x2+21xy-49y2 18xy+19y2 0 38xy2-30y3 -9xy2+44y3 0 y4 0 | | -8xy-23y2 x2+16xy+17y2 0 -22y3 xy2+y3 0 0 y4 | 3 o4 : A-module, quotient of A i5 : C = resolution N 3 8 5 o5 = A <-- A <-- A <-- 0 0 1 2 3 o5 : ChainComplex i6 : d = C.dd 3 8 o6 = 0 : A <----------------------------------------------------------------------------- A : 1 | -49x2-45xy-27y2 -32x2+23xy-50y2 x3 x2y-44xy2-44y3 20xy2-33y3 y4 0 0 | | x2+21xy-49y2 18xy+19y2 0 38xy2-30y3 -9xy2+44y3 0 y4 0 | | -8xy-23y2 x2+16xy+17y2 0 -22y3 xy2+y3 0 0 y4 | 8 5 1 : A <---------------------------------------------------------------------------- A : 2 {2} | -38xy2+32y3 -9xy2+23y3 38y3 21y3 35y3 | {2} | 43xy2+2y3 36y3 -43y3 13y3 -33y3 | {3} | -36xy-4y2 29xy-44y2 36y2 -7y2 25y2 | {3} | 36x2+43xy+29y2 -29x2+44xy-19y2 -36xy-39y2 7xy-7y2 -25xy+10y2 | {3} | -43x2+43xy+27y2 -39xy+4y2 43xy-45y2 -13xy+26y2 33xy+22y2 | {4} | 0 0 x+18y 50y -44y | {4} | 0 0 -46y x+34y -43y | {4} | 0 0 -16y -19y x+49y | 5 2 : A <----- 0 : 3 0 o6 : ChainComplexMap i7 : s = nullhomotopy (x^3 * id_C) 8 3 o7 = 1 : A <------------------------- A : 0 {2} | 0 x-21y -18y | {2} | 0 8y x-16y | {3} | 1 49 32 | {3} | 0 -21 -3 | {3} | 0 30 -6 | {4} | 0 0 0 | {4} | 0 0 0 | {4} | 0 0 0 | 5 8 2 : A <---------------------------------------------------------------------------- A : 1 {5} | -4 21 0 4y -47x-46y xy+32y2 -5xy-14y2 36xy+10y2 | {5} | -44 19 0 -7x-37y -27x-18y -38y2 xy+7y2 9xy+36y2 | {5} | 0 0 0 0 0 x2-18xy+41y2 -50xy+2y2 44xy-48y2 | {5} | 0 0 0 0 0 46xy+13y2 x2-34xy-24y2 43xy-30y2 | {5} | 0 0 0 0 0 16xy+4y2 19xy+47y2 x2-49xy-17y2 | 5 3 : 0 <----- A : 2 0 o7 : ChainComplexMap 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 i9 : s^2 5 3 o9 = 2 : A <----- A : 0 0 8 3 : 0 <----- A : 1 0 o9 : ChainComplexMap

## Ways to use nullhomotopy :

• "nullhomotopy(ChainComplexMap)"

## For the programmer

The object nullhomotopy is .