This function currently just finds the elements whose boundary give the product of every pair of cycles that are chosen as generators. Eventually, all higher Massey operations will also be computed. The maximum degree of a generating cycle is specified in the option GenDegreeLimit, if needed.
This is an example of a Golod ring. It is Golod since it is the Stanley-Reisner ideal of a flag complex whose 1-skeleton is chordal [Jollenbeck-Berglund].
i1 : Q = ZZ/101[x_1..x_6]
o1 = Q
o1 : PolynomialRing
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i2 : I = ideal (x_3*x_5,x_4*x_5,x_1*x_6,x_3*x_6,x_4*x_6)
o2 = ideal (x x , x x , x x , x x , x x )
3 5 4 5 1 6 3 6 4 6
o2 : Ideal of Q
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i3 : R = Q/I
o3 = R
o3 : QuotientRing
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i4 : A = koszulComplexDGA(R)
o4 = {Ring => R }
Underlying algebra => R[T , T , T , T , T , T ]
1 2 3 4 5 6
Differential => {x , x , x , x , x , x }
1 2 3 4 5 6
isHomogeneous => true
o4 : DGAlgebra
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i5 : isHomologyAlgebraTrivial(A,GenDegreeLimit=>3)
Computing generators in degree 1 : -- used 0.0629331 seconds
Computing generators in degree 2 : -- used 0.0548155 seconds
Computing generators in degree 3 : -- used 0.0539423 seconds
o5 = true
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i6 : cycleList = getGenerators(A)
Computing generators in degree 1 : -- used 0.00417878 seconds
Computing generators in degree 2 : -- used 0.033909 seconds
Computing generators in degree 3 : -- used 0.034744 seconds
Computing generators in degree 4 : -- used 0.0169907 seconds
Computing generators in degree 5 : -- used 0.0154139 seconds
Computing generators in degree 6 : -- used 0.0143611 seconds
o6 = {x T , x T , x T , x T , x T , -x T T , -x T T , -x T T , -x T T , -
5 4 5 3 6 4 6 3 6 1 6 1 3 5 3 4 6 3 4 6 1 4
------------------------------------------------------------------------
x T T + x T T , - x T T + x T T , x T T T , x T T T - x T T T }
6 4 5 5 4 6 6 3 5 5 3 6 6 1 3 4 6 3 4 5 5 3 4 6
o6 : List
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i7 : tmo = findTrivialMasseyOperation(A)
Computing generators in degree 1 : -- used 0.00400312 seconds
Computing generators in degree 2 : -- used 0.0331259 seconds
Computing generators in degree 3 : -- used 0.034539 seconds
Computing generators in degree 4 : -- used 0.0033691 seconds
Computing generators in degree 5 : -- used 0.0032959 seconds
Computing generators in degree 6 : -- used 0.0033097 seconds
o7 = {{3} | 0 0 0 0 0 0 0 0 0 0 |, {4} | 0 0 0 0 0
{3} | 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0
{3} | 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0
{3} | 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0
{3} | 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0
{3} | 0 0 0 0 0 0 0 -x_6 0 0 | {4} | 0 0 0 0 0
{3} | 0 0 0 0 0 0 0 0 0 -x_6 | {4} | x_6 0 0 0 0
{3} | 0 0 0 0 0 0 -x_6 0 0 0 | {4} | 0 0 x_6 0 0
{3} | 0 0 0 0 0 0 0 0 -x_6 0 | {4} | 0 0 0 0 0
{3} | 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0
{3} | 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0
{3} | 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0
{3} | 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0
{3} | 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0
{3} | 0 0 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 0
{3} | 0 0 0 0 0 0 0 0 0 0 |
{3} | -x_5 0 x_6 -x_6 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 -x_6 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 |
------------------------------------------------------------------------
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 x_6 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 x_6 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_6 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_5 0 x_6 0 -x_5 0 -x_6 0
------------------------------------------------------------------------
0 |, {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |,
0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
0 | {5} | 0 0 0 0 0 0 x_6 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 0 x_6 |
0 | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
0 |
0 |
x_6 |
0 |
0 |
0 |
0 |
0 |
0 |
------------------------------------------------------------------------
0, 0}
o7 : List
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i8 : assert(tmo =!= null)
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Below is an example of a Teter ring (Artinian Gorenstein ring modulo its socle), and the computation in Avramov and Levin’s paper shows that H(A) does not have trivial multiplication, hence no trivial Massey operation can exist.
i9 : Q = ZZ/101[x,y,z]
o9 = Q
o9 : PolynomialRing
|
i10 : I = ideal (x^3,y^3,z^3,x^2*y^2*z^2)
3 3 3 2 2 2
o10 = ideal (x , y , z , x y z )
o10 : Ideal of Q
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i11 : R = Q/I
o11 = R
o11 : QuotientRing
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i12 : A = koszulComplexDGA(R)
o12 = {Ring => R }
Underlying algebra => R[T , T , T ]
1 2 3
Differential => {x, y, z}
isHomogeneous => true
o12 : DGAlgebra
|
i13 : isHomologyAlgebraTrivial(A)
Computing generators in degree 1 : -- used 0.0175823 seconds
Computing generators in degree 2 : -- used 0.0383864 seconds
Computing generators in degree 3 : -- used 0.0369069 seconds
o13 = false
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i14 : cycleList = getGenerators(A)
Computing generators in degree 1 : -- used 0.00333274 seconds
Computing generators in degree 2 : -- used 0.0241599 seconds
Computing generators in degree 3 : -- used 0.0237199 seconds
2 2 2 2 2 2 2 2 2 2 2
o14 = {x T , y T , z T , x*y z T , x*y z T T , x y*z T T , x*y z T T ,
1 2 3 1 1 2 1 2 1 3
-----------------------------------------------------------------------
2 2 2 2 2 2
x*y z T T T , x y*z T T T , x y z*T T T }
1 2 3 1 2 3 1 2 3
o14 : List
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i15 : assert(findTrivialMasseyOperation(A) === null)
Computing generators in degree 1 : -- used 0.0031395 seconds
Computing generators in degree 2 : -- used 0.0235375 seconds
Computing generators in degree 3 : -- used 0.0234815 seconds
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