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 0.0226403 seconds
Computing generators in degree 2 : -- used 0.0556077 seconds
Computing generators in degree 3 : -- used 0.05476 seconds
o5 = true
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i6 : cycleList = getGenerators(A)
Computing generators in degree 1 : -- used 0.00411556 seconds
Computing generators in degree 2 : -- used 0.0342778 seconds
Computing generators in degree 3 : -- used 0.0348851 seconds
Computing generators in degree 4 : -- used 0.0171347 seconds
Computing generators in degree 5 : -- used 0.0155993 seconds
Computing generators in degree 6 : -- used 0.0144926 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 0.00403312 seconds
Computing generators in degree 2 : -- used 0.0334307 seconds
Computing generators in degree 3 : -- used 0.0348346 seconds
Computing generators in degree 4 : -- used 0.00336544 seconds
Computing generators in degree 5 : -- used 0.00336474 seconds
Computing generators in degree 6 : -- used 0.00332508 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
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i13 : isHomologyAlgebraTrivial(A)
Computing generators in degree 1 : -- used 0.0178258 seconds
Computing generators in degree 2 : -- used 0.0390094 seconds
Computing generators in degree 3 : -- used 0.0371699 seconds
o13 = false
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i14 : cycleList = getGenerators(A)
Computing generators in degree 1 : -- used 0.0032578 seconds
Computing generators in degree 2 : -- used 0.0244363 seconds
Computing generators in degree 3 : -- used 0.0238189 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 0.0031939 seconds
Computing generators in degree 2 : -- used 0.0237312 seconds
Computing generators in degree 3 : -- used 0.0236771 seconds
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