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 0.0427664 seconds
Computing generators in degree 2 : -- used 0.104891 seconds
Computing generators in degree 3 : -- used 0.283785 seconds
o5 = true
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i6 : cycleList = getGenerators(A)
Computing generators in degree 1 : -- used 0.00762528 seconds
Computing generators in degree 2 : -- used 0.0626935 seconds
Computing generators in degree 3 : -- used 0.0663255 seconds
Computing generators in degree 4 : -- used 0.0317309 seconds
Computing generators in degree 5 : -- used 0.0290174 seconds
Computing generators in degree 6 : -- used 0.0273852 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 0.00785371 seconds
Computing generators in degree 2 : -- used 0.063603 seconds
Computing generators in degree 3 : -- used 0.0676253 seconds
Computing generators in degree 4 : -- used 0.00649151 seconds
Computing generators in degree 5 : -- used 0.00647297 seconds
Computing generators in degree 6 : -- used 0.00640177 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 0.0309339 seconds
Computing generators in degree 2 : -- used 0.067071 seconds
Computing generators in degree 3 : -- used 0.0639033 seconds
o13 = false
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i14 : cycleList = getGenerators(A)
Computing generators in degree 1 : -- used 0.00594464 seconds
Computing generators in degree 2 : -- used 0.0432913 seconds
Computing generators in degree 3 : -- used 0.0437561 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 0.00593169 seconds
Computing generators in degree 2 : -- used 0.0432427 seconds
Computing generators in degree 3 : -- used 0.0434993 seconds
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