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 0.0459034 seconds
Computing generators in degree 2 : -- used 0.308998 seconds
Computing generators in degree 3 : -- used 0.103657 seconds
o5 = true
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i6 : cycleList = getGenerators(A)
Computing generators in degree 1 : -- used 0.00805163 seconds
Computing generators in degree 2 : -- used 0.0675243 seconds
Computing generators in degree 3 : -- used 0.0717954 seconds
Computing generators in degree 4 : -- used 0.034462 seconds
Computing generators in degree 5 : -- used 0.0312089 seconds
Computing generators in degree 6 : -- used 0.0290932 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 0.00841773 seconds
Computing generators in degree 2 : -- used 0.0681196 seconds
Computing generators in degree 3 : -- used 0.0715122 seconds
Computing generators in degree 4 : -- used 0.0067577 seconds
Computing generators in degree 5 : -- used 0.00656404 seconds
Computing generators in degree 6 : -- used 0.00653901 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 0.0328019 seconds
Computing generators in degree 2 : -- used 0.0711007 seconds
Computing generators in degree 3 : -- used 0.0670145 seconds
o13 = false
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i14 : cycleList = getGenerators(A)
Computing generators in degree 1 : -- used 0.00614452 seconds
Computing generators in degree 2 : -- used 0.0454456 seconds
Computing generators in degree 3 : -- used 0.0454712 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 0.00612331 seconds
Computing generators in degree 2 : -- used 0.0454061 seconds
Computing generators in degree 3 : -- used 0.0455461 seconds
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