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
|
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
|
i3 : R = Q/I
o3 = R
o3 : QuotientRing
|
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
|
i5 : isHomologyAlgebraTrivial(A,GenDegreeLimit=>3)
Computing generators in degree 1 : -- used 0.0123102 seconds
Computing generators in degree 2 : -- used 0.0304168 seconds
Computing generators in degree 3 : -- used 0.0295442 seconds
o5 = true
|
i6 : cycleList = getGenerators(A)
Computing generators in degree 1 : -- used 0.00225094 seconds
Computing generators in degree 2 : -- used 0.0194281 seconds
Computing generators in degree 3 : -- used 0.0187985 seconds
Computing generators in degree 4 : -- used 0.00951319 seconds
Computing generators in degree 5 : -- used 0.00867706 seconds
Computing generators in degree 6 : -- used 0.00802516 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
|
i7 : tmo = findTrivialMasseyOperation(A)
Computing generators in degree 1 : -- used 0.00221723 seconds
Computing generators in degree 2 : -- used 0.0188057 seconds
Computing generators in degree 3 : -- used 0.0198722 seconds
Computing generators in degree 4 : -- used 0.00188501 seconds
Computing generators in degree 5 : -- used 0.00187024 seconds
Computing generators in degree 6 : -- used 0.00187903 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
|
i8 : assert(tmo =!= null)
|
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
|
i11 : R = Q/I
o11 = R
o11 : QuotientRing
|
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 0.00909283 seconds
Computing generators in degree 2 : -- used 0.0205902 seconds
Computing generators in degree 3 : -- used 0.0196334 seconds
o13 = false
|
i14 : cycleList = getGenerators(A)
Computing generators in degree 1 : -- used 0.00178236 seconds
Computing generators in degree 2 : -- used 0.0135782 seconds
Computing generators in degree 3 : -- used 0.0129343 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
|
i15 : assert(findTrivialMasseyOperation(A) === null)
Computing generators in degree 1 : -- used 0.00169657 seconds
Computing generators in degree 2 : -- used 0.0129392 seconds
Computing generators in degree 3 : -- used 0.0131612 seconds
|