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.
Golod rings are defined by being those rings whose Koszul complex KR has a trivial Massey operation. Also, the existence of a trivial Massey operation on a DG algebra A forces the multiplication on H(A) to be trivial. An example of a ring R such that H(KR) has trivial multiplication, yet KR does not admit a trivial Massey operation is unknown. Such an example cannot be monomially defined, by a result of Jollenbeck and Berglund.
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 0.0534107 seconds Computing generators in degree 2 : -- used 0.313592 seconds Computing generators in degree 3 : -- used 0.115788 seconds o5 = true |
i6 : cycleList = getGenerators(A) Computing generators in degree 1 : -- used 0.00909714 seconds Computing generators in degree 2 : -- used 0.0754685 seconds Computing generators in degree 3 : -- used 0.078926 seconds Computing generators in degree 4 : -- used 0.0386125 seconds Computing generators in degree 5 : -- used 0.0351505 seconds Computing generators in degree 6 : -- used 0.032991 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 0.00931498 seconds Computing generators in degree 2 : -- used 0.0763948 seconds Computing generators in degree 3 : -- used 0.0799356 seconds Computing generators in degree 4 : -- used 0.00760736 seconds Computing generators in degree 5 : -- used 0.00761305 seconds Computing generators in degree 6 : -- used 0.00758191 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 0.0375317 seconds Computing generators in degree 2 : -- used 0.080881 seconds Computing generators in degree 3 : -- used 0.0770566 seconds o13 = false |
i14 : cycleList = getGenerators(A) Computing generators in degree 1 : -- used 0.00708537 seconds Computing generators in degree 2 : -- used 0.0519593 seconds Computing generators in degree 3 : -- used 0.0521431 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 0.00710529 seconds Computing generators in degree 2 : -- used 0.0520062 seconds Computing generators in degree 3 : -- used 0.052309 seconds |