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 0.0183454 seconds Computing generators in degree 2 : -- used 0.0483199 seconds Computing generators in degree 3 : -- used 0.0450211 seconds o5 = true |
i6 : cycleList = getGenerators(A) Computing generators in degree 1 : -- used 0.0030504 seconds Computing generators in degree 2 : -- used 0.027966 seconds Computing generators in degree 3 : -- used 0.0286399 seconds Computing generators in degree 4 : -- used 0.0142837 seconds Computing generators in degree 5 : -- used 0.0124749 seconds Computing generators in degree 6 : -- used 0.0112753 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 0.00313919 seconds Computing generators in degree 2 : -- used 0.026795 seconds Computing generators in degree 3 : -- used 0.0846394 seconds Computing generators in degree 4 : -- used 0.00267416 seconds Computing generators in degree 5 : -- used 0.00241577 seconds Computing generators in degree 6 : -- used 0.00260813 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 0.0137466 seconds Computing generators in degree 2 : -- used 0.029518 seconds Computing generators in degree 3 : -- used 0.0280394 seconds o13 = false |
i14 : cycleList = getGenerators(A) Computing generators in degree 1 : -- used 0.00249203 seconds Computing generators in degree 2 : -- used 0.0175172 seconds Computing generators in degree 3 : -- used 0.0179309 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 0.00249745 seconds Computing generators in degree 2 : -- used 0.0178103 seconds Computing generators in degree 3 : -- used 0.0182083 seconds |