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 0.0153653 seconds Computing generators in degree 2 : -- used 0.0384254 seconds Computing generators in degree 3 : -- used 0.0374798 seconds o5 = true |
i6 : cycleList = getGenerators(A) Computing generators in degree 1 : -- used 0.0027456 seconds Computing generators in degree 2 : -- used 0.023166 seconds Computing generators in degree 3 : -- used 0.0243068 seconds Computing generators in degree 4 : -- used 0.0116652 seconds Computing generators in degree 5 : -- used 0.0103621 seconds Computing generators in degree 6 : -- used 0.0095598 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 0.00274331 seconds Computing generators in degree 2 : -- used 0.0234201 seconds Computing generators in degree 3 : -- used 0.0422475 seconds Computing generators in degree 4 : -- used 0.00221921 seconds Computing generators in degree 5 : -- used 0.00221371 seconds Computing generators in degree 6 : -- used 0.00220577 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 0.0113451 seconds Computing generators in degree 2 : -- used 0.0241937 seconds Computing generators in degree 3 : -- used 0.0225605 seconds o13 = false |
i14 : cycleList = getGenerators(A) Computing generators in degree 1 : -- used 0.00203422 seconds Computing generators in degree 2 : -- used 0.0151673 seconds Computing generators in degree 3 : -- used 0.0151606 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 0.00204274 seconds Computing generators in degree 2 : -- used 0.0151533 seconds Computing generators in degree 3 : -- used 0.0152216 seconds |