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 0.0336279 seconds Computing generators in degree 2 : -- used 0.202794 seconds Computing generators in degree 3 : -- used 0.0744735 seconds o5 = true |
i6 : cycleList = getGenerators(A) Computing generators in degree 1 : -- used 0.00573738 seconds Computing generators in degree 2 : -- used 0.0471519 seconds Computing generators in degree 3 : -- used 0.049712 seconds Computing generators in degree 4 : -- used 0.0245507 seconds Computing generators in degree 5 : -- used 0.0228848 seconds Computing generators in degree 6 : -- used 0.0209035 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 0.00588074 seconds Computing generators in degree 2 : -- used 0.0477878 seconds Computing generators in degree 3 : -- used 0.0507428 seconds Computing generators in degree 4 : -- used 0.00470074 seconds Computing generators in degree 5 : -- used 0.00482608 seconds Computing generators in degree 6 : -- used 0.00486862 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 0.0236453 seconds Computing generators in degree 2 : -- used 0.0517176 seconds Computing generators in degree 3 : -- used 0.0483961 seconds o13 = false |
i14 : cycleList = getGenerators(A) Computing generators in degree 1 : -- used 0.00440424 seconds Computing generators in degree 2 : -- used 0.0326177 seconds Computing generators in degree 3 : -- used 0.0330099 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 0.00438 seconds Computing generators in degree 2 : -- used 0.0326517 seconds Computing generators in degree 3 : -- used 0.0326672 seconds |