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DGAlgebras :: findTrivialMasseyOperation

findTrivialMasseyOperation -- Finds a trivial Massey operation on a set of generators of H(A)

Synopsis

Description

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 0.010441 seconds
Computing generators in degree 2 :      -- used 0.0254645 seconds
Computing generators in degree 3 :      -- used 0.0413979 seconds

o5 = true
i6 : cycleList = getGenerators(A)
Computing generators in degree 1 :      -- used 0.00137888 seconds
Computing generators in degree 2 :      -- used 0.0142681 seconds
Computing generators in degree 3 :      -- used 0.0141437 seconds
Computing generators in degree 4 :      -- used 0.00724066 seconds
Computing generators in degree 5 :      -- used 0.00622645 seconds
Computing generators in degree 6 :      -- used 0.00595744 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 0.00135976 seconds
Computing generators in degree 2 :      -- used 0.0137248 seconds
Computing generators in degree 3 :      -- used 0.0329271 seconds
Computing generators in degree 4 :      -- used 0.00155034 seconds
Computing generators in degree 5 :      -- used 0.00121518 seconds
Computing generators in degree 6 :      -- used 0.00114904 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 0.00669455 seconds
Computing generators in degree 2 :      -- used 0.0147311 seconds
Computing generators in degree 3 :      -- used 0.0327921 seconds

o13 = false
i14 : cycleList = getGenerators(A)
Computing generators in degree 1 :      -- used 0.00113651 seconds
Computing generators in degree 2 :      -- used 0.00981747 seconds
Computing generators in degree 3 :      -- used 0.0163426 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 0.00293794 seconds
Computing generators in degree 2 :      -- used 0.021851 seconds
Computing generators in degree 3 :      -- used 0.0208097 seconds

Ways to use findTrivialMasseyOperation :