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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 0.0426883 seconds
Computing generators in degree 2 :      -- used 0.104695 seconds
Computing generators in degree 3 :      -- used 0.279965 seconds

o5 = true
i6 : cycleList = getGenerators(A)
Computing generators in degree 1 :      -- used 0.00766799 seconds
Computing generators in degree 2 :      -- used 0.062659 seconds
Computing generators in degree 3 :      -- used 0.066181 seconds
Computing generators in degree 4 :      -- used 0.0317489 seconds
Computing generators in degree 5 :      -- used 0.0290075 seconds
Computing generators in degree 6 :      -- used 0.0277233 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 0.00788535 seconds
Computing generators in degree 2 :      -- used 0.0635126 seconds
Computing generators in degree 3 :      -- used 0.0675292 seconds
Computing generators in degree 4 :      -- used 0.00650023 seconds
Computing generators in degree 5 :      -- used 0.0064895 seconds
Computing generators in degree 6 :      -- used 0.00641196 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 0.0309624 seconds
Computing generators in degree 2 :      -- used 0.067097 seconds
Computing generators in degree 3 :      -- used 0.0640216 seconds

o13 = false
i14 : cycleList = getGenerators(A)
Computing generators in degree 1 :      -- used 0.00595286 seconds
Computing generators in degree 2 :      -- used 0.0432885 seconds
Computing generators in degree 3 :      -- used 0.0436127 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 0.00591948 seconds
Computing generators in degree 2 :      -- used 0.0433398 seconds
Computing generators in degree 3 :      -- used 0.0436369 seconds

Ways to use findTrivialMasseyOperation :