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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 0.0113931 seconds
Computing generators in degree 2 :      -- used 0.0266916 seconds
Computing generators in degree 3 :      -- used 0.0871529 seconds

o5 = true
i6 : cycleList = getGenerators(A)
Computing generators in degree 1 :      -- used 0.0022126 seconds
Computing generators in degree 2 :      -- used 0.016123 seconds
Computing generators in degree 3 :      -- used 0.0215798 seconds
Computing generators in degree 4 :      -- used 0.0101705 seconds
Computing generators in degree 5 :      -- used 0.00705363 seconds
Computing generators in degree 6 :      -- used 0.00716723 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 0.00179328 seconds
Computing generators in degree 2 :      -- used 0.0164332 seconds
Computing generators in degree 3 :      -- used 0.02317 seconds
Computing generators in degree 4 :      -- used 0.00278912 seconds
Computing generators in degree 5 :      -- used 0.0016489 seconds
Computing generators in degree 6 :      -- used 0.00162243 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 0.00802711 seconds
Computing generators in degree 2 :      -- used 0.0169417 seconds
Computing generators in degree 3 :      -- used 0.0164975 seconds

o13 = false
i14 : cycleList = getGenerators(A)
Computing generators in degree 1 :      -- used 0.00146661 seconds
Computing generators in degree 2 :      -- used 0.011375 seconds
Computing generators in degree 3 :      -- used 0.0104105 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 0.00222611 seconds
Computing generators in degree 2 :      -- used 0.0119025 seconds
Computing generators in degree 3 :      -- used 0.0164836 seconds

Ways to use findTrivialMasseyOperation :