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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 0.0105942 seconds
Computing generators in degree 2 :      -- used 0.0297762 seconds
Computing generators in degree 3 :      -- used 0.0554161 seconds

o5 = true
i6 : cycleList = getGenerators(A)
Computing generators in degree 1 :      -- used 0.00177406 seconds
Computing generators in degree 2 :      -- used 0.014651 seconds
Computing generators in degree 3 :      -- used 0.015638 seconds
Computing generators in degree 4 :      -- used 0.0076537 seconds
Computing generators in degree 5 :      -- used 0.00684372 seconds
Computing generators in degree 6 :      -- used 0.00669455 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 0.00170875 seconds
Computing generators in degree 2 :      -- used 0.015089 seconds
Computing generators in degree 3 :      -- used 0.0154683 seconds
Computing generators in degree 4 :      -- used 0.00149574 seconds
Computing generators in degree 5 :      -- used 0.00143532 seconds
Computing generators in degree 6 :      -- used 0.00143408 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 0.00731225 seconds
Computing generators in degree 2 :      -- used 0.0161953 seconds
Computing generators in degree 3 :      -- used 0.0148121 seconds

o13 = false
i14 : cycleList = getGenerators(A)
Computing generators in degree 1 :      -- used 0.00131821 seconds
Computing generators in degree 2 :      -- used 0.00981164 seconds
Computing generators in degree 3 :      -- used 0.00985147 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 0.00132538 seconds
Computing generators in degree 2 :      -- used 0.0102517 seconds
Computing generators in degree 3 :      -- used 0.0101475 seconds

Ways to use findTrivialMasseyOperation :