homologyClass -- Computes the element of the homology algebra corresponding to a cycle in a DGAlgebra.

i1 : Q = QQ[x,y,z]

o1 = Q

o1 : PolynomialRing

i2 : I = ideal (x^3,y^3,z^3)

             3   3   3
o2 = ideal (x , y , z )

o2 : Ideal of Q

i3 : R = Q/I

o3 = R

o3 : QuotientRing

i4 : KR = koszulComplexDGA R

o4 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,3
      Differential => {x, y, z}

o4 : DGAlgebra

i5 : z1 = x^2*T_(1,1)

      2
o5 = x T
        1,1

o5 : R[T   ..T   ]
        1,1   1,3

i6 : z2 = y^2*T_(1,2)

      2
o6 = y T
        1,2

o6 : R[T   ..T   ]
        1,1   1,3

i7 : H = HH(KR)
Finding easy relations           :  -- used 0.013947s (cpu); 0.0127573s (thread); 0s (gc)

o7 = H

o7 : PolynomialRing, 3 skew commutative variable(s)

i8 : homologyClass(KR,z1*z2)

o8 = X X
      1 2

o8 : H