This function determines if the canonical map from ambient R --> R is Golod. It does this by computing an acyclic closure of ambient R (which is a DGAlgebra), then tensors this with R, and determines if this DG Algebra has a trivial Massey operation up to a certain homological degree provided by the option GenDegreeLimit.
i1 : R = ZZ/101[a,b,c,d]/ideal{a^4+b^4+c^4+d^4} o1 = R o1 : QuotientRing |
i2 : isGolodHomomorphism(R,GenDegreeLimit=>5) Computing generators in degree 1 : -- used 0.0252808 seconds Computing generators in degree 2 : -- used 0.0223272 seconds Computing generators in degree 3 : -- used 0.0216048 seconds Computing generators in degree 4 : -- used 0.0202896 seconds Computing generators in degree 5 : -- used 0.0043049 seconds o2 = true |
If R is a Golod ring, then ambient R → R is a Golod homomorphism.
i3 : Q = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4} o3 = Q o3 : QuotientRing |
i4 : R = Q/ideal (a^3*b^3*c^3*d^3) o4 = R o4 : QuotientRing |
i5 : isGolodHomomorphism(R,GenDegreeLimit=>5) Computing generators in degree 1 : -- used 0.034457 seconds Computing generators in degree 2 : -- used 0.0452335 seconds Computing generators in degree 3 : -- used 0.0841472 seconds Computing generators in degree 4 : -- used 0.204768 seconds Computing generators in degree 5 : -- used 0.64651 seconds o5 = true |
The map from Q to R is Golod by a result of Avramov and Levin.
Currently, it does not try to find a full trivial Massey operation on acyclicClosure(Q) ** R, it just computes them to second order. Since there is not currently an example of a ring (or a homomorphism) that is not Golod yet has trivial product on its homotopy fiber, this is ok for now.