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
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i2 : isGolodHomomorphism(R,GenDegreeLimit=>5)
Computing generators in degree 1 : -- used 0.0057621 seconds
Computing generators in degree 2 : -- used 0.00505894 seconds
Computing generators in degree 3 : -- used 0.00486581 seconds
Computing generators in degree 4 : -- used 0.00452954 seconds
Computing generators in degree 5 : -- used 0.000928241 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 0.00866159 seconds
Computing generators in degree 2 : -- used 0.0114074 seconds
Computing generators in degree 3 : -- used 0.0301225 seconds
Computing generators in degree 4 : -- used 0.030982 seconds
Computing generators in degree 5 : -- used 0.172649 seconds
o5 = true
|
The map from Q to R is Golod by a result of Avramov and Levin.