isGolodHomomorphism -- Determines if the canonical map from the ambient ring is Golod

Synopsis

Usage:

isGol = isGolodHomomorphism(R)
Inputs:
- R, a quotient ring,
Optional inputs:
- GenDegreeLimit => ..., default value infinity, Option to specify the maximum degree to look for generators
- TMOLimit => ..., default value infinity, Option to specify the maximum degree to look for generators when computing the deviations
Outputs:
- isGol, a Boolean value,

Description

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)

o2 = true

If R is a Golod ring, then ambient R $\rightarrow$ 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,TMOLimit=>3)

o5 = true

The map from Q to R is Golod by a result of Avramov and Levin; we can only find the trivial Massey operations out to a given degree.

Ways to use isGolodHomomorphism :

isGolodHomomorphism(QuotientRing)

For the programmer

The object isGolodHomomorphism is a method function with options.