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

## Synopsis

• Usage:
isGol = isGolodHomomorphism(R)
• Inputs:
• R, ,
• 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, ,

## 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 .