This function determines if the Koszul complex of a ring R admits a trivial Massey operation. If one exists, then R is Golod.
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 : isGolod(R)
Computing generators in degree 1 : -- used 0.0172619 seconds
Computing generators in degree 2 : -- used 0.0152652 seconds
Computing generators in degree 3 : -- used 0.0146825 seconds
Computing generators in degree 4 : -- used 0.0136385 seconds
o2 = true
|
Hypersurfaces are Golod, but
i3 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4}
o3 = R
o3 : QuotientRing
|
i4 : isGolod(R)
Computing generators in degree 1 : -- used 0.017898 seconds
Computing generators in degree 2 : -- used 0.0825719 seconds
Computing generators in degree 3 : -- used 0.0380325 seconds
Computing generators in degree 4 : -- used 0.0347644 seconds
o4 = false
|
complete intersections of higher codimension are not. Here is another example:
i5 : Q = ZZ/101[a,b,c,d]
o5 = Q
o5 : PolynomialRing
|
i6 : R = Q/(ideal vars Q)^2
o6 = R
o6 : QuotientRing
|
i7 : isGolod(R)
Computing generators in degree 1 : -- used 0.021449 seconds
Computing generators in degree 2 : -- used 0.0587542 seconds
Computing generators in degree 3 : -- used 0.0797285 seconds
Computing generators in degree 4 : -- used 0.210192 seconds
o7 = true
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The above is a (CM) ring minimal of minimal multiplicity, hence Golod.