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 |
i2 : isGolod(R) Computing generators in degree 1 : -- used 0.00617837 seconds Computing generators in degree 2 : -- used 0.00536656 seconds Computing generators in degree 3 : -- used 0.00510073 seconds Computing generators in degree 4 : -- used 0.00477022 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 0.0230925 seconds Computing generators in degree 2 : -- used 0.0143105 seconds Computing generators in degree 3 : -- used 0.0132551 seconds Computing generators in degree 4 : -- used 0.0120342 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 0.00760519 seconds Computing generators in degree 2 : -- used 0.0220425 seconds Computing generators in degree 3 : -- used 0.048049 seconds Computing generators in degree 4 : -- used 0.0626525 seconds o7 = true |
The above is a (CM) ring minimal of minimal multiplicity, hence Golod.
Currently, it does not try to find a full trivial Massey operation for the ring R, it just computes them to second order. Since there is not currently an example of a ring that is not Golod yet has trivial product on H(KR), this is ok for now.