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 0.00757313 seconds Computing generators in degree 2 : -- used 0.00669628 seconds Computing generators in degree 3 : -- used 0.00637219 seconds Computing generators in degree 4 : -- used 0.00593456 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 0.0216278 seconds Computing generators in degree 2 : -- used 0.0182044 seconds Computing generators in degree 3 : -- used 0.0169921 seconds Computing generators in degree 4 : -- used 0.0153141 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 0.00995221 seconds Computing generators in degree 2 : -- used 0.0286947 seconds Computing generators in degree 3 : -- used 0.0502296 seconds Computing generators in degree 4 : -- used 0.0802282 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.