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) 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) 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) o7 = true |
The above is a (CM) ring minimal of minimal multiplicity, hence Golod. The next example was found by Lukas Katthan, and appears in his arXiv paper 1511.04883. It is the first known example of an algebra that is not Golod, but whose Koszul complex has a trivial homology product.
i8 : Q = ZZ/101[x_1,x_2,y_1,y_2,z,w] o8 = Q o8 : PolynomialRing |
i9 : I = ideal {x_1*x_2^2,z^2*w,y_1*y_2^2,x_2^2*z*w,y_2^2*z^2,x_1*x_2*y_1*y_2,x_2^2*y_2^2*z,x_1*y_1*z}
2 2 2 2 2 2 2 2
o9 = ideal (x x , z w, y y , x z*w, y z , x x y y , x y z, x y z)
1 2 1 2 2 2 1 2 1 2 2 2 1 1
o9 : Ideal of Q
|
i10 : R = Q/I o10 = R o10 : QuotientRing |
i11 : isHomologyAlgebraTrivial koszulComplexDGA R o11 = true |
i12 : isGolod R o12 = false |
Note that since the Koszul complex is zero in homological degree beyond the embedding dimension, there are only finitely many Massey products that one needs to check to verify that a ring is Golod.
The object isGolod is a method function.