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DGAlgebras :: isGolod

isGolod -- Determines if a ring is Golod

Synopsis

Description

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 0.0336687 seconds
Computing generators in degree 2 :      -- used 0.0295985 seconds
Computing generators in degree 3 :      -- used 0.0285303 seconds
Computing generators in degree 4 :      -- used 0.0268627 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 0.0359265 seconds
Computing generators in degree 2 :      -- used 0.0777278 seconds
Computing generators in degree 3 :      -- used 0.0726721 seconds
Computing generators in degree 4 :      -- used 0.0660741 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 0.0400333 seconds
Computing generators in degree 2 :      -- used 0.114734 seconds
Computing generators in degree 3 :      -- used 0.16488 seconds
Computing generators in degree 4 :      -- used 0.341258 seconds

o7 = true
The above is a (CM) ring minimal of minimal multiplicity, hence Golod.

Caveat

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.

Ways to use isGolod :